ABSTRACT

EXOTIC NUCLEAR REACTION MECHANISMS: MULTISTEP PROCESSES

IN HEAVY-ION INDUCED TRANSFER REACTIONS AND COMPLEX

PARTICLE EMISSION IN THE 12C+l4N INTERACTION

David L. Hanson

Yale University 1975

This dissertation presents the results of a study of two distinct

classes of nuclear reactions which cannot be adequately understood within the

the first-order distorted-wave-Bom-approximation (DWBA) model for direct

reactions: (1) heavy-ion induced two-nucleon transfer reactions on rare-earth

nuclei; and (2) heavy-ion induced multinucleon (A>4) transfer reactions in

the 2s -Id shell.

Recently, Ascuitto and Vaagen have predicted that, for heavy-ion

induced two-nucleon transfer reactions between highly collective deformed/nuclei, the interference between direct and multistep transfer routes should

produce dramatic and unambiguous alterations in the usual bell-shaped

angular distributions which characterize heavy-ion direct transfer reactions

at energies not too far above the Coulomb barrier. To test these predictions,154 12 14we have measured angular distributions for the reactions Sm( C, C),

182W(12C ,14C), 184W(12C ,14C), 186W(12C ,14C), ^ W ^ C .^ B e ) , and Iftfi Ifi 1ft

W( O, O) at the Brookhaven National Laboratory (BNL) Tandem Facility

using the BNL QDDD magnetic spectrometer. The most significant results

emerging from this study include: (1) the first conclusive evidence that multi-

step inelastic transitions can significantly alter angular distributions, in the

vicinity of the grazing angle, through interference between direct and indirect

transitions; (2) the first observation of a strong influence on particle transfer

reactions from Coulomb excitation; and (3) the observation of a systematic

variation in the shape of the 2+ angular distribution across the rare-earth

region which depends on the structure and distribution of deformed orbitals

around the Fermi level of these nuclei.

The second part of this dissertation concerns compound processes12' 14in heavy-ion reactions, with particular emphasis on C+ N induced reactions.

Forward-peaked angular distributions observed in the reaction ^C(^4N,^Li)29Ne

have been variously interpreted as arising from direct 8-nucleon transfer

and from compound nucleus formation followed by complex particle emission.

To resolve this question, we have measured light particle and gamma-ray 12 14

production from c+ N reactions at low energies and find that the resulting

cross sections are well reproduced within the statistical compound model.

Using the same parameters, we extend this analysis to include all data for6 7 7 9 12 14complex particle ( Li, Li, Be, Be) production for C+ N reactions at

energies up to E = 7 2 .5 MeV. We find that, at all but the highest c.m.energies, both shapes and magnitudes of the measured cross sections can be

accounted for in terms of statistical compound nucleus formation and decay,

provided that the dynamical limit on angular momentum in the entrance

channel predicted by the Bass model is imposed. At the highest energies,

significant departures from the statistical compound reaction mechanism

are noted.

EXOTIC NUCLEAR REACTION MECHANISMS:

MULTISTEP PROCESSES IN HEAVY-ION INDUCED TRANSFER REACTIONS

AND COMPLEX PARTICLE EMISSION IN THE 12C+14N INTERACTION

A Dissertation

Presented to the Faculty of the Graduate School

of

Yale University

in Candidacy for the Degree of

Doctor of Philosophy

by

David L. Hanson

December 1975

%

TO MY FAMILY

*

ACKNOWLEDGEMENTS

I would like to take this opportunity to express my appreciation

to the members of the Wright Nuclear Structure Laboratory community

for providing a stimulating and enjoyable atmosphere which contributed

much to the success of this work.

In particular I wish to thank my thesis advisor Professor

Karl A , Erb, who played an essential role in initiating the two-nucleon

transfer experiments and who provided many good ideas and much useful

advice, both professional and personal, along the way. I am also

indebted to Dr. Robert G. Stokstad, presently at Oak Ridge National

Laboratory, who served as my advisor for two years and contributed

much work, insight, and encouragement to the study of compound reactions

contained in this thesis. I also wish to thank Dr. Catherine Olmer for

her cheerful collaboration over these years and Dr. Martin W. Sachs

for his considerable help with experimental and computer-related problems.

It has been my good fortune to work with Professor Robert J. Ascuitto

and Dr. Jan S. Vaagen, who generously shared their enormous insight

into nuclear reactions, offered much encouragement and advice to the

experimentalists, and took the trouble to involve me in a serious way in

the theoretical analysis of the transfer data. Finally, I wish to thank

Professor D. A . Bromley, who; as Laboratory Director and senior member

of the Heavy-ion Group, has provided support and continuing encouragement

in all aspects of this work.

In addition, I wish to thank the members of the WNSL professional

and secretarial staff for their efforts, in particular, Harriet Comen for

typing, Gail Mercer for the excellent drafting of many of the figures

in this dissertation, George Saportin for machine shop support, and

Kenzo Sato, John Benjamin, and the Accelerator staff for their cooperation

in providing beams for the experiments performed at WNSL.

Special thanks are extended to our colleagues at Brookhaven

National Laboratory, Drs. James J. Kolata and Daniel J. Pisano,

for their invaluable assistance in data taking, and to the entire BNL

Tandem Facility staff for a series of superbly smooth and trouble-free

runs.

Finally, Mrs. Donna Belli is to be commended for her fast

and reliable typing of the manuscript.

TABLE OF CONTENTS

ABSTRACT

ACKNOWLEDGEMENTS

CHAPTER I. INTRODUCTION 1

A . Nuclear Reaction Mechanisms and Heavy-Ions as

Nuclear Probes 1

B. Multistep Processes in Transfer Reactions 4

C. Complex Particle Emission in Compound Reactions 11

D. Overview 14

CHAPTER H. TWO-NUCLEON TRANSFER REACTIONS: THEORY 16

A . Introduction 16

B. Coupled-equations for Inelastic Scattering 19

C. DWBA, CCBA, and the Source Term Method (STM) 28

1. The Distorte d-Wave-Bom Approximation (DWBA) 28

2. The Coupled-Channels-Bom Approximation (CCBA) 32

3. The Source Term Method (STM) • 34

D. STM Equations and Two-Nucleon Transfer Form Factors 38

E. Nuclear Structure for Deformed Nuclei: Single Particle Orbits

and Residual Interactions 42

1. Single-Particle Orbits for Deformed Rotational Nuclei 42

2. Residual Interactions and the BCS Pairing Model 47

Appendix IIA1. Form Factors for Inelastic Scattering 53

Appendix HA2. Derivation of Radial Non-Recoil Form Factor 58

Appendix HA3. Expansion of Deformed Orbits on a Sturmian Basis 71

CHAPTER IE. TWO-NUCLEON TRANSFER REACTIONS: EXPERIMENTAL

PROCEDURE AND PRESENTATION OF DATA 79

A . Introduction 79

B. Beam Production and Transport 82

C. Targets and Charged Particle Detection System 85

D. Electronics and Data Acquisition System 92

E. Data Analysis, Normalization, and Errors 94

F. Presentation of Data 100

Appendix HIA1. The BNL QDDD Magnetic Spectrometer 106

CHAPTER IV. TWO-NUCLEON TRANSFER REACTIONS: THEORETICAL

ANALYSIS 116

A . Introduction 116

B. General Features of Heavy-Ion Transfer Reactions 119

1. Classical Trajectories and L-Space Localization 120

2. Kinematic Restrictions: Transferred Angular Momentum

and Q-Value 127

C. Outline of Calculation and Determination of Parameters 130

D. An Example: l86W(12C ,14C)184W 140

E. Comparison of (*2C ,44C) Reaction on 154Sm, 482W , 184W, and186W : The Role of Underlying Nuclear Structure 143

, 154 12 14 152 ^ 4 _ j1. Sm( C, C) Sm : Strong Direct Quadrupole

Transition 145_ 182„r/12<_ 14 180 A T . .. . _2. W( C, C) W : An Intermediate Case 150

3. * 84W(12C ,*4C)*82W : Weak Quadrupole Transfer Strength 153

4. *88W(*2C ,14C)*84W : (Almost) Pure Nuclear-Coulomb

Interference 155

F. *86W(*2C ,* 8Be)*88Os : The Role of Projectile Binding Energy 158

G. 186W(160 , 180)184W : Projectile Dependence of the Reaction

Mechanism 161

H. Conclusion 165

CHAPTER V. COMPLEX PARTICLE EMISSION IN STATISTICAL

COMPOUND REACTIONS 167

A . Introduction 167

B. The Statistical Model: Formalism 171

C. Gamma-Ray and Light Particle Production in 12O l4 N

Interactions at Low Energy 176

D. Entrance Channel Limiting Angular Momenta: Models 188

E. Complex Particle Emission and Limiting Angular Momenta

in Heavy-Ion Induced Statistical Compound Reactions 197

CHAPTER VI. CONCLUSIONS AND PROJECTIONS 215

A . Introduction 215

B. Transfer Reactions 216

C. Compound Reactions 221

REFERENCES 223

1

A. Nuclear Reaction Mechanisms and Heavy-ions

as Nuclear Probes

The study of nuclear reaction mechanisms has been

a central theme of nuclear physics since its earliest days

(Bo36, Op35, Bu 5 0 ) . This is easy to understand because it

is the reaction mechanism (i.e. the complex of nuclear inter­

actions between target and projectile ultimately leading to

a transfer of nucleons) which determines the type of nuclear

information which may be obtained from a given reaction.

Obviously, an accurate description of the reaction process

is required if the detailed predictions of nuclear structure

models are to be meaningfully related to experimental reac­

tion data.

Nuclear reactions have traditionally been divided

into two conceptually simple and distinct classes on the

basis of the time scale on which the interaction takes place.

These catagorles include: (1) direct reactions, which Involve

only a few degrees freedom of the target-projectile system

and occur in a time comparable to the transit time of the

projectile through the interaction region (10“ 2^s e c ) ; and

(2) compound reactions, in which the projectile is absorbed

by and shares its energy with the target, the compound

system eventually ( 'vlO- ^7 sec) coming to

CHAPTER I INTRODUCTION

2

thermodynamic equilibrium and decaying by emission of a

particle or gamma-ray. Such a classification has consider­

able validity for reactions at moderate energies. As

bombarding energy is Increased still further, the number of

channels open for the decay of the highly excited compound

nucleus becomes enormous. At some point, lifetimes of the

compound states become comparable to direct reaction times

and the distinction between direct and compound processes

becomes much less clear. These two simple reaction models

have been adequate to describe most light-ion induced

reactions, and indeed, because of this simplicity, a wealth

of nuclear structure information, including much of our

knowledge of low-lying states and nuclear level densities,

has been obtained from such reactions. However, light-ion

projectiles are limited in the charge, mass, and angular

momentum which they bring into an interaction.

The introduction, in the last fifteen years, of

heavy-ions as nuclear probes has revealed a new dimension

in the richness and variety of nuclear reaction mechanisms,

and has made possible the study of nuclear structure prev­

iously inaccessible to investigation using lighter projectiles.

It is such "exotic" heavy-ion Induced reactions which form

the subject of this thesis.

The term "exotic" refers here to those nuclear reaction

mechanisms whose effects cannot be adequately understood

within the simplest realistic direct reaction theory, the

first-order dis torted-wave-Born approximation (DWBA)^ or

within a statistical model for compound reactions which

ignores the effects of large angular momentum. This thesis

will present the results of a study of two distinct classes

of "exotic" nuclear reaction mechanisms: (1) multistep

inelastic processes in heavy-ion induced two nucleon trans­

fer reactions on deformed nuclei in the rare earth region;

and (2) compound nucleus formation and decay by complex

particle emission in heavy-ion induced multi-nucleon fA >4

transfer reactions in the s-d shell.

Further background and motivation for the study of

these problems will be given in the remaining sections of

this chapter.

4

B. Multistep Processes In Transfer Reactions

"Multi9tep processes" are generally considered to

be those processes which excite only a limited number of

nuclear degrees of freedom, but which involve interactions

of higher than first order. Important examples of such

processes include: (1) multistep inelastic excitation in­

volving the population of an excited state through several

intermediate inelastic excitations; (2) transfer-plus-

inelastic processes involving direct transfer accompanied

by inelastic excitation of one or both of the particles in

the initial and/or final fragmentation; and (3) sequential

transfer, in which the reaction proceeds by direct and

transfer-plus-inelastic processes through one or more

virtual Intermediate fragmentations to the final state.

The mechanisms of greatest concern in the two-nucleon

transfer reactions studied as part of this thesis are (1)

and (2). Processes of the second type affect heavy-ion

induced transfer reactions in several ways. They introduce

additional transfer routes which may often carry more

intrinsic transfer strength, by reason of Improved parentage

with the final state, or which may be kinematically more

favored than the direct transfer route. Also, the angular

distribution associated with a multistep transfer route may

be considerably broadened as a result of the enhanced

L-space localization of the reaction in a narrow band of

5

partial waves which participate strongly in both inelastic

transitions and transfer. In configuration space the reaction

is localized in a transparent ring and an analogy to the

two-slit diffraction problem is appropriate. Interference

between contributions from opposite sides of the nucleus

may lead to high frequency oscillations superimposed on a

broad diffraction envelope whose width is determined by the

width of the region of maximum transfer strength. Thus,

the narrower "slits" associated with the multistep process

will produce a broader distribution than the wider "slits"

for the direct reaction. Corrections are also Introduced

by de-excitation processes of the first type involving real-

plus-virtual inelastic scattering in the entrance and exit

channels. The magnitude and shape of the observed angular

distributions will, of course, ultimately be determined by

a coherent combination of all transitions Including the

direct.

Multi-step processes have been studied extensively

for light-ion induced reactions on the rare-earth nuclei

(T a 70,A s 71,T a 71,A s 72,A s 7 2 b ,K i 71,A s 74), where strong

inelastic coupling is expected on the basis of the highly

collective nature of these rolatlonal deformed nuclei.

While a full coupled-channels Born approximation analysis,

including all Important inelastic transfer routes, yields

Improved agreement with the experimental data in terms of

6

the absolute magnitude and shape of the angular distri­

butions compared to a DWBA analysis, the differences are

not dramatic and are somewhat sensitive to the optical

model parameters used to generate the distorted waves. A

typical example is the study by King et a l . (K172, K173)186of the W(p,t) reaction populating the ground state

rotational band in shown in Fig. 1-1. Thus, while

definite evidence exists for the Importance of multi-

step processes in light-ion reactions, such evidence,

taken by Itself is not entirely unamblglous or convincing.

One might expect that the effect of these multistep

processes involving Inelastic excitations would be more

pronounced and easily Identifiable for heavy-lon induced

transfer reactions because of two unique characteristics of

the heavy-ion projectile. First, the semi-classical nature

of the heavy-ion projectile leads to particularly simple

bell-shaped angular distributions for direct one-step

transfer on heavy nuclei at energies not too far above the

Coulomb barrier. Secondly, the large mass and charge

which the heavy-lon brings into a collision enhances the

probability for nuclear and Coulomb inelastic excitation in

both the initial and final fragmentation, and thus Increases

the chances of the transfer going by an indirect route.

Nuclear-Coulomb interference is certainly a prominent

feature of heavy-ion

Fig.I-1. Ground state band angular distributions, DWBA, and CCBA186calculations for the W(p,t) reaction at E = 18.0 Mev

(K173).

1000

0° 40° 80° 120° 160°CENTER OF MASS ANGLE

Fig. 1-1

7

inelastic scattering data at energies not too far above the

Coulomb barrier (Ch73, Fo73), as Illustrated by the work of

Ford et al. (Fo73) on the 29®Pb(XXB ) reaction shown

in Fig. 1- 2 .

Indeed, the possibility of observing the influence

from indirect Coulomb excitation on heavy-ion ‘induced trans­

fer cross sections through the interference between nuclear

and Coulomb Inelastic transitions has been the subject of

speculation since the earliest days of heavy-ion physics

(Br59, A872, G o 7 4 ) . However, prior to the work presented

in this thesis, no experimental evidence for such an effect

has been reported. A number of heavy-ion transfer reactions

have been studied prior to this work for which multi-step

processes involving intermediate inelastic transitions

appear to be important (Ma73, Le74, Bo74, Bo74a, Y a 7 4 ) .

Yagi et al. (Ya74), for example, have measured cross

sections for the (*2C , 4C) reaction at 78 M e V

and find that two 0+ states and a removal-type 2+142quadrupole vibrational state in Nd(N ■ 82) , expected

to be excited strongly in a direct two-neutron pickup reaction,

have bell-shaped angular distributions characteristic of

direct reactions with heavy-ions. A second 2+ state con­

sisting of a particle-hole quadrupole vibration forbidden in

direct pick-up has an essentially flat angular distribution

which rises at forward angles. Such broadening of the angular

Fig. 1-2. Angular distributions and DWBA calculations for

208 11 11 208Pb( B, B') Pb(3 ,2.61 Mev) inelastic

scattering at an incident energy of 72.2 Mev

(Fo73). Additional (dot-dash) curves show the

effect of including only Coulomb excitation or

nuclear excitation alone in the DWBA calculation.

Nuclear-Coulomb interference results from a

coherent combination of the two (solid and dashed

line s).

(da

/d£

l)c

m (m

b/s

r)

® c .m . W «q )

Fig . 1-2

8

distribution could result from L-space localization in

the inelastic-plus-transfer route as discussed earlier.

While anomolous angular distributions characteristic of

multistep processes have been observed in all of these

studies, no conclusive evidence has been found for des­

tructive interference between direct and two-step processes

or between nuclear and Coulomb inelastic transitions in

any heavy-ion transfer reaction prior to the work described

in this thesis.

The experiments which we have performed were based

on the suggestion of Ascuitto and Tlagen, who predicted

(As74a) in 1974 that, for heavy-ion-induced two-nucleon

transfer reactions between highly collective rotational

nuclei of the rare-earth region, the interference between

direct and multistep transfer routes should produce dramatic

and unambiguous alterations in the usual bell-shaped angular

distributions which characterize direct heavy-ion transfer

reactions. The original predicted angular distributions

for the , C ) reaction at two different energies

are shown in Figs. 1-3 and 1-4. It may be seen that the

interference effects should be most prominent, in this case,

for bombarding energies not too far above the Coulomb

barrier where both nuclear and Coulomb orbits play an im­

portant role in the transfer process.

9

Any experimental test of these predictions faces

the formidable (and not unrelated) problems of measuring

small (1-100 yb/sr) cross sections with a heavy-ion

energy resolution of better than 100 keV. We have taken

advantage of the excellent energy resolution and large

solid angle characteristics of the Brookhaven National

Laboratory (BNL) QDDD magnetic spectrometer operating at

the BNL Dual Tandem Facility to overcome these difficulties,

and have measured (Er74, As75, Ha75a, Ha75b) angular

distributions for the reactions 186W ( 12C ,1AC ) , 184W (12C , 14C ) ,182 .,12 14 154 .12 14 186 16 18W( C, C ) , Sm( C, C ) , W( 0, 0), and

186W ( 12C, 19Be) at bombarding energies approximately 20 MeV

above the Coulomb barrier. The most significant results

emerging from this study include: (1) the first conclusive

evidence that multistep inelastic transitions can signifi­

cantly alter angular distributions, in the vicinity of the

grazing angle, through Interference between direct and

indirect transitions; (2) the first observation of a strong

influence on particle transfer reactions from Coulomb

exltation; (3) the observation of a systematic variation

in the shape of the ground state band 2+ angular distri­

bution across the rare earth region which depends in a

sensitive way on the structure and distribution of de­

formed orbitals around the Fermi level of these nuclei;

(4) the observation of a dependence of the reaction mechanism

on the nuclear structure and binding energy of the transferred

1 0

nucleons in the light system. Because of this sensitivity

to underlying nuclear structure, heavy-ion Induced two-

nucleon transfer reactions are potentially useful as a

spectroscopic tool for studying correlations in deformed

nuclei. However, because of the complex nature of the

reaction mechanism, the information derived from such

studies must of necessity be Indirect.

Fig. 1-3. Full CCBA angular distributions for transitions to the

0+,2+ ,4+ members of the ground state band in *84W12

for the 186W(12C ,14C) reaction at E C= 70 MeV,

predicted by Ascuitto and Vaagen (As74a) in advance

of the present measurements.

12CFig. 1-4. A similar prediction at E = 110 MeV.

Fig. 1-3

Fig. 1-4

11

C. Complex Particle Emission In Compound Reactions

Another characteristic of heavy-ions, the large

angular momentum which they can bring into a collision, is

an essential feature for the understanding of heavy-ion

induced compound reactions. For example, until quite recen­

tly it was generally believed that the probability for the

decay of any statistical compound nucleus by heavy particle

(e.g. ^Li .^Be.^Be) emission should be negligibly small

because the factor for "preformation" of a large cluster

would be severly reduced. While the qualitative predictions

based on such an argument were in agreement with observa­

tions on compound reactions involving light projectiles

and/or heavy targets, the idea of a "preformation factor",

as pointed out by Cohen (C06O) , is an intuitive concept

based on m acroscopic analogies, such as the vanishingly

small probability for the spontaneous emission of a

snowflake from a drop of water, and is thus not applicable

in the microscopic, quantum mechanical realm. Because of

the degenerate Fermlon nature of the compound nuclear

system, the emission of complex particles is not discrimi­

nated against with respect to light particle emission except

on the basis of available phase space, which is governed

by centrifugal and Coulomb barriers, reaction thresholds,

and the density of states in all residual nuclei as a

function of excitation energy and spin. Thus, because

12

relatively low barriers and large angular momenta favor

compound decay by complex particle emission rather than by

light particle or gamma-ray emission, heavy-ion reactions

on p and s-d shell nuclei which involve the transfer of

more than four nucleons may take place either by direct

transfer or by statistical compound nucleus formation and

decay through the emission of a complex particle. A

careful analysis of the reaction mechanism is then required

to determine what Information may be extracted from the data

for such a reaction.

The second part of this thesis will concern compound

processes in heavy-ion reactions, with particular emphasis

on 12C + induced reactions. The original motivation

for this study came from conflicting interpretations of the12 1 ix f\ 20C ( N , L^) Ne reaction mechanism. Forward-peaked

angular distributions and the (2J + 1) dependence of cross

section magnitudes observed for the ^ N e reac­

tion have been variously Interpreted as arising from direct

8-nucleon transfer (Ma71, Na71) and from compound nucleus

formation followed by complex particle emission (Be73). To

resolve this question, we have measured light particle and12 14gamma ray production from C + N reactions at low ener­

gies and find that the resulting cross sections are well

reproduced in both shape and absolute magnitude within the

statistical compound model in which angular momentum and

parity is strictly conserved. Using the same parameters,

13

we have extended thle quantitative analysis to Include all

data for complex particle ,7L ^ ,^ B e ,^Be) production12 14for C + N reactions at energies up to E ■ 72.5 MeVc .m.

(Ha74). We find that, at all but the highest energies,

both shapes and magnitudes of measured cross sections can be

accounted for in terms of statistical compound nucleus for­

mation and decay, provided that the dynamical limit on angu­

lar momentum in the entrance channel predicted by Bass

(Ba73, Ba74) is imposed. At the highest energies, signi­

ficant departures from the statistical compound reaction

mechanism are noted.

14

1). Overview

This dissertation is organized into two relatively

independent parts concerned, respectively, with the mech­

anisms for heavy-ion Induced direct and compound reactions.

Chapters Il-IV contain a discussion of multistep processes

in heavy-ion Induced reactions, while Chapter V is concerned

with complex particle emission in heavy-ion Induced compound

reactions on light nuclei.

Chapter II presents the theoretical background for a

CCBA analysis of data for heavy-ion induced two-nucleon

transfer reactions on deformed rotational nuclei. We con­

sider in some detail the coupled-channel treatment of

inelastic scattering, the source term formulation of the

CCBA, the simple BCS pairing model for Intrinsic structure

of deformed rotational nuclei, and the structure of the

transfer form factors entering the source term. In Chapter

III, the instrumentation and experimental techniques used

in obtaining the two-nucleon transfer data are described,

and the data are presented. Chapter IV begins with a dis­

cussion of those general features of heavy-ion direct and

multistep reaction processes, such as the seml-classlcal

nature of the heavy-ion projectile, L-space localization

of transfer reactions, and the kinematic dependence of

cross section magnitudes, which do not depend on nuclear

structure. In the remainder of the chapter, the results

15

of a CCBA analysis of the reactions ^S m (*2C ,*4c) t

182W ( 12C,14C ) , 1 (12c ,14 c ) , 188W ( 12C,1A C ) , 186W ( 12C ,10B e ) , and

186w(16o,18o) are presented.

Chapter V contains a complete discussion of ^2C + *4ji

induced compound reactions over an energy range of

Ec # m . “ 6.0 - 72.5 MeV. The experimental arrangement used to

obtain data on gamma-ray production at low energies is des­

cribed, and the results of a statistical model analysis of

this and associated light particle production data are presented.

Dynamical models for heavy-ion complete fusion based on the

liquid-drop model are discussed and their predictions of an

entrance channel angular momentum limit are tested in a

complete statistical compound analysis of complex particle

emission data for ^2C + *4^ reactions.

Finally in Chapter VI, the principal results of this

study are reviewed, and suggestions for Improvements in the

present analysis, and for the extension of this work to new

areas are presented.

16

CHAPTER II. TRANSFER REACTIONS:THEORY

A . Introduction

In this and the following two chapters, we present

a study of the role of multistep process in heavy-ion in­

duced two-nucleon transfer reactions on rare-earth nuclei.

Such higher-order processes should be important for these

permanently deformed nuclei because of their high degree

of inelastic collectivity, and indeed, the experimental

angular distributions presented in Chapter III for transi­

tions to low-lying excited states of the ground band show

dramatic differences in width and structure from the trad­

itional bell-shaped angular distributions observed for the

corresponding ground state transitions. To reproduce even

the main features of these data, it will be necessary to

calculate a transition amplitude which is the coherent

sum of contributions from all physically important direct

and multistep inelastic transitions. As background for an

anaylsis of the experimental data, this chapter will

review the physical basis and essential formalism for

those nuclear reaction and structure models needed to carry

out a realistic multistep calculation of transfer cross

sections, focussing in particular on: (1) the coupled-

channels approach to nuclear reaction theory; and (2) the

pairing model description of intrinsic structure for rota­

tional deformed nuclei.

17

First, the coupled-equations for inelastic scat­

tering are derived and the problem of truncation of the

model space, as it relates to the optical model description

of scattering, is discussed. The basic ideas and under-

lying assumptions of the distorted-wave-Born approximation

(DWBA) and the coupled-channel-Born approximation (CCBA)

treatment of nuclear transfer reactions are then presented.

It is shown that the source term method (STM) for CCBA

analysis involves the solution of a system of coupled-

equations related in a conceptually simple way to the

coupled-equations of inelastic scattering. The structure

of the two-nucleon transfer form factor describing the

overlap between initial and final systems is then discussed.

General features of deformed rotational nuclei are noted

and the Sturmian expansion method for efficiently generating

single particle orbitals in a deformed well is presented.

Finally, the simple BCS pairing model, used to construct

the intrinsic states of the identical-rigid-rotor m o d e l , is

outlined.

Wherever possible, only essential results have been

included in main text with details left to the Appendices

at the end of the chapter. The notation used throughout the

chapter is standard, but will be discussed here briefly totu ^avoid any confusion. (x) represents an ordinary spher­

ical harmonic. Vector coupling of two tensors of rank

X ,X ' is denoted by

- T (XmX'm'|AM)X"X™! mra’ * X

where (XmX'm'lAM) is a Clebsch-Gordon coefficient.

Finally the shorthand notation J = /5J+l' is frequently

used.

19

B. Coupled-EquatIons for Inelastic Scattering

A realistic model for heavy-ion induced transfer

reactions on highly collective nuclei must include the

effects of inelastic transitions in both the target and heavy

residual nucleus, and for a completely accurate description,

must also treat excitations of both the projectile and light

reaction product. In the analysis employed in this work,

only excitations of the heavy systems are considered, but

it will be demonstrated that the main features of the data

are w e 11 reproduced at this level of approximation. This

section will present the formalism for a coupled-channels

description of nuclear and Coulomb inelastic scattering on

heavy rotational nuclei. These results will then be applied

directly in the CCBA analysis of transfer reactions in

the next section.

Following the treatment of Glendennlng (0167), we

consider the inelastic scattering of a particle P from

a nucleus A. We assume that Information on the microsco­

pic or macroscopic structure of the target nucleus, in­

cluding wave functions for the nuclear states, may be

generated within a particular nuclear model, so that a

scattering experiment represents a test of this model

description. The eigenfunctions ^0j(A) of the nucleus A

are determined by

(HA - • 0 ( I I ‘ 1 )

20

where is the model Hamiltonian for the nucleus of

mass A with internal coordinates labelled by A , J

is the total angular momentum, and the label a includes

all other quantum numbers relevant to a description of the

nuclear state.

solution is defined, with appropriate boundary conditions,

given by the sum of the model Hamiltonian H., the kinetic

energy T of the scattered particle, and the interaction

V between the scattered particle and nucleus, with E

denoting the bombarding energy.

wave functions $a j(A), we first construct the spin-orbit

functions for the scattered particle (considered for the

present argument to be a nucleon cluster with spin s), as

where r represents the polar coordinates of r and x

is the spin function of the scattered particle.

The full scattering problem for which we seek a

by

(H - E)u/(T/X) = 0 (11 - 2)

with the Hamiltonian

H = HA + T + V(r,l) (II-2a)

To express this solution in terms of the nuclear

It is convenient to label the whole collection of quantum

numbers defining the intrinsic state of the nucleus and

scattered particle and their relative angular momenta by

c = aJ£sj before the collision, and c ’= a'J'fc's'j' after

the collision. We can now write an expansion for the

solution of (II-2) with angular momentum I and parity tt as

By inserting the expansion (II-3) for ^ into the

Schrodinger equation (II-2), taking an inner product with

we arrive at a set of coupled-equatlons for the radial

functions u(r) of the scattered cluster which, for each

total angular momentum and parity I11 of the system and

for each channel c', takes the form

M(11 — 3)

Ctt I

where

and

respect to <{>M" and making use of the orthronormality

22

Here,t = /. H + t ( » « )' \ d r 2 r 2

and

where an integration over all internal coordinates and the

polar angles of r leaves a function of radius r ° |r|.

coupled-equations to manageable size while still retaining

the relevant physics of the scattering problem, it is

necessary to radically truncate the model space of (II-3)

to include only states having large cross sections, plus

any other states of interest.

channels neglected in this truncation may be included

through a modification of the effective nuclear interaction.

We follow the procedure of Feshbach (Fe58, Fe62) in defining

projection operators P and Q such that P projects

the full wave function ^(r.A) onto the truncated space

to be used in the calculation. 0 is defined by

Operating on the original equation (II-2) leads to a set

of coupled-equations

To reduce the problem of numerical solution of these

We now consider how some of the effects of the

P + Q - 1.

(E - H Qp )»Q - HQpTp

23

and

<* - v ^ p ' ’V q ( u - 5 b ) ,

where

Hpp = Q H P , and Vp = P'f'-

A formal solution to (II-5a) may be written, using the

operator representation of the Green's function, as

\y m .____ i_ ft \tf

0 E-HQQ+ie Qp p

Similarly, a formal solution for the truncated wave function

Tp can be found by inserting the solution for Y q into

(11- 5 b ) :

'V/(E - T - V)4' => 0 (II-6)

where%V = + V

and T and V are defined as in (II-2a), with

Vpn «* P V Q • Equation (II-6) is of conceptual rather than

practical value. While it represents an exactly equivalent

reformulation of (II-2) in terms of the truncated spaoe and

shows that the effect of the neglected channels enters

through the second term of the effective interaction

potential V, it is still necessary to have a description

of all channels in the Q-space to determine the effective

potential V. In practice, the effective interaction•\j

potential V, which according to (II-6a) should be complex-

24

valued, energy-dependent, and non-local, is replaced by a

simplified model potential U ("optical-model potential").

Typical optical potentials are based on a parametrlzatlon

of elastic scattering data and correspond to the extreme

case in which P projects out only the elastic channel.

However such an elastic scattering optical potential will

not be a good model for the exact effective interaction V

in general whenever there is strong coupling to Important

channels in the Q-space.

In the problem of Interest in this work, i.e.

scattering from highly collective rotational nuclei where

coupling to inelastic channels is very strong,

the model space P must be enlarged to explicitly include

in the coupled-equations (II-4) not only elastic scattering

hut also inelastic scattering to all important states

within the ground state rotational band and to other states

of interest which are strongly coupled to that band

(0167, 0168). The diagonal matrix elements Vc »c « will be

similar for all channels in P and may be adequately

described in terms of an optical model potential for elastic

scattering on a nearby spherical nucleus. The off-diagonal

matrix elements are determined from the non-spherical

multipole components of a complex phenomenological potential

for. the deformed nucleus, of the type given in Appendix IIA1.

Assuming that the scattered particle P remains in

a 0+ ground state, the inelastic coupling elements

V ^ c*(r) between members of the ground state band for a

doubly-even rotational nucleus, derived in Appendix IIA1,

take the form (for c'#c")

Vc ?c"<r > ■ I Ac ' c " (A)£ ^ ! ' c " <r> (II‘7>A>0

where A*fc „(A) is a geometry factor carrying the selec­

tion rules |L-L'|< A £ L+L' and L+A+L' = even for the

relative angular momentum t! between P and A, and Ar T i »t (r) are the intrinsic radial inelastic form factors.v c cFor a multipole transition of rank A between states of

spin and J 2 « the radial form factor becomes

25

a l la 2 2 1^0 0 0/VA (r) (II-8)

where v /\(r) “ NA^r + CA^r (II-8a) , is the sum of the rank A multipole terms of the nuclear and Coulomb potentials.

In particular,

« (~ Ro>n , 3nVN (r)NA (r) " I — Ti Ba ----------------- (II"9)A n=l n! A 3rn

3l iso that, e.g.---- ^^(r) ■ -RQ B2----- + higher order terms.3r

Similarly (G167)I_ n

CA (r) = -1Zp ZA e 2

2 A+l R c

-I (A+n-3)! e c(n)n=l n! (A-2)! A

A+3I gc (n)

1,-1 nl (A+3-n) !

( 1 1 - 1 0 )

26

so that

23

<

> $ c + . ...2

T.n Chapter IV, considerable use will be made of the

fact that the overall sign (or phase) associated with a

given transfer route is a product of the signs for the

individual direct and Inelastic transitions making up that

route. The sign convention determined from such a product-

type distribution of transfer strength applies, of course,

to a particular trajectory and assumes that a fixed phase

convention consistent with the macroscopic model descrip­

tion for Intermediate inelastic transitions has been estab­

lished for the nuclear state functions used in calculating

the transfer form factors. The Interference phenomena

influencing the shapes of the transfer angular distribu­

tions will then be determined in a transparent way by the

relative signs and strengths of the dominant transfer

routes. It is of considerable practical importance, then,

to he able to determine the asymptotic sign of the inelastic

form factor for a given nuclear or Coulomb multipole trans­

ition directly from a knowledge of the g ’s and simple

arguments based on (II-9) and (11-10). It can be seen

immediately from (II-9) and (11-10) that, at least in the

surface region of the nucleus, nuclear and Coulomb potential

27

terms of multipole order 2 and of lowest order in B.Awill have opposite signs whenever the nuclear and Coulomb

B's are of the same sign (which is almost always the case).

The overall sign of an inelastic form factor for a transi­

tion of multipole A will then be determined by the sign of

the sum of lowest order nuclear and Coulomb terms in/J i A J 2\multiplied by the sign of the geometry factor I I, which\0 0 Of

is given by (-1)8 where 2g - + A+ J 2 •

The real parts of the intrinsic Inelastic form factors V^(r,fS), A = 2,4 for 0+ -v 2+ and 0+ -»■ 4+ transitions

for a typical case of heavy-ion scattering on a heavy d e ­

formed nucleus are shown in Fig. II-l. The nuclear and

Coulomb contributions to the inelastic quadrupole (A ■ 2)

form factor for the 0+ 2+ transition have oppositeKTsigns (since $2 anc* B 2 always have the same sign). In

the surface region, the negative nuclear component dominates

while ir the assymptotic region, it is the slowly decaying

(1/r3) positive Coulomb component which is important. As

will be discussed in Chapter IV, inelastic nuclear-Coulomb

interference may take place when trajectories from these

two regions In r-space scatter into the same angle.

Fig. n-1. Intrinsic inelastic form factors for heavy-ion scattering

on a heavy deformed nucleus.

I

Fig. n-i

28

C. DWBA, CCBA, and the Source Term Method

1. The Distorted-Wave -Born Approximation (DWBA)

Now we consider the more general collision problem

in which nucleons are transferred between target and projec­

tile and the fragmentation in the initial and final systems

is not the same. The full collision problem for a reaction

of the general form A(a,b)B is described by the Schrodinger

equation

(H - E)^ = 0.

Assuming the interaction be tween the partners of each

fragmentation a+A and b+B to be of finite range, the

Hamiltonian for each fragmentation i=a,b may be written

as a sum

H " T i + V i

of the kinetic energy T^ of the system and the interaction

potential acting between the two fragments. If we

let aa and bp label the channels, or states of relative

and internal motion, of the initial and final systems,

respectively, then the eigenstates of T , for example,a

will be determined from T a<f>aa = E<t>a a *

From elementary quantum mechanics, the cross section

for the transfer process (aqt) -*■ (bp) is given by

29

do UaUR k b

“ a a . b e " I V . ^ I ( I I - U ) -

Here k is the relative momentum and y the reduced mass

of the initial and final systems, NQ p is a statistical

weighting factor resulting from antisymmetrization , and

Taaibp is the transition amplitude between channels aa

and bR defined by

T aa,bP = < + b S !V b! > (II-12a)

= < W bP_ > |Va |*ac, > (II-12b).

We have written (II-12a) in the "post" form and

(II-12b) in the "prior" form, which differ mainly in the

channel to which the stripping (pickup) interaction ^b(a)

refers. 4; ( + ) (^ (- ) ) are eigenfunctions of H with incoming

(outgoing) plane waves.

We again project the exact wave function onto

a severely truncated model space ^p = PV which satisfies

the equation (II-6)

n,(E - T - V)Vp - 0

n,and replace the complex, non-local effective interaction V

referred to the truncated space by a local complex optical

model potential U.

30

The eigenfunctions for the potential Ub which describes

the scattering of b from B within this truncated model

space, for example, will be defined by

(Tb + Ub - E) ^ bB - 0 (11-13).

It may be shown (Au70) that the transition amplitude can

be put into the form

T aa , bB

for stripping and

T aa,b„ (II-14b)

for pickup.

Considering the pickup reaction (B+x)+a -*■ B+(jC+a) , for

example, we may break up the pickup interaction V ft into

parts corresponding to the interaction V ^ a between target

core B and projectile a and the Interaction V betweenX 3

the transferred nucleons x and the projectile a, so that

T . = < C I v„ + V - U (11-15).aa.bg bg ' Ba Txa a>s aa

A considerable simplification results if we choose the

optical potential Ua to cancel most of Vg a and approxi­

mate the complete n't- ) by the eigenfunctions ofbB

the optical potential for scattering of b on B. The

result

T HP " |v I E (+) > (11-16)aa,bf> bB a«

31

is referred to as the "distorted waves approximation" for

the transition amplitude. The term "distorted-waves-Born

approximation" (DWBA) refers to the special case of a

pure direct transfer in which P projects out a single

channel in each fragmentation and U fl and are

chosen to be elastic scattering optical potentials.

In the DWBA, the distorted waves refer to only

a single channel and thus may he written, for the pickup

reaction A(a,b)B, as the simple products

of target (<J>) and projectile (<f>) internal wave func­

tions and wave functions of relative motion (x)• The

DWBA transition amplitude for pickup may then be written

(II-17a)

(II-17b)

DWBAa,B (11-18).

32

2. The Coupled-Channels-Born Approximation (CCBA)

By extending the truncated model space P to include

several channels in the reaction model it is possible to

take into account explicitely the effects of inelastic

scattering in the initial and final systems. Following an

approach originally formulated by Penny and Satchler (Pe64),

we start again with the distorted wave approximation for

the transition amplitude (11-16)

hut in this case the effective potentials U and Ua b

must be coupled-channel optical potentials with diagonal

components derived from the usual elastic scattering optical

potentials and the off-diagonal elements determined from

some model for inelastic scattering, as discussed in

Sec tion B .

The generalized distorted masses are now sums of

wave function products for all states i in P:

C (+) = E 4>1 <f>1 X 1(+) (11-19)a £ A a a

Substituting this expression into the defining equation

for the distorted waves of potential U

(Ta - E + Ufl) f^+) - 0

and making use of the usual orthogonality relations, we

arrive at a set of coupled-equations for the relative

33

wave functions:

(

and similarly for the generalized distorted waves of the

This formulation includes not only direct transfer from

initial to final state, but all possible inelastic-Plu8~

transfer processes (involving transfer to first order)

within the model space P which connect the initial and

final states.

numerical problem which requires the generation of distorted

waves from two sets of coupled equations and the calcula­

tion of overlap integrals between each pair of incoming and

outgoing channels.

final partition ). The CCBA transition amplitude

then has the form

As it stands, the full CCBA is a difficult

34

3. The Source Term Method (STM)

The "source term method" developed by Ascuitto and

Glendenning (As69) represents a more convenient way of

including inelastic couplings In the transfer problem.

The transfer amplitude is evaluated by solution of only

two sets of coupled equations and the need for evaluating

overlap integrals is avoided entirely. This method has

been used in performing the calculations presented in

Chapter IV of this study.

The coupled-equations of the source term method for

the pickup reaction A(a,b)B may be derived as follows.

The total wavefunction ¥ is first projected onto the two

fragments A+a and B+b using the operator P

%where V is the complex effective potential corresponding

to the truncated space defined by P. If we put (11-22)

P

(11-22).

The truncated wavefunction ¥ P satisfies the equation

(E - T - V)¥p = 0 (11-23)

into (11-23), approximate ^ by

V ^ Ua + ub + vx

35

and apply the othogonality relation

( 11 -2 4)

in addition to the usual orthogonality relations in taking

These equations resemble the coupled-equations for elastic

scattering of Sec. B, hut have an additional term containing

the transfer interaction V and thus treat both inelasticX fl

and transfer processes to all orders. If we assume, as in

C C R A , that transfer is weak relative to inelastic processes

and need only be retained to first order, then

i i an inner product with respect to ♦ A 4>J aA aarrive at the two sets of coupled-equations

and $ <f> ,we

. - I X i' (+)i*i' a

- If

f (+)(II-25a)

and

(II-25b)

and the last term in (II-25a) can be

neglected. The coupled-equations then become

36

( T - ■ * i c+>

" ' J w ( i i - 2 6 a )

- E + O M X K O ) >SW ( i i - 2 6 b >

■ • J t , »r(+) - [ p a

where

Pf l * < * f t l v „ l . ‘ * i > x‘ <+>

Now the entrance channel equations (II-26a) are identical

with the usual system of coupled-equatlons for inelastic

scattering derived in Section B and describe the feeding

of channel 1 by inelastic processes from the other

channels 1'. However, in the final fragmentation, the

channel f is fed from the other channels f' by inelastic

transitions, but it is also fed by the transfer reaction

from the entrance channels 1. Thus, in addition to the

usual Inelastic coupling elements, the coupled-equations

(II-26b) contain a source term E p w h i c h describesi

the production of particles in channel f by transfer.

To obtain the transition amplitude for the transfer

reaction, the homogeneous entrance channel coupled-equa-i(+)tions (II-26a) are first solved for the x * subjectfl

to the boundary conditions that only the elastic channel

has an incoming wave while all channels may have outgoing

waves. Then the source terms are constructed from the

37

solutions x * Finally, the inhomogeneous equations a

(II-26b) describing the final system are solved subject to

the boundary conditions that there be only outgoing waves.

The S-matrix element, or complex amplitude for the outgoing

relative radial wave *n tbe asymptotic region, is

related to the transition amplitude by (Go64)

S ■ <"> - 2*1 Taa,bB aa,bB aa,bfi

and the cross section may be obtained directly.

While the orthogonality assumption (11-24) used in

the derivation of the source term coupled-equatlons is

open to some question (Ra67, 0h70, H o 7 1 ) , nevertheless, it

has been shown that the CCBA transition amplitude can

be exactly derived from the source term coupled-equatlons

(G171) and that the cross section obtained from the STM

equations in the absence of inelastic coupling In the

entrance and exit channels reduces exactly to the DWBA

result (As69).

38

D. STM Equations and Two-Nucleon Transfer Form Factors

In this section, we write down the coupled-equations

and source term for a particular example, the two-neutron

stripping reaction A(T,P)B, and then discuss the structure

of the transfer form factor entering the source term.

The coordinate set {R, r ^ , r 2) used in this

analysis is shown in Fig. II-2. The effects of recoil are

partly included by using the scaling approximation ° $

and rp ■ £ R for the channel radii. Further simplification

is possible if the light systems T and P are assumed

to remain in 0* ground states.

By analogy to (11-23) and (11-26) , we write the

Internal wavefunctions for the Initial system having total

angular momentum I as a product

<f>t7Tl(R,A,T) 5c a J — r— a T ■ + t T T -+• M

(R> 9 <(,t (T) L 0 * A (A) ( H - 2 7 )I

The eigenfunction of the total Hamiltonian H may then

be written in terms of the (T-A) relative motion wave

functions u fc7T (R) as

m.M 1 r 1 1 H i u ^ |* T " “ i u (R)*” . t (R,X,*) (11-28)TTl R ^ ' I t TTI

We may then project out the coupled-equations for the radial

functions u t to obtain the result

Fig. n -2 . Coordinate system used in calculation of the two-nucleon

transfer form factors.

Coordinate system

r r

x. = r .a . 1 ~i 1

\

//

/

X . = £ . 0 . i i i

The channel radii i ^ .X p are related to the vector set X 2 } by

N 1 N 2 X t = S + ( Y ' £1 + "t X 2 ^

A N1 N2

(A etc. ) .

Fig. n -2

r 7TI t ttT irl t.irl^TtCRJ+Vtj-CRl-E^u,.1 (R)- - I V t t ,(R)ut , (R) (11-29)

where V**,(R) - ^ ( R , t ,f) | V (R , A ,T) | j. ( R ,* )/>

■ i V (A)^ A A ' (R) ( H - 3 0 )A>0 tc AA

with the inelastic coupling elements defined as in Section

39

B, and , / j AT (fcT+l)T (R) = J L _ [_-d ♦

2v t V dR

We arrive in a similar manner at a set of coupled—equations

for the (P-B) relative motion in the final fragmentation

{ v o + vJ p (R) ■ e p} u p <*>iri fi*I „ tjFl

X ',P P ' <R) V (R) - I Opt (R> (” - 3 DP f P t

where the transfer coupling elements to the (T-A) channels

(source term) are given by

P (R) ■ pt

r < * p p i (R-®>? ) ivSi * » i , f R>I ' f ) - ^ v — — }>a r

A t TTl J- - f u^1W I (£r) I A p t (J) Fab(Jr)

B * A j Pt AB A (11-32)

IJA~ JBI<J<JA + J B

and A71 is a geometry factor. • The finite-range stripping

interaction is approximated by the sum V8 - V(x1 )+ V(x2 )

of the interactions between the light core P and the

transferred particles and N 2< A similar set of coupled*

40

equations describe the pickup reaction. The corresponding

source term for the pickup reaction B(P,T)A will take

the form:

P ^ O O ■p i ,

R (R,A,T) |vP |*M (Ar ,S,?)_!^_---x tirl PTTl B Ar '

B Dir I . i n J A B- - • * ( |R ) I At p (J )FBA( iR) (11-33)A P

JB"JA i J- JB+JA

The radial non-recoil stripping form factors

for the reaction A(T,P)B are defined (As74a, As75a) byA

f ( R ) —7T- ( J * 0 J 0 | J B0 ) V ? . uA, , v B . , Ip?l B 3 v | n | n V I ri I TT t tt

f . „ ( r ) — — ( j . o j o I J r,o)< 4 uA v B Xb ” ( r );

»B(11-34)

This expression consists of a sum over the deformed orbitals

vftir of the intrinsic state for the heavy system and over

the shell-model orbitals t - (n^J,^jT ) of the light system. A BU , . , V . are standard occupation probability ampli-V I ft I TT V | 0 | IT

tudes obtained from a constant pairing-matrix BCS calcu­

lation to be discussed in Section E, and the p* are

parentage (admixture) amplitudes for the light system

orbitals t. The "elementary" form f actor tvSin (B >measures the probability for two neutrons to be transfered

from the projectile configuration [t fl t]^ to the heavy

nucleus configuration E [(vSln), 0 (v0it),i t] . TheJ , J ' 31 3

contribution which each deformed two-particle configuration

makes to the transfer amplitude depends on the ability of

that configuration to assimilate the two neutrons given

their spatial correlation in the projectile. A detailed

derivation of the radial transfer form factors (As74a,

As75a) used in the theoretical analysis of Chapter IV is

presented in Appendix IIA2.

42

E. Nuclear Structure: Single Particle Orbits in

Deformed Nuclei

In this section we discuss the bound state problem

for deformed rotational nuclei and outline the basic

features of the simple BCS pairing model used to construct

the intrinsic states of the heavy system which appear in

the transfer form factor of Section D.

1. Rotational Nuclei

Nuclei in the rare-earth region are characterized

by their large permanent deformations and by the highly

collective nature of their Internal motion. Both of

these properties have their origin in the effects of the

long range part of the residual interaction between nucleons

which becomes dominant over the short range residual inter­

actions when there are large numbers of nucleons outside

of a closed shell (M06O, N a 6 5 ) .

The lowest lying collective states for such deformed

nuclei are rotational. It is assumed in the "strong-

coupling approximation" introduced by Bohr and Mottelson

(Bo52, Bo53) that, because of the large moment of inertia

of these heavy nuclei, the rotation of the nuclear surface

will be slow compared to the velocities of the individual

nucleons. The Hamiltonian for the system will then take

43

the form

H - H p + H,

where Hp describes the motion of the particles In the

body-fixed frame and Hp describes the rotation of that

frame. The total wavefunctlon of the system, known as the

"adlobatic” or "Bohr-Mottelsen" wavefunctlon, can be

written as a product of the Intrinsic and rotational

wavefunctions in the form

Euler angles d,<p,\p which define the orientation of the

body fixed axes relative to the space fixed axes. K is

the projection of the total angular momentum J of the

system on the body fixed symmetry axis. The intrinsic

wavefunctions defined by

are axially symmetric about the body-fixed axis and symmetric

under reflection in a plane perpendicular to the symmetry

axis .

The eigenvalues of the "Bohr-Mottelson" wavefunctions

are (Bo52, Bo53)

<J>(uj) T--------- (w)$INT+(-i)JMK 4"(1 + <3k 0 ) l/2l MK K

J+Kt?J (W)*INT | (11-35) M-K -K

The (^-functions are eigenfunctions of H . w labels theK

(11-36)

E - _ [J(J+l)2 l

9 P 2K ] + Er (II-37).

44

For each intrinsic state or mode of excitation of the

nucleus within the body-fixed system there should be a

rotational band consisting of a set of states

J ■ K, K + 1, K + 2,... with energy spaclngs determined

by the J(J+1) rule for a quantized rigid rotor with a

moment of inertia I. For K ■ 0, only even or odd

members of the band are allowed, depending on the parity

of the intrinsic state. In even-even deformed nuclei,

the lowest intrinsic state will have all nucleons paired

to angular momentum zero, so that K11 - 0+ . Also, the

first excited intrinsic states for even-even deformed

nuclei will in general be collective vibrational states.

In the present work, angular distributions have been

measured for a number of two-nucleon transfer reactions

populating members of the ground state rotational band

up to J17 ■ 6+ in Sm and W Isotopes, as well as.for

the lowest (2+ ) member of the y-vibrational band in

populated in the ^ ,^ C ) reaction. Y " v ibrations

consist of vibrations away from axial symmetry in the

intrinsic system.

The intrinsic Hamiltonian for the deformed system

may be written as the sum of an average independent particle

Hamiltonian plus residual interactions

45

(11-38)

where

By choosing the average field U to be deformed, most of

the long-range part of the residual Interaction may be

included in the single-particle Hamiltonian. For the

approximate structure calculations and BCS pairing calcu­

lations performed in the present study, the single-particle

energies and wavefunctions of the deformed system have

been generated within a deformed Woods-Saxon well of the

form used by Ogle et a l . (0g71) :

component of the isospln operator, equals +1 for neutrons

charge density bounded by the (sharp) deformed nuclear

surface. The Woods-Saxon shape is defined by

U - V(r) + V S0 U , ; , o ) + 1/2(1- t 3)V COUL (11-39)

where the central potential has the form

V (r ) = - [V - T 3V 1 (N-Z)/4A] w(r) (II-39a)

and the spln-orblt potential Is given by

. -»•.V s o (r,p,0 ) (II-39b)P

Here, a is the Pauli spin operator and , the third

and -1 for protons. VCOUL 18 taken to be a uniform

46

v(r) - [l + exp (j a ^ ) J-1

where the nuclear surface is expanded in spherical harmonics

as

I1 +J a V 9>]R(0) - cRp |jL +^E B) (0)

The constant c in this expression is chosen to make the

volume of the nucleus rT , and thus depends on theB's.

In the structure calculations with fixed well depth,

the single-particle eigenvalue problem

Ho<f> _ ■ e _ <() ^V^TT VftlT VJlTT

is solved by expanding the wavefunctions <J> of thevntt

deformed Woods-Saxon well on a cylindrical harmonic

oscillator basis

NnN-Al<t> ~ “ I W l«1T;NnNzA>V^TT' M M A PP z

and then diagonal1zing the resulting matrices to obtain

the elgenenergle8 c - . In this expression, A is theVWTTprojection of the orbital angular momentum on the body-

fixed symmetry axis, and N ,N are the number of oscll-P zlator quanta perpendicular and parallel, respectively,

to the intrinsic symmetry axis, with N ■ N q + N z . The

single-particle states are customarily labelled by the

asymptotic quantum numbers of the dominant cylindrical

component: N Al*z

In the Sturmian expansion method (described in

Appendix IIA3) used to generate the actual radial wave-

function8 which enter the calculation of the transfer form

factors, the eigenenergy 18 fixed, the depth of a

Woods-Saxon well of construction similar to (11-39) is

varied, and a system of coupled-equations is solved for

all bound states with e - The physical solution

having well depth and gross structure similar to that

obtained from the fixed well-depth calculation is then

selected from the many mathematical bound state solutions.

2. Residual Interactions and the BCS Pairing Model

In this analysis, we will be interested only in

calculating the intrinsic structure for the ground state

rotational band, so it is sufficient to consider only the

short-range components of the residual force. The simplest

model for a short range force, the delta-function inter­

action, has the deslreable feature that it gives the lowest

energy to those states having the maximum number of parti­

cles coupled to angular momentum zero (Mo60, Na65) . A

mathematically more tractable short-range interaction

turns out to be the "pairing force" suggested by Bohr,

Mottelson and Pines (Bo58). The pairing Hamiltonian may

be written as

47

48

PAIR

where ^avfiir are tlie creation (distruction) opera­tors for a particle In deformed orbit | v I2tt > , related to

the corresponding time-reversed operators by

H PAIR reproduces the basic features of the delta-function

force. The only non-zero matrix elements occur between

states with pairs of particles coupled to angular momentum

zero and all matrix elements have the same sign. In the

absence of a more sophisticated pairing theory, the pairing

strength G is generally taken to be a constant, with

the implicit assumption that the slngle-partlcle basis will

be limited to orbitals in some symmetric region around the

Fermi level.

The Hamiltonian for particles occupying the set of orb i­

tals v»(vQir} of Interest around the Fermi level is then

assumed to have the form

3r* . (- i ) W 2 - a atVflTT v - R tt

V

(11-41) .

Some progress can be made toward finding the solution of

lowest energy for this Hamiltonian if the pairing inter­

action can be written in a form appropriate for a system

of noninteracting particles or quasiparticles. This may

49

be done by performing a Bogoliubov-Valatin transformation

(Bo58a, B o 5 8 b , Va58)

a + - U a + - V a (II-42a)v v v v v '

“v " Uv ® + + Vv av (II-42b)

If we require the normalization

U2 + vj - 1 (11-43)

then the ''quasiparticle" operators a + ,o + »a.a will satisfy

the usual Permion anticommutation relations. The vacuum

state of these quasiparticle operators defined by

> m avQirl >0 ® *8 related to the vacuum stateof the particles | 0 > by

| Q P V > “ n (Uv + Vv a+v av ) | 0 ] > (11-44).<uv + v

If we expand the Hamiltonian H in terms of quasiparticle

operators and neglect all terms which contain more than two

q u a s l p a r t i d e operators (terms corresponding to quaslpartlcle

interactions), we arrive at a quasiparticle Hamiltonian

^APPROX which, however, does not commute with the number

operator N. As a result, this approximate Hamiltonian

will have no valid eigenstates in a subspace of Hilbert

space corresponding to a fixed number of particles, while

the state of lowest energy for the full Hilbert space will

be the trivial solution corresponding to the particle vacuum

| 0> .

50

To obtain a nontrivial solution, it is necessary to solve

a new eigenvalue problem corresponding to the operator

H - XN, where the Lagtange multiplier X is determined

subject to the constraint that the average value of the

number operator N be equal to the number of fermions

(protons or neutrons) n present in the system.

In the approximation in which all interactions

between quasiparticles are neglected, we require a solution

to the eigenvalue problem for the operator

H - XN - 2 I (e.,-A)V2APPROX v

- G(E U V )2 + GF. V4,V v v V V

+ T. (ev -A)(u2-vJ)(crfa + a M * v ) v

+ 2G £ u V T. u V (a t ia .)+ a tJS )) v v' v' v 'v v v v v

+ "Vf 'V,I pi IV

- G z U V Z (U2-vJ)(0+ 3 + 3f a ) ( 1 1 - 4 5 )v ’ v ' v ’ v v

+ 2T ( e v - X ) u v Vv (0i^ + a v a v )

This eigenvalue problem can be solved exactly if we choose

the coefficients U _ and V such that the termsVfi lT VflTT

containing o and act vanish, i.e. choose

n 2 _ v 2 - 2UvniTVVftTr(e vn,IT“ X) fTT-46' lUvn, VvniI ------------------------- -r -------- ( I I 4 6 ) .

v ' Q ’ t t 1 v ' ft 1 tt ' v ' f t 1 tt '

51

Introducing the notation

h ' 0 I uvn *vvnir ( 1 1 - 4 7 )v H tt

and Evfhr " V ( e vn7r-X )2+ A 2t (11-48)

and using (11-43) and (11-46), we get

UvfiTT " T/zfl + -- — ) (11-49)V EvQir 'f Evf!ir_ ^ \

1/2L l ' *vn- / ( I I - 5 0 )v v!Jir

The "chemical potential" X is defined by

< C QPV| n |q p v > - n

which reduces to 2 r V - n orv!!u

I ( ‘ ■ ( H - 5 1 )vftn V vfiir •

Thus the solution in terms of Uvfiir and Vvfjir is completely

determined once the so-called "gap parameter" A is fixed.

The state of lowest energy for A + 0 and positive

G (attractive pairing force) is the state of zero quasi­

particles defined by (11-49) and (11-50). This state, de ­

noted by

, B C S ^ > " VRU (I,vShT + VvfJlTat;niT a+fj* , 0 > (11-52)

52

is an approximate solution to the pairing problem, found

originally by Bardeen, Cooper, and Schrieffer (BCS) in

connection with work on superconductivity in metals (Ba57),

which may be used to describe the average intrinsic ground

states of heavy deformed nuclei. The physical significance

of the various parameters should be clear from (11-47)-

(11-51) and the definition of V _ . The effect of the

short-range residual Interactions, as described by the

BCS ground state solution, is to alter the occupation of

single particle levels by scattering particles across the2Fermi surface. The quantity V measures the probabi-v O tt

lity that a pair of particles occupies the time-reversed

deformed single particle orbitals vfiir and v-fiir while2vftir is the Pr°babllity for these orbitals to be empty.

The Lagrange multiplier corresponds to the Fermi energy of

the nucleus and the diffuseness of the Fermi surface is

proportional to the "gap parameter" A

In the construction of transfer form factors

(Appendix I1A2) used in the analyses of Chapter IV, we

have assumed that the initial and final doubly-even rare-

earth nuclei are Identical rigid rotorB and have chosen the

intrinsic states of the ground-state band for these nuclei to be

identical BCS vacuum states.

53

APPENDIX IIA1 FORM FACTORS FOR INELASTIC SCATTERING

We follow closely the treatment of Glendennlng

(G167) to arrive at a macroscopic model description for

inelastic scattering on deformed rotational nuclei.

Explicit expressions are derived for the inelastic coupling

elements V c^c"^r (which appear in the STM coupled-A Kequations) and their associated form factors F , „(r).c c

Assuming that the orientation of the target nucleus

does not change significantly during the collision process

(adiabatic approximation) , the full interaction Hamiltonian

H = T + T + V(r ,10)

may be replaced by

H a - T + V ( r ,w) (IIA1-1)

where the rotational energy T^ is neglected. Here u>

are the Euler angles defining the orientation of the

nucleus in the space-fixed (lab) frame.

We consider only excitations between members of the

same rotational band, and express the effective interaction

V between the scattered particle and deformed nucleus in

terms of a local non-spherical complex optical potential

which carries in it the effects of the neglected intrinsic

excitations and their rotations associated with the coupling

to other bands.

54

Inelastic excitation of rotational states involves

interaction of the projectile with non-spherleal compo­

nents of the potential field. It is therefore convenient

to parametrize the half-density radius of the deformed

nucleus by

R - RQ + 6 R ( 0 ’ ')

[l + T a Y (0* (IIA1-2)- R0 11 + T. a. YX , K

where the coordinates refer to the body fixed

symmetry axis, and to assume, as in the case of spherical

nuclei, that the strength of the optical potential depends

only on the distance r - R from the nuclear surface.

A simple Woods-Saxon volume potential for the nuclear

field would then take the form

VN U C L (r"R) " V o fP< r > + iW0 fi(r)-1

where

fx (r) “ 1 + exp\ aox . J

Expanding the nuclear optical potential in a Taylor aeries

about the spherical shape givesCO

VN U C L (r_R) = V N U C L < r ' R o ) + 1 (n r } (IIA1-3)n*l

We note that

f e ) " ^ X(C( 0 \ * ’> (IIA1-4)X K

and use the addition theorem of spherical harmonics to obtain

55

( n )6 Y (©' ,4»') (IIA1-5)

AK AK AKwhere the generalized deformation parameters are

defined by the recursion relation

.(*»> . . . . I

AK Xy<Y 0 0 0

wi th( 1 ) ~ „ _

6 = a , a ■ p , and A " /2A+1'AK Ak AO AThen we may write the nuclear potential as

% U C L (r-R) ’ VN U C L (r-|(‘» + I N A K (r)YA K (0 '''t,')l\.(11A 1-7)

where

N, (r) - I «<» 31A r ' ' n I «„V(r)

A K V nol n! Ak »rn

Similarly, it is possible to write the Coulomb

potential for a charge distribution p(r)

, 2 f P lV jL J I r-r * IVC0U L^r " z z ’e z j , Hb v-t-, dr'

in terms of the as

VC0IIL(^ ) " V C0UL<r> + ^ CA K (r) < H A l - 8)

For the detailed form of C (r), we refer to (G167).AK-

56

By summing the nuclear and Coulomb potentials and

transforming from the body-fixed to the laboratory frame

using the P-rotation functions, we finally arrive at

where " VA K " ^AK + ^A k ' ^*r8t term *-n thisexpression is the usual spherical optical potential and has

only diagonal matrix elements. It is the second, non-

spherical part that is responsible for excitations of the

deformed nucleus from one rotational state to another.

The inelastic coupling elements then have the form

(IIA1-10).

Here the Inelastic form factor is defined by

(w » r ) | |V| I*

(IIA1-11).

57

Using the Bohr and Mottelson adiabatic wavefunction

for a permanently deformed nucleus (Bo52,Bo53)A

V m , <“ 'r) ■ i T u ^ ) 1 ' 2 )we get

^ ? c ..(r, - v A K < r > < ((">r , I >0 (IIA1-12)

The reduced matrix elements, which express the

probability for the transition, are evaluated as Integrals

over three p-functions, and take the formP A +p A

$ I | I I $a ' J ' K ' M ' 1' l + <5 1 1AO

■ J'J" (K' - K" - 0yo o ofor the ground state band of doubly-even rotational nuclei.

For the relevant case of a target and projectile both

with spin zero and positive parity, the channel spin I is

J u 81 the orbital angular momentum of the projectile. Then

£',J' for all channels are governed by the selection rules

-* -*• 9 + 9 '?. *■ a ' + j ' , (- l ) - -ft *

and the geometry factor for the coupling elements will be

given by

T+J'+A ( V A 9"'

0 0 0A^ c „(A) = (-1)

58

APPENDIX IIA2 DERIVATION OF RADIAL NON-RECOIL FORM

FACTOR F J (R)AB

The radial non-recoil stripping form factors

Fa b (R) f°r the reaction A(T,P)B are defined in terms

of the coordinates of Fig. II-2 by (As74a, As75a)

FMAMB(r) f d 3r d3r / < * MA(A)i*MB(B)\ ,AB ■ ' 1 2 \ N A B / A

V s < V <P>> ,f0+(T)> p ) o lo 2

■ S / W " j b V p a b (r )j (iia2- 1)

where-*■ M J ~

f a b <r )j - r a b (r> y j <R >-The finite-range stripping interaction V g is approximated

by the sum V s ■ V(x^) + V( x 7) of the interactions between

the light core P and the transferred particles N^ and

N 7 . Vs is derived from the interaction V(x^) that binds

the orbital An identical interaction is used for

p i c k u p .

The light system overlap

< 4>0+(p ). <J>0+ (T) * O p T (x ! ’ x 2 0

is a tensor of rank zero. Since the light system configur­

ations were limited to pairs of equivalent orbitals, the

parentage expansion of the light system overlap takes the

simple form

59

O r (x , X )° - Ib [p .Tl rb (x ,x )° (IIA2-2)PT 1 2 o t tt tt 1 2 0

where

♦ tt ■ 8 ♦nI ET ,T <*2),0

« , (x) = R . (r) i f (x) » R (r)[Y (r)9 X, / (cf) Jm.1w« nm (T..1<p T^T nT T T T 1/2 j,

and<t>

T -iT ^ -yjy "TThe parentage expansion coefficients are defined by

B [P.T] - * - J P l|fA + 0 ♦ IT l7 0'I>tt ^ 0+ ^ t tO ° Jwith

A JtO ° T T [aJ 0 3 t ’o (IIA2-3)constructed from creation operators for the spherical

single-particle basis used in the microscopic model for

the nuclear states of the light system.

Eq. IIA2-1 is then obtained by a tensor expansion of

the heavy systems overlap in terms of the transferred total

angular momentum J :

JThe heavy systems overlap may be written as

Q u i ( W " “P T < ^ % B|at (xi )a+ (x2} I*a A ^AB 1 2 j p B 1 2 A (IIA2-4)

We first tensor expand the external (lab) frame

creation operator a^ ( X ) as

a+(X) = I a^"(R)'j£"(X) (IIA2-5)£ .1 m

60

wherem (x) - [yp ( x ) » x . , (o)r

t y i j v"' ' ' ' H 2 ' ' ‘ jWe then express the radial tensor creation operators

a+^(R) in terms of radial creation operators referring to

the intrinsic (body-fixed) frame

aj"<!0 CHA2-ft>

where cj ■■ (<t>,8,i[i) are the standard orientation angles

and n is the projection of total angular momentum on the

intrinsic symmetry axis. After some manipulation using

(IIA2-5.6), the heavy systems overlap function can be put

into the form (As74a, As75a)

° a » ( W • j " « “ j - ( Ri ' V

% i ( V * % ' r ‘ V ’ ? ( , I A 2 - 7 )

where £ and £' are restricted by the parity condition

( _ D £+£ 1 =TTA --nB , and

pABJ (r ,r ) - (-1) A B I (1 P. 1 'n ' | J fl+fi’ )1 2 P Q 1

BJ,B

(IIA2-8)Restricting the discussion to even-even ground-state-

band members for systems A and B, described by the

adiabatic Bohr-Motte1 son state-functions

61

m * / T j . d j a

A 4* M A f 0 ( « - ) | A >(IIA2-9)

__ J T j b j r _^ B 8 " 4V PM B >0^aiM B V 5'

we obtain

< ^ $ M B j a+ Q ( R ) a+ ^ ' ( R )P p J ( oj) « | $ M a > T M !^ B 1 £1 1 i \ r 2 L M . n + n ' a 'v j Jt

A A

. U l ^ ~ U v ' 3 » P JJ B (_8"2 7 MB-° m -!

• <lB|'>+t” < V ‘V ' j , < ,y |A > (1IA2-10)

,n+nf P m .,o

B B

JA 'I A

Jj7 ( V a jm| j b V (j a o j o 1 V ) < B 1 * M (Ri)4 r i ' (r 2> 1 A> « s i ',-n

The heavy system overlap thus becomesA

< X , (V V ■ * (Ja V M IJ B M B> T ( V J 0 IJ B 0)J B

m I t I 7 T ' ■ ^ ’-!>IJ 0 ^ ? 1 » . T <R1-R 2>[>iil(xi>(IIA2-11)

where the radial factors*

/"7n (R , R ) = < C B | a + Q (R )a + _f! (R ) I A 7 >r j i n ' y 1 2 ' £1 1 £ \ r 2 1

depend only on the choice of intrinsic states for A and

B, Since IT * II_ ■ + for all band members, (-1)^ -(-1)^.A a

In the adiabatic Bohr-Mottelson model, intrinsic

states are constructed from a set of deformed orbitals

62

where denotes the orbital energy and the radial

multipole functions. In practice, these radial functions

are generated in a convenient way using the Sturmian

expansion method described in Appendix IIA3.

To calculate the intrinsic f actor ? ' j ’ *

we expand the intrinsic frame radial creation operatprs

in terms of t*- '■’“ formed orbital creation n-eratorstnV | ft | IT

d ^ and the radial multipole functions as

a+f2(R) = I R VftTT ( r ) d+ ft - (_!)«■ (IIA2-13)H V £.1 V | ft 1 1T

where v runs over all possible deformed orbitals con­

taining the proper parity and components. For the

calculation described in this work, the operators ^ | it

are defined with respect to an axially symmetric deformed

Woods-Saxon basis which carries the average deformations

B for systems A and B. Then*

. (R , R ) = < ^ B | a + n (R )a+_ 9 ,(R ) | A >J I J U ' I 1 1 2 ^ 1 11 1 2. ' .1 • 2 1 ^

*

vv

_ vftTT v'fin' i' + i'-ft ^ + ft + ft“ I R (R ) R (R ) (-1) B dT . . dT , . . A

. K i.1 1 V j ’ 2 ' v | ft | ir v' | ft | ir1

(IIA2-14)where we have used the result that, since on some standard

spherical basis,VftlT „ VftTT V - f t l T 1+S.-1/2 VftTT

% (R> ' I V - j W 0 C.»l ' (_1> C " ^ ’

63

it must follow that

The time-reversed deformed orbital creation operators are

defined by

. n H 2-0. +-Qd + . , - (-1) (IIA2-15),v J n | tt v | n | tt

In the calculations presented in this work, we assume that

the intrinsic states for systems A and B are built up

from time-reversed pairs of deformed orbitals. This leads

to the diagonal form

< B ' d + v|n|ud J , !n|Tr ’A ^ > “ 6 w ' < B , p + v | n | J A >(IIA2-16)

with the time-reversed pair creation operator defined by

B v I n I IT ^ V I S I I TT d V 1 1 I TT

With the simplifying assumption of identical deformations,

the intrinsic radial factor collapses to a single sum

< » l ( I I A 2 - 1 7 )

The heavy system overlap can now be writtenA

0 ^ B( x 1 , x 2 ) - I ( j a maj m | J bmb) ^ ( J aOJOIJB0)J JB

' m ' r

y / i + i v** vf2ir vQtt r *. 4 / BI Pf A > ’ R <R,) R (*2) V <X 1 > ® ^ (X2>v x v | n | tt ' t'.r L ^ j 1 o v y J j

(IIA2-18)

64

Since J is even, E “ 2 1 and we arrive at the simplen | n |

formA

O a b (X1 ’ X2) ° ( J a ma j m Ij bmb ) ^ ( J aOJO|JbO)J B

< 1

B lp In. IA > ( R <X1 ’X 2>*J (IIA2-19)v I ft j ir ^ J- j

where

m V / 2 1 J + * ~ n“ jij- T ^ T u o j ' - a l " ) ' - ”

r.I V^TT VftlT r ^ ~1 M^R i ' R r j.‘R 2)Ly t1 (*i> * V r ^ l L

- ^ “ " ( K 2 )Rvn’' (itpfV (x2 ) e X <x2 > > \ ij j'j ' L xj 1 i'] ■ 1 J j j

’ I TTT777 < ] n r - S ! 1 | J O H - i ) 1 ' + * , - °j>j ’ 33 (IIA2-20)

•{*v’ ( * x ) * ^ : c * 2)[yt j (i1) » y t .r ( * 2 ) ] " 1

This last step is carried out at this point but is

possible only because the assumption of equivalent orbitals

in the light system makes the direct and (-) exchange

terms in ( Q ^ q ^ identical in the elementary form

factor (IIA2-25). We have now written the overlap function

0 U l . * 2 > " - ,1, < B lP\ , | n | » * © v ! J r <Xl-X2 ^J V I n I TT

in the form of a superposition of individual deformed

orbital contributions each weighted by an overlap factor

s.In evaluating the overlap factor <^B|P+v |q )n I^ ,

65

we assume that the intrinsic states are direct products of

deformed BCS-vacuum for protons and neutrons. If

operates on the neutron orbits, then the proton overlap is

given simply by the overlap between the two BCS ground

states for protons

l * B C S > A . " J T (UPA+VPA d+P d+p ) | 0 > (IIA2-21a)

^ B C S > B " J ' (UP B+VP B ^+P -+P) | 0 > (IIA2-21b)where p labels the set of quantum numbers {vPir} ,

the prime indicates that only orbits with P > 0 are in­

cluded, and U,V are standard pairing occupation proba-2 2bility amplitudes normalized such that U p + Vp *■ 1.

Then it is easy to show, using the completeness relation

for the vacuum |0 >< 0| “ 1, that

" B ^ ♦ g r s ^ B C S ^ A

- n'(u AUp B+VpAVpB)P

Simlarly, for the neutron overlap, we calculate

b < ^ B C S I TT4* V?n | TT I J

a 0N UA VBv P tt v | P | it v | P | tt ’

where

0N o - n fu A U B+ V A V B .vPtt , i , V n n n n )

n^v I P I it 'ni*v | P |

Finally, for neutron transfer, we have

<'b |P+ ,. i IaV * - UA V B 0P0N r> . (IIA2-22)\ 1 v | P | it / vPtt vPtt vPtt

In general, 0P , ^ 1 » so that

*S B | . . | a \ UA VB\ 1 V I P I TT / — vPtt vPtt

66

In this approximation, the full heavy system overlapbecomes

C ^ B (x1 ,x2) » I(j a m a j m |j b m b ) ^ (j a o j o |j b o )J Tg

J u U C | B | 1 . V v | S > | „ 6 >v ! ! „ ( ! t l - X 2 > ! ? • < 1 1 * 2 - 2 3 )

Now that the tensor expansion of the overlaps has

been carried out, we return to the problem of evaluating

the form factor

FA B < R >J f * rld r 2<\ ^ A B ( Xl»X 2^j* vS^ r ^ X l ,X2^0

Inserting the expressions for the light and heavy systems

ov e r l a p s , we getA

j * MFA n (R)” = ^ (J.OJOlJ.O) I U 10 1 vf i o | X 3 [ P , T ] i y (R)AB J j b A 1 b v in *^ v In w I I t tt ^ tvair j

° i a 2

(IIA2-24)

where

(R ) j f d 3r d 3r^|5vftTr (^l»^2^J* ^ ^tt^x l ,x2^0

(IIA2-25)

* J- , ’-^1 j o ) ( - D J , + £ , ' nj>j i+6j:J.f \ - r v S I n ~ v f i u ~J d 3rld 3r2 < j R tj ( R j j y ^ U x ) « ^ . j ■ ( R2 >''n • , ■ (X 2>Jj ,

V S lst (r1)y * X t ( r 2 )y,T J l (x2 ) ] ° >

Taking the radial stripping interaction to be

vS ■ V 2T .1T (rl) + ’ l j ' ' ! 1 T T T T

wi th V £ . (r) - V (r ) + 1/261 v (r) , <5t°j(j+l)- T 90

f (£+1)— 3 / 4 ,

67

we can rewrite this as

a a r - Q U o x - ! , ^ ' -n

/ d 3r i d 3r 2< : [ r ^ ( R i ^ j C X i ) « *][?]. <R2 ) y 4 ,j t (X2>]j ,

[ \ jT<r2,Rt <r1)VM T(Bi ) 9 * . <r*) ’'tlJ l ( i *>] S > 2

+ / ■ ' 3r 1 <l3r2 C ^ VJJ <Rl > ' , l J <*l> « Rr j - < « 2 > V : | ' ( *2>[|j •

[ V ' i > y * t i T ( * l > * v V T ( r 2 ) R t < r 2 ) \ j T < ; 2 ) ] S > J

We use the result(1IA2-26)

^ [ ^ 2 (b1> * xl / 2 (c' l )J j 8 | j t ' ( * 2 3 8 X l /2 lo23J j i ] ? •

f V ' 2 ’ 8 xl / 2 ( ° l [ | JT® [’

|rt (ii

l / 2' u l ' | J . ” |Y 2T ( r 2> 8 X 1/2( o 2 )J

M r- — LL'J (-1) L G ) e Y , (r

T 1)] l • ^ ( ^ > 1

(1IA2-27)where

A A A

’ * Tj T = j j t L( -1)L+1t +1/2 p 1 i/2,

|jlx Jt ^

r j TLw ^ +i/ 2 j * ' ^ j ’ ; ; : f wZj J x ^ 1

L L’ J

jj.

. W(j • Jl • ,1T JtT ; 1/2 L')J 1 J ’ J

J T J T ®L L' J

(-1) (IIA2-28).

68

to obtain

/~r + M * 2/?1= I i^6 (jo.r-Q I j o x - D - r + R ' - n

j j_ j .■i j

• (-1)M I r, U 'JL L ’ ‘J * ’!'!,!,

• { i > . i L . ( « , ] - » ♦ . i ; . < * ) ] ■ ; }

(IIA2-29).Here we have

- J d 3r [ R ^ 7r(R)*Yp(R) g R t <r >Y i!,T ^ >

and

m J d3r l R ^ n (R)*Yp (R) 0 V £ < r ) I M r)Y t U ) ] L

These integrals may be written as

I v ”

where

(IIA2-30)

l + l^+L , . 5t

o n o8 u T i " i ^ " T +L ( - u * A 7 ^ ? t ( t h L

69

The ra d i a l i n t e g r a l s are e v a l u a t e d u s i n g s t a n d a r d F o u r i e r

t r a n s f o r m a t i o n t e c h n i q u e s , in w h i c h the ra d i a l f u n c t i o n s

are gi ve n by

00

- / dk k2 C (k)* v k ) n (k,,)

V r , VftTT V- J dk k2 x t 1 (k> x t (k).iL ( « )

(IIA- 31a)

(I IA 2- 31 b)

w h e r e

OOVftlT FP f 0 vftTT

V i (k) = ^ J 0d R i R i 1p/k R l>R aj ( R l )

00X t (k ) J d r lr l 21Jl( k r l ) R t ( r l )

ooxY(k) = J d r ;lr l 2.1Jl( k r l )V (r1)T?t ( r 1 )

T T q *• f . I f

for i = 1 ,2 .

F i n a l l y , u s i n g the re s u l t

^ ~ f L 4-r J \

h / r ) » v (k> r J - (-d j ( „ „ „ ] ’ > > (i i *2-32).

we ma y w r i t e the " e l e m e n t a r y " form f a c t o r s in the form

O r O (r)*M = I T T (1 n 1 '-n I JO) (-1)jtVftTT J ^ 1 + Ai r

* y r L L 'J [ r v <r M „ M * t (R). •*, 11?. ’ 1 ' lT .1T | VftTT?) ,t i VftlT J, ’ L

t; A+ p vaire,i ,c i C i . r r . t <*>(.'] y j (r)

( I I A 2 - 3 3 ) .

70

The geometry factor G has the formg l l 'j — l l 'j j _m l l I / l L' j

^ U M ’U I t * M t L L'(-l) /v HT 1 i - i i T \ 0 0 0

A A ( 1 1 ' JAir' LL'(i.0 L'o|jO) | .1T1 " T 0 I sJJt^ s j . J ^

\ L L ’ j ' T(I I A 2 - 34)

with S defined as

<;^T _ / i\ 1t+ ^t+i72 i £+ f-T+ *- * ?££ L (_1) 1 m ^ T * -T

" l? L \ ( 1 £ 1/2'^ ^ * 1 (IIA2-35)0 0 0 J 1t L

In summary, the full radial non-recoil form factor

may be expressed in the formA

FI f B(R) ■ f ‘ w ^ bV r (v j ° i j Bo)j j b

J, Uv I r? | TT Vv I ° I TT E Bt t (P .T I TT t J

w i th

r 2 1' 2 ■« * j. P 1 Otvn.(,) ’ J r i+ 4 i r <ln l-n l J 0 >

L L ’ C , s l 1 ' ' 1 ' M t { RvSlitt.1 , t t R) LRv SJnf . ' i ' , t <R) 1 '

(I1A2-36)

71

APPENDIX IIA3 Expansion of Deformed Orbitals on a

Sturmian Basis

In this Appendix, we review the important features

and technical details of the "Sturmian" method (An70,

Ga73, As75b) for construction of the deformed orbitals

<J> I . (X;e ) which carry the structure of the intrinsicV | P. | TT VQir

state of the heavy deformed system in the transfer form

factors.

We consider the problem of the expansion of a

deformed neutron orbit on a Sturmian basis U £ ^ ( r ) gener­

ated at the energy of the orbit • We wish to generate

the deformed neutron orbitVpTT

\ | n | J i x) " , ! | „ | ~ ! d ^ ~ [V ; > • xa /2e o i a (HA3-D

as a normalized physical solution to non-spherical eigen­

value problem

( r )

T(r) + V S 0 (X :G-S0) ' p

■ ( H A 3 - 2 ) .

Here the neutron is bound in a deformed axially symmetric

Woods-Saxon potential with geometry g = (ro ,a0 ) and

deformation P - (B 2P 4 »p 6 » * • ' • and the ener8y solutionsare grouped in 1^1^ sets corresponding to the conserved

72

q u a n t u m n u m b e r s S2 ( p r o j e c t i o n of total a n g u l a r m o m e n ­

tum on the s y m m e t r y a x i s ) a n d tt (rarity), w i th all a d d i t i o n a l

q u a n t u m n u m b e r s l a be ll ed by the index v .

In the expansion procedure used in this work, the

eigenvalue problem (IIA3-2) is first solved using the

program DEF2NT of Sorensen and Ascuitto. From this

calculation, we obtain for each ftir combination a set

of single-particle energies ant* coefficients for the

expansion of the corresponding orbitals <f> . , (X) on aV I fi I TT

cylindrical oscillator basis

Uv M " (x)> ’ * . C s V ; '»■>>..* i> •N ,N z , A

The dominant £j-components for an equivalent expansion

on a spherical oscillator basis may then be determined

from the tables of Chi (Ch66). This knowledge of the

single particle energies and gross structure of all orbi­

tals to be Included in the intrinsic states may then be

used in connection with the Sturmian method to carry out

a more exact and convenient construction of these orbitals.

Suppose that it is desired to bind the orbital

. . (X) at e „ , where c _ may correspond to theV | n I IT \)QlT \jS7tt

single-particle energy obtained from the rough calculation

above, or may assume some other value, such as half-the-

two-neutron separation energy, for reasons of convergence

discussed in the text of Chapter IV. Now we fix the energy

73

in the eigenvalue problem (IIA3-2) and solve for

the well depths V 0 , m ■ 1,2,3,... which produceID

mathematical bound solutions corresponding to this energy.

The desired physical solution is then identified from among

the many mathematical solutions by its gross structure and

approximate well depth as determined from the rough calcu-

lat ion ab o v e .

The solutions to (IIA3-2) are determined in the

Strumian method as follows.

By inserting the expansion for ♦ into the eigenvalue

problem (IIA3-2) and taking an inner product with respectA 0to spin orbit function [Y? 1(r) 0 we arr*ve at

a set of coupled-equatlons for each solution mPir of

(IIA3-2)

where

-K2 d2 ■K2 i<l .+ l ) __S0. _S0.U <r;Ev n n ) 2S 5T72 + “ '2^77 + v n.1(riG >

C „ v P tt

w i th

„ S 0 , , ,,S0 f 1 > ) j , 2 1 d SO,v 0 , (r ) “ V f i * - — F ( r ; g )£.1 o ( ( *+1 ) < j " r dr

* 2)

74

and

FJJ •'<r :BR) " < [Y (r) ° X l / 2 (o>Jj,, F(?;RB )[Y£(r) 0 X l / 2 (^ > ^

I F, (r ; f B) G?, . (A)X-0,2,.. A n

Outside of the effective range of the Woods-Saxon potential,

Fjj,(r;gB) and VB^*(r;G^®) vanish, and the equations

decouple to take the form

■ 0

with

I (r • - - * 1 i l , -fi2*(ft + l)2..1 ,£:v O tt 2m d r 2 2mr2 Gv P.tt

The radial amplitudes u (r) all approach the form ofa J

a spherical Hankel function with a decay factor2tnevOTP Near the origin, the angular

dependence of F ( r ;gB) disappears, so that only the

diagonal term on the right of (IIA3-3) remains and approa­

ches a constant Vm ^ . The equations again decouple and

the surface-peaked spin orbit term vanishes , leaving

. / \ mPiT . . *L n .(r •E )u (r) - 09. 1 v O t t VJ

with

75

We note now that In the limit of zero deformations

(8 0), the operator becomes self-adjoint, and the

system of coupled-equations (IIA3-3) reduces to a simple

Sturm-Lionvi11e problem. It becomes possible to convert

the system of coupled-equations (IIA3-3) into a matrix

diagonalization problem by expanding the solutions of

(IIA3-3) on a spherical Sturmian basis generated from

L (r;e )u (r) - V F(r;g)u (r ) (IIA3-4)£j v O tt nf. j nf. j nf.j

where the geometry *g ■ "g(g,8) ■ CrQ t^0 ) is chosen to

give an optimum fit to the monopole term in the expansion

of F(r;g8). This procedure has the advantage that, since

the monopole term does not in general deviate too much

from the Fermi form, most of the diagonal parts of (IIA3-3)

are contained in (1IA3-4), which may be conveniently

solved using existing numerical codes.

An essential feature to note here is that, in the

asymptotic region, (IIA3-3) and (IIA3-4) are equivalent.

Thus each spherical Sturmian basis function u R^j(r)

carries the same asymptotic slope as the radial multipole

amplitudes u ^ ^ C r ) , this slope being determined entirelyA j

by the orbit energy • Since each basis function

carries the correct asymptotic slope, convergence in the

surface region and beyond is essentially a convergence in

76

magnitude. Thus the Sturmlan expansion method should lead

to rapid convergence whenever the multipole expansion of

the deformed orbital is dominated by a few strong multi­

pole amplitudes (£j v a l u e s ) . This is in contrast to the

situation for the harmonic oscillator expansion where the

orbital energy eigenvalue obtained from the oscillator

dlagonallzation may be unrelated to the binding energy of

the transferred particle, so that a large number of terms

is generally required to achieve convergence to the correct

slope in the tail region.

Liouville theory to form a complete countable infinite

Bet whose eigenfunctions are orthogonal with respect to

the weight function F(r;g) with a normalization given by

We now expand the radial amplitudes on a Sturmlan

basis

insert this expansion in (IIA3-3), and use (IIA3-4) to

obtain the equations

The basis states

,00

d r u «t . 1 < r ) F ( r ; e ) V t i < r )

1 (IIA3-5).

(IIA3-6)

77

— m f2 tt mfiTT

I, V * . i F ( r ; ° V t j <r)c„ ' * j

v „ na I, { r ' ^ » v t ' r (r)cl T f ( I I A W )

.1 ’ > I n I

A matrix equation for the expansion coefficients may be

obtained by a projection onto the Sturmian basis using

the orthogonality relations (IIA3-5)

■ ' , D. I , I <un * r F i r < r ; E B > V t \ i ' ) c " ' i i \ i ’

j ’>n

or

infill r> _ fi mfiTc = v > y r cnf. j mfiTT n < j ' > | Q | n .1 » n ' i ' n ' f> ' j '

The resulting matrix equation

fP* mn-rr fpa^mQitC - V F C

may be formulated as an eigenvalue problem

FCn' - v j r C " <»«-’>ml (IT

for the matrix F where the eigenvalues are the inverse

well depth and the elgencolumns are the sets of expansion

coef f icients.

It may be seen that the matrix F depends on the

energy through the construction of the Sturmian

78

basis functions un ^^(r). A matrix problem of practical

size Is achieved by truncating expansions (IIA3-1) and

(IIA3-6) so that they Include only a limited number of

oscillator numbers N ■ 2(n-l) + £ (thus limiting j and

n) and by introducing an additional limit j jmax

when appropriate. In a practical calculation, the radialU V^ 1Tmultipole components R * M are calculated for eachr

orbital and written on tape, to be entered as numerical

input to the form factor calculation. Since the energy

of each orbital enters the calculation as an input parameter,

there is no difference in complexity between binding the

orbitals at their eigenenergles or at some common energy

of interest, such as half-the-two-nucleon separation energy

of the transferred particles.

Deformed proton orbitals are constructed by a

similar procedure, except that the generating equation

(IIA3-2) contains a deformed Coulomb potential term.

79

E X P E R I M E N T A L P R O C E D U R E AND P R E S E N T A T I O N OF DATA

A. I n t r o d u c t i o n

M e a s u r e m e n t of cross s e c t i o n s for h e a v y - i o n - i n d u c e d

t w o - n u c l e o n t r a n s f e r r e a c t i o n s on r a r e - e a r t h nu c l e i i m p o s e s

ra t h e r se v e r e r e s t r i c t i o n s on the b e a m p r o d u c t i o n and

p a r t i c l e d e t e c t i o n s y s t e m s employ ed . The p r i n c i p a l p r o b l e m

is that of o b t a i n i n g e n e r g y r e s o l u t i o n of b e t t e r than

100 ke V r e q u i r e d to r e s o l v e h e a v y - i o n r e a c t i o n p r o d u c t s

p o p u l a t i n g m e m b e r s of the ground state r o t a t i o n a l band in

the r e s i d u a l nu c l e i involved. T a n d e m van de G r a a f f a c c e l ­

e r a t o r s are c a p a b l e of p r o d u c i n g b e a m s of a d e q u a t e e n e r g y

w i t h an en e r g y sp re ad (AE/E) of b e t t e r than 0. 02 % and

are t h e r e f o r e not a l i m i t i n g factor in o b t a i n i n g good

e n e r g y r e s o l u t i o n . A m o n g p o s s i b l e d e t e c t i o n s y st em s, the

least e x p e n s i v e p o t e n t i a l so lu t i o n , a c o u n t e r - t e l e s c o p e

p a r t i c l e i d e n t i f i e r s y s t e m u s i n g solid state d e t e c t o r s ,

w i ll not give h e a v y - i o n e n e r g y r e s o l u t i o n of b e t t e r than

about 200 keV.

The second s e r i o u s r e s t r i c t i o n on the c h o i c e of a

d e t e c t i o n s y s t e m is the small size of the cross s e c t i o n s

to be m e a s u r e d (1 - lO O y b / s r ) . In p r i n c i p l e , e n e r g y

r e s o l u t i o n of b e t t e r than 5 keV could he o b t a i n e d in a

p a r t i c l e - g a m m a c o i n c i d e n c e m e a s u r e m e n t i n v o l v i n g a Ce(Li)

CHAPTER III TWO-NUCLEON TRANSFER REACTIONS:

80

gamma d e t e c t o r and s i l i c o n p o s i t i o n s e n s i t i v e d e t e c t o r , but

the low d e t e c t i o n e f f i c i e n c y of Ge(Li) d e t e c t o r s and

c o m p l e x i t y of a n a l y s i s r e q u i r e d to o b t a i n an a n g u l a r d i s t r i ­

bu ti on from p a r t i c l e - g a m m a c o i n c i d e n c e data m a ke this an

u n a t t r a c t i v e so lu tion.

The t r a d i t i o n a l m e t h o d for o b t a i n i n g e x c e l l e n t e n e r g y

r e s o l u t i o n for b o th light and h e a v y ions has b e e n to use a

, b r o a d - r a n g e m a g n e t i c s p e c t r o g r a p h . H o w e v e r , si m p l e m a g ­

nets of the B r o w n e - R u e c h n e r type (Rr56) are not able to

c o m p e n s a t e for k i n e m a t i c b r o a d e n i n g and m u st e m p l o y a small

a n g u l a r a c c e p t a n c e and solid angle to m a i n t a i n good r e s o l u ­

tion. W h i l e count r a t e s are low and long term s t a b i l i t y is

r e q u i r e d , h e a v y - i o n m e a s u r e m e n t s of the type c o n s i d e r e d

here be c o m e p r a c t i c a l w h e n m a ny such m a g n e t s are c o m b i n e d

into a m u l t i g a p s p e c t r o g r a p h (E n6 2, K o 7 Q ) . ^

In r e c o g n i t i o n of the I n c r e a s i n g i m p o r t a n c e and

p o t e n t i a l of h e a v y - i o n r e s e a r c h , w o rk was begun around 1967

on the d e v e l o p m e n t of m a g n e t i c s p e c t r o m e t e r s of the Q D D D

( q u a d r u p o l e plus three dipo le ) type.

^ In recent tests by V . Lanf or d, K. Frb, W. C a l l e n d e r , and the au th or u s i n g p o s i t i o n - s e n s i t i v e gas p r o p o r t i o n a l c o u n ­ters, d e v e l o p e d by L a nf or d, in the Y a le m u l t i g a p m a g n e t i c s p e c t r o g r a p h , e n e r g y r e s o l u t i o n of 80 kcV was o b t a i n e d for the * <’ ,S m ( 1 2 c t14c) r e a c t i o n at = 50 Me V witha 50 iic/cm^ a e t a l i c 5m target on a 30 g g / c m ^ ca r b o n hacking.

81

By u s i n g a m o r e c o m p l i c a t e d s v s t e m of m a g n e t i c

lenses to c o m p e n s a t e for k i n e m a t i c b r o a d e n i n g and to c o r r e c t

for a b b e r a t i o n s up to fourth order, these m a g n e t s are able

to meet the r e q u i r e m e n t s of broad e n e r g y rang,e, e x c e l l e n t

e n e r g y r e s o l u t i o n , and large solid angle impo se d by h e a v y -

ion t r a n s f e r m e a s u r e m e n t s .

All of the t w o - n u c l e o n t r a n s f e r data p r e s e n t e d in

this thesis were taken at the B r o o k h a v e n N a t i o n a l L a b o r a t o r y

T a n d e m F a c i l i t y (Th74) u s i n g one of the first such m a g n e t s

to come into o p e r a t i o n in N o r t h A m e r i c a , the BNL O D DD

m a g n e t i c s p e c t r o m e t e r (Le70). The fact that all of the

an g u l a r d i s t r i b u t i o n s were m e a s u r e d in f o u r t e e n r u n n i n g

days over a five m o n t h pe r i o d is e v i d e n c e for the e n o r m o u s

p r o d u c t i v e p o t e n t i a l of such an i n s t r u m e n t .

82

B. Beam P r o d u c t i o n and T r a n s p o r t

The T a n d e m Van de Gr a a f f f a c i l i t y at B r o o k h a v e n

N a t i o n a l L a b o r a t o r y (Th74) c o n s i s t s of two MP ta n d e m

a c c e l e r a t o r s , H i gh V o l t a g e E n g i n e e r i n g M P -6 and MP-7,

a r r a n g e d i n - l i n e (Fig. 1 1 1 -1). W h i l e these a c c e l e r a t o r s may

be o p e r a t e d i n d e p e n d e n t l y as c o n v e n t i o n a l t w o - s t a g e t a n d em s

to p r o v i d e be am s for two d i f f e r e n t e x p e r i m e n t s , the u n i q u e

f e a t u r e of in t e r e s t here is the a b i l i t y of these two

a c c e l e r a t o r s to o p e r a t e t o g e t h e r as a t h r e e - s t a g e s y s t e m

to p r o v i d e be am s of h i g h e r e n e r g y than those

a v a i l a b l e from any other p r e s e n t l y o p e r a t i n g tandem. For

t h r e e - s t a g e o p e r a t i o n , the c h a r g i n g and c o n t r o l s y s t e m s of

M P -6 are c o n v e r t e d to n e g a t i v e ion o p e r a t i o n of the h i gh

v o l t a g e te rm inal, and M P -6 is used as an i n j e c t o r of

n e g a t i v e ions into MP-7.

By o p e r a t i n g in the t h r e e - s t a g e c o n f i g u r a t i o n , a

90 MeV 1 ^ 0 2 + beam was o b t a i n e d w i t h 2 00-3 00 nA on ta rg et

for the ^ 8^ W (^ ^ 0 , ^80) m e a s u r e m e n t . An O H ~ n e g a t i v e ion

b e am is p r o v i d e d by a d i r e c t e x t r a c t i o n d u o p l a s m a t r o n

s o u r c e l o c a t e d in si de the n e g a t i v e high v o l t a g e t e r m i n a l of

M P - 6 . T h e s e n e g a t i v e ions are a c c e l e r a t e d to ground,

p i c k i n g up about 7 M e V in e n er gy , and are i n j e c t e d into

MP-7, wh e r e the h i g h - v o l t a g e te r m i n a l is at a p o s i t i v e

p o t e n t i a l of about 10.3 M V . The 7 M e V n e g a t i v e ions

Fig. ra-1. Physical layout of BNL Dual Tandem Van de Graaff

Facility (Th74).

BNL TANDEM VAN DE GRAAFF FACILITY

83

gain 10.3 M e V in be in g a c c e l e r a t e d to the p o s i t i v e te r m i n a l

w h e r e e l e c t r o n s are st r i p p e d off in p a s s a g e t h r o u g h a thin

(2-10 U g / c m 2 ) c a r b o n foil s t r i p p e r located in the terminal.

B e c a u s e of the a d d i t i o n a l e n e r g y ga i n e d in a c c e l e r a t i n g by

M P - 6 , the b e a m has a h i g h e r a v e r a g e c h a r g e state after

s t r i p p i n g than it w o u l d h a v e for e q u i v a l e n t t w o - s t a g e

o p e r a t i o n , and as a result, will gain e v e n m o re e n e r g y in

the final a c c e l e r a t i o n stage c o m p a r e d to t w o - s t a g e o p e r a t i o n .

F i na ll y, the 90 «eV c o m p o n e n t of the b e a m is s e l e c t e d

by an a n a l y z i n g m a g n e t and d i r e c t e d into the O D D D beam

line by a s w i t c h i n g magn et .

C a r b o n be am s used in these e x p e r i m e n t s w e r e p r o d u c e d

by a M i d d l e t o n - t y p e n e g a t i v e - i o n source, m a n u f a c t u r e d by

F x t r i o n Corp., and a c c e l e r a t e d by t w o - s t a g e o p e r a t i o n of the

M P - 7 . A C 5+ b e am at 65 Me V w i t h an a n a l y z e d e l e c t r o n i cj

beam current of 500 nA was produced for the 154 10 l /

S m ( C , C ) m e a s u r e m e n t u s i n g s i n g l e foil s t r i p p i n g and

a t e r m i n a l v o l t a g e of 10.8 M V . The p r o d u c t i o n of a 70 MeV

c a r b o n b e a m r e q u i r e d for a n u m b e r of the e x p e r i m e n t s was

a c c o m p l i s h e d by the use of a dual foil s t r i p p e r s y s t e m

(Th7 4 a ).

Beam t r a n s p o r t plays a ra th er c r i t i c a l role in

o b t a i n i n g pood e n e r g y r e s o l u t i o n w i t h the Q D D D . For the

Q D D D s p e c t r o m e t e r to p r o v i d e o p t i m u m c o m p e n s a t i o n for

k i n e m a t i c b r o a d e n i n g (the v a r i a t i o n of e n e r g y w i th angle for

84

r e a c t i o n p r o d u c t s of a gi ve n O - va lu e) w h i l e o p e r a t i n g at a

large solid angle, it is n e c e s s a r y to m a i n t a i n a o n e - t o - o n e

c o r r e s p o n d e n c e b e t w e e n the p o s i t i o n of the r e a c t i o n p r o d u c t s

e n t e r i n g the magnet, a p e r t u r e and their an gl e of arri va l. This

r e q u i r e m e n t pl a c e s r e s t r i c t i o n s on b o th the c o n v e r g e n c e

angle (±0.07 deg) and the spot w i d t h (±1.4 m m ) of the

beam at the target. All slits and c o l l i m a t o r s have b e en

remo ve d from the Q D D D b e am line to p r e v e n t d e g r a d a t i o n

of the e n e r g y r e s o l u t i o n t h r o u g h slit s c a t t e r i n g . The

b e am is vi e w e d on a quartz, ta r g e t l o c a t e d lust b e y o n d the

s w i t c h i n g m a g n e t and a l l i g n e d on the o p t i c a l axis of the

beam line by s t e e r i n g until a n o n - s t e e r i n g c o n d i t i o n is

a c h i e v e d for the image q u a d r u p o l e s . The b e am is then

fo c u s s e d to a 10:1 v e r t i c a 1- 1o - h o r i z o n t a 1 as pe ct at the

qu ar ts target. Final f o c u s s i n g and s t e e r i n g is d o ne w i t h a

target room q u a d r u p o l e b e t w e e n the qu a r t z target and s c a t ­

tering chamber. The b e am is r o u g h l y c e n t e r e d and f o cu ss ed

to a 3 x 5 mm spot on a zinc s e l i n i d e sq u a r e at the

target p o s i t i o n and final c e n t e r i n g is a c c o m p l i s h e d by

s t e e r i n g onto a gold spot target.

C. T a r g e t s and C h a r g e s - P a r t i c l e D e t e c t i o n S y s t e m

All of the t a rg et s used in these m e a s u r e m e n t s w e r e

e v a p o r a t e d o n to thin (20-30 Jig/cm2) c a r b o n foils. T h e s e

foils, p u r c h a s e d from Y i s s u m R e s e a r c h and D e v e l o p m e n t

C o r p o r a t i o n , w e r e f l o a t e d f r om gl as s sl i d e s and m o u n t e d on

c i r c u l a r ta rg et fr a m e s w i th 3/8 inch and 1/2 inch

d i a m e t e r a p e r t u r e s . T h e s e frames w e re then c l a m p e d in a

s p e c i a l l y d e s i g n e d t a r g e t - m a k i n g jjg c o n s i s t i n g of a

m o u n t i n g frame to a c c u r a t e l y p o s i t i o n three ta rg et frames

with r e s p e c t to a r e m o v a b l e m a s k h a v i n g three h o r i z o n t a l

k n i f e - e d g e sl it s of d i m e n s i o n 2.4 m m x 12.4 mm. The

m o t i v a t i o n b e h i n d the use of h o r i z o n t a l line t a r g e t s will

d i s c u s s e d later in this section.

The s a m a r i u m metal t a r g e t s used in this e x p e r i m e n t

w e r e m a de by r e d u c t i o n of Sn ^O ^ w i t h Zr m e t a l p o w d e r

and e v a p o r a t i o n of the r e s u l t i n g Sm onto a 20 p p / c m 2

c a r b o n b a ck in g. The Sn^O^, p u r c h a s e d from Oak Ri dg e

N a t i o n a l L a b o r a t o r y , was i s o t o p i c a l l y e n r i c h e d to 98.697 in ^ ^ S m . Th e s t o i c h i o m e t r i c ratio of Sm to Zr is

d e t e r m i n e d by the c h e m i c a l b a l a n c e e q u a t i o n

2 S m 20 3 + 3Zr 3Z r 02 + 4Sm.

The e v a p o r a t i o n was i n i t i a t e d by r e s i s t a n c e h e a t i n g of the

S n ^O ^ - Zr m i x t u r e in a t a n t a l u m c h i m n e y in a v a c u u m of

about 5 x 1 0 ” ^ torr. Up to nine t a rg et s of t h i c k n e s s

86

3 0 -7 0 p g / c m ^ could be p r o d u c e d in a si n g l e e v a p o r a t i o n w i th

a 60 mg ch a r g e of Sm 2 0^ - Zr m i x t u r e and w i t h the c a r b o n

foils located a p p r o x i m a t e l y 5 in c h e s from the c h i m n e y to

pr ev en t heat damage.

X 82W, and X 8 8 W h o r i z o n t a l line t a r g e t s were

p r o d u c e d by e v a p o r a t i n g i s o t o p i c a l l y e n r i c h e d W 0^ onto

20 -3 0 U g / c m " c a r b o n b a c k i n g s . I s o t o p i c e n r i c h m e n t of the

WO^ s a mp le s was as follows: 18^W, 97.06%; 94.22%;

1 ft 9 — SW, 94.32%. At the p r e s s u r e of a p p r o x i m a t e l y 10 torr

used here, ^ 0^ has an e v a p o r a t i o n t e m p e r a t u r e of about

1350° (as c o m p a r e d to 2600° for m e t a l l i c t u n g s t e n ) ,

and thus could be e a s i l y e v a p o r a t e d by r e s i s t a n c e h e a t i n g

in a t a n t a l u m ch i m n e y . T y p i c a l l y , the t a r g e t foils w e re

l o ca te d 4 inches from the c h i m n e y and a ch a r g e of about

50 mg of W 0^ was r e q u i r e d to p r o d u c e a set of 100 U g / c m

thick targets.

T a r g e t t h i c k n e s s e s w e r e d e t e r m i n e d r o u g h l y w i t h an

a l ph a p a r t i c l e guage by m e a s u r i n g the d i f f e r e n c e in e n e r g y

2ioloss of 5.30 M e V al ph a p a r t i c l e s from a Po so ur ce

in a c a r b o n foil b a c k i n g and in an i d e n t i c a l b a c k i n g

c o n t a i n i n g the strip of e v a p o r a t e d target m a t e r i a l . Such

an a p p r o x i m a t e d e t e r m i n a t i o n of target t h i c k n e s s is a u s e f u l

i n d i c a t i o n of the e n e r g y r e s o l u t i o n that m i g h t be e x p e c t e d

for a gi ve n target, but a p r e c i s e k n o w l e d g e of ta rg et

t h i c k n e s s is not n e c e s s a r y in our m e t h o d for o b t a i n i n g a b s o ­

lute cross s e c t i o n n o r m a l i z a t i o n .

87

H e a v y - i o n r e a c t i o n p r o d u c t s w e re d e t e c t e d in the

focal plane of the BNL Q D D D m a g n e t i c s p e c t r o m e t e r (Le70)

u s i n g a 5.0 cm long by 1.0 cm h i g h s i l i c o n solid state

p o s i t i o n s e n s i t i v e d e t e c t o r (PSD) loca te d near the c e n t r a l

ray. The Q D D D s p e c t r o m e t e r , shown in Fig. III-2, c o m b i n e s

the c h a r a c t e r i s t i c s of h i gh r e s o l u t i o n , large solid angle,

large d i s p e r s i o n , and broad e n e r g y range ne ed ed to c a r r y

out h e a v y - i o n m e a s u r e m e n t s u s i n g o n - l i n e d e t e c t o r syst em s.

F u r t h e r d e t a i l s on the d e s i g n f e at ur es and o p e r a t i n g c h a r a c ­

t e r i s t i c s of the BNL Q D D D may be found in A p p e n d i x IIIA1.

For all m e a s u r e m e n t s p r e s e n t e d in this work, the

Q D D D ma g n e t was o p e r a t e d at a solid an gl e of 8 msr,

c o r r e s p o n d i n g to an a z i m u t h a l a c c e p t a n c e angle of ± 2 . 2 9 ° .

This solid an gl e r e p r e s e n t e d a c o m p r o m i s e b e t w e e n c o u n t i n g -

rate c o n s i d e r a t i o n s , on the one hand, and r e s o l u t i o n r e q u i r e ­

me n t s and a n g u l a r d e f i n i t i o n on the other. D u r i n g the

o p e r a t i n g pe ri od w h en our d a ta was taken, o n l y q u a d r u p o l e

c o r r e c t i o n s for k i n e m a t i c b r o a d e n i n g were m a d e r o u t i n e l y

w i t h the m u l t i p o l e elem en t. T y p i c a l e n e r g y r e s o l u t i o n of

100 keV or b e t t e r was a c h i e v e d at all ex c e p t the most

b a c k w a r d an g l e s w i t h a c a r b o n b e am and r a r e - e a r t h t a r g e t s

2of t h i c k n e s s 75 H g / c m on c a r b o n h a c k i n g s . T a r g e t -

re l a t e d c o n t r i b u t i o n s to e n e r g y r e s o l u t i o n will be d i s c u s s e d

in some d e t a i l at the end of this section.

Fig. m -2 . Schematic diagram of the BNL QDDD magnetic spectrometer.

DIPOLE 2

CHAMBER

BROOKHAVEN QDDD MAGNETIC SPECTROMETERFig. HI-2

88

The PSD used to d e t e c t the h e a v y - i o n r e a c t i o n

p r o d u c t s c o n s i s t s of a long s i l i c o n s u r f a c e b a r r i e r d e t e c ­

tor with a r e s i s t i v e c o a t i n g e v a p o r a t e d o n t o the b a ck face.

An e n e r g y signal E, p r o p o r t i o n a l to the total e n e r g y

d e p o s i t e d , is t a k e n from the front face of the d e t e c t o r .

The r e s i s t i v e layer at the back of the d e t e c t o r acts as a

c u rr en t d i v i d e r so that w h e n one end of the back face, of

l e n g t h L, is g r o u n d e d and the i n c i d e n t c h a r g e p r o d u c i n g

p a r t i c l e a r r i v e s at a d i s t a n c e X from the g r o u n d e d end,

a p o s i t i o n si gn al p r o p o r t i o n a l to ( X /L )* E m a y be taken

from the ot h e r end of the back. B e c a u s e of the large

d i s p e r s i o n of the Q D D D m a g n e t , e n e r g y r e s o l u t i o n is not

l i mi te d by the p o s i t i o n r e s o l u t i o n (<v 1.0 mm) of the

PSD.

S e v e r a l c o n s i d e r a t i o n s led to the use of h o r i z o n t a l

line targ et s, ra t h e r t h an area spot ta r g e t s , in these

m e a s u r e m e n t s . Since the Q D D D has a v e r t i c a l m a g n i f i c a t i o n

of a p p r o x i m a t e l y 3, it is n e c e s s a r y to r e s t r i c t the

v e r t i c a l e x c u r s i o n of the beam to a 2.4 mm h i gh strip to

i nsure that all r e a c t i o n p r o d u c t s e n t e r i n g the m a g n e t are

c o l l e c t e d on a 1 cm h i g h p o s i t i o n s e n s i t i v e de te c t o r .

(As a part of the p r e p a r a t i o n for e a ch e x p e r i m e n t , the

v e r t i c a l p o s i t i o n of the d e t e c t o r in the c a m e r a box is

also ad j u s t e d to m a x i m i z e yield.) Spot t a r g e t s will, of

course, serve to r e s t r i c t the e f f e c t i v e h e i g h t of the beam,

but if the beam spot is fo c u s s e d o f f - c e n t e r in the h o r i z o n t a l

89

d i r e c t i o n or is large e n o u g h to co ve r the target spot

s e r i o u s d e g r a d a t i o n in e n e r g y r e s o l u t i o n ma y r e s u l t from

"edge e f f e c t s ' , i.e. the d i f f e r e n c e in e n e r g y loss b e t w e e n

r e a c t i o n p r o d u c t s g e n e r a t e d near the c e n t e r of the target

and n e ar the edge may be s u b s t a n t i a l , p a r t i c u l a r l y for

large angle s c a t t e r i n g . T h e s e edge e f f e c t s are e l i m i n a t e d

by u s i n g a line target e x t e n d i n g ac r o s s the w i d t h of the

target frame o p en in g. The use of a line ta rg et also m a k e s

it p o s s i b l e to m i n i m i z e the c o n t r i b u t i o n of e n e r g y loss to

o v e r a l l r e s o l u t i o n . T h i s m a y be d o ne by r o t a t i n g the

target t h ro ug h an an gl e such that the e n e r g y loss of the

p r o j e c t i l e going t h r o u g h the target in the beam d i r e c t i o n

is just equal to the loss of the r e a c t i o n p r o d u c t going

t h r o u g h the target at the d e t e c t i o n angle. For all

m e a s u r e m e n t s p r e s e n t e d in this work, the target was r o t a t e d

by 30° from no r m a l to the beam to wa rd the m a g n e t a p e r t u r e

For the (^2C,^4c) r e a c t i o n in the t u n g s t e n r e g i o n at the

e n e r g i e s c o n s i d e r e d , this has the ef fe ct of r e d u c i n g the

e n e r g y loss d i f f e r e n c e to zero for ^4^ r e a c t i o n p r o d u c t s

d e t e c t e d at about 0 T .„ = 45°, but the c o n t r i b u t i o n fromLA d

e n e r g y loss at m o r e b a c k w a r d an g l e s r e m a i n s s i g n i f i c a n t .

In a d d i t i o n to ma g n e t a b e r r a t i o n s and e n e r g y loss,

im po rt an t c o n t r i b u t i o n s to e n e r g y r e s o l u t i o n re s u l t from

s t r a g g l i n g in the target m a t e r i a l and c a b o n foil b a ck in g.

Ta bl e III-l c o n t a i n s a s u m m a r y of i n f o r m a t i o n on t a r g e t -

90

- r e l a t e d c o n t r i b u t i o n s to o b s e r v e d e n e r g y r e s o l u t i o n for

s e l e c t e d runs w i th v a r i o u s c o m b i n a t i o n s of beam , target,

and d e t e c t i o n an gl e . Some t y pi ca l s p e c t r a are sh ow n in

Figs. III-3, III-4, and III-5. The i m p r o v e m e n t in r e s o l u ­

tion a c h i e v e d by u s i n g the lower-Z c a r b o n b e am and a

m e t a l l i c , ra t h e r than oxide, ta rg et is o b vi ou s. The

i n t r i n s i c r e s o l u t i o n of the O D DD s p e c t r o m e t e r o p e r a t i n g

at a solid angle of 8 msr , af te r target e f f e c t s have

been taken into ac c o u n t , a p p e a r s to he about 1 5 - 3 0 k e V ,

i o 18 6with the e x c e p t i o n of the one case i n v o l v i n g C on W,

w h e r e an e x c e s s i v e ri p p l e on the a c c e l e r a t o r t e r m i n a l

v o l t a g e may have m a de an a d d i t i o n a l c o n t r i b u t i o n . Such

r e s o l u t i o n is m o re than a d e q u a t e to ca rr y out t r a n s f e r

r e a c t i o n s to e v e n - e v e n nu cl ei in the r a r e - e a r t h r e g i o n and

op en s the p o s s i b i l i t y for even m o re d e m a n d i n g s t u d i e s in

the f u t u r e .

TABLE HI-1. Energy Resolution Obtained with the BNL QDDDa

ReactionBeam

Energy(MeV)

LabAngle

Calculated Straggling ^ (keV)

Isotope O C Total

Maximum C Energy Loss

Difference (keV)

ExperimentalEnergy

ResolutionFWHM

(keV)

154Sm<12C ,14C> 65.0 44° 26 0 18 31 0 48 d

182W(l 2C ,14C) 70.0 60° 23 13 22 35 10 59

184W(12C ,14C) 63.0 76° 23 13 22 34 27 75

186W(12C ,14C 70.0 47° 18 10 22 30 1.0 90®

186W(160 . 160) 90.0 30° 33 19 22 44 16 70f

186W(180 , 180) 90.0 67° 32 18 22 43 29 96g

18W 0Be, 70.0 51° 21 12 22 33 39 90

a QDDD operated at A 6 = ± 40 mr, A<p = ± 50 mr, solid angle 0 = 8 msr. Samarium target was metallic Sm2 2 on 20 (i g/cm carbon backing. Tungsten targets were WO on 30 ji g/cm carbon backings.

b lStraggling calculated using the expression (Co66) 6 E = 0 .9 3 6 z[(Z /A )£X ] (keV), where z = average charge of

/ 2projectile, Z = target charge, A = target mass number, A X = target thickness (u g/cm ). Total straggling is

given by 6 TQT = ^ISO T O P E + ^OXYGEN + 6 ^CARBONo

■>

Energy loss difference between projectile passing through target in beam direction and reaction product passing

through target at detector angle.

Fig. m -3 . Fig. m -4 . Fig. m -13 . g Fig. HI-5.

in o C 4 * 4- 154o /12_ 14 _ 152Fig. m -3. Spectrum for the reaction Sm( C, C) sm at12

E = 6 5 MeV and < ^ 3 = 44°.

T7- TTT A a 4 f *. .. 186„,.12 14_ 184Fig. m -4. Spectrum for the reaction W( C, C) W at12

E = 7 0 MeV and 0 _ = 47°.LAB

Fig. HI-5. Spectrum for the reaction 133W(13Ot130)134W at10o

E = 9 0 MeV and ©T A = 6 3 .6 °.LAB

CO

UN

TS

POSITION CHANNEL

Fig. m -3

CO

UN

TS

3 0

20

1 0

,86W ( I2C.,4C),84W

E = 70 MeV

S l a b = 4 7 •

4 +3 6 4 kev

n J .

2 * '

III kev

+ i - O f

0 +

g.s.

+

in— 90 keV

111

H10 2 0 3 0 4 0 5 0

POSITION CHANNEL

+6 0 70

Fig. m-4

CO

UN

TS

10 20 30 40 50

POSITION CHANNEL

Fig. ni-5

92

D. E l e c t r o n i c s and Data A c q u i s i t i o n S y s t e m

Fig. Ill-f is a s c h e m a t i c d i a g r a m of the t y pi ca l

e l e c t r o n i c s a r r a n g e m e n t used in ta ki ng d a ta w i th one

p o s i t i o n s e n s i t i v e d e t e c t o r in the focal pl an e of the

Q D D D and a s i l i c o n solid state m o n i t o r d e t e c t o r in the

s c a t t e r i n g cham be r. W h e n the P S D is c h o s e n to be s u f f i ­

c i e n t l y thick to stop all heavy p a r t i c l e s of in t e r e s t and

d e t e c t o r no is e is a c c e p t a b l y low, a d e q u a t e p o s i t i o n r e s o l u ­

tion over a l i mi te d range of e n e r g i e s can be o b t a i n e d

d i r e c t l y from the r a w - p o s i t i o n - t i m e s - e n e r g y (X-E) signal,

and a r e l a t i v e l y si m p l e c o n f i g u r a t i o n results.

In the BNL T a n d e m data a c q u i s i t i o n system, each

event of in te r e s t , such as the g e n e r a t i o n of a pair of E

and X ’E pu l s e s w h e n a p a r t i c l e s t r i k e s the PSD, will

have a s s o c i a t e d w i t h it an event line and m a y have one or

m o re d e v i c e s c o n n e c t e d to that event line by a d e v i c e

coupler. In the a r r a n g e m e n t of Fig. I I I - 6 , an i n c o m i n g E

signal is a m p l i f i e d , and the r e s u l t i n g b i p o l a r pu l s e is fed

into a si ng le c h a n n e l a n a l y z e r w h i c h g e n e r a t e s the event

t r i g g e r pulse. The a s s o c i a t e d E and X • F. u n i p o l a r p u l s e s

are d e l a y e d by about 1 v sec w i t h r e s p e c t to the l e a d i n g

edge of the event t r i g ge r pu ls e and fed into a n a l o g - t o -

d i g i t a l co nv e r t e r s .

Data a c q u i s i t i o n i 9 c o o r d i n a t e d by a d a ta c o l l e c t i o n

i n t e r f a c e (DCI) c o n s i s t i n g of ev en t t r i g g e r inpu ts , d e v i c e

c o u p l e r s , and a p u s h b u t t o n i n t e r - c o n n e c t Ion m a t r i x w h i c h

c o n n e c t s event lines w i t h their a s s o c i a t e d de v i c e s . W h e n

an event line c o n t r o l r e c e i v e s a t r i g g e r signal, it t r a n s ­

m i t s the s i g n a l to all d e v i c e s a t t a c h e d to that line. The

event line then r e m a i n s in a b u sy st at e u n t i l all d e v i c e s

have r e s p o n d e d w i t h a " c o n v e r s i o n c o m p l e t e " m e s s a g e , at

w h i c h time the DCI s e r v i c e s all ev en t li ne s on the b a s i s of

a p r e a s s i g n e d p r i o r i t y . D a t a from all d e v i c e s a t t a c h e d to

e a ch event line are read into a n a l y z e r s in core of a S I G M A 7

c o m p u t e r .

All event t r i g g e r pu l s e s are sc al ed and st or ed d u r i n g

each run, and the event t r i g g e r p u l s e s g e n e r a t e d by the b e am

c u r r e n t i n t e g r a t o r c o n n e c t e d to the F a r a d a y cup se rv e as a

m e a s u r e of i n t e g r a t e d charge.

A n a l y z e r s and ev en t s c a l e r s st or ed in core of the

S I G M A 7 may be d i s p l a y e d on the s c r e e n of the d i s p l a y te rm i n a l ,

and ga te s ma y be set and s i m p l e p e ak s u m m i n g c a r r i e d out

u s i n g a light pen system. In the c o n f i g u r a t i o n s h o w n in

Fig. I I I - 6 , a gate has been set on a slice ( u s u a l l y a si ng le

E chan ne l) of I n t e r e s t in the t w o - d i m e n s i o n a l E vs. X*E

s p e c t r u m by t a g g i n g w i t h the light pen. E V E N T 4 p o s i t i o n

s i g n a l s f a l l i n g w i t h i n the gate are then st o r e d in a one-

d i m e n s i o n a l a n a l y z e r . At the end of each run, this o n e ­

d i m e n s i o n a l p o s i t i o n s p e c t r u m is p l o t te d and all a n a l y z e r s

and s c a l e r s are filed on d i s c d a ta sets.

93

Fig. in -6. Diagram of electronics and BNL Tandem data acquisition

system configuration for a simple experimental arrange­

ment involving a PSD, monitor detector, and beam

current integrator.

P S D

M O NITO R

DATA COLLECTION INTERFACE 1INTER-CONNECTION MATRIX

E V E N T DATA COLLECTIONTRIGGERS DEV,CES

1 2 3 4 5 6 7 81□ 1 • •2 □ E 2 •3 □ V 34 □ E 4

" IS

¥5 ■67 • 7 •8 • 8

DISPLAYTERMINAL

♦LIGHT PEN

SIGMA 7

COMPUTER

FARADAY —INTE­ EVENT 3

CUP — GRATOR TRIGGER

DISCSTORAGE

F i g . H I - 6

94

E. Data A n a l y s i s , N o r m a l i z a t i o n , and Errors

W h e r e e n e r g y r e s o l u t i o n was s u f f i c i e n t l y pood, as

1 0in the case of C - i n d u c e d re a c t i o n s , yi e l d s for p a r t i c l e

groups could be o b t a i n e d e i t h e r by d i r e c t c o u n t i n g for

c o m p l e t e l y r e s o l v e d states or by si mp le h a n d - f i t t i n g of

G a u s s i a n s d e r i v e d from the line shape of the s t r o n g l y

p o p u l a t e d 0+ gr ou nd state peak. In g e n e r a l the s p e c t r a

were clean and no b a c k g r o u n d s u b t r a c t i o n was n e c e s s a r y .

In the case of ^ 8 2 y ^ 1 2 C ,^ 4 c) data, h o w e v e r , p a r t i c l e

18 2gr ou ps c o r r e s p o n d i n g to a Ta c o n t a m i n a n t in the W

target fell in the re g i o n of the ^ ® ^ W ( 4 + ) peak and made

e x t r a c t i o n of 4+ y i e l d s i m p o ss ib le .

A m o r e e l a b o r a t e p e a k - f i t t i n g p r o c e d u r e was r e ­

qu ir ed to a n a l y z e the data for the *fi0- i n d u c e d r e a c t i o n

1 8 6 y (1 , 1 8q) w here i n c r e a s e d s t r a g g l i n g in the WO^

target and C b a c k i n g r e d u c e d r e s o l u t i o n to about 100 keV

at forward an gl es and 130 keV at m o re b a c k w a r d angles.

In this case, the p e a k - f i t t i n g p r o g r a m S E S A M E (C172) was

used to p e r f o r m a s i m u l t a n e o u s l e a s t - s q u a r e s fit of G a u s s i a n s

to the three peaks In the sp ec tr um . Peak p o s i t i o n s d e t e r ­

m i ne d from an e n e r g y c a l i b r a t i o n and peak w i d t h s o b t a i n e d

from the shape of the st r o n g 0+ g.s. eroup were c o n s t r a i n e d

in the f i t t i n g p r oc ed ur e.

Er ro r bars shown on the data po i n t s are e i t h e r s t a t ­

isti ca l er r o r s in the case w h e r e p a r t i c l e gr ou ps are c l e a r l y

95

re so lved, or, w h e r e m o re e l a b o r a t e a n a l y s i s was r e q u i r e d ,

the f i t t i n g error g e n e r a t e d by the least s q u a r e s m i n i m i z a ­

tion p r o c e d u r e , w h i c h is in g e n e r a l s o m e w h a t l a r g e r than the

s t a t i s t i c a l error.

A b s o l u t e r e a c t i o n cross s e c t i o n s w e r e o b t a i n e d by

n o r m a l i z i n g the m o n i t o r e l n s t l c yield to e l a s t i c s c a t t e r i n g

m e a s u r e d at a forward angle (OjAR % 30°) wh e r e the cr os s

s e c t i o n is p r e d o m i n a t l y R u t h e r f o r d . D e p a r t u r e s from pure

R u t h e r f o r d s c a t t e r i n g w e re i n v e s t i g a t e d w i t h D W U C K D W BA

c a l c u l a t i o n s for the e l a s t i c s c a t t e r i n g channel. O n l y for

the i n i t ia l m e a s u r e m e n t on Sm w h e r e the e l a s t i c n o r m a l i ­

za t i o n run was taken at © L A B = was a n Y s i g n i f i c a n t

c o r r e c t i o n ('W7 ) n e c e s s a r y .

In this n o r m a l i z a t i o n p r o c e d u r e it is n e c e s s a r y to

know the r e l a t i o n s h i p b e t w e e n the ch a r g e state f r a c t i o n for

r e a c t i o n p r o d u c t s and that for forward s c a t t e r e d e l a s t i c

pa rt ic le s. The ratio of ^ 2 C ions (70 Me V i n c i d e n t energy)

d e t e c t e d in the 5+ and 6+ ch ar ge st at es at © l ab = 37.5°

f o l l o w i n g e l a s t i c s c a t t e r i n g from was d e t e r m i n e d to

be f(5+ ) | f ( 6 + ) = 0.032. The c o r r e s p o n d i n g ratio for

65 Me V 1 2 C 's s c a t t e r e d from at © L A B = was

found to he 0.^39. The a g r e e m e n t (to w i t h i n a few perc en t)

b e t w e e n these r e s u l t s and the ta bu l a t e d ch a r g e state f r a c ­

tions p r e s e n t e d in M a r i o n and Y o u n g (wa 6p ) s u g g e s t s that an

e q u i l i b r i u m ch a r g e state d i s t r i b u t i o n is r e a c he d at even

96

the most forward an gl es for the range of target t h i c k n e s s

and b o m b a r d i n g e n e r g y e m p l o y e d in these m e a s u r e m e n t s .

B e c a u s e the t w o - n e u t r o n p i c k u p r e a c t i o n s on the s a m a r i u m

and tu n g s t e n i s o t o p e s have small Q - v a l u e s and, a c c o r d i n g

to M a r i o n ( M a 6 8 ) the a v e r a g e ch a r g e st at e is a sl o w l y

v a r y i n g f u n c t i o n of e n e r g y (and t h e r e f o r e of angle for

these he av y n u c l e i ) , we ma v assume that the ch a r g e state

1A 18fr a c t i o n for C ( 0 ) r e a c t i o n p r o d u c t s is the same, to

w i t h i n a few perc en t, as that for f o r w a r d - s c a t t e r e d

12 i^C( 0) e l a s t i c pa r t i c l e s . The k i n e m a t i c c o n d i t i o n s are

such that this w i ll also be true for the 1 B e r e a c t i o n

p r o d u c t s from the large n e g a t i v e Q - v a l u e (12C , 19Be)IRA

r e a c t i o n on W at 70 M e V i n c i d e n t energy.

M o n i t o r p o s i t i o n , target angle, and the O D D D solid

angle are fixed t h r o u g h o u t data taking, and a k n o w l e d g e of

I n t e g r a t e d b e a m c u r r en t and target t h i c k n e s s se r v e s as a

rough c o n s i s t a n c y check, hut Is not e s s e n t i a l to the

n o r m a 1 i z a t i o n pr o c e d u r e . This n o r m a l i z a t i o n p r o c e d u r e is

quite r e p r o d u c i b l e and yi e l d s a b s o l u t e cross s e c t i o n s that

agree w i t h i n s t a t i s t i c s for o v e r l a p p o i n t s ta ke n w i t h

d i f f e r e n t targ et s over a time snan of s e v e r a l months.

The n o r m a l i z a t i o n co n s t a n t used to o b t a i n a b s o l u t e

cross s e c t i o n s from r e a c t i o n p a r t i c l e y i e l d s is d e f i n e d

as follows:

97

N = E L A S T I C

w h e r e

Y Q D D D f _ _ U l l / s r ) ______________ ^E L A S T I C d T. AR T . A R ' V C O U N T S/ MO N I TOR C O U N T

Y M O N I T O R dfiC M ( e CM)E L A S T I C

d O k _dfl^ CM^ “ e l a s t i c s c a t t e r i n g d i f f e r e n t i a l

E L A S T I Ccross s e ct io n, at the a v e r a g e

c e n t e r - o f - m a s s angle P of the O D D D ,c . m .

d e t e r m i n e d from R u t h e r f o r d s c a t t e r i n g form ul a

or (at b a c k w a r d angles) from DWBA c a l c u l a t i o n ;

Y O D D D = Q D D D e l a s t i c s c a t t e r i n g yield at 0 ;E L A S T I C LAB

Y M O N I T O R = m o n i t o r e l a s t i c s c a t t e r i n g yield;

E L A S T I C

and d!JLAB<°LAB>j o = l a b - t o - c e n t e r - o f - m a s s solid angled' CM<U Cm '

c o n v e r s i o n factor.

The a b s o l u t e d i f f e r e n t i a l cross s e c t i o n for a r e a c t i o n

p r o d u c t s y i e l d s is then given by:O D D D JR E A C T I O N

do _ y q d d d^ = N R E A C T I O N LAB LAB >

drR E A C T I O N Y M O N I T O R d n C M * R CM^

E L A S T I C

(lib/sr)

The a b s o l u t e er ro r on the n o r m a l i z a t i o n of the r e a c t i o n

cross s e c t i o n s is e s t i m a t e d to be <_ 207. The s t a t i s t i c a l and

s y s t e m a t i c er r o r s that w e re c o n s i d e r e d in a r r i v i n g at this

e s t i m a t e of u n c e r t a i n t y are listed in T a b l e II-2.

98

TABLE III-2 . Estimate of Uncertainty in Absolute Cross Sections

The normalization factor used to determine absolute cross sections

is

do , . ELASTICN = dfl ( C M ' CM

YELASTIC d f i LAB( ^LAB)

YMONITOR d O CM( 0CM)

Uncertainties in calculated elastic cross sections

(asumed to be Rutherford) resulting from:

do , ELASTICdp CM

o. a1. Uncertainty in scattering (magnet) angle ( ± 0 .3 )

2. Deviation from pure Rutherford scattering 15

3. Assumption that scattering angle equals angle of average

cross section integrated over magnet aperture

4. Maximum uncertainty in beam energy of 100 keV

YELASTICUncertainties in factor -------------------- resulting from:

y m o n it o rQ5. Elastic peak count analysis

c6. Monitor peak count analysis

7. Variations in monitor angled

e8. Deadtime corrections

4%

5%

5%

2%

4%

4%

4%

0%

Additional contributions from:

9. Difference in charge state fraction between elastics

and reaction products 5%

99

TABLE DI-2 < cont.).

Estimated absolute error < 20%.

Combined effect of movement of beam spot on target and anomolous

rotation of scattering chamber with magnet.

b o- i * 154 12 0 14 . 186 12 1 4 ^ .Significant only for Sm( C, C) and W( C, C) runs where

normalization run was taken near the elastic grazing angle; corrections

of 7% were based on DWBA calculations.

c +Includes statistics and contribution from tail of 2 inelastic peak.

d Resulting from anomolous rotation of scattering chamber with magnet.g

Count rates were limited such that deadtime < 10%, and deadtime

corrections were made automatically by DC I operated in Common

Deadtime Mode.

100

F. ’.’r e s e n t a t i o n of Data

Final a b s o l u t e cr os s s e c t i o n s and er r o r s for the

t w o - n u c l e o n t r a n s f e r a n g u l a r d i s t r i b u t i o n s m e a s u r e d as

part of the e x p e r i m e n t a l work of this thesis are p r e s e n t e d

in T a b l e s III-3 t h r o ug h III-9 and in Figs. III-7 t h r o u g h

111 -1 2 .

101

154 12 14 152 CTABLE m -3. Sm( C, C) Sm E = 65.0 MeV

Absolute Differential Cross Sections

6 c .m .(deg) 0

g .s .

do IdO| c .m .

(/X b/sr)

122 keV4

367 keV

25.05 0.44: 0.4 11.9± 2.2 12.8± 2.4

32.46 20.2± 2.9 22.44:3.0

39.96 8.72± 1.9 42.34:4.2 17.94:2.7

47.42 49.04: 3.2 64.2± 3.6 11.6± 1.5

54.97 170± 6 35.14:2.8 4.144: 0.95

62.33 2344: 6 25.84:1.9 3.704:0.73

69.49 1394: 6 41.0± 3.1 6.94:1.3

77.50 61.14; 3.7 61.74:3.8 11.14:1.6

85.10 20.04: 1.8 28.24:2.2 11.84:1.4

92.45 6.1± 1.1 14.14:1.7 4.704:0.96

102

TABLE in-4. 182W(12C ,14C)18°W E ° = 70.0 MeV

Absolute Differential Cross Sections

6

da .dQ| c .m . (fib/sr)

c .m . + +(deg) 0 2

g .s . 103 keV

39.95 3 .O i l . 3 6 . O i l . 9

47.44 S.Oi 2.3 10. 7 i 2.6

55.10 1 9 .8 i 2.5 1 6 .l i 2 .3

62.41 8 5 .4 i 5.0 10 . 3 i 1.6

69.90 132± 8 2 5 .8 i 4 .4

74.97 115± 13 3 1 .4 i 7.8

79.96 104i 10 3 7 .4 i 6.3

88.25 4 6 .2± 4 .4 2 8 .5 i 3 .6

100.05 6. 8± 1.9 4 .5 i 1.6

102a

TABLE HI-5. 184W(12C ,14C)182W E C = 63.0 MeV

Absolute Differential Cross Sections

6 c.m .(deg) 0

g .s .

dadO(M

c .m .b/sr)

100 keV4

329 keV

59.77 9 .O i l . 6 6 . O i l . 3 9 .5 i l . 6

69.84 2 8 .9 i 3 .0 2 2 .5 i2 .9 1 6 .0 i2 .2

75.25 104i 8 1 1 .l i 3 .8 3 0 .2 i4 .4

79.98 143i 9 4 4 .1 i5 .4 1 2 .2 i2 .9

89.98 117i 9 121i 10 6 .7 i 3 .6

100.29 5 3 .8i 6 .4 7 7 .7 i8 .0 1 3 .7 i3 .2

103

TABLE m -6 . 18W V 84w e 12°

Absolute Differential Cross

= 70.0 MeV

Sections

0 c .m . +

do i . dO| c .m .

( |ib/sr)

+ +(deg) 0 2 4

g .s . 111 keV 364 keV

39.96 7 .5 ± 1.3 1 2 .1± 1.7 9 .8 ± 1,5

45.54 20. 6± 2.1 22 . 7± 2.2 1 6 .7± 1 .9

49.96 3 7 .5± 2.7 2 5 .9± 2.2 20. 6± 2.0

54.49 6 7 .5± 4 .4 2 5 .0± 2.7 2 4 .5± 2.7

59.11 148±6 1 5 .1± 1.9 1 9 .4± 2.1

63.37 259± 6 2 2 .8± 2.3 1 4 .6 ± 1 .5

67.02 308± 9 4 5 .7± 3 .4 6 .4 ± 1 .3

70.21 281± 9 7 8 .0± 4.6 1 .92± 0.72

75.13 250± 9 113± 6 5 .9 ± 1.3

79.03 192± 8 115 A 6 9 .6 ± 1 .7

88.03 7 9 .3± 4.5 8 5 .6± 4.7 1 7 .8± 2.1

96.85 3 3 .6± 2.5 3 4 .6± 2.5 1 3 .0± 1 .5

103a

TABLE m -7 . 186W(12C ,14C)184W E C - 70.0 MeV

Absolute Differential Cross Sections

6 c.m . +

da id f2| c .m .

( yb/sr)

+(deg) 6

748 keV2y

904 kev

39.00 1.86± 0.77 1.16± 0.67

49.06 4 .7 ± 1.3 6 .9 ± 1 .5

59.00 1 2 .2± 2 .4 9 .4 ± 2.1

68.94 8 .9 ± 1.9 3 .5± 1.2

78.97 5 .4 ± 1.3 1 . 02± 0.56

90.25 2.32± 0.79 2.85± 0.85

100.46 1.61± 0.80 0.40± 0.40

104

TABLE HI-8. 186 12 10„ 188^ W( C, Be) Os 12cE = 7 0 .0 MeV

Absolute Differential Cross Sections

6c .m .(deg) +

0g .s .

dadfl c .m .

i/sr)

2+155 keV

35.08 2.92± 0.71 3.10± 0.73

42.46 6.91± 0.89 3 .68± 0.64

47.70 14. O i l . 2 2 .51± 0.51

48.53 1 4 .8 ± 1.8 2.44± 0.74

53.55 22. 5± 2.0 1.34± 0.51

58.49 2 5 .3± 2.0 3 .26± 0.74

68.46 2 7 .O i l . 4 6.60± 0.70

78.51 11 .3±0 .9 3.44± 0.51

90.05 1 .58± 0.59 1 .1 3 i 0.51

105

TABLE m-9. 186W(160 , 180)184W E ° = 90.0 MeV

Absolute Differential Cross Sections

® c .m . +

do.dn| c .m .

( jx b/sr) + +

(deg) 0 2 4g .s . 111 keV 364 keV

57.08 0.76± 0.53 4 .6 ± 1 .3 1.51± 0.75

64.52 5 1 .5± 3.3 15 .3± 2.1 6 . 5± 1.15 5 .4± 3.1 1 2 .7± 1.7 6 .5 ± 1 .0

68.39 100± 6 3 4 .0± 4 .0 10. 5± 2.2

71.95 187± 12 46 .0± 7.1 5 .5± 2.2

72.00 207±9 4 5 ,0± 4 .6 7. 3± 1.7

76.03 240± 22 9 0 .3± 9 .3 1 .55± 0.89

79.88 181± 10 122± 8 0.87± 0.65

84.00 148± 11 8 1 .3± 11 2. 4± 1 .3

90.96 5 5 .9± 6.8 6 9 .6± 8.0 5 .1± 2.2

97.31 1 2 .0 i 2 .3 2 0 .4± 3.1

101.80 1.74± 0.72 1 .7 ± 1.1 2 .05± 0.86

Fig. HI-7. Experimental angular distributions for the reaction

1540 .12 14 152 4 t,12C „ „ TTSm( C, C) Sm at E = 6 5 MeV.

Fig. in-8. Experimental angular distributions for the reaction

182W(12C ,14C)18°W at E C= 70 MeV.

Fig. IE-9. Experimental angular distributions for the reaction

184W(12C ,14C)182W at E C= 63 MeV.

Fig. m -10 . Experimental angular distributions for the reaction

186W(12C ,14C)184W at E C= 70 MeV.

Fig. HI-11. Experimental angular distributions for the reaction

186W(12C ,10Be)188Os at E C= 70 MeV.

Fig. m-12. Experimental angular distributions for the reaction

186W(160 , 180 )184W at E °= 90 MeV.

(JS

/qr

/) u

‘°

(u

p/-

°p

)

100

10

100

154SmC2C. 14C) 152Sm•

,2CE - 65.(3 MeV_

• «-----

i-----

•0 +.s.

> r~r. ”

g

__ _

♦

... i... .....----- ----- t

..... .

> 2 + •

1 2 2 k e V it

• — «k4 ► — • ' ---- 1

•— - — ♦•

♦ ♦1 '*• •

. 4-

♦4

567 keV►

t- ^ ♦•J----- j ----

1 1 tT

----- ----- ----- --- -----

10

10

20 30 40 50 60 70 80 90 100ac.m.

F i g . m -7

(dc

r/d

&)cm

(fjLb/sr)

100

10

10

I

182 vV(I2C, l4C )l£50wl2C

= 7 0 .0 MeV

4 ♦ ,b•

0 +g.s.

♦

1t

{ i1

<i

•

It

I

r~ '' ■

2 +T T

103 kbV1 _

3 0 4 0 5 0 6 0 7 0 8 0 9 0 100 110

e.cm.

F i g . n i - 8

(do

-/d

&)cm

(/

xb/s

r)

1 0 0

10

100

10

100

10

l84Wl(l2C,'‘*c)182\2r

W E = 63.0 MeV

4• —1 l

4► 0 * 9 s.

------ A------1t

>4»

•A

i► 2+

h i 100 keV1 iT

.4329 keV

♦

------ ii

------ ► ►. . - <►

40 50 60 70 80 90 100 110

^c.m. ( d e g )

Fig. HI-9

(JS/q

r/) w

°(UP/-op)

1000

100

10

100

10

100

10

10

10

0.1

I86vV (l2<; . l4c ) l84\ ,2CN E = 7 0 MeV

•*r •i • '

'• c) +

s.»

q-•

-------' i>2 +

■>I ll l\C V

-------1---•

• • ♦ *•

<» ♦

4 +3 6 4 keV ----

* • ♦ ♦▼

— ♦

T

6 +>

CO eV

-------.

J

2 *>

leVy u 4 »

< T— ,—

t

20 30 40 50 60 70 80 90 100 110ftc.m.

Fig. HI-10

(do

-/d

&)cm

(pb/sr)

l86W ('2C ,l0Be)l880s E C = 7 0 .0 MeV

♦ ♦•

44

t 1 g.s.

t4►

f

1 t I

1 t 155 keV 441

30 40 50 60 70 80 90

® c . m .

Fig. ra-Il

(dcr/dii),

100

10

16,l86W ( V 80),84W E° = 90.0 M e V

0 +

g.s.

\.o

S

100

Eo

10

s

to

I

-f

l — 4~

2 *

I keV

t

- t

4 +364 keV

t:

4 0 50 6 0 70 8 0 9 0 100 110ac.m.

Fig. m -12

106

A p p e n d i x 1 1 1 A 1: The HNL O D DD M a g n e t i c S p e c t r o m e t e r

We b e g i n this A p p e n d i x w i th a d i s c u s s i o n of some

gene ra l c o n s i d e r a t i o n s a f f e c t i n g the d e s i g n of m a g n e t s for

h e a v y - i o n r e a c t i o n m e a s u r e m e n t s (He74) and then d e s c r i b e

how the O D D D s p e c t r o m e t e r d e s i g n m e e t s these p a r t i c u l a r

r e q u i r e m e n t s .

I d ea ll y, a m a g n e t i c s p e c t r o m e t e r to he used for he a v y -

ion e x p e r i m e n t s w i t h an o n - l i n e d e t e c t o r s y s t e m should

c o m b i n e the c h a r a c t e r i s t i c s of h i g h r e s o l u t i o n , la rg e solid

angle, large d i s p e r s i o n , and br oa d en e r g y range. For a

n u m b e r of r e a s o n s , these r e q u i r e m e n t s ma y be met s i m u l t a n ­

e o u s l y o n ly w i t h a v e r y la r g e (and t h e r e f o r e e x p e n s i v e )

system.

ESince the r e s o l v i n g po we r RP = — =r of a s p e c t r o -

6 E

me t e r is in g e n e r a l p r o p o r t i o n a l to the ra t i o D/M, w h e r e

D is the d i s p e r s i o n of the m a g n e t and M the m a g n i f i c a t i o n

in the r e a c t i o n plane, it is i m p o r t a n t in the d e s i g n of a

s p e c t r o m e t e r s y s t e m to m a k e the ra ti o D/ M as large as

p o s s i b l e . The d i s p e r s i o n D r e q u i r e d for the s y s t e m will

he d e t e r m i n e d by the type of p a r t i c l e d e t e c t i o n s y s t e m to

be used. W h e n n u c l e a r e m u l s i o n s are used for p a r t i c l e

d e t e c t i o n , the focal p l a n e m u st be kept as c o m p a c t as

p o s s i b l e to m a k e o p t i c a l s c a n n i n g p r a c t i c a l . H o w e v e r , the

p o s i t i o n r e s o l u t i o n for o n - l i n e p a r t i c l e d e t e c t o r s , such as

107

gas p r o p o r t i o n a l c o u n t e r s and s i l i c o n p o s i t i o n s e n s i t i v e

d e t e c t o r s , Is t y p i c a l l y on the o r d e r of 1 mm so that it

is n e c e s s a r y to i n c r e a s e d i s p e r s i o n to insure that r e s o l u ­

tion is not limi te d by the de te c t o r .

The fact that large d i s p e r s i o n is r e q u i r e d for the

use of o n - l i n e d e t e c t o r s m e a n s that if the m a g n e t is also

to co ve r a broad e n e r g y range, then it must have a long

focal pl an e and be v e r y large. Also, if a large solid

an gl e is to be a c h i e v e d , it is n e c e s s a r y to p r o v i d e

c o m p e n s a t i o n for k i n e m a t i c b r o a d e n i n g (i.e. k i n e m a t i c

v a r i a t i o n of p a r t i c l e e n e r g y w i th angle). T h is is e s p e c i a l l y

i m p o r t a n t for h e a v y - i o n or light target e x p e r i m e n t s , but

b e c o m e s d i f f i c u l t in a s y s t e m f e a t u r i n g large d i s p e r s i o n ;

the amount of c o r r e c t i o n ne e d e d is r o u g h l y p r o p o r t i o n a l to

DM, so that if D/ M is held c o n s t a n t , the d i f f i c u l t y of

k i n e m a t i c c o m p e n s a t i o n i n c r e a s e s as . In the Enge

split--pole d e s i g n (Sp67) for e x am pl e, c o m p e n s a t i o n for

k i n e m a t i c b r o a d e n i n g is a c h i e v e d by d i s p l a c e m e n t of the

focal surface. Ho w e v e r , for s p e c t r o m e t e r s w i th large M

and 0, such as the ODDD, an u n a c c e p t a b l v large and

e n e r g y - d e p e n d e n t shift of the focal pl an e wo ul d be r e q u i r e d

to o b t a i n the n e c e s s a r y c o m p e n s a t i o n , and a n o t h e r m e t h o d

must he e m p l o y e d to m a k e this c o r r e c t i o n .

A n o t h e r e s s e n t i a l c o n s i d e r a t i o n in the d e s i g n of a

s p e c t r o m e t e r is the c o r r e c t i o n of a b e r r a t i o n s r e s u l t i n g from

108

f r i n g i n g fields and m e c h a n i c a l i m p e r f e c t i o n s , as w e ll as from

k i n e m a t i c b r o a d e n i n g , w h i c h r e p r e s e n t a p r i m a r y l i m i t a t i o n

on the r e s o l v i n g po we r of any magn et . For a s p e c t r o m e t e r

of la rg e solid angle, the most i m p o r t a n t a b e r r a t i o n s will

be those w h i c h d e p e n d on the angle of d e v i a t i o n from the

c e nt ra l ray in the h o r i z o n t a l and v e r t i c a l d i r e c t i o n s . In

the d e s i g n stage, such a b e r r a t i o n s may be i n v e s t i g a t e d by

ray t r a c i n g c a l c u l a t i o n s . The p o s i t i o n and an gl e of any

ray r e l a t i v e to sore c e n t r a l ray ma y be d e s c r i b e d at the

m a g n e t a p e r t u r e by a v e c t o r X , and then the c o r r e s p o n d i n g

p o s i t i o n of the ray at the focal pl an e w i ll be gi ve n to first

order by X = VX^ wh e r e V is the t r a n s p o r t m a t r i x for

the s n e c t r o m e t c r , c o n s i s t i n g of a p r od uc t of m a t r i c e s of

the in di vi du al e l e m e n t s w h i c h m a k e up the sy s t e m (P e 61).

H i g h e r o r d e r e f f e c t s may be i n v e s t i g a t e d by u s i n g a ray

t r a c in g p r o g r a m to solve the L o r e n t z force e q u a t i o n

d ( m v ) / d t = qv x B p o i n t - b y - p o i n t based on a s e m i - e m p i r i c l e

d e s c r i p t i o n of the m a g n e t i c fields B ( x, y, z) for the system.

O nce a s p e c t r o m e t e r has b e e n bu il t, a b e r r a t i o n s ma y

be i d e n t i f i e d t h r o u g h p h y s i c a l ray t r a c i n g by l o c a t i n g an

a p p r o p r i a t e point so ur ce of p a r t i c l e s at the beam spot and

o b s e r v i n g w h e r e p a r t i c l e s w h i c h pass t h r o u g h a hole In a

m a s k at the m a g n e t a p e r t u r e st ri ke the focal plane (Lo73).

The p o s i t i o n of any ray e m e r g i n g from a point so u r c e w i t h

c o o r d i n a t e s (0 ,A ) at the m a g n e t e n t r a n c e a p e r t u r e ma y be

109

c h a r a c t e r i z e d at any point Z al ong the c e n t r a l ray by a

d i s t a n c e X(0 ,«|>,Z) of the ray r e l a t i v e to the c e n t r a l ray

(0 = 0,<j> =■ 0). This d i s t a n c e will be giv en in terms of a

set of t r an sf er or a b e r r a t i o n c o e f f i c i e n t s up to fourth

orde r by

X(0 , < p , Z ) = (X/0)0 + ( X / q 2 ) 0 2

+ ( X / r 3 ) o 3 + ( x / 0 * ) o * + ( X/<t,4 )-bA

+ ( X /0 24» )0 2(f, +( X/0 2(J) 2 )0 2rt> 2 -

Ray tr a c i n g has sho wn that these a b e r r a t i o n c o e f f i c i e n t s

(X/0 ), (X/0 2 ), etc. v a r y l i n e a r l y as a f u n c t i o n of Z

in the v i c i n i t y of the focal su rf ace , so that ray tr a c i n g

m e a s u r e m e n t s s p e c i f y i n g the beam p r o f i l e in two pl an es

Z^ and Z 2 , p e r p e n d i c u l a r to the c e n t r a l ray and lying

in front of and b e hi nd the e x p e c t e d focal plane, are

ad e q u a t e to r e c o n s t r u c t the beam pr o f i l e at the focal plane.

It turns out that some a b e r r a t i o n c o e f f i c i e n t s do

not af fe ct r e s o l u t i o n and no eff o rt need be m a de to co rr ec t

for them. In ge ne ral , a b e r r a t i o n s a f f e c t i n g the v e r t i c a l

focus of a s p e c t r o m e t e r ma y n e g a t i v e l y effe ct c o l l e c t i o n

e f f i c i e n c y but will have little effect on re so lu t i o n .

The same is true for a b e r r a t i o n s a f f e c t i n g the an g l e of

c o n v e r g e n c e of the be am in the v e r t i c a l d i r e c t i o n and

a b e r r a t i o n s w h o s e initi al v e c t o r e l e m e n t s are small e n o u g h

that their c o n t r i b u t i o n is n e g l i g i b l e . Ho wev er, w h en a

110

large s p e c t r o m e t e r a p e r t u r e is used, there will a l wa ys

r e ma in a b e r r a t i o n c o e f f i c i e n t s wh ic h will affect r e s o l u t i o n

and the shape and p o s i t i o n of the focal pla ne and wh ic h

must be c o r r e c t e d for.

Si nc e the f o c u s s i n g a c t i o n of a s p e c t r o m e t e r is

d e t e r m i n e d to first ord er by the size and e n t r a n c e an gl e of

the m a gn et pole piece s, it is d e s i r e a b l e in c o r r e c t i n g for

hi gh er or der a b e r r a t i o n s to have a v a i l a b l e d e v i c e s w h i c h do

not have m a t r i x e l e m e n t s in first order, but wh i c h do have

second and h i g h e r or de r m a t r i x el eme n ts . C o r r e c t i o n s for

a b e r r a t i o n s can be ma de u s i n g m u l t i p o l e e l e m e n t s or by

m a c h i n i n g c i r c u l a r c u r v a t u r e s on the b o u n d a r i e s of d i p o l e

ma gn et s. In p r i n c i p l e , each mag,net edge or m u l t i p o l e

el em en t could be used to c a nc el one a b e r r a t i o n . If in

pr ac t i c e , d e v i c e s of si mi la r f u n c t i o n are spaced too c l o s e l y

togeth er, their e f f e c t i v e n e s s in c o r r e c t i n g for a b e r r a t i o n s

will be limited. The p r o b l e m of c o r r e c t i n g for large h o r i ­

zont al and v e r t i c a l a p e r t u r e s is a p a r t i c u l a r l y d i f f i c u l t

one and a si ng le d i p o l e m a g n e t w i t h two d e g r e e s of f r e e d o m

is not in ge ne ra l able to c o m p e n s a t e for a b e r r a t i o n s in both

d i r e c t i o n s . The a d d i t i o n a l d e g r e e s of f r e e d o m gai n ed in the

Enge split pole (Sp67) by s o l i t t i n g the d i p o l e in two resu lt

in a larger u s e a b l e solid angle, but such a s o l u t i o n is still

not a d e q u a t e for a broad ra nge s p e c t r o m e t e r w i th large

di sp er s i o n .

Ill

In the Q D DD s p e c t r o m e t e r de si gn, the m a n y d e g r e e s

of fr ee do m need ed to correc t for a b e r r a t i o n s up to four th

or der in 0 and <(> are rea l iz ed by s p l i t t i n g the d i p o l e

into three s e p a r a t e e l e m e n t s and by p r o v i d i n g a d d i t i o n a l

m u l t i p o l e m a g n e t el eme nts . The BNL OD DD s p e c t r o m e t e r ,

de s i g n e d by Enge and L e Vi ne (Le70) and bu ilt by S c a n d i t r o n i x

of Up p s a l a Sweden, is shown in Fig. III-2. C h a r a c t e r i s t i c s

of the s p e c t r o m e t e r are given in Ta bl e I I I A 1- 1. This

i n s t r u m e n t is a n e c e s s a r i l y large and c o m p l e x s o l u t i o n to the

di f f i c u l t p r o b l e m of a c h i e v i n g a v e ry large solid an gl e and

high r e s o l v i n g power in c o m b i n a t i o n wi t h a large d i s p e r s i o n

and broad e n er gy range. The Q u a d r u p o l e el em en t just be yo nd

the m a g n e t a p e r t u r e p r o v i d e s f o c u s s i n g in the v e r t i c a l

d i r e c t i o n and p r o d u c e s a v e r t i c a l c r o s s o v e r or a s t i g m a t i c

image of the target spot b e t w e e n the first d i p o l e e l e m e n t s

a p p r o x i m a t e l y at the p o s i t i o n of the m u l t i p o l e elemen t.

The v e r t i c a l rays are br o u g h t to se con d focus at the focal

su rf ac e by the c o m b i n e d a c t i o n of the f r i n g i n g fiel ds of

Dip o le 2 and D i p o l e 3. Th e r e is no i n t e r m e d i a t e or c r o s s ­

over focus of rays in the h o r i z o n t a l pla ne a n y w h e r e b e t w e e n

the targ et and focal plane. The first a d v a n t a g e of this

op ti ca l c o n f i g u r a t i o n is that d i s p e r s i o n i n c r e a s e s w i th the

numb er of h o r i z o n t a l or v e r t i c a l i n t e r m e d i a t e i m ag es formed.

A se co nd a d v a n t a g e of an i n t e r m e d i a t e v e r t i c a l focus

i m p o r t a n t to s u c c e s s f u l c o r r e c t i o n of a b e r r a t i o n s is that

the aspect ra tio of the b e a m c h a n g e s d r a s t i c a l l y t h r o u g h o u t

the i n st ru me nt . T h i s p r o v i d e s a m e a n s for s e p a r a t e l y c o r r ­

e c ti ng for a b e r r a t i o n s in the m e d i a n pl an e and the t r a n s v e r s e

plane.112

In the v i c l n t y of the v e r t i c a l c r o s s o v e r , the

v e r t i c a l d i m e n s i o n s of the b e am w i ll be v e r y small, and

the shape of the po les at the exit of D i po le 1 and the

e n t r a n c e of D i po le 2 will have a s i g n i f i c a n t ef fe ct on ly

on m e d i a n pl an e (<t> = 0 ) a b e r r a t i o n c o e f f i c i e n t s X/0 ^,

X / 0 3 , etc. O D DD d i p o l e s are of c o n v e n t i o n a l d e s i g n with

the e n t r a n c e and exit b o u n d a r i e s curved to co rr ect for

a b e r r a t i o n s and to co nt ro l the shape of the focal surface.

The field cl am ps lo ca te d just in front of the pole edges

play an e s s e n t i a l role in s h a p i n g the f r i n g i n g fields at

the d i p o l e e n t r a n c e s (Hu70) and pr o v i d e a m e a n s for final

c o r r e c t i o n of a b e r r a t i o n c o e f f i c i e n t s in the m e d i a n plane.

In the ab s e n c e of field clam ps, s a t u r a t i o n of the m a g n e t i c

field at the pole edges of a se ct or m a g n e t wo u l d re su lt in

ex t e n d e d f r i n p i n g fiel ds and a loss of v e r t i c a l f o c u s s i n g

power. In ord er to r e du ce the s a t u r a t i o n effect, ro und ed

pole edges of the type first su gg e s t e d bv P o g o w s k i for

e l e c t r o s t a t i c e l e m e n t s are used in the ODDD. The field

c l am ps t h e m s e l v e s are m a g n e t i c short c i r c u i t i n g e l em en ts ,

plac ed about one gap w i d t h from the pole edges, w h i c h are

d e s i g n e d to limit and shape the f r i n g i n g fi eld s at the

e n t r a n c e and exit of m a g n e t i c lenses. The field clamp,

w h o s e " m a g n e t i c c o n d u c t i v i t y " is hi gh er than that of air, is

able to c o ll ec t field lines from the l o w- fi el d s t r e n g t h

tail of the f r i n g i n g field, w h i c h u s u a l l y ex te nd s far

113

out of the ma g n e t , and lead them away rrom the ce nt er of

the clamp, thus t e r m i n a t i n g the d e f l e c t i n g field much

ea rl ier than In the case of an u n s h i e l d e d magnet.

A third a d v a n t a g e in hav i ng an i n t e r m e d i a t e v e r t i c a l

c r o s s o v e r point b e t w e e n the first two di p o l e s is that a

m a g n e t i c m u l t i p o l e el em ent may he located at this point,

wh er e it has m i n i m a l effect on v e r t i c a l fo cu ss in g, to pr o v i d e

c o m p e n s a t i o n for k i n e m a t i c b r oa de ni ng . Since k i n e m a t i c

b r o a d e n i n g is to first ord er a d e f o c u s s i n g effect , it can

be c o m p e n s a t e d for by f i r s t - o r d e r focussing, a c t i o n in the

ho riz ont al plane p r o v id ed by the qtiadrupole field of the

m u l t i p o l e elemen t. Thus, instead of p r o v i d i n g for k i n e m a t i c

c o m p e n s a t i o n hv a large p h ys ic al d i s p l a c e m e n t of the focal

plane, a quadru pole fo cu s s i n g el em en t is used to re tu rn the

focal plane to its o r i g i n a l p o si ti on , and the h i gh er

m u l t i p o l e fiel ds may then he used to e l i m i n a t e some of the

r o t a t i o n s and c u r v a t u r e ch a n g e s in the focal s u r f a c e i n t r o ­

duced by k i n e m a t i c s and to c o m p e n s a t e for some of the

ad d i t i o n a l a b e r r a t i o n s in t r o d u c e d wi th the k i n e m a t i c c o r r e c ­

tions. D u ri ng the peri od wh en we used the BNL s p e c t r o ­

me ter , only q u a d r u pole c o r r e c t i o n s for k i n e m a t i c b r o a d e n i n g

were made ro ut in ely . Wh il e the m u l t i p o l e el e m e n t cannot

pr ov id e pe rf ect k i n e m a t i c c o m p e n s a t i o n , the loss in e n er gyd£

r e s o l u t i o n for the d i f f i c u l t case of K = d_9 = 0.3 isP

e s t i m a t e d by L c Vi ne (Le70) to be o n ly about 507 . with k i n e ­

ma tic c o r r e c t i o n c o m p a r e d to a fa cto r of 1on wi th ou t.

114

The m u l t i p o l e el em en t w h i c h p r o v i d e s k i n e m a t i c

c o m p e n s a t i o n was d e s i g n e d by F.nge ( W i 7 2 ) and c o n s i s t s of

two p a r a l l e l bar' m a g n et s, each e n c i r c l e d by a set of

coils, s e pa ra te d by i n s u l a t i n g teeth. Each coil is s u b ­

di vi de d into four s e c t i o n s of w i n d i n g s , one for each m u l t i ­

pole field, and the n u mb er of turns for each se c t i o n is

var i ed from coil to coil to p r o d u c e the a p p r o p r i a t e field;

e.g. the q u a d r u p o l c coil s e c t i o n s all have the same n u mb er

of turns so that the m a g n e t i c force wi ll i n c r e a s e l i n e a r l y

from the c e nt er out, w h i l e the s e x t a p o l e s e c t i o n s have turn

n u m b e r s w h i c h i n c r e a s e l i n e a r l y from the c e nt er out. Thus,

in the case of h e a v y - i o n r e a c t i o n s w h o r e the full solid

angle of 14.7 msr is bein g used, the m u l t i p o l e el em en t

can s i m u l t a n e o u s l y pr o v i d e dipole , q u a d r u p o l e , se x t a p o l e ,

o c t a p o l e , and d c c a p o l e fiel ds and should he able to make

the ve ry large k i n e m a t i c and a b e r r a t i o n c o r r e c t i o n s

n e c e s s a r y for a d e q u a t e re so l u t i o n . An e l e c t r o s t a t i c

v e r t i c a l d e f l e c t o r has been locat ed b e t w e e n Dip o le 2 and

Di po le 3 to p r o v i d e a d d i t i o n a l p a r t i c l e i d e n t i f i c a t i o n .

P a r t i c l e s w i th d i f f e r e n t m a s s - t o - c h a r g e rati os m a y he

s e p a ra te d, with limited r e s o l u t i o n , by p a s s i n g th ro ug h the

t r a n s v e r s e e l e c tr ic field ap pli ed by this elemen t. The

e l e c t r o s t a t i c d e f l e c t o r was not used in the cou r se of our

m e a s u r e m e n t s . In the a b s e n c e of h i gh er or de r a b e r r a t i o n s ,

the na tu ra l line shape for the O D DD wi ll he H a u s s i a n at

the point wh er e the first or d e r a b e r r a t i o n c o e f f i c i e n t

115

v a n i s h e s [(X/0) = 0], r e f e r r e d to as the G a u s s i a n - i m a g e -

point. Ray tr ac ing m e a s u r e m e n t s on the M u n i c h Q3D

(L0 7 3 ) I n d i c a t e that the sma l le st h a l f - w i d t h for the be am

o c c u r s cl ose to the G a u s s i a n - i m a g e - p o i n t w h i l e the s m a l l e s t

base wi dt h o c c u r s some cm's from this point. Ray tr a c i n g

m e a s u r e m e n t s are in p r o g r e s s on the R r o o k h a v e n ODDD.

Fig. T I I -13 shows a typi cal l i n e s h a p e at the focal

pl an e near the ce nt ra l ray for the BNL ODDD. The peak

is the pr od uc t of elasti c s c a t t e r i n g of bO H eV ions

from a W 0 3 line target on a c a rb on foil backin g.

TABLE IIIAl-1 . BNL QDDD Characteristics

Orbit Radius

Angular Range

Angular Acceptance

Solid Angle

Magnetic Gap Width

Magnetic Field Strength

Length of Focal Plane

Radius of Curvature of Focal Plane

Energy Range, E /E .b max mmMaximum Proton Energy

Angle of Incidence at Focal Plane

Energy Resolution at 3-13 kG

99 cm max 81 cm min

- 20° to +160° continuously

± 7 0 mr in vertical direction ± 60 mr in horizontal direction

14.7 msr (square with comers cut off)

72 mm

2700 - 16000 G

250 cm

215 cm

1.51

121 MeV at 16000 G

o o _ ,

22 -48 from normal

E/dE = 4000 for 14.7 msr, 10% energy range, 0.75 mm target spot size

E/dE = 4000 for 3.9 msr, 50% energy range, 0.75 mm target spot size

Dispersion

Magnification along Focal Plane

D/M

Magnification in Transverse Direction

Approximate Weight of Iron and Copper

A x /A p = 5*8 - 11.5 cm /%

0.78 - 1 .4

7 .5 - 8 .5 cm /%

2.9 - 3 .5

90 Tons

16 186Fig. HI-13. Elastic peak for the scattering of O on W at 16

E = 9 0 MeV. The QDDD was positioned at 0 ^ ^ = 30C

and operated at a solid angle of 8 m sr.

CO

UN

TS

16000

14000

12000

10000

8000

6000

4000

2000

l60 + '86W E L A S T IC PEAK

,60_ E = 90 MeV

Slab * 30° j\QDDD SLIT SETTINGS: | \

POSITION CHANNEL

Fig. Ill-13

116

A N A L Y S I S

A . I n t r o d u c t I o n

This chapter, will pr es en t the r e s u l t s of a fi nit ei

range n o n - r e c o i l c o u p l e d - c h a n n e l - B o r n - a p p r o x i m a t i o n (CCBA)

a n a l y s i s of the tw o- nu c l e o n t r a n s f e r da ta p r e s e n t e d inI

C h a p t e r III, based on the t h e o r e t i c a l m e t h o d s o u t l i n e d in

C h a p t e r II. Th os e s e m i - c 1 as s i c a l and k i n e m a t i c fe at ur es

of the h e a v y - i o n i n t e r a c t i o n w h i c h s i m p l i f y our u n d e r s t a n d i n g

of the data wi ll be e m p h a s i z e d t h r o u g h o u t this d i s c u s s i o n .

In p a r t i c u l a r , the r e l a t i v e m a g n i t u d e s , slopes, and signs of

the m u l t i p o l e t r a n s f e r form fa ct ors d i s c u s s e d in C h a p t e r II

w i ll d e t e r m i n e the role of v a r i o u s c l a s s i c a l o r b i t s in d i re ct

and m u l t i s t e p t r a n sf er p r o c e s s e s , and may be used in a

q u a l i t a t i v e w a y to pr ed ic t bo th the shap es and the effe ct

of p a r a m e t e r c h a n g e s on a n g u l a r d i s t r i b u t i o n s . Thus, for

h e a v y - i o n r e a c ti on s, it is p o s s i b l e to m a ke a d i r e c t c o n n e c ­

tion b e t w e e n the e s s e n t i a l ph y s i c s of the p r o b l e m and the

o u t c o m e of an i n v o l v e d q u a n t u m - m e c h a n i c a l C C BA c a l c u l a t i o n .

Such insi ght is not g e n e r a l l y a v a i l a b l e in l i g h t - i o n induce d

r e a c t i o n s .

T w o - n u c l e o n t r a n s f e r r e a c t i o n s in vo lv e an a d d i t i o n a l

d e g r e e of c o m p l e x i t y c o m p a r e d to o n e - n u c l e o n tra ns f er , since

they are s e n s i t i v e to t w o - n u c l e o n c o r r e l a t i o n s in the o v e r ­

laps b e t w e e n in it ial and final n u c l e a r states. T r a n s i t i o n s

CHAPTER IV TWO-NUCLEON TRANSFER REACTIONS: THEORETICAL

117

between states, 9uch as BCS ground states, which involve

a large number of similar coherent two-nucleon configurations

will be favored. However, in the absence of multistep

processes, such nuclear structure information will be re­

flected for heavy-ion reactions only in the magnitudes of

the resulting bell-shaped angular distributions. It is the

interference structure introduced into the angular distri­

butions by multistep processes which allows us to follow

small systematic changes in the underlying microscopic

structure across a range of nuclei. The systematic change

in the microscopic distribution of transfer strength has

been studied for the (X2C , ^ C ) reaction across the rare-

earth region in this analysis. In addition, the reactions

an(j (lf>o7 B0) on the same target

nucleus have been examined for O-value and projectile

dependence of the reaction mechanism.

Because the shapes of the experimental angular

distributions result from a delicate balance in the

competition between direct and indirect transfer routes

involving both nuclear and Coulomb inelastic transitions,

the data provide a stringent test for any nuclear pairing

model. Certain fundamental problems with the simple constant

pairing BCS model for intrinsic states became apparent

during the analysis and will be discussed in the following

sections.

118

The codes used in this analysis for the construction

of two-nucleon transfer form factors and for the coupled-

channels calculations were developed by R. J. Ascuitto and

J. S. Vaagen (As70, As75a, As75b). The code DEF2NT

used to determine the gross structure and pairing ampli­

tudes for deformed orbitals of the rare-earth nuclei was

written and tested by Bent Sorensen and R. J. Ascuitto.

119

B. General Features of Heavy-ion Transfer Reactions

Certain general features of the angular distribu­

tions observed in these experiments are common to all heavy-

ion transfer reactions and may be understood in terms of

kinematics, Q-values, and the semi-classica1 nature of the

interaction between complex nuclei. Indeed, to the extent

that these considerations depend on the strong absorption

of interior parti'al waves and the surface nature of the

transfer reaction, they also hold for some light-ion

reactions, notably alpha-particle Induced reactions (Au70).

It has been pointed out (G174) that the effects of variations

in bound state wave functions and interaction potentials

are easier to predict for such reactions than for proton-

induced reactions because the concept of a classical

trajectory is now meaningful, and the effect of parameter

modifications on the contributions from the various classical

orbits is often intuitively obvious. This section will

concentrate on those features of heavy-ion angular distri­

butions which do not depend on nuclear structure for their

understanding.

120

1. Classical Trajectories and L-space Localization

A semi-classical description of heavy-ion scattering

(G174, Sc74) may be based on the following two characteris­

tics of heavy-ion interactions:

(1) the extension of the wave-packet of the heavy-

ion, given by the deBroglie wave length of the relative

motion ■ 'fi/pv, is small compared to the sum of the

nuclear radii R^+R2 (X -*-0 in the classical limit); and

(2) large angular momentum values L contribute to the

cross section (L -*■ <®, AL -*• 0 in the classical limit).

When these conditions are met, the heavy-ion projectile may

be considered (at least for the purpose of qualitative

arguments) to have a well-defined trajectory which samples

a high density of angular momentum values as a function of

impact parameter.

The total interaction for such a system may be des­

cribed in terms of a real effective potential of the type

V 0 (r) - V„ (r) + V„ (r) + * 2A U + D2yr2

where

and

VoV (r) - ----------—n ,. r-Ro1+exp -----

,, / \ |zz 1 e 2 _ _Vc (r) » — , r > Rc

[’ - ( k ) 2] ^ e r < Rc

121

with Rc - rcA 21/3(Al < A 2> and Ro “ ro(A l1/3+A21/3) » shown in Fig. IV-la for several values of £. The

distance of closest approach for a system of given energy

E and angular momentum £ is determined in the simple

classical picture as the largest radius r for which E

is equal to the potential V^(r). As shown in Fig. IV-la

and TV-lb, a discontinuity occurs in the distance of closest

approach at £c , the critical angular momentum for which,

at the given energy, the barrier in V^ is just surmounted.

Trajectories with £ > I remain outside the nucleuscwhile those with £ < £c plunge deeply in.

Fig. IV-2a shows a plot of some typical classical

trajectories for a heavy-ion potential of the type V^(r).

The circle represents the half-density radius of the Woods-

Saxon potential. The grazing trajectory having the maximum

deflection angle for non-penetrating orbits is labeled by g.

In classical Rutherford scattering, the distance of closest

approach of the two particles is

( - . . . ( I ) ] -

The classical grazing angle , taken as an average over

initial and final orbits for a transfer reaction, will be

given by the expression

\ -1°GR * 2 8l"'1

2E(R1+ R 2) _ -jz!z 2e /

122

where E Is the center-of-mass energy. The grazing tra­

jectory g will be scattered to an angle forward of the

classical (Coulomb) grazing angle (dotted line)

because of the influence of the nuclear field. The orbits

labelled 1,2, and 3 sample very different parts of the

potential, but all scatter into the same angle forward of

the grazing angle.

Information concerning all classical scattering

trajectories may be conveniently summarized in terms of a

"deflection function", a plot of scattering angle as a

function of angular momentum (or impact parameter). A

typical deflection function for heavy-ion scattering is

shown in Fig. IV-2b, where the location of orbits from

Fig. IV-2a has been marked. Orbits of Region I are deflected

through small angles by the repulsive Coulomb field. With

decreasing L, the nuclear field comes into play, decreasing

the angle of deflection below what it would be for pure

Coulomb scattering, until the grazing orbit, which scatters

to the maximum angle possible for nonpenetrating orbits, is

reached. The orbits of Region II skim the edge of the

attractive nuclear potential. The deflection function has a

singularity at the critical angular momentum £c , corres­

ponding to the top of the barrier where the net radial

force is zero. Orbits with £ n, circle the nucleus for

varying periods before escaping. Plunging orbits of Region III

Fig. IV-1. a) Real effective interaction potential V (r) for heavy-ion

scattering from a heavy nucleus, displayed for several

values of the orbital angular momentum L b) Plot

showing discontinuity in distance of closest approach

at critical angular momentum L of classical model.

Fig. IV-2 . a) Classical orbits corresponding to various impact

parameters for potential of type V.(r) (see text).

The circle represents the half-density radius for the

nuclear matter distribution, b) Classical deflection

function with position of orbits from a) indicated.

o) H E A V Y -IO N P O T E N T IA L

b) D ISTANCE OF CLOSEST APPROACH

Fig. rv-l

a:LlIh~LU

<QC

2

O

2

b) DEFLECTION FUNCTION

ORBITAL ANGULAR MOMENTUM

Fig. IV-2

123

such as 3 in Fig. IV-2a are bent to negative angles while

trajectories corresponding to even lower angular momenta

are deflected to large backward angles.

The role which these various trajectories play in

heavy-ion transfer reactions may be most easily understood

in terms of the physical optics of the nuclear system.

The grazing angle, or maximum deflection angle for non­

penetrating orbits (identified with the trajectory g ) ,

in general corresponds to the peak in the angular distribu­

tions for heavy-ion transfer reactions involving direct

processes. This results because the grazing angle trajectory

goes through a localized region of the nuclear surface

where direct transfer probability is large while absorption

into other channels is small. More penetrating orbits,

which may scatter either forward or backward of this angle,

are attenuated by absorption, while more distant orbits,

which may scatter only forward of the grazing angle, make

a reduced contribution to transfer because of the exponential

decrease of the bound state wave functions with Increasing

radius. Thus, heavy-ion transfer reactions may be thought

of as being localized in r-space to a ring in the surface

region of the nucleus. Many features of heavy-ion transfer

angular distributions may then be understood by an analogy

to the two-slit diffraction problem of physical optics.

A "two-slit" interference pattern centered around

zero degrees and damped at more backward angles may result

124

from the interference between orbits from opposite sides of

the ring (orbits 2 and 3) scattering into the same

angle. This is possible because orbits 2 and 3 may make

approximately equal contributions to transfer at very forward

angles while the effect of orbits of type 3 will be

greatly reduced at more backward angles as a result of

increased absorption (G174). Such forward-angle oscillations

in the transfer angular distributions are not important for

the range of energies and angles considered in this study,

but will be a prominent feature for other reactions

(e.g. 116Sn (180, 160) at 100 MeV (As74a]).

By extending this analogy still further, it may be

seen that the bell-shaped envelope of the grazing peak for

a direct heavy-ion transfer reaction corresponds to the

"single-slit" diffraction pattern whose width will be in­

versely proportional to the width of the region of maximum

transfer strength (the r-space ring). The angular spread

of the grazing angular distribution for a reaction localized

to a region of width A£ in L-space is restricted by the

uncertainty principle to be

A0 = h -

The L-space localization and, therefore, the angular width

will depend on both the number of nucleons transferred and

the reaction mechanism.

125

Two-step reaction processes are even more localized

L-space than direct reactions, since the two-step reaction

amplitude is the product of an inelastic form factor (which

nentially decreasing transfer form factor. In contrast to

the situation for Coulomb excitation, where hundreds of

partial waves may be required to achieve convergence in

the inelastic cross section, the two-step processes consid­

ered here require in general less than one hundred partial

waves for convergence. Additional diffractive spreading of

the wave-pocket as it passes through the reduced L-window

results in a wider envelope for the two-step angular distri­

bution (routes A in the sketch below) compared to that

of a one-step process (route B) .

As a consequence of this diffractive spreading, interference

between the various transfer routes will occur over a much

larger angular interval and the resulting interference

structure will vary more slowly than that of the very forward

for A = 2 and large L j n , goes as 1/R3) and an expo-

126

angle diffractive oscillations. These features should be

generally useful in identifying two-step processes involving

intermediate inelastic transitions.

Referring to Fig. IV-2a and the sketch above, we

note the possibility of nuclear-Coulomb interference in the

inelastic part of the two-step transfer routes A between the

large-impact-parameter Coulomb orbit 1 and the low-impact-

parameter nuclear orbit 2 which both scatter into the same

angle forward of the grazing angle. This effect will be

seen to have important consequences for the shape of the

186w ^12q p14 q ) 2+ angular distribution, where the direct

quadrupole (J = 2) transition B is extremely weak.

2. Kinematic Restrictions: Transferred Angular Momentum

and Q-value

In the first part of this section we discussed

factors determining the general shape of the angular

distributions for heavy-ion transfer reactions. We now

consider some general kinematic factors which influence

the absolute and relative magnitudes of the transfer cross

sections.

It has long been recognized (Bu66, Br72, A172)

that the transfer cross sections for surface reactions

involving strong absorption depend strongly on the reaction

Q-value. This dependence is intimately, related to the sharp

L- and r-space localization of the transfer process

discussed earlier. Because the transfer strength for heavy-

ion reactions is concentrated in a narrow ring in r-space,

the maximum transfer cross section will result for that

Q-value which corresponds to equal distances of approach

before and after transfer and leads to a continuous trajectory

for a grazing orbit passing through this ring. Assuming that

the particles in the initial and final fragmentation have

spin zero, so that the transferred angular momentum is

AL = 0, the optimum Q-value close to the Coulomb barrier

will be given by (Bu71, A172)

128

which is zero for two-neutron transfer reactions. For

fixed L- transfer, the transfer cross section will decrease

as the Q-value moves away from ^OPT" Similarly, for a

fixed Q-value, the cross section will vary as a function of

L-transfer, and the favored L-transfer will correspond to

the difference in orbital angular momentum between the

initial and final trajectories having the same impact

parame te r .

Table IV-1 summarizes the most important dynamical

parameters for the reactions studied. None of the two-

neutron transfer reactions involve a large angular momentum

mismatch, and differences in the absolute and relative

cross sections for the various reactions studied at equiva­

lent bombarding energies will thus reflect differences in

nuclear structure of the deformed system. However, the small

cross sections measured for the *® (12c ,1OBe) reaction

are a consequence of both a large Q-value mismatch and

nuclear structure effects.

TABLE IV-1. Reaction Systematics

Two-nucleon Separation Energies

Reaction Ground State Q-value (MeV)

LightSystem

(MeV)

HeavySystem

(MeV)

LabBombarding

Energy(MeV)

Exp.Grazing

Angle

FavoredL-transfer

(Ex =0.0)

(*)

Optimum Q-value (A L=0)

(MeV)

154Sm(12C ,14C)

18W 4c)

184W(12C ,14C)

186W(12C ,14C)

186W( 16o ,18o>

186W ( 12C ,10Be)

-0.667

-1.808

-0.482

-0.161

-0.773

-14.136

13.123

13.123

13.123

13.123

12.189

27.187

13.845

14.700

13.603

12.952

12.952

13.204

65.0

70.0

63.0

70.0

90.0

70.0

62

70

82

69

76

64

2.1

3.1

1.7

3.4

1.8

1.6

0.0

0.0

0.0

0.0

0.0

-20 .7

130

C . Outline of Calculation and Determination of Parameters

In this section, we gather together the final pairing,

deformation, and optical model parameters used in this

analysis, and indicate their role at various points in the

calculation. Attention in the remainder of the chapter will

then be focussed on the physical interpretation of the

experimental cross sections.

The full calculation of two-nucleon transfer form

factors and CCBA cross sections presented in this study

involves a number of discrete steps. These will be outlined

briefly here :

(1) In a preliminary calculation using the program

DEF2NT , the deformed single-particle valence orbitals of

the initial and final intrinsic states are generated within

a deformed Woods-Saxon well of fixed depth. The single­

particle potential well and nuclear deformation parameters

are defined in Chapter II and listed in Tables IV-2 and

IV-3. The nuclear deformations used are taken from analyses

of the best available experimental data and are scaled to

the radius parameter rQ = 1.25 fm using the scaling pres­

cription R8 = RoBo of Blair (B163, He73). Output from

this preliminary calculation includes the single-partic1e

energies ^s.p. ant* coefficients of expansion on a cylin­

drical oscillator basis for all orbitals in the Woods-

Saxon well. The structure information for the 20-32 valence

131

orbitals located In a symmetric region about the Fermi level

which compose the intrinsic state is then written on tape.

For later reference, the dominant £j components of the

orbitals are Identified from their principal cylindrical

asymptotic quantum numbers using the tables of Chi(Ch66).

The effect of the residual pairing interactions is

determined from a BCS calculation in which the pairing

force is taken to be constant and of the form

G- is adjusted to reproduce the neutron (An) and proton

(Ap) pairing gaps given by Ogle et al. (0g71), and the

resulting calculation determines the Fermi level X and

occupation probability amplitudes bv^7r * V ^or tbe

active valence orbitals. The number and range of binding

energy of the valence orbitals, as well as pairing gaps,

force constants, and Fermi level for each system, are

listed in Table IV-4. In addition, this preliminary

calculation generates transfer form factors for the

corresponding light-ion (p,t) or (n,^He) reactions,

which give a rough indication of the relative strengths

of the corresponding heavy-ion form factors. Such infor­

mation can be useful in making a quick and inexpensive

survey of systematics over a large mass region.

v Q tt(2) The radial wave functions IRjj (r) are cal­

culated for each valence orbital using the STURMIAN

132

program. The orbitals are generated from a deformed

Woods-Saxon potential of variable depth whose nuclear

deformation parameters are taken to be the average of

those of the target and residual nucleus given in Table

IV-3. For the proton bound state problem, the Coulomb

potential is that of a deformed uniform charge distri­

bution with a sharp surface and quadrupole deformation

P 2C . For a given orbital i, the well depth is varied

and all bound state solutions with eigenenergy equal to

the orbital single-particle energy are found. The

orbitals may be conveniently bound at their individual

single-particle energies e g p> from (1), or, at some

common energy. The desired physical solution is then

identified from this set as the solution having approxi­

mately the same well-depth and Nilsson structure as was

obtained in the rough calculation of (1). During the

second execution of the Sturmian program (or the first if

the necessary information i9 already available), the

index number specifying the physical solution of interest

is entered as input and the radial functions correspon­

ding to that solution are written on tape. Single particle

wave functions for the light system are also generated at

this time. The fixed bound state potential parameters

for both light and heavy systems are given in Table IV-5.

(3) Programs STEP 1 and STEP 2 are now executed.

133

These combine the radial wave functions and pairing ampli­

tudes UyQn , (from (1)) for each orbital and the

Cohen-Kurath (Co70) expansion coefficients for the light

system (Table IV-6) with the appropriate geometry factors

to give the complete radial non-recoil form factors

for each orbital. These partial form factors

are printed out, and the total form factor F^(R) summed

over all orbitals is punched out and serves as input to

the transfer source term program.

(4) Finally, the coupled-channeIs calculation of

the reaction cross sections is carried out using the main

programs, R E A C , SOURCE, and CROSS. First, the inhomo-

geneous coupled-equations describing the elastic and

inelastic scattering in the entrance channel are solved

using the program REAC. Distorted waves describing the

relative motion in both the entrance and exit channels are

calculated using the spherical optical potential of

Becchetti et al. (Be74) for the scattering of carbon on

lead, with the real radius parameter increased by up to

3% to bring the calculated ground state grazing angle

into agreement with the experimental results (Table IV-7).

The nuclear deformation parameters used for determining

the inelastic matrix elements are the same as those used

in the bound state calculation (Table I V - 3 ) , but hexa-

decapole deformations are now included in the deformed

134

Coulomb potential. Experimental Coulomb B's have been

scaled to the Coulomb radius parameter r * 1.127 fm incsuch a way as to preserve the appropriate reduced electric

multipole transition matrix elements. Except where noted,

all inelastic transitions connecting the 0+ , 2+ , 4 +

states in both the target and residual nucleus are included

in these calculations. The solutions just obtained for

the entrance channel inelastic scattering problem are

then used, together with the multipole transfer form

factors calculated earlier, to construct the transfer

source terms with the program SOURCE. The inhomogeneous

coupled-equations for the final fragmentation are then

solved in a second application of REAC to give the

S-matrix elements, or scattering amplitudes, describing

the asymptotic relative motion in the outgoing channels.

Finally, the cross sections for the transitions of interest

are calculated from these S-matrix elements using the

program CROSS.

The computations of Step (1) can be run at WNSL

using the IBM 360/44 computer. For reasons of required

core size and computational economy, the calculations of

steps (2)-(4) are performed on the NYU CDC6600 computer.

All of the calculated angular distributions pre­

sented in this chapter have been individually normalized

to the data. In general, the theoretical cross sections

are low in absolute magnitude by a factor of 10 or more,

135

while the relative magnitudes of the angular distributions

all agree with the data to within better than a factor

of 1.5. This discrepancy between theoretical and

experimental cross sections is common to many reaction

calculations (Ba74), and may arise from the optical poten­

tial used to describe the distorted waves in the scattering

problem or in the present case, from the size of the con­

figuration space used in describing the light system over­

laps. However, these possibilities have not been extensi­

vely explored in the present calculations.

The approach taken in this analysis has been to

use the high quality experimental data as a testing ground

for the simple constant pairing BCS nuclear structure

model and the multistep reaction mechanism. At this stage

in our experience it was considered important to attempt

to reproduce the systematics of a large range of data

using the best independently derived parameters, rather

than to explore the sensitivity of the calculations to

variations in individual parameters with the goal of

extracting information on these parameters from our data.

Such questions as whether the calculated angular distri­

butions are sensitive to factor-of- two changes in the

nuclear and Coulomb B^s, for example, remain unanswered

at this point, but sufficient experience has now been

gained to answer such questions in a meaningful way in

the future.

TABLE IV-2. Single-particle Well Parameters

Vo v i Vso r0

rc a

(MeV) (MeV) (MeV) (fm) (fm) (fm)

-51 .0 132.4 32.0 1.25 1.25 0.60

TABLE IV-3. Deformation Parameters

Nucleus 0 N P 2 8 4 ^ 66C * 2

sc8 4 Ref.

152Sm 0.238 0.047 - 0.012 0.260 0.056 He68

154Sm 0.259 0.051 -0.018 0.285 0.066 He68

180W 0.219 -0.069 0.0 0.285 -0.127 Ap70

182w 0.219 -0.069 0.0 0.285 -0.127 Ap70

184W 0.221 -0.087 0.0 0.266 -0.107 Le74b

! 8«w 0.209 -0 .088 . -0.008 0.250 -0.108 Le74b

i 186w 0.200 -0 .075 -0.008 0.236 0.0 As74a, Le74b

L 188os 0.200 -0 .075 -0.008 0.236 0.0 As74a,Le74b

136

Table IV-4. Residual Interaction Parameters

Nucleus

(MeV)

A aP

(MeV)

G

(MeV)Gi

(MeV)

X

(MeV)

Number of Valence Orbitals

Range of Binding Energies

(Mel7)

152Sm 1.07 0.216 38.0 - 6.20 20

154Sm 1.07 0.211 38.0 -5 .85 20 (-1 .392)-(-8 .126)

180W 0.73 0.140 29.0 -6.796 32

182W 0.68 0.139 29.5 -6.271 32 (-1.227)-(-10.884)

184W 0.73 0.145 31.5 -5.590 31 (-1 .07)-(-10 .80)

186W 0.80 0.148 32.5 -5.513 31 (-1 .13)-(-10 .16)

J ' 116W 0.85 0.201 32.35 -6.081 23

l_1880s 0.87 0.198 32.56 -4 .684 23 ( -1 .297)-(-9 .321)

a From Ref. Og71.

138

Table IV-5. Sturmian Bound State Potential Parameters

Nucleus

12.

16.O

154oSm

182W

184w [186w J

O(fm)

1.25

1.18

1.25

a

(fm)

0.65

0.64

0.60

Vso(MeV)

5.81

5.81

8. 0

TABLE IV-6. Parentage Expansion Coefficients for Light Systema

$ +(12C) = 0.8754 , -1 ^ , -1 01 /2 1 /2 0

+ 0.4834-1 -1

1P3/2 ® 1P3/200 4. +(14C)

0

Cl Based on Cohen-Kurath admixtures (Co70).

TABLE IV-7. Optical Parameters Used in CCBA Analysis3,

Standard Parameters Used for All Reactions13

V aR Wri 3I r

c(MeV) (fm) (MeV) (fm) (fm) (fm)

40.0 0.45 15.0 1.31 0.45 1.127

Real Radius Parameter c

154Sm(12C ,14C) 182W(12C ,14C) 184W(12C ,14C) 186W(12C ,14C) 186W(12C ,10Be) 186W(160 , 180)

rR(fm) 1.33 1.35 1.37 1.35 1.33 1.35

£L The same parameters were used in the entrance and exit channels for all reactions shown.

b 12 208Optical model parameters of Becchetti et al. (Be74) for elastic scattering of C on Pb.

Real radius parameter of Becchetti et al. (Be74) r = 1.31 fm increased by 3-5% to bring calculateda

grazing peak into agreement with the experimental results. A similar shift in the grazing peak could

be produced by increasing, for example, the real diffuseness parameter of the optical potential for the

exit channel.

v£)

140

D. An Example: 18 6W (1 2C ,U C)184W

It should be emphasized that the qualitative fea­

tures of the experimental angular distributions for heavy-

ion induced two-nucleon transfer reactions in the deformed

rare-earth region may be understood in a simple way in

terms of the relative magnitudes and signs of the inelastic

and transfer form factors for the most important competing

transfer routes. To illustrate this in detail, we take as

an example the reaction at 70 MeV.

From Fig. IV-3, which shows the intrinsic form

factors for two-neutron transfer, it may be seen that the

J * 0 (monopole) form factor is large and positive in

the surface region (R ^ 9 fm) , while the J ■ 2 (quadrupole)

form factor in the corresponding region is quite weak and

positive. The J ■ 4 form factor, by contrast, is nega­

tive at the surface.

Fig. IV-4 presents the experimental angular distri­

bution for the 0+ transition together with the corres­

ponding CCBA calculation and a schematic picture of the

most important transfer routes leading to the 0+ ground

state in ^ 8 4 ^ Here the cross section is dominated by

the strong direct monopole transfer, while the indirect

routes are relatively weak. This strong direct transition

leads to a bell-shaped angular distribution peaked near

the classical grazing angle.

Fig. IV-3. Total intrinsic transfer form factors for the reaction

X88W(X2C ,X4C)484W calculated using deformed orbitals

bound at their single-particle energies (solid line) and

at half-the-two-neutron separation energy (dotted line).

The orbitals in the light system were bound at half-

the-the-two-neutron separation energy in each case.

Fig. IV-3

141

The most important transfer routes for the 2"

transition are shown in Fig. IV-5. Because the quadru-

pole transfer form factor is small in the surface region,

the direct 2+ transfer is weak and not very important.

As a consequence, two-step processes involving a mono­

pole transfer and inelastic excitation dominate the transfer

cross section. The minimum in the 2+ angular distri­

bution arises from destructive interference between small-

impact-parameter nuclear and large-impact-parameter Coulomb

excitation contributions to the Inelastic transitions.

On the left are shown the classical orbits for nuclear and

Coulomb scattering into the same angle 0 forward of the

grazing angle. The quadrupole inelastic form factors for

these two orbits, shown on the right, have opposite signs

and destructive interference results. This example for

^8BW, j[n fact, represents the first observation of strong

influence on particle transfer reactions from Coulomb

excitation, and will be discussed in more detail in the

next section.

The interference minimum in the 4+ angular

distribution, shown in Fig. IV-6, results mainly from

destructive interference between direct and indirect

transitions. The direct J *» 4 transfer form factor

is negative in the asymptotic region and is reduced in

strength by approximately a factor of 12 compared to

the monopole form factor. Angular momentum matching

142

conditions at this bombarding energy favor an L-transfer

of 4 units of angular momentum, so that the direct

transition is well-matched. Among the many indirect

routes feeding the 4+ state, only those involving

relatively weak quadrupole transfer and going through the

Coulomb part of the inelastic-scattering potential interfere

constructively with the direct transition. However, the

dominant two-step transfer route, consisting of a strong

direct monopole transition followed by an inelastic

transition to the 4+ state (or vice versa), is of

comparable strength to the direct transition, has an

overall positive phase, and interferes destructively with

the direct transfer route. The direct transition is

characterized by a sharply peaked, bell-shaped angular

distribution, similar to that of the 0+ transition,

whereas the indirect transitions are characterized by

rather broad angular distributions because of their en­

hanced L-space localization. The resultant interference

minimum occurs near the grazing angle where the cancellation

between direct and indirect transitions is most efficient.

These 4+ data contain the first conclusive evidence that

multi-step inelastic transitions can significantly alter

angular distributions, in the vicinity of the grazing angle,

through interference between direct and indirect transitions.

Fig. IV-4 . Diagram showing dominant role of strong direct transfer

in determining the shape of the angular distribution for 0+

transition in 488W(X2C ,X4C)X84W reaction. Upper panel

shows data and angular distribution resulting from full

CCBA calculation.

Fig. IV-5. Diagram illustrating dominant role of nuclear-Coulomb

interference in determining shape of angular distribution

for 2+ transition in the X86W(X2C ,X4C) reaction.

Fig. IV-6. Diagram illustrating dominant role of interference between

direct and two-step transitions in determining shape of

angular distribution for 4+ transition in the X88W(12C ,X4 C)

reaction.

I86W (I2C ,I4C )I84W E ,2c = 70 MeV

GROUND STATE TRANSITION

DIRECT

(S T R O N G )

MULTI-STEP

(W E A K )

4 * ■

A-2

2 *

0A - 2

4*

2*

0*

S TR O N G D IR E C T T R A N S F E R

<=0 " B E L L - S H A P E D " a n g u l a r d i s t r i b u t i o n

Fig. IV-4

I86W (I2C ,'4C)I84W E,2c = 70MeV

CO\X)3

aT3

b■o

Fig. IV-5

-O

ciX3

b-o

l86W (l2C, l4C)l84W E,2c= 70MeV

0 c m < d«g)

4 * TRANSITION

DIRECT NEG. FF

4 *--j- y - j - j - 4 *

T C X # ’ 'o+-i-i— 7 J - L — o*A A - 2

MULTI-STEPNEG FF POS FF(WEAK) (STRONG)

4 * -------- , , ----.--- 4 * 4 * -------- ----,---- 4*

J v | <•*•> 5 (+)2*--- .-- J~-2,--1----2* 2*------ -- $— 2*

(+>£ I0+--- 2 -----0* 0* v (♦) ^ C----------- 0+

A A - 2 A ' V V A - 2J = 0

INTERFERENCE BETWEEN DIRECT AND MULTI-STEP TRANSFER

o MINIMUM IN ^ NEAR 6GRAZING

Fig. IV-6

143

E. Comparison of (12C,1AC) Reaction on 182W,

and The Role of Underlying Nuclear

Structure

The usefulness of heavy-ion induced two-nucleon

transfer reactions as a spectroscopic tool depends pri­

marily on the sensitivity of these reactions to details

of the underlying structure description of nuclei. To

explore this sensitivity across the rare-earth region we

have measured angular distributions for the (12C,1AC) reaction on I54gm> 182w, and populating members of the

ground state bands of 1 8 0 ^ lB2w and

respectively. Fig. IV-7 shows experimental and theore­

tical ground state and 2+ angular distributions for

three of these reactions. The ground state angular dis­

tributions all can be seen to have a simple bell shape

characteristic of direct heavy-ion transfer reactions at

energies not too far above the Coulomb barrier. In

addition, these bell-shaped angular distributions all »peak at roughly the same angle, indicating that the kine­

matic conditions are similar for all three reactions.

These ground state data are typical of most heavy-ion

transfer data in that nuclear structure effects appear,

if at all, only through the cross section magnitudes.

In contrast, the 2+ data exhibit marked structure, with

the 2+ experimental angular distributions becoming

144

progressively less symmetric about the grazing angle as

target mass increases, with the forward angle peak b e ­

coming systematically smaller. These features are well-

reproduced within the CCBA and can be understood in

terms of the microscopic structure of the intrinsic

transfer form factors.

In the remainder of this section, each of these

reactions will be discussed in turn, with emphasis placed

on understanding the detailed distribution of transfer

strength in terms of the structure and distribution of

deformed orbitals around the Fermi level of these rare-

earth nuclei .

Fig. IV-7 . Experimental and calculated (CCBA) angular distributions

for the 0+ and 2+ ground state band members, for a

series of rare-earth nuclei. The CCBA calculations

(solid curves) include all orders of inelastic scattering

and all possible first-order simulteneous two-neutron

transfer routes connecting the 0+ , 2+ , and 4+ ground-

state-band members of the target and residual nucleus.

Intrinsic transfer form factors were calculated with the

half-the-two-neutron separation energy prescription and are

based on average deformation and pairing-gap parameters

for these nuclei. The theory curves have been normalized

individually to the data although relative cross sections are

reproduced to within a factor of 1 .5 . The dashed curve

in Fig. IV-7a and c shows the effect of omitting the

quadrupole transfer form factor. The dashed curve in

Fig. IV-7b corresponds to the same CCBA calculation as the

solid curve except the intrinsic transfer form factors used

are those labelled (1/2 S) x (EE) in Fig. IV-12.L H

d<r/

dilcm

{/j

Lb/sr

)a b c

Fig. IV -7

Fig. IV- 8. Comparison of total intrinsic monopole (J=0) and quadrupole

154 12 14(J=2) transfer form factors for the reactions Sm( C, C)

186 12 14and W( Cf C) calculated using deformed orbitals bound

at their single-particle energies (solid line) and at half-

the-two-neutron separation energy (dotted line). In both

cases the orbitals in the light system were bound at half-

the-two-neutron separation energy.

R(fm)

Fig, rV-8

145

1. 154Sm(12C , 14C ) 152Sm: Strong Direct Quadrupole

Translt ion

Experimental angular distributions for the

4^A S m ( 1 , l^c) reaction populating the 0+ , 2+ , and 4+

members of the ground state rotational band in

are shown together with CCBA calculations in Fig. IV-9.

Both the 2+ and 4+ angular distributions are broadened

compared to the 0+ angular distribution and show a

pronounced interference minimum near the 0+ grazing peak.

As noted in Sec. D, an angular distribution of this type

may arise when the direct and indirect transfer routes

are of comparable magnitude and opposite phase. The

interference minimum will then occur near the grazing

angle where cancellation between direct and indirect

transitions is most efficient. The intrinsic transfer

form factors F^(P) for the reaction 454sm (12c , 14 q ) are

shown in Fig. IV-10. The J = 2 form factor is in fact

comparable in strength to the J = 0 form factor, while

the J ** 4 is reduced by only about a factor of 10.

We can gain further insight into the features of

the angular distributions and the corresponding transfer

form factors from a study of the microscopic structure of

these form factors. As discussed in Chapter II, an

intrinsic finite-range transfer form factor corresponding

to angular momentum transfer J can he expressed in terms

146

of the individual deformed orbital contributions as

FJ (R) = Z < A+2|P+ ] A > ^ J (R) ,V | Q | TT VftTT VQlT

where P+ creates a pair of particles in time-reversed

deformed orbitals. If the intrinsic states in this

expression are described as deformed BCS vacua, the matrix

element reduces essentially to UA vA + a product ofVftlT vflu

pairing amplitudes which expresses the distance of the

deformed orbital from the Fermi level (Yo62). The partial

form factors carry all information about the

specific properties of the deformed orbitals and how they

participate in the reaction (e.g. their Nilsson structure).

It has been found (As75a) for rare-earth nuclei that, as

a general rule, signs and magnitudes of the J-th-orbital

moment correspond closely to asymptotic signs and to some

extent to relative magnitudes of the partial form factors

for angular momentum transfer J. Thus, for the quadrupole

form factor positive contributions correspond to deformed

orbitals with prolate quadrupole moment whereas negative

contributions correspond to an oblate quadrupole moment.

Fig. IV-11 shows the weighting factor UvflTr * Vv Q tt

and the partial form factors _ **(R ■ 11.2 fm) for the

l-^Sm(12c,14c) reaction as a function of the single

particle orbital energy. Solid (open) circles represent

positive (negative) contributions. Each deformed orbital

Fig. IV-9. Experimental and calculated (CCBA) angular distributions

+ + + for the 0 , 2 , and 4 members of the ground-state

rotational band in 452Sm for the reaction 4!*4Sin(42C ,44C) 12c

at E = 6 5 MeV. The theory curves have been normalized

to the data. The CCBA calculations include all orders of

inelastic scattering and all possible first-order simultaneous

two-neutron transfer routes connecting the 0 # 2 f and 4

ground state band members of the target and residual

nucleus. Intrinsic transfer form factors were calculated

with the half-the-two-neutron separation energy prescription

for binding and are based on average deformation and

pairing-gap parameters for these nuclei.

(js/qr/)

u,0(up

/-°p)

®c.m.

Fig, IV- 9

Fig. IV-10. Total intrinsic transfer form factors for the reaction

154 12 14 152Sm( C, C) Sm calculated using deformed orbitals

bound at their single-particle energies (solid line) and at

half-the-two-neutron separation energy (jdotted line). The

orbitals in the light system were bound at half-the-two-

neutron separation energy in each case.

R(fm)

Fig. IV-10

Fig. IV-11. Individual deformed orbital contributions to the total

intrinsic transfer form factors at 11.2 fm for the

154 152Sm- Sm system. Solid (open) circles represent

positive (negative) contributions. Each deformed orbital

is labelled by twice its P -value and by its parity.

154X labels the position of the Sm Fermi level and S/2

labels half-the-two-neutron separation energy in the

heavy system.

E„„(MeV)

Fig. IV-11

147

is bound at its single particle energy and is labelled by

twice its fi-value and by its parity. A labels the

position of the ^"^Sm Fermi level and S/2 labels half-

the-two-neutron separation energy in the heavy system. We

note that all deformed orbitals contribute constructively

to the monopole form factor. The orbitals around the

Fermi level for Sm are predominantly prolate and make

large constructive partial form factor contributions

resulting in a strong J “ 2 form factor. However, the

orbits in the region of the Fermi level have both positive

and negative hexadecapole moments and sum to a relatively

weak J ■ 4 form factor.

One problem which arises with the use of a constant

pairing matrix element in the pairing calculation is that

particles may scatter into orbitals far from the Fermi

level with finite probability. This lack of rapid conver­

gence in the UV product of pairing amplitudes, evident

in Fig. IV-11, means that loosely bound orbitals, which

slope less rapidly in the asymptotic region, will be

weighted too heavily in the total wave function admixture.

This is a serious problem since it is these loosely bound

orbitals which often make the largest partial form factor

contributions (see Fig. IV-11) and, if included, will have

a significant influence on the total form factors, particu­

larly on the slope of these form factors in the asymptotic

148

region moat important to Coulomb excitation. While a more

exact treatment of pairing would result in sharper and

more rapidly convergent UV distributions, the use of a

simple constant pairing BCS model requires a truncation

in the number of orbitals included. In the calculations

shown in this study, we have included 20 orbitals around

the Fermi level in Sm and about 30 orbitals for the

W-region. The cut-offs have been made at natural gaps in

the level sequences, and the number of orbitals included

correspond roughly to twice the number of valence particles.

This particular truncation scheme was found to give an

convergent description of (p,t) reactions in the rare-

earth region (As75c) and has been adopted on that basis.

One way to attempt to compensate for the limi­

tations of the constant pairing BCS model and obtain a

wave function with a more realistic slope in the asymptotic

region is to bind the orbitals in both the light and heavy

systems at one-half-the-two-nucleon separation energy.

This method reduces the asymptotic slopes and partial

form factors of the surface orbitals, and in general,

leads to improved fits to experimental angular distribu­

tions, particularly with respect to the depth of the 2+

interference minimum (see Figs. IV-7b and IV-23). This

procedure for binding has been used in CCBA calculations

shown in Figs. IV-7 and IV-9. The effect of the two

149

different binding prescriptions on the slope and magnitude

of the intrinsic transfer form factors may be seen in

Fig. IV-10. Form factors corresponding to the solid curve

were calculated with the light system orbitals bound at

one-half- the- two-neutron separation energy and the heavy

system orbitals bound at their single particle energies,

while the dotted lines correspond to the case where both

light and heavy system orbitals are bound at half-the-

two-neutron separation energy. The monopole and quadru-

pole transfer form factors may be seen to decay more

rapidly in the asymptotic region for the half-the-two-

neutron separation energy binding prescription. It should

be mentioned that binding the heavy system orbitals at

half-the-separation energy of the transferred nucleons,

while reducing the contribution of the surface orbitals,

has the unfortunate effect of increasing, rather than

reducing, the contributions from the most highly bound

orbitals which may be sizeable for some systems. Thus,

the artificial convergence introduced by this binding

procedure is not equivalent to the convergence in the UV

factor that would result from a more exact pairing

calculat i o n .

150

2. 182W ( 12C,14C ) 180W: An Intermediate Case

Experimental 0+ and 2+ angular distributions

for the 182W ( 12C , 1AC) reaction at a bombarding energy

of 70 MeV are given in Fig. IV-7b. The solid and dashed

lines correspond, respectively, to full CCBA calcula­

tions using transfer form factors based on half-the-

separation energy and exact binding prescriptions. As

can be seen, the half-the-two-neutron separation energy

prescription produces the best fit to the 2+ experi­

mental angular distribution, with the most pronounced

difference coming in the region of the interference mi n i ­

mum. Inspection of the intrinsic transfer form factors

Fd (F) given for both binding prescriptions in Fig. IV-12

shows that the J ■ 2 form factor is now reduced by a

factor of 5 compared to the J ■ 0 form factor at 11 fm.

Thus the direct J ■ 2 transition is weakened, and nuclear-

Coulomb interference in the dominant two-step route has

begun to compete with direct-two-step interference in

determining the shape of the 2+ angular distribution.

The increased importance of Coulomb excitation introduces

both an asymmetry with respect to the grazing peak, so

that the 2+ interference minimum now appears 'v 6° forward

of the grazing angle, and an increased sensitivity to the

slopes of the transfer form factors. The relative strength

of nuclear and Coulomb two-step transfer routes will

151

depend on which trajectories scatter into a given angle

and therefore will vary strongly as a function of both

energy and angle. However, for a given bombarding energy,

the strength of the nuclear-Coulomb interference governing

the shape of the 2+ angular distribution will be

sensitive to the slope of the J = 0 transfer form factor.

This is particularly true for that limited range of angles

around the Interference minimum where the transfer-plus-

inelastic transitions for the nuclear and Coulomb trajec­

tories are of comparable strength. For the gently sloping

monopole transfer form factor calculated using the exact

binding prescription, a larger fraction of transfer strength

is concentrated in large Impact parameter orbits Important

for Coulomb excitation, and a very deep nuclear-Coulomb

interference minimum is produced in the 2+ angular dis­

tribution (Fig. IV - 7 b ) . The Increased slope of the mono­

pole transfer form factor calculated within the half-the-

two-neutron separation energy prescription reduces the

Importance of Coulomb excitation and the resulting nuclear-

Coulomb Interference minimum is reduced in depth.

We can continue to follow the systematics of the

microscopic distribution of transfer strength in Fig. IV-13,

which shows partial form factors as functions of energy

for the T82W (12c , reaction. Again, each deformed

orbital is bound at its single-particle energy. The Fermi

152

level has now moved up in energy from the Sm case to

A 'v* 6.8 MeV. Again, all deformed orbitals contribute

constructively to the monopole form factor. However, for

the quadrupole form factor, there is now some cancellation

among orbitals around the Fermi level, but prolate orbitals

contribute somewhat more strongly than oblate orbitals,

and the moderately strong and positive form factor shown

in Fig. IV-12 results.

Fig. IV-12. Total intrinsic transfer form factors for the reaction

182 12 14 180 , , * j . „ * „W( C, C) W calculated using deformed orbitals

bound at their single-particle energies (solid line) and at

half-the-two-neutron separation energy (dotted line). The

orbitals in the light system were bound at half-the-two-

neutron separation energy in each case.

Fig. IV-13. Individual deformed orbital contributions to the total intrinsic

182 180transfer form factors at 11.2 fm for the W - W system.

See caption for Fig. IV-11.

R(fm)

Fig. IV-12

E„n„IMeV)

Fig. IV-13

153

3. *8^ W (I 2C ,^ C ) 1 8 ZW : Weak Quadrupole Transfer Strength

The I8AW (12 c ,14c) reaction was measured at a

bombarding energy of 63 MeV, and thus the kinematic

conditions are somewhat different than for the other three

(I2C,I^C) reactions discussed in this section. In Fig.

IV-14, which shows the experimental angular distributions

and corresponding CCBA calculations,the ground state

grazing peak is shifted backward by more that 10° com­

pared to the other tungsten examples. While a direct

comparison of the shapes of the angular distributions is

not particularly informative in this case, it should be

noted that the minimum in the 2+ data occurs forward

of the grazing angle, indicating the importance of

nuclear-Coulomb interference in the Inelastic part of the

dominant two step-transfer route and the depletion of

direct quadrupole transfer strength at the upper end of

the rare-earth region. Again, the 4 + angular distri­

bution has a minimum near the grazing peak, characteristic

of interference between direct and two-step transitions.

The continuing trend of decreasing quadrupole

transfer strength toward the upper end of the rare-earth

region may be followed in the intrinsic transfer form

factors F^(R) of Fig. IV-15, in which the target and

projectile orbitals have all been bound at half-the-two-

neutron separation energy. For the ^8 2W (12 c ,14C ) reaction,

154

the J ■ 2 form factor at 11 fm (calculated using the

same binding prescription) is reduced by a factor of 5

compared to J * 0, while for the l®^W(12c,14c) reaction,

the reduction factor is 10. This trend will continue

into 1®AW as the cancellation of contributions from

prolate and oblate orbitals from around the Fermi level

becomes even more efficient.

Fig. IV-14. Experimental angular distributions for the 0+ , 2+ , and 4+

182members of the ground state rotational band in W for1 9

184 12 14 Cthe reaction W( C, C) at E = 6 3 MeV. The CCBA

calculations shown include all orders of inelastic scattering

and all possible first-order simultaneous two-neutron transfer

routes connecting the 0+ and 2+ ground state band members

for the target and residual nucleus. Inclusion of the 4+

band members should move the 2+ angular distribution

backwards by about 2 °. Intrinsic transfer form factors were

calculated with the half-the-two-neutron separation prescription

for binding.

Fig. IV-15. Total intrinsic transfer form factors for the reaction

184 12 14 182\V( C, C) W calculated with orbitals in both the

light and heavy systems bound at one-half-the-two-neutron

separation energy.

(do

Vd

&)cm

(/

ib/s

r)

® c . m . M e g )

Fig. IV-14

F (R

)

Q c . m .

Fig. IV-15

155

4. 188W ( 1 , 1 4c )184w . (Almost) Pure N u c 1ear-Coulomb

Interference

The details of the interference between direct and

multistep transfer routes for the IB b y (12 c ,14c) reaction

have been discussed in Sec. D; here we present only a

summary of the data and details of the form factors as

they relate to systematics of the distribution of multi­

pole transfer strength throughout the rare-earth region.

Fig. IV-16 shows angular distributions for 0+ , 2+ ,

4 + , and 6+ members of the ground state rotational band

and the 2+ member of the y-vibrational band in ^84w,

together with CCBA calculations for the 0+ , 2+ , and 4+

states which are in good agreement with the data. The 6+

angular distribution is bell-shaped and contains no evi­

dence of an interference minimum, while the J = 6 transfer

form factor (Fig. IV-3) is reduced by only a factor of 15

compared to the monopole form factor. This suggests that

the 6+ member of the ground state band is populated

primarily by the direct J = 6 transition. A CCBA calcu­

lation which includes Inelastic transitions to all members

of the ground state rotational hand up to 6+ has not yet

been performed, nor has a calculation of the 2+ ^ angular

distribution.

Inspection of the intrinsic transfer form factors

FJ (R) for the 18flW (12c , 14 c) reaction (Fig. IV-3) shows

156

that the J = 2 form factor with both light and heavy

system orbitals bound at hal f- the-two-neutron-separation-

energy is reduced by a factor of 30 from the J ■ 0 form

factor at 11 fm and by an even larger factor far out

on the tail of the wave function. Thus the direct quadru­

pole transition is extremely weak for this reaction, and

the 2+ state is populated predominantly through two-step

processes, with the interference minimum forward of the

grazing angle arising mainly from destructive nuclear-

Coulomb interference in the inelastic part of the dominant

two-step route. A c o m p a r i s o n 'of the consequences of neglec­

ting the direct quadrupole transfer route in the CCBA

calculations for (12c,14c) on 154sro ancj 186^ is given

in Figs. IV-7a and c, respectively. For the

1 S m ( 1 , l^C) reaction where the J = 2 form factor is

strong (Fig. IV-8) , the 2+ interference minimum arises

mainly from destructive interference between direct and

two-step processes, so that a calculation which omits

the direct transfer route is unable to reproduce the char­

acteristic features of the angular distribution. For

where the quadrupole form factor is extremely weak, the

effects of neglecting the direct route in the calculation

are much less severe, but by no means negligible.

The connection between the intrinsic transfer form

factor strength and the structure of the orbitals parti­

cipating in the transfer is made in Fig. IV-17. For the

157

J => 2 form factor, as the neutron number increases from

182-180^ to 186-184y^ tbe Fermi level moves up in energy

to X 2: 6 MeV where the UV product weighting, along with

the corresponding changes in nuclear deformation, causes

the cancellation between prolate and oblate neutron orbital

contributions composing the quadrupole form factor to

become extremely efficient. In addition, the monopole

transfer strength is further enhanced, and, as a result,

the 2+ angular distribution corresponds to that of an

almost pure two-step process characterized by a dramatic

nuclear-Coulomb interference minimum forward of the grazing

angle. Again, loosely bound orbitals make large contri­

butions to the form factors, and an artificial convergence

must be introduced into the problem by binding orbitals

in the light and heavy systems at one-half-the-two-neutron-

separation-energy in order to obtain the agreement shown

in Fig. IV-16.

Fig. IV-16. Experimental and calculated (CCBA) angular distributions12

for the reaction 186W(12C ,14C)184W at E C= 70 MeV.

The theory curves have been normalized to the data. The

CCBA calculations include all orders of inelastic scattering

and all possible first-order simultaneous two-neutron transfer

■f -froutes connecting the 0 , 2 , and 4 ground state band

members of the target and residual nucleus. Intrinsic

transfer form factors were calculated with the ha If-the-two-

neutron separation energy prescription for binding and are

based on average deformation and pairing-gap parameters

for these nuclei.

Fig. IV-17. Individual deformed orbital contributions to the total intrinsic

1 1transfer form factors at 11.2 fm for the W - W

12system and a C projectile. See caption for Fig. IV-11.

1000

100

10

100

10

100

■o

I

10

I

10

I

0.1

86W (,2C .,4C)'84 wl2CE - 70 MeV

30 40 60 70 80 90 100 IKe,c m.

Fig. IV-1C

E w (MeV)

Fig. IV-17

158

F • ^ 8 ^ W ( ^ Z C , ^ B e ) ^ 8 R Os R e a c t i o n : The Role of P r o j e c t i l e

B i n d i n g En e r g y

In the ^ 8BW ( ^ 2 C ,I^C) r e a c t i o n just d i s c u s s e d , the

b i n d i n g of the t r a n s f e r r e d n e u t r o n s is s u f f i c i e n t l y w e a k in

both ca r b o n and t u n g s t e n that the c o r r e s p o n d i n g s l ow d e c a y

of the t r a n s f e r form fa ct or a l l o w s h i g h - L (large impact

p a r a m e t e r ) C o u l o m b e x c i t a t i o n c o n t r i b u t i o n s to i n t e r f e r e

c o h e r e n t l y w i t h l o w e r - L n u c l e a r c o n t r i b u t i o n s to the i n e l a s ­

tic s c a t t e r i n g c o m p o n e n t of the t w o - s t e p proc es s. Si nc e in

this e x a m p l e the i n t r i n s i c J ■ 2 t r a n s f e r form factor,

a s s o c i a t e d w i t h the di r e c t f e e d i n g of the 2+ st at e of

is al m o s t an or de r of m a g n i t u d e w e a k e r than the

J = 0 form factor, the c o r r e s p o n d i n g a n g u l a r d i s t r i b u t i o n

is d o m i n a t e d by the t w o - s t e p p r o c e s s and he nc e p o s s e s s e s

the fa m i l i a r n u c l e a r - C o u l o m b i n t e r f e r e n c e m i n i m u m c h a r a c ­

t e r i s t i c of pure i n e l a s t i c s c a t t e r i n g .

In o r d e r to u n d e r s t a n d how the r e l a t i v e i m p o r t a n c e of

Co u l o m b e x c i t a t i o n is i n f l u e n c e d by the b i n d i n g e n e r g i e s of

the t r a n s f e r r e d n u c l e o n s , we have s t ud ie d the r e a c t i o n

186W (12c,10Be) 1 8 8 0s , at a b o m b a r d i n g e n e r g y of 70 MeV.

In this case the en e r g y r e q u i r e d to re m o v e two p r o t o n s from

12c('v27 MeV) is about twice that to re m o v e two n e u t r o n s

from l^C, w h i l e the b i n d i n g of two p r o t o n s in 188os is

about the same as that for two n e u t r o n s in 1 8 BVJ(r('13 MeV) .

In a d d i t i o n , this h e a v y - i o n t w o - p r o t o n t r a n s f e r r e a c t i o n

p r o v i d e s a test for the m i c r o s c o p i c s t r u c t u r e d e s c r i p t i o n

159

of p r o t o n o r b i t a l s for w h i c h there is no l i g h t - i o n c o u n t e r ­

part (here, we a s s u m e the (3He,n) r e a c t i o n to be i m p r a c t i c a l ) .

The ^ 8 6 ^ ^ 1 2 c ^ 1 0 g e ) g+ and 2+ a n g u l a r d i s t r i b u t i o n s

t o g e t h e r w i t h C C B A c a l c u l a t i o n s are p r e s e n t e d in Fig. IV-18.

In the ( ^ C , T D g e ) re ac t i o n , the large two pr o t o n

b i n d i n g in the p r o j e c t i l e c a u s e s the t r a n s f e r form factors

to d e c a y r a p i d l y (Fig. IV-19), thus r e d u c i n g the p r o b a b i ­

lity for t r a n s f e r from d i s t a n t orbits. In Fig. IV-20,

the role of C o u l o m b e x c i t a t i o n is c o m p a r e d for the 2+

t r a n s i t i o n in the ( 1 2 c t14c) and (l ^C jl ^B e) r e a c t i o n s .

In each case, for a pure t w o - s t e p p r o c e s s i n v o l v i n g e i t h e r

n u c l e a r or C o u l o m b i n e l a s t i c t r a n s i t i o n s alone, the c a l ­

c u l a t e d a n g u l a r d i s t r i b u t i o n is b e l l - s h a p e d , but b r o a d e n e d

as a result of e n h a n c e d L - s p a c e l o c a l i z a t i o n in the two-

step process. H o w e v e r , for the pure t w o - s t e p p r o c e s s in

w h i c h both n u c l e a r and C o u l o m b e x c i t a t i o n p r o c e s s e s act

c o h e r e n t l y , C o u l o m b e x c i t a t i o n may be seen to play a m u ch

r e d u c e d role In the (* ^C ,* ^B e) r e a c t i o n as c o m p a r e d to

the t w o - n e u t r o n t r a n s f e r re ac ti on .

In a d d i t i o n , b e c a u s e of the large n e u t r o n e x c e s s

in r a r e - e a r t h n u c l e i , t h e t w o - p r o t o n t r a n s f e r r e a c t i o n

s a m p l e s d e f o r m e d o r b i t a l s from lower sh e l l s as c o m p a r e d

to the t w o - n e u t r o n t r a n s f e r re a c t i o n s . Thus, w h i l e there

Is a d r a m a t i c c a n c e l l a t i o n of p r o l a t e and o b l a t e n e u t r o n

o r b i t a l s (Fip. TV-17) w h i c h r e s u l t s in the d e p l e t i o n of

q u a d r u p o l e t r a n s f e r s t r e n g t h for the (*^C,*4c) r e a c t i o n ,

Fig. IV-18. Experimental and calculated (CCBA) angular distributions

"1“ *4*for the 0 and 2 members of the ground state rotational

188band in Os for the two-proton transfer reaction

12188W(12C ,1^Be) at E 70 MeV. The theory curves

have been normalized to the data, but the relative cross

section calculated within the CCBA is in agreement with

the experimental result. The CCBA calculations include

all orders of inelastic scattering and all possible first-

order simultaneous two-proton transfer routes connecting

the 0+ and 2+ ground state band members for the target

and residual nucleus. Intrinsic transfer form factors were

calculated with the deformed proton orbitals bound at half-

the-two-proton separation energy.

Fig. IV-19. Total intrinsic transfer form factors for the two-proton

186 12 10 188 transfer reaction W( C, Be) Os calculated with

orbitals in both light and heavy system bound at one-half-

the-two-proton separation energy.

(js/q-W) W0(UP/-°P)

®c.m.

Fig. IV-18

R(fm)

Fig. IV-19

Fig. IV-20. Comparison of the relative importance of two-step

nuclear-Coulomb interference and direct-two-step (nuclear)

interference in the population of the 2 ground-state

rotational band members of the nuclei l8 4 w (Fig. W-20a)

188and Os (Fig. IV-20b). In both cases the CCBA

calculations included the 0+ and 2+ states of both the target

and residual nucleus.

400

200

100

40

20

10

6

20

10

4

2

I

.6

.■ '■ I

* * ‘ ■ *« ^(0)

/ . / /s x

/> •

/ / *r/

•/ / • •

// / ♦

- xn •

/

* * • *

a < 4 / ♦

m.• X.• X• t \

•- V/Js

V • A

•

L.\ 2+

III kbV

• V••••

•

l 8 6 W ( l 2 C , l 4 C ) l 8 4 W E =

------------------------------ DIRECT ♦ T W 0-

— — — — — TWO-STEP (NUC

------------- ---------------- TWO-STEP (NUC

.................................. TWO-STEP (CO

7 0 MeV

STEP

:l e a r + COULOMB)

: l e a r ONLY)

ULOMB ONLY)

(b)>

•< s'

. ^\ ss' k

• t \ N v1 1 1 1.

7t T \ \ \k

( > 2+155 keV

' \

,86W ( l2C, l0Be) l880s E = 7 0 MeV

40 50 60 70 80 90 100

^ c . m .

Fig. IV-20

160

the same phenomenon does not occur for the two-proton

transfer reaction (I2C,^©Be). The dominance of prolate

proton orbitals around the Fermi level has the effect of

producing a large intrinsic J = 2 transfer form factor,

almost half the strength of the J = 0, and of the same

sign (Fig. IV-19). Since the intrinsic J * 2 two-

proton transfer form factor is strong and of the same sign

as the J = 0, the resulting minimum in the 2+ angular

distribution arises mainly from destructive interference

between a strong direct and two-step (nuclear-dominated)

process. The stronger binding in the two-proton transfer

reaction combined with the large negative Q-value ('''-17 MeV)

result in cross sections which are considerably smaller

than for the two-neutron transfer reaction. In an explor­

atory measurement, an upper limit of 'v-2yb/sr was placed

on the peak cross section for the 4 + transition and no

attempt was made to obtain a complete angular distribution.

Figs. IV-21 and IV-22 show absolute S-matrix elements

associated with different two-step routes leading to the

2+ states of the final nuclei for the (^2C , ^ C ) and (12c ,10b c ) reactions on ma y be seen that the

two-step (nuclear + Coulomb) S-matrix element has a

stronger Coulomb tail for the two-neutron case, a result

which is reflected in the more dominant role of nuclear-

Coulomb interference for the (^2C,^4c) reaction.

Fig. IV-21. Absolute S-matrix elements associated with different

two-step routes leading to the 2+ state in X84W for the

186 12 14 X2Creaction W( C, C) at E = 7 0 MeV. Note the strong

Coulomb tail for the two-step (nuclear + Coulomb) two-

neutron transfer S-matrix element.

Fig. rv-22. Absolute S-matrix elements associated with different two-step

+ 188routes leading to the 2 state in Os for the reaction

186W(12C ,10Be) at E c = 70 MeV.

A • •

10

8

f tI : ‘ ii :'

S f x 4.41*10'

a • •f :

af:

a

ih

P :

a

/ : /'/

/ •'// .*

r:*x/.'

20 30

‘ I

11it

'i

\

*in " “ out ” ^

,86W (,2C. ,4C ) ,84W

E = 70 MeV

it\

•I

\i

T W O - S T E P(NUCLEAR* COULOMB)

— — TWO-STEP(NUCLEAR ONLY)

; T W O - S T E P• (COULOMB ONLY)

i\I x

1 '\*.\*.

4 0 50 6 0 70

L;n (tl)Fig. IV-21

10 r9 -

8 -

7 -

6 -

5 -

4 -

3 -

10 20

1\

>

iS f x 1.64-10

out i

Il»

ihn

i!

186W (I2C. ,0Be) 188Os

E = 70 MeV

TWO-STEP(NUCLEAR*COULOMB)

TWO-STEP(NUCLEAR ONLY)

30

\\ N

\ ", — L^.4 0 50 6 0 70

Lin no

Fig. IV-22

161

G. I8BW ( 1^0,1 ® 0 ) R e a c t i o n : Projectile Dependence

of Reaction Mechanism

In order to investigate the dependence of the reac­

tion mechanism on the nuclear structure of the projectile, we have studied the reaction 1 ( 1 6 o ,1^0)184y at

90 MeV bombarding energy. Experimental angular distri­

butions and the corresponding CCBA calculations are

shown in Fig. TV-23.

In comparing the shapes of angular distributions

for the and (1^0 ,1®0) reactions, we notez 1z 2 e 2that the Sommerfeld parameter ri ° ------- for the heavier

160 projectile is larger, so that from semi-classical

arguments, the density of partial waves per unit impact

parameter increases, i-space localization becomes less

severe, and one would expect the envelope of the angu­

lar distributions to be narrower, as is the case here.

From an experimental point of view, studies using an 1^0

projectile are more difficult than those with ^ ZC, both

because of the increased straggling and energy loss of

in the target, and because of the finer angular resolu­

tion required to define the resulting narrow interference

structure.

As may be seen from Figs. IV-24 and IV-25, the

direct quadrupole (J * 2) transition is extremely weak

162

for this system as It Is for the 1 3 ^W(12C , 14C) reaction.

Indeed, from a comparison of Figs. TV-17 and IV-25,

it may be seen that the microscopic structure of the in­

trinsic transfer form factors is very similar for the

(12c,14c) and (*80,^80) reactions on Again,

the 2"*" angular distribution will be most sensitive to

the slope far out on the tail of the monopole (J ■ 2)

transfer form factor through inelastic nuclear-Coulomb

interference in the dominant two-step transfer route.

The dashed line in Fig. IV-23 shows the results of

a CCBA calculation in which the valence orbitals of the

heavy system are bound at their eigenenergles. The mono­

pole transfer form factor is quite strong in the asymptotic

region containing the Coulomb orbits most important for

inelastic n u c 1 ear-Coulomb interference, and a very deep

interference minimum results which is not reflected in

the data. The effect of introducing an artificial conver­

gence into the problem by binding the orbitals of both

the light and heavy systems at one-half-the-two-neutron

separation energy is shown by the solid lines of Fig. IV-23.

While agreement with the 2 + angular distribution is

generally improved using this prescription for binding,

the significant interference minimum predicted by the

CCBA calculations does not appear in the data. This dis­

crepancy suggests the need for: (1) an improved structure

Fig. IV-23. Experimental and calculated (CCBA) angular distributions for

the 0 , 2 , and 4 members of the ground-state rotational1 fi104 1 I f i 1 Q O

band in \V for the reaction W( O, O) at E = 90 MeV.

The CCBA calculations include the 0+ ,2+ ,4+ ground-state-

band members of the target and residual nucleus (with the

exception of the dotted curve which includes only the 0+ and

2+ band members). The solid line corresponds to a CCBA

calculation made with intrinsic transfer form factors calcu­

lated using 31 deformed orbitals bound at half-the-two-neutron

separation energy. The dashed line represents a CCBA

calculation based on 24 deformed orbitals bound at their

single particle energies. In each case, the orbitals in the

light system were bound at half-the-two-neutron separation

energy. The dotted curve shows the effect of neglecting

Coulomb excitation in the inelastic part of the two-step

transfer routes for the 2+ transition.

4

2

I

0 .44

200

100

4 0

20

10

4

2

I

20

10

4

2

I

0 .4

0.2

0rc.m.

Fig. IV-23

Fig. IV-24. Total intrinsic transfer form factors for the reaction

^88W(^80 ,^ 80 )184W calculated with orbitals in both light

and heavy (31 orbitals) systems bound at their single-particle

energies (solid lines) and at half the separation energy of the

transferred neutrons (dotted lines). The dashed lines

represent the form factors calculated with the light system

bound at half-the-two-neutron separation energy and 24

deformed orbitals bound at their eigenenergies.

Fig. IV-25. Individual deformed orbital contributions to the total intrinsic

186 - l84wtransfer form factors at 11.2 fm for the W

1 0and an O projectile. See caption for Fig. IV-11.

system

R ( f m )

Fig. IV-24

Fig. IV-25

163

description for the light system; (2) an improved pairing

description for the heavy system; and/or (3) additional

forward angle data taken in fine angular steps to define

more precisely the presence or absence of any interference

structure.

In any case, while the agreement of the CCBA cal­

culations with the data might benefit from an improved

pairing description, nu c 1ear-Cou1omb interference still

plays an essential role in determining the shape of the

2+ angular distribution. This is illustrated in Fig. 1V-23

where the dotted line shows the effect of neglecting

Coulomb excitation in the inelastic transitions by zeroing

the Coulomb p's in the coupled-channel calculation. The

resulting cross section is in violent disagreement with

the data at angles forward of the grazing angle where

nuclear-Coulomb interference is most effective.

Because the direct J = 4 transition is comparable"

in magnitude and opposite in sign to the indirect route

involving the strong monopole transition (as it was for the

( 12C , lAC) reaction), the minimum in the 4+ angular

distribution results from direct-two-step interference with

the inelastic transitions going predominantly through low-

impact parameter nuclear orbits. For this reason, we can

expect the 4+ angular distribution to be relatively

insensitive to details in the tail of the wave function,

164

and the data are In fact, well reproduced by calculations

In which the single particle orbits are bound both at

their eigenenergies and at half the two-neutron separation

ene r g y .

165

H. Conclusion

In this study of transfer reactions on deformed

rotational nuclei, we have noted the varied and crucial

role which Coulomb excitation plays in determining the

shapes of the ground state band 2+ angular distribu­

tions. It should be emphasized once again that it is

Coulomb excitation which makes it possible to observe

small continuous variations in quadrupole transfer strength

across the rare-earth region. If nuclear excitation alone

were important, a decrease in direct quadrupole transfer

strength would be reflected only in a change in depth of

the 2+ interference minimum at the grazing angle, and

angular distributions for ^82W (12c , 4AC) and 2c , 1^ C)

might be indistinguishable. It is the changing competition

between direct-two step (nuclear-dominated) interference,

affecting the angular distribution near the grazing angle,

and inelastic nuclear-Coulomb interference, most effective

at forward angles, which introduces an additional asym­

metry and provides an unambiguous signature for the change

in transfer strength.

We have also noted that, in a multistep transfer

process, the importance of Coulomb excitation depends on

the transfer strength associated with large impact para­

meter orbits. Thus, unlike pure inelastic scattering,

Coulomb excitation accompanying a transfer process is

166

governed not only by the electric multipole moments of

the nucleus being excited, but also by the binding

energies associated with the transfer form factors.

It is the sensitivity to the slopes of transfer

form factors introduced by Coulomb excitation which has

the most important implications for the use of these

transfer reactions as a spectroscopic tool. When the

direct quadrupole transfer strength is weak, as it is in

the tungsten region, the sensitivity of these transfer

reactions to the slope of the form factors in the tail

region is greatly enhanced by strong nuclear-Coulomb

interference in the inelastic part of the dominant two-

step transfer route. So 2+ angular distribution data

present a very sensitive test for nuclear wave functions

generated within a nuclear structure model. We have found

that, although the simple constant pairing BCS-model for

the microscopic structure of identical rigid rotors can

generate form factors which reproduce the main features of

the observed angular distributions, the data are of suffi­

cient quality to reveal discrepancies with the nuclear

model. The resolution of these problems presents a chal­

lenge for any more realistic pairing model.

167

CHAPTER V COMPLEX PARTICLE EMISSION

IN STATISTICAL COMPOUND REACTIONS

A. Introduction

One useful way to classify nuclear reactions is in

terms of the time delay which occurs between the arrival of

the projectile within the "range of interaction" of the

target nucleus and the time of emergence of the first particle

(or gamma ray) following the interaction. At a given inci­

dent energy, this time delay will in general depend on the

degree of overlap or the classical "distance closest

approach" for the target and projectile and the average

interaction time will in general decrease with increasing

energy for all trajectories.

In the earlier discussion of heavy-ion induced direct two-nucleon transfer reactions, the emphasis was on those

classical trajectories (or quantum partial waves), associated

with the region near or just inside the nuclear surface, for

which the projectile is not strongly absorbed. In a classical

picture, the attractive nuclear force for such orbits is not

sufficiently strong to overcome the repulsive Coulomb and

centrifugal forces and "capture" the projectile, so that the

resulting interaction must necessarily take place in a

time on the order of the nuclear transit time and involve the

transfer of a relatively few nucleons (A < 4).

168

In this chapter, we will be concerned with those

heavy-ion reactions which involve closer trajectories and a

larger degree of overlap , and which may therefore lead to

the transfer of a large number of nucleons between target

and projectile. Such multinucleon transfer reactions may

have a sizeable direct component arising in the larger

partial waves when kinematic and nuclear structure

conditions are particularly favorable, but most of the

incident flux will go into compound nucleus formation. In

the model of the compound nucleus of interest here, the

initial kinetic energy of the system is shared by a series

of two-body interactions until statistical or thermodynamic

equilibrium involving a large number of nuclear configur­

ations is reached. The lifetime of such a compound state

depends in detail on the barriers and associated phase space

for decay and will thus depend strongly on excitation energy

of the compound nucleus, varying from (typically) 103^sec

at low energies to 10 “ 1 gec at high excitation energies

where the distinction between direct and compound reactions

dissappears.

For light-ion reactions, these traditional catagories

of direct and compound reactions must be extended to include

"pre-equilibrium emission" processes which take place on an

intermediate time scale. In the pre-equilibrium decay models

of Griffin (Gr66) and Blann (B168), the light projectile

shares its energy with the target by a series of two-body

interactions, and at each stage in the approach to equilibrium,

169

characterized by an "exciton" number, the compound system

may decay by emitting a particle. vost of the emission

takes place early in the process while there is still

enough energy to escape, and the evidence for the operation

of such pre-equilibrium emission processes consists of an

enhanced yield in the high energy part of the energy spectrum

for many (p,p) and (p,a) reactions on intermediate mass

n u c l e i .

At the other extreme, multinucleon transfer between

very heavy targets and projectiles may take place in

"strongly-damped" or "deep-inelastic" collisions (Hu74). In such

a collision, the two fragments orbit with orbital angular

momentum greater than the critical angular momentum for

fusion, and before separation, a large part of the kinetic

energy of the system is converted into internal energy by

nuclear friction while large numbers of nucleons are

transferred. Such processes may have important applications

in the production of super-heavy nuclei.

Heavy-ion multinucleon transfer reactions may yield

very different, but equally valuable, information depending

on the mechanism for the reaction. A reaction which involves

the simultaneous transfer of many nucleons in a direct process

may he useful as a tool to Investigate nuclear correlations

(M171), while compound reactions mav be used to selectively

populate high spin states in residual nuclei (St 74, K175)

170

under certain conditions. The extraction of nuclear infor­

mation from data for such reactions, then, requires that

the reaction mechanism be identified. One approach to

the problem of determining whether a reaction mechanism

is predominantly direct or compound is to make theoretical

estimate of the cross section expected on the basis of

statistical compound nucleus formation alone using a

Hauser-Feshbach-type statistical model code. As the following

discussion will show, such Hauser-Feshbach calculations are

able to predict the magnitude of the compound contribution

to a reaction to within a factor of two or better, and are

used extensively in the following analyses.

171

B. The Statistical Model: Formalism

The Hauser-Feshbach expression for energy-averaged

reaction cross sections may be derived based on the following

assumptions:

(1) total flux is conserved in the interaction;

(2) the Bohr Independence Hypothesis holds, according

to which "the mode of disintegration of the com­

pound system depends only on its energy, angular

momentum, and parity but not on the specific way

in which it has been produced" (B152) allowing

the compound nucleus cross section expression to

be factored into formation and decay terms wit h­

out interference;

and (3) time reversal invariance holds, permitting the

application of the reciprocity theorem.

The Hauser-Feshbach expression for the angle-inte­

grated cross section averaged over compound nucleus

fluctuations, is given by (FebO, Vo64)

where unprimed quantities refer to the incoming channel c,

primed quantities to the exit channel c ' , and the sum in

the denominator runs over all possible outgoing channels.

( v - l )

172

The quantum numbers of each channel c are

c » (a , I , i , s , £ , J ,tt ) , where a labels the pair of particles

and their state of excitation, T and i are the ground

state spins of the target and projectile, s is the channel

the total angular momentum (J = £ + s) and parity. ka is

the wave number of the incident channel. The transmission

potential appropriate to each pair of particles. This

expression may be seen to consist of the product of a term

for the total cross section for formation of the compound

nucleus and a term giving the ratio of the phase space for

decay to a particular state «' to the total phase space

available for decay of the compound nucleus.

The corresponding expression for the energy-averaged

statistical compound angular distribution Is then

where Z is the Z-coefficient defined in Ref. (Fe 6 0) and

of-mass angle between the outgoing particle and the direction

o.f the incident beam. In the statistical model analyses to

be described, F.qs. V-l and V-2 were evaluated using the

spin (s = I + i), £ the orbital angular momentum, and

coefficient may be calculated from the optical-model

x

the P^(cos 0) are Legendre polynomials, with 0 the center-

173

Hauser-Feshbach code STATIS fSt72) written by R. G. Stokstad.

When compound nucleus formation followed by the

emission of two particles becomes important, it is neces­

sary to evaluate an expression of the form

tation energy for populating all states of excitation energy

F. *, spin J , parity tt in the initial compound nucleus,

p ( E* ' , J ' , tt ' ) Is the density of states ( E * ' , J ' , tt ' ) in

the residual nucleus, and

is the branching ratio for the decay of a state (E*,J,tt)

in the initial compound nucleus to a state ( E * ' , J ' , ft ' ) in

the residual nucleus. Summation is over all states in the

compound nucleus which are energetically allowed to decay

to the given level in the residual nucleus. Extension to

multiple particle emission is straightforward.

Statistical model, calculations involving multistep

decay were carried out with the code BREAKUP (Ha75)

written by the author specifically for applications where

gamma decay is not important but where it is essential to

(E* 1

( V- 3 )•B(E*,J,tt ; E* ' ,J' ,tt')

where a (E*) is the total cross section per unit exci-J , TT

T. T n 'Z ' , a ' Z '

B(E*,J, tt;E*’ ,J' ,tt * ) = " J ’ .u '

174

consider many particle decay channels.

In the codes STATIS and BREAKUP, the effects of

spin-orbit forces are neglected, so that the transmission

coefficients for a given orbital angular momentum are

equal for all channel spins s allowed by angular momentum

conservation (J = Z + s). As a rule, only proton optical

potentials include a spin-orbit term and, for the proton

channel, the transmission coefficient for a given Z with

the spin-orbit term omitted will be approximately the

average of the Z + 1/2 and Z - 1/2 coefficients calcu­

lated with such a term present. Thus, the neglect of spin-

orbit forces represents no major restriction. In addition,

neither y-decay nor isospin are treated explicitly, but

these restrictions are not expected to be Important for

calculations in the 2sld shell. For these light nuclei,

the Coulomb and angular momentum barriers for particle

decay are relatively low. Thus, the electromagnetic decay

of highly excited compound states will be a weak process,

in general, compared to particle emission, and y-decay

may be neglected. Similarly, the proton and neutron

threshold resonance reactions, from which the level

density parameters are derived, are sensitive only to

states of lowest isospin. Because such states must be2 6populated in the Al compound nucleus formed through

175

12 14the C + N entrance channel, and because isospln

appears to be conserved up to rather high excitation ener­

gies for the compound reactions of interest here (see Sec. E ) ,

no malor problems should result from the neglect of isospin

in the present case.

1ITo evaluate the quantity [ 7 T.n ] T

c' , s" ,1Tin the denominator of Eqs. (V-l)-(V-3), representing the

total number of channels open for the decay of the compound

nucleus, the known energy, spin, and parity of all discrete

levels up to a certain cutoff excitation energy E areCUTread into the program. From ® q u t UP tC> tb8 allowed energy, the sum over discrete states is replaced by

an integral over states in the continuum calculated using a

level density formula R i v e n by (La63, Vo64 , Gr72):

n . C L L t i J . _ _ _ _ _ _ _1/4 5/4 2 3 / 21 2 a (U + t ) ( 2 a )

exp 2 ( a U ) 1/2_ (J + l/ 2^ 22o T

( V-4 )

0 (U,.T,tt) = 1 / 2 0 (U,J).

In Eq . (V-4) the quantity U = F. - A = at^ - t is the

excitation energy corrected for pairing energy A, t is

the nuclear temperature, and = 1 t/h^ is the spin cutoff

factor with I the rigid-hody moment of inertia. In all

calculations to be described, the values for the single part­

icle 1evel-density parometer, a, are those given by Facchini

and Saetta- enichella (Fa68) who use the pairing energy correc­

tions of Cilhert and Cameron (G165).

176

19 1 /C. Gamma-ray and Light Particle Production in C + N

Interactions at Low Energy

The original motivation for our interest in 12 14C + N reactions at low energies came from early gamma-

ray measurements of Almqvist et al. (A160). Their spectra

for gamma rays produced in the bombardment of by1220 MeV C ions show prominent lines for the de-excitation

of the first 2+ and 4+ states in 2^Mg but little evi­

dence for transitions in the other reaction channels energeti-2 6cally available for the decay of the Al compound nucleus

(Fig.V-1). This apparent discrepancy with the compound model

led to speculation by Almqvist et al. that the observed

gamma transitions might result from the preferential popula­

tion of states in 2Sfg b y tbe direct transfer of a ^ 2C

cluster in the reaction.

The original gamma-ray measurements were made with

low resolution using a Nal detector. Recent measurements

by Nomura et al. (No69) using a Ge(Li) detector are more

consistent with a compound nucleus picture, but still do

not distinguish between the contributions to the 24Mg

transition yields from successive p-n emission, which

should be dominant for a statistical compound nuclear

reaction mechanism, and deuteron emission.

To make a definite identification of the reaction

mechanism for these reactions, we have measured both gamma-

ray and charges particle yields in the ^2C + system

at low energies. A complete statistical model analysis of

Fig. V -l . Reaction channels which are energetically available for the

26Al compound nucleus. Indicated gamma transitions were

observed in the present experiment.

177

this data has served both to confirm the statistical compound

nature of the reaction mechanism and to establish a consis­

tent set of parameters for llauser-Feshbach calculations at

higher energies to be described in Sec. E.

1 2Absolute y-ray yields were measured using a C

beam and nitrogen gas target over a range of energies

14 < 5. 33 MeV. The experimental configuration of

detectors and target for the y-ray measurements is shown in

Fig. V-2. The target consisted of a n i t rr. gen-f 1 11 ed gas

cell, 1.4 cm long, with a 0.51 me/cm^ nickel entrance

window. The gas pressure was typically 174 Torr corres­

ponding to a thickness of 0.40 mg/cm^. A liquid nitrogen

cold trap with a cylindrical hole along the beam axis was

located just before the gas cell to prevent carbon build-up

on the beam side of the gas cell entrance window and also

to serve as a mount for a beam-defining tantalum collimator.

The beam was stopped by a tantalum insert within the gas

cell. The gas cell itself served as a Faraday cup, with

electrical isolation maintained by a teflon insulating

spacer between the beam pipe and gas cell, and an electron

suppressor grid, operated at 1000 volts negative bias and

located just before the gas cell entrance. Background

contributions to the y-ray spectra were identified both

by in-beam measurements made with the gas cell evacuated

and by room background measurements with the beam off. The

Fig. V -2 . Gas cell used in gamma-ray measurements.

TEFLON INSULATING SPACER

ELECTRON SUPPRESSOR GRID

GAS CELL

Ge (L i)

DETECTOR

GAS LINE

TO GAS BOTTLE AND ROUGHING PUMP

Fig. V -2

178

number of incident C ions was determined both from the1 2integrated beam current and from the number of C ions

backscattered from the nickel foil entrance window into a0silicon surface barrier monitor detector fixed at 154 .

Gamma-rays were detected with a 36 cm^ Ge(Li) detector/O

placed at 0 with respect to the beam and located 6.3 cm

from the center of the gas cell. Signals from the Ge(Li)

detector preamp were fed into an Ortec 450 Research

Amplifier and then stored in 4096 channels of a Northern

Scientific multi-channel analyzer. Energy calibrations

were made periodically with a set of standard gamma-ray

sources, and amplifier gain was set to observe a range of

gamma-ray energies up to about 4 MeV. Typical energy

resolution for the Ge(Li) detector - electronics system

was 3 keV(FWHM) for pulser generated peaks. The

absolute photopeak efficiency of this Ge(Li) detector

was determined in a subsequent measurement with calibrated

y-ray sources positioned at the center of the target

volume and all effective attenuators (tantalum beam stop

and gas cell hack) in position. A typical ga-nma ray spec­

trum covering an energy range up to E = 2 M e V at 30 MeV

bombarding energy is shown in Fig. V-3.

Gamma ray peak yields were extracted using a light-

pen interactive peak fitting program MPSFIT, which per­

forms a simultaneous least-squares fit of Gaussians for

1 2

179

up to six peaks designated by the user, with linear or

quadratic background subtraction.

The energies and assignments of observed y-ray

transitions which originated in the 12C + interaction

are listed in Table V-l. Listed are only those transitions

whose photopeaks were sufficiently large and free of back­

ground to permit extraction of a peak area. Several peaks

in the y-ray spectrum are Doppler-shifted and Doppler-

broadened, as may be seen in Fig. V-3. For those transitions

proceeding from states having lifetimes less than a nano­

second, the observed gamma-ray energies are larger by an

amount consistent with the Doppler shift expected on the

basis of the reaction kinematics, the length of the gas cell,

and the long stopping times of ions moving in a gas.

Excitation functions for y-ray transitions in a

number of residual nuclei are shown in Figs. V-4 and V-5.

The data points are plotted at the energy of the beam at

the center of the gas cell, which was determined using the

stopping power tables of Northcliffe and Schilling (No70).

The horizontal bars indicate, for each bombarding energy, the

range of energy available in the entrance channel as the

projectile transverses and loses energy in the gas cell.

The vertical error bars include contributions from uncer­

tainties in detector efficiency, the number of incident ions,

A

12TABLE V -l . Gamma-ray Transitions from C +

aTransition Light ReactionProducts

14N Interaction

_TRANE y(MeV)

MeanLifetime

Doppler Broadened and Shifted?

E OBsby(MeV)

18F[ 1 .122(5+) -* 0.937(3+)3 2a 0.185 218 ns No 0.185

21Na[ 0.332(5/2+) -♦ g .s .(3 /2 +)] n + a 0.332 14 ps Yes 0.340

21Ne[0.350(5/2+) -» g .s .(3 /2 +)] p + a 0.350 20 ps Yes 0.360

25 4* + Mg[ 0.975(3/2 ) -» 0.585(1/2 )] P 0.390 14 ps Yes 0.403

24Na[ 0.473(1+) -» g .s .(4 +)] 2p 0.473 29 ns No 0.471

22 f + + ^Na[ 0.583(1 ) -» g .s . (3 )] a 0.583° 352 ns No 0.583

25Mg[ 0 .585(l/2+) -» g .s .(5 /2 +)] P 0 .585° 4.9 ns No 0.585

18F[ 0.937(3+) -» g .s .( l+)] 2a 0.937 68 ps Nod 0.937

24Mg[ 1.369(2+) -* g .s .(0 +)] (P + n; d) 1.369 1.8 ps Yes 1,413

21Ne[ 1.746(7/2+) -» 0.350(5/2+)] p + a 1.396 230 fs Yes 1.442

24Mg[ 4.123(4+) -+ 1.369(2+)] (P + n; d) 2.754 55 fs Yes 2.836

24Mg[ 4.239(2') -» 1 .369(2 ')] (P + n; d) 2.870 100 fs Yes 2.964

cl Energies, spins, parities, and lifetimes are from F. Ajzenberg-Selove, Nucl. Phys. A190,1(1972) and P. M.. Endt and C. van der Leun. Nucl. Phys. A214,1(1973).U 1 ' 0d Observed gamma-rays from spectrum for E(C12) = 30 MeV. These states are unresolved.

This state is fed predominantly by the long-lived 5+ state at 1.22 MeV.

180

Fig. V -3. Gamma-ray spectrum for C+ N induced reactions measured

12C oat E = 3 0 MeV with Ge(Li) detector at 0 with respect to beam.

12 14

CHANNEL NUMBER

181

and background subtraction.

The absolute y-ray yields have been computed on the

assumption of isotropic angular distributions. The error

introduced by this assumption is probably not significant

for the present purpose. Assuming that the nuclear spin

of the residual nucleus is completely aligned perpendicular

to the beam axis, as would be approximately the case for

proton emission from a system with entrance channel spin

of zero and very large angular momentum, maximum corrections

of +250% and - 18% would need to he applied to the cross

sections for the 2^~ -*■ 0"*" and A-*" -► 2+ transitions in 9 /Mg, respectively. However the fact that these mea sure­

ments, involving a system with entrance channel spin HI,

are done at relatively low bombarding energies (for which

a maximum of 10 units of angular momentum are brought in)

and that these states are reached by either successive evapor­

ation of two particles or emission of a deuteron suggest that

any such alignment of the residual nucleus will be strongly

attenuated. If the coefficients for the P 2 and P^ terms

in the expression for angular correlation were attenuated by

60 and 80%, respectively, the above corrections are altered

to -14 and -22%. Such corrections are sufficiently small

that they may he neglected for the present work. The effects

of cascade summing , which removes events from the photo­

peaks, have been estimated and found to be less than 10%.

182

The gamma-ray measurements were made with a thick

target (typically 2 MeV) , and thus the experirental cross

sections are already energy averaged and may he compared

directly with statistical model predictions. However,

additional complications enter the calculation of gamma-ray

intensities because gamma-ray feeding from higher excited

states, and successive evaporation of protons, neutrons, and

a-particles may all contribute to the yield of a given gamma-

ray transition. The successive evaporation of particles

was calculated without additional approximations using the

statistical model code BREAKUP (Ha75) to evaluate

EH. V-3. Optical model and level density parameters used

in these calculations were taken from the literature and are

listed in Tables V-2 and V-3, respectively. These same

parameters are used in calculations for *2C + reaction

cross sections at higher energies, to be described later.

Calculated cross sections at the lower energies are much

less sensitive to both limiting angular momentum for fusion

in the entrance channel and the value of the spin cutoff,

used in determining the angular momentum dependence of the

level density in the residual nuclei, than those at higher

bombarding energies, but are more sensitive to the optical

parameters for the ^ 2C + entrance channel. Thus

calculations at low energies may serve as a consistency

check for parameters to be used in calculations at high

12 14TABLE V-2. Optical-model Parameters for the C + N Reactions

Channel V iR =r A3 o o ao w R.=r.A3l l a.l Rc Ref.(MeV) (fm) (fm) (MeV) (fm) (fm) (fm)

25 ,A1 + n 48.2-0.3E c.m . 3.66 0.65 11.50a 3.66 0.47 0.0 Pe6325™ Mg + p 52.2-0.3E c.m . 3.66 0.65 11.50a 3.66 0.47 3.66 Pe6324Mg + d 61.2 4.08 0.57 17.40b 3.14 0.85 4.08 Pe63a22Na + a 54.4 4.76 0.53 9.80b 4.76 0.53 3.92 Sa6521 5 Na + He 54.4 4.76 0.53 9.80b 4.76 0.53 3.92 Sa6521Ne + 5 Li 35.5 1.42AT* 0.92 7.94a 1.71AT* 0.89 6.79 Be6920Ne + 6 Li 35.5 1.42At = 0.92 7.94a 1.71At ^ 0.89 6.79 Be6919F + 7 Be 35.4 1.74At 3 1.05 11.50a i2.13AT3 0.62 6.79 Be 6918F + 8 Be 35.4 1.74At 3 1.05 11.50a 2 .1 3 A ^ 0.62 6.79 Be6914n + 12c 14.0 1.35(A ^+A 23 ) 0.35 0.4+0.IE b c.m . 1.4(A1^+A23); 0.35 6.58 Re 7 3

DSurface absorption potential. Volume absorption potential.

12 14TABLE V-3. Level Density Parameters for the C + N Reactions

Residual Nucleus 25ai 25Mg 24Mg 2 2 mNa 2 1 XTNa 2 1 x tNe 2 0 x tNe 19f 00 14n

a/Aa 0.148 0.148 0.149 0.167 0.152 0.152 0.152 0.152 0.152 0.152Ab(MeV) 2.67 2.46 5.13 0.0 2.67 2.46 5.13 2.67 0.0 0.0

ECUTC<MeV) 5.06 5.00 10.07 4.36 6.51 5.77 9.50 5.94 4.96No. of discrete

levels18 18 28 19 20 20 19 23 22

Level density parameters from Ref. Fa68.b Pairing energies from Ref. Gi65.Q Excitation energy in residual nucleus above which level density formula is used.

185

e n e r g i e s wh e r e a d d i t i o n a l a n g u l a r m o m e n t u m d e p e n d e n c e

e n t e r s the problem.

In these m u l t i - s t e p d e c a y c a l c u l a t i o n s w i t h B R EA KU P,

the p o p u l a t i o n d i s t r i b u t i o n of the I n t e r m e d i a t e c o m p o u n d

n u c l e u s is c a l c u l a t e d as a f u n c t i o n of e x c i t a t i o n energy,

a n g u l a r m o m e n t u m and parity, and then th es e q u a n t i t i e s

are c o n s i d e r e d e x p l i c i t l y in the s t a t i s t i c a l m o d e l c a l c u ­

l a t i o n of the d e c a y of the i n t e r m e d i a t e system. T h e o r e t i c a l

cross s e c t i o n s for the i n t e n s i t y of a y - r a y t r a n s i t i o n in,

0 /for ex a m p l e , Mg were o b t a i n e d by first c a l c u l a t i n g the

se p a r a t e cross s e c t i o n s for the p o p u l a t i o n of i n d i v i d u a l

st at es by p r o t o n - n e u t r o n , n e u t r o n - p r o t o n , and d e u t e r o n

em is s i o n . K n o w n or e s t i m a t e d g a m m a - r a y b r a n c h i n g r a t i o s

w e r e then folded w i th the sums of the s e p a r a t e c r o s s s e c t i o n s

to o b t a i n the total ga mm a ray i n t e n s i t y for a g i v e n t r a n s i ­

tion.

The e x p e r i m e n t a l and p r e d i c t e d r e s u l t s are c o m p a r e d

in Figs. V-4 and V-5. The s t a t i s t i c a l m o d e l c a l c u l a t i o n s

are able to r e p r o d u c e b o th the sh a p e s and a b s o l u t e m a g n i t u d e

of the e x c i t a t i o n f u n c t i o n s for l o w - l y i n g g a m m a - r a y t r a n s i t i o n s

for w h i c h yi e l d s could be ex t r a c t e d . At the lower e n e r g i e s ,

w h e r e the cr o s s s e c t i o n s c h a n g e ra p i d l y , the e f f e c t i v e

e n e r g y at w h i c h th e o r y and e x p e r i m e n t sh o u l d be c o m p a r e d no

lo ng er c o r r e s p o n d s to the ce n t e r of the gas cell. Ra t h e r

this e f f e c t i v e e n e r g y m o v e s cl o s e r to the i n i t ia l e n e r g y at

Fig. V-4. Measured excitation functions for gamma-ray transitions in24 25 24Mg, Mg, and Na. The solid lines are Hauser-Feshbachpredictions.

Fig. V-5. Measured excitation functions for gamma-ray transitions in22 18 21 21Na, F, Na, and Ne. The solid lines are Hauser-Feshbach predictions.

TRANSITION CROSS SECTION (mb)

4.0 60

80 10.0

120 14.0

16.0 160

4.0 60

80 10.0

12.0 14.0

I6j0 160

Ec.m.(MeV)

TRANSITION CROSS SECTION (mb)

186

the e n t r a n c e of the gas cell. The d i s c r e p a n c y b e t w e e n the

p r e d i c t e d v a l u e s and some of the d a ta at h i g h e r e n e r g i e s

(Figs. V-4 (d) and V- 5 (c)) v e r y p r o b a b l y a r i s e s from u n c e r ­

t a i n t i e s in the b r a n c h i n g ra ti os for st a t e s at h i g h e x c i ­

ta ti on e n e r g y in the r e s i d u a l nu cl ei or from a n g u l a r

c o r r e l a t i o n effects.

C h a r g e d p a r t i c l e a n eu la r d i s t r i b u t i o n s for the

p o p u l a t i o n of l o w - l y i n g st at es in ZZNa, 2A pg, and 2 ^Mg

w e re also m e a s u r e d w i t h an b e a m on a 40 y g / c m 2

n a t u r a l c a r b o n foil at b o m b a r d i n g e n e r g i e s of 20 and 25

MeV. D e t a i l s of m e a s u r e m e n t , d a t a , and s t a t i s t i c a l mo de l

p r e d i c t i o n may he found e l s e w h e r e (0174). All of the

( n o n - e n e r g y - a v e r a g e d ) e x p e r i m e n t a l a n g u l a r d i s t r i b u t i o n s

show an a p p r o x i m a t e s y m m e t r y about 90° in the c e n t e r - o f -

mass, have a b s o l u t e cross s e c t i o n s w h i c h are r e p r o d u c e d to

w i t h i n a fa ct or of 1.5 or b e t t e r by H a u s e r - F e s h b a c h c a l c u ­

lati on s, and f l u c t u a t e about the 1/ s in O shape of the

e n e r g y - a v e r a g e d t h e o r e t i c a l a n g u l a r d i s t r i b u t i o n s . T h e s e

fe a t u r e s are st r o n g e v i d e n c e that the r e a c t i o n m e c h a n i s m for

19 -I /light p a r t i c l e p r o d u c t i o n in the ~C + N i n t e r a c t i o n at

low e n e r g i e s is p r e d o m i n a n t l y c o m p o u n d n u c l e a r in origin.

The total e x p e r i m e n t a l cross s e c t i o n for the I 2C ( ^ N , d ). 24

r e a c t i o n p o p u l a t i n g the 1. 36 9 MeV 2 state in Mg is

1.01 mb at a b o m b a r d i n g e n e r g y of 25 MeV, c o m p a r e d to

187

a total cross s e c t i o n of 200 mb for the

2^ M g (1 . 3 6 9 (2+ ) -*• g . s . ( 0 + )) y ~ r a y t r a n s i t i o n at this energy.

Thus, s t a t i s t i c a l m o d e l c a l c u l a t i o n s ba se d on i n d e p e n ­

d e n t l y d e r i v e d t r a n s m i s s i o n c o e f f i c i e n t s and level d e n s i t y

p a r a m e t e r s are able to r e p r o d u c e a b s o l u t e m a g n i t u d e s and

g e n e r a l sh a p e s of the a n g u l a r d i s t r i b u t i o n s for light

p a r t i c l e p r o d u c t i o n as well as the o b s e r v e d y - r a y y i e l d s w h i c h

re s u l t from c o m b i n e d single and s u c c e s s i v e p a r t i c l e em is si on .

Si nc e the s t a t i s t i c a l m o d e l can a c c o u n t for the w e a k d e u t e r o n

yiel ds , as w e ll as the I n t e n s e y - r a y t r a n s i t i o n s o b s e r v e d

9 /in Mg, it ma y be c o n c l u d e d that s t a t i s t i c a l p,n e v a p o r -

2 6at io n from the Aj£* c o m p o u n d n u c l e u s is the m a i n p r o c e s s

p o p u l a t i n g the l o w - l y i n g st a t e s in ^ M g , and that the

9 Ap r o m i n e n t Mg y - r a y lines o b s e r v e d by A l m q v i s t et al.

(A160) do not ar is e from the di r e c t t r a n s f e r of a

cl u s t e r in the 2 C ( , d ) 2^Mg re ac t i o n .

188

D. E n t r a n c e C h a n n e l L i m i t i n g A n g u l a r M o m e n t a : M o d e l s

The state of h i g h e s t a n g u l a r m o m e n t u m ^ c u t

a c o m p o u n d n u c l e u s w i th e x c i t a t i o n e n e r g y E* w i l l h a ve a

spin a p p r o x i m a t e l y equal to the s m a l l e r of the four q u a n ­

tities J h r a Z ’ J Y R A S T ’ J M A X ’ and J CRlT* J ^RAZ is the

a n g u l a r m o m e n t u m of the g r a z i n g p a r t i a l wave, d e f i n e d as

the p a r t i a l w a ve for w h i c h Tp ■= 0 .5 . ^ Y R A S T ’ t*ie ^rast

* 2cutoff, is gi ve n by E* = — - J (J +1) , and r e p r e s e n t sy y

the m a x i m u m a n g u l a r m o m e n t u m w h i c h the c o m p o u n d n u c l e u s can

s u p p o r t if all e x c i t a t i o n e n e r g y goes into the r o t a t i o n a l

m o t i o n of a u n i f o r m rigid s t r u c t u r e h e a v i n g the m o m e n t of

i n e r t i a of the e q u i l i b r i u m c o m p o u n d n u c l e u s at e x c i t a t i o n

e n e r g y E*. Jj*AX t*ie m a x ^muin a n g u l a r m o m e n t u m a l l o w e d

for a g i v e n b o m b a r d i n g e n e r g y as a re su lt of d y n a m i c a l r e s ­

t r i c t i o n s in the e n t r a n c e channel. F i n a l l y , isCRIT

the c r i t i c a l (a bs ol ut e) l i m i t i n g a n g u l a r m o m e n t u m in the

e n t r a n c e c h a n n e l for w h i c h the a t t r a c t i v e n u c l e a r force

b e c o m e s equal to the r e p u l s i v e C o u l o m b and c e n t r i f u g a l

forces, and r e p r e s e n t s the h i g h e s t a n g u l a r m o m e n t u m for

w h i c h c o m p l e t e fu s i o n may occur.

For low b o m b a r d i n g e n e r g i e s , the e n t r a n c e c h a n ne l

ca nn ot br i n g in e n o u g h a n g u l a r m o m e n t u m to p o p u l a t e the

Yr as t le ve ls and the g r a z i n g a n g u l a r m o m e n t u m d GRAZ

e f f e c t i v e l y d e t e r m i n e s the li m i t s on c o m p o u n d n u c l e u s

fo rm at io n. Thus in this regi on , the o p t i c a l p o t e n t i a l

189

a d e q u a t e l y p a r a m e t r i z e s the c o m p o u n d n u c l e u s f o rm at io n.

For a range of i n t e r m e d i a t e en er g i e s ,

it ma y be p o s s i b l e to form a c o m p o u n d n u c l e u s in w h i c h the

Y r a s t levels are p o p u l a t e d , but u s u a l l y this limit is

re a c h e d f o l l o w i n g p a r t i c l e e v a p o r a t i o n . At still h i g h e r

b o m b a r d i n g e n e r g i e s , c o m p o u n d n u c l e u s f o r m a t i o n is l i m i t e d

to v a l u e s of a n g u l a r m o m e n t u m J C U T — M I N ^J M A X ’ J C R I T ^ '

In this case, 3 cUT g e n e r a l come f r om a d y n a m i c a l

n u c l e a r mo d e l and ma y be a p p l i e d in the e n t r a n c e c h a n ne l

of the s t a t i s t i c a l m o d e l code to e s t i m a t e c r o s s s e c t i o n s

or fix o t h e r p a r a m e t e r s . C o n v e r s e l y , if all o t h e r p a r a ­

m e t e r s in a c a l c u l a t i o n are well d e t e r m i n e d , the a n g u l a r

m o m e n t u m c u t o f f ma y be a p p r o x i m a t e l y d e t e r m i n e d by a l l o w i n g

J c to v a r y to give a best fit to the data.

In this s e ct io n, we will o u t l i n e b r i e f l y the m a in

f e a t u r e s of those d y n a m i c a l m o d e l s m o s t c o m m o n l y used for

d e t e r m i n i n g e n t r a n c e c h a n n e l a n g u l a r m o m e n t u m li m i t s for

c o m p l e t e f u s i o n of h e a v y - i o n s .

190

In the m o d e l of W i l c z y n s k i (W173) the n u c l e a r force

a c t i n g b e t w e e n the two n u c l e i is taken to be the d e r i v a ­

tive of the s u r f a c e p o t e n t i a l e n e r g y b e t w e e n two s p h e r i c a l

liquid d r o p s of h a l f - d e n s i t y ra di i and R 2 « For a

n u c l e a r d e n s i t y d i s t r i b u t i o n of the form

for r > R ^ + R 2> w h e r e y^» a n d Y 2 are tbe s u r ^ ace t e n s i o n

c o e f f i c i e n t s of d r o p s 1 and 2, r e s p e c t i v e l y . The a t t r a c ­

tive n u c l e a r force o b t a i n s its m a x i m u m v a l u e at the s e p a r a t i o n

d i s t a n c e r ® + R 2 c o r r e s p o n d i n g to the n u c l e a r s a t u r ­

a t i o n d e n s i t y in the r e g i o n of o v e r l a p :

The c r i t i c a l (a bs ol ut e) l i m i t i n g o r b i t a l a n g u l a r

m o m e n t u m ^CRIT tben d e t e r m i n e d by the c o n d i t i o n that

the a t t r a c t i v e n u c l e a r force equal the r e p u l s i v e C o u l o m b

and c e n t r i f u g a l forces:

1

the n u c l e a r force F(r), as a f u n c t i o n of the d i s t a n c e

b e t w e e n c e n t e r s of the two n u cl ei , is gi ve n w i t h i n the

liquid drop m o d e l by

F(r)

- 2tt(y1+Y2) ( r + R 1~ R 2 ) ( r ± R2-Ri)r n 2

r l + e x p { ( r - R ^ - R 2) / 2a}

F ( R X+ R 2 ) -2it ( Y 1 + Y 2 ) r 1 r 2

191

2l.(Y 1+ Y 2 ) B 1»2r 1+r2 ” (R1+ R 2)2 ^ r ^ r - T ^

w h e r e p is the r e du ce d mass. The d i s t i n g u i s h i n g f e at ur e

of this mo d e l is that it p r e d i c t s a c r i t i c a l l i m i t i n g o r b i t a l

a n g u l a r m o m e n t u m w h i c h is i n d e p e n d e n t of the b o m b a r d i n g

e n e r g y hut ma k e s no s t a t e m e n t about a n g u l a r m o m e n t u m limits

at e n e r g i e s b e l o w the c r i t i c a l b a r r i e r heig ht . In e v a l u a t i n g

the c r i t i c a l a n g u l a r moiientum ^ c R I T ^or s y sten,s*1/3

W i l c z y n s k i takes the h a l f - d e n s i t y radii as R = r 0 A , w i th

r = l.ll fm, and the s u r f a c e t e n s i o n c o e f f i c i e n t s as o n - z i" 7 = 0 . 9 9 f 1 - 1.78 (' A^' * ) ^ 1 MeV fm 2 , in a g r e e m e n t w i th the

e x p r e s s i o n for the n u c l e a r s u r f a c e e n e r g y d e r i v e d by M e y e r s

and Sw in t e c k i (Me 67).

The P.ass m o d e l (Ba73, B a 7 4 ) for l i m i t i n g a n g u l a r

m o m e n t u m is s i m i l a r in m a n y r e s p e c t s to that of W i l c z y n s k i ,

but s e v e r a l a d d i t i o n a l a s s u m p t i o n s lead to an e n e r g y

d e p e n d e n c e of 2. . The m a i n f e a t u r e s of the m o d e l aremax

p r e s e n t e d in Fig. V - 6 . A g a i n the c o l l i d i n g nuclei are

a s su me d to be sp he ri ca l in the o u t e r part (r > R 2 ) °

the i n t e r a c t i o n rep ion and to lose their i d e n t i t y in the

inner r e g i o n (r < R i 2 ) • The cutoff d i s t a n c e

P ^2 = + s chosen to yield the s a t u r a t i o n1 2

d e n s i t y of n u c l e a r m a t t e r in the o v e r l a p region, w i th the

point of half m a x i m u m d e n s i t y for each fr a g m e n t given by

r = 1.07 fm(Ho57) for nuclei w i th A < 30 and bv o

r = 1.0 fm for A < 30. The n u c l e a r c o n t r i b u t i o n to the o

192

e f f e c t i v e p o t e n t i a l in the e x t e r n a l r e g i o n is t a k e n to be

12a A ,s 1

!/3 1/3 r-Rexp 12

the ch an ge in s u r f a c e e n e r g y of the two f r a g m e n t s due to

their m u t u a l I n t e r a c t i o n , w h e r e an e x p o n e n t i a l d e p e n d e n c e

of the s p e c i f i c s u r f a c e e n e r g y on local f r a g m e n t s e p a r a t i o n

is assumed. d is the range of the i n t e r a c t i o n and a iss

the s u r f a c e e n e r g y p a r a m e t e r of the liquid drop mo d e l m a ss

formula. The total e f f e c t i v e p o t e n t i a l in the e x t e r n a l

region, c o n s i s t i n g of Co u l o m b , c e n t r i f u g a l and n u c l e a r

parts, is then gi v e n by

„ , 3 Z 3Z 9e2 *2Z2V, (r) = 1 2 + -----x- -------- 2 V r 2r L l 2r

This m a y be r e w r i t t e n as

xR

<. | - r " * “r 12

V 0 (r) = a a ? /3 a 1/3S 1

y r ;12 7"12+ — 75— - z exp

r z

r-R 12

w h e r e the shape of the p o t e n t i a l , and thus the fu s i o n p r o ­

pe r t i e s of the system, are d e t e r m i n e d by the three d i m e n -

s i o n l e s s p a r a m e t e r s

2 e 2 z 1 z 2z 7. e

R 1 2 a sA 1^ A , 1 ' 3 A 1'r77I 7 r 'f3( A 1 W 3 +A2

y =---

171J

A 1+ A 2__________ = __ A l+A 2___________

2^ l 2R 122a sA l 1/ l A 21/3 2m 0r 02a s A x* 1 3A 2* 1 3 1 /3+ A 2 1 ^

z =

1 2('ro t A l

W 3 + A - 1 / 3')

193

In a c l a s s i c a l p i c t u r e , as shown in Fig. V - 6 ,

the i n ci de nt e n e r g y E must ex c e e d the f u s i o n b a r r i e rcm

BpuS £ * ven by the m a x i m u m v a l u e of V o (r), for the f r a g ­

m e n t s to p e n e t r a t e to r = Rj^ and f u s i o n take place.

S u r f a c e (direct) r e a c t i o n s can take place if E cm e x c e e d s

the i n t e r a c t i o n t h r e s h o l d &JNT a"d the n u c l e i came w i t h i n

the ra ng e of their n u c l e a r i n t e r a c t i o n . E x c e l l e n t a g r e e m e n t

w i t h e x p e r i m e n t a l f u s i o n and i n t e r a c t i o n b a r r i e r s for a

large range of i n c i d e n t e n e r g i e s and t a r g e t - p r o j e c t i l e

c o m b i n a t i o n s is a c h i e v e d w h e n the v a l u e s of a d j u s t a b l e p a r a ­

m e t e r s in the m o d e l are taken to be d = 1.35 fm, “ p 12^

2d = 2.70 fm, and a g = 1 7 .0 MeV.

C o n d i t i o n s w h i c h m u st be met for f u s i o n to take place

in the Bass m o d e l are: ( l ) t h a t the f r a g m e n t s p e n e t r a t e to the

cu t o f f ra di us ^ 12 ’ p r o d u c i n g an i n c o m p l e t e l y fu se d s y s t e m

("cont ac t state") w h i c h may d e c a y by ei t h e r c o m p l e t e fu s i o n

or s e p a r a t i o n , and (2) that the net force in the c o n t a c t

state be a t t r a c t i v e . R e f e r r i n g to Fig. V - 6 , then, for i n c i ­

dent e n e r g y E , c o m p l e t e f u s i o n w i ll take pl ac e for p a r t i a l cm

w a v e s Z < J?C U T * but not for I > &cux + 1 si nc e b a r r i e r

p e n e t r a t i o n to R-^ is not p o s s i b l e in a c l a s s i c a l picture.

Thus the a n g u l a r m o m e n t u m cu to ff will i n c r e a s e w i t h i n c r e a s i n g

in c i d e n t e n e r g y up to ^-FUSION CUTO FF * w h e r e the b a r r i e r

for the c r i t i c a l or l i m i t i n g a n g u l a r m o m e n t u m , d e f i n e d by

0 , is e x c e e d e d and the e f f e c t i v e force3 V C I U T ( r )9r Ir = R i 2

at the cu t o f f r a d i u s *s no lo ng er a t t r a c t i v e .

Fig. V-6. Essential features of the Bass Model (Ba73, Ba74) for entrance channel limiting angular momenta.

BASS M ODEL

V.(r) . - J L osA ’ A|eK|2

3 » 3 _ \ d '

Vc'uV

2 A i 2 r

"CONTACT STATE"

— CUTOFF RADIUSj. j.

R = r (A 3 + A 3 )12 0 ' I 2

YIELDS SATURATION D EN SITY OF NUCLEAR

M ATTER IN O V E R LA P REGION

NUCLEAR FRICTION CONVERTS ORBITAL ANGULAR

M OM ENTUM TO IN TE R N A L ANGULAR M OM ENTUM OF FRAGMENTS

Fig. V-6

194

The novel f e a t u r e of the Bass mo d e l Is that e f f e c t s

of n u c l e a r f r i c t i o n in the c o nt ac t state are taken into

account. Part of the o r b i t a l a n g u l a r m o m u n t u m £ is

a s su me d to he c o n v e r t e d into i n t e r n a l a n g u l a r momemturn of

the f r a g m e n t s by s u rf ac e i n t e r a c t i o n s , and the c e n t r i f u g a l

force is c a l c u l a t e d using a r e d u ce d e f f e c t i v e o r b i t a l a n g u l a r

m o m e n t u m ^ p p p = f * £ < £• A v a l u e of f = 5/7 is o b t a i n e d

from a c l a s s i c a l e s t i m a t e a s s u m i n g rigid r o t a t i o n and c o m p l e t e

e q u a l i z a t i o n of the s u rf ac e v e l o c i t i e s of the two fr ag me nt s.

One of the s i g n i f i c a n t p r e d i c t i o n s of the m o d e l then is that,

for s y st em s w i th a C o u l o m b p a r a m e t e r x < 1 (for x> 1, only

V Q(r) has a m a x i m u m at r > an(l fu s i o n should not occur),

the l i m i t i n g a n g u l a r m o m e n t u m for fu s i o n sh ou ld i n c r e a s e w i t h

b o m b a r d i n g e n e r g y from the fu si on t h r e s h o l d up to

z l z 2e ^ -x 1 1 dE FU S I O N C U T O F F = ' r ~ ^1+ ~ 2 x 2 ~ x R]~2* 3nd t h e °

s a t u r a t e at a m a x i m u m v a l u e given by

0 1 -x£ = ---

C RIT 2 f 2y

m o ro 2 a s A i A / 3A 24/3 (ft11/ 3+ A 21/3). /l-x

•b2 A ^ + A 2 y f 2

A third m o d e l , the " c r i t i c a l d i s t a n c e of a p p r o a c h 1’

model, put forward by G a l i n et al. (Ga74) is e s s e n t i a l l y

e q u i v a l e n t to the Bass m o d e l in its d e s c r i p t i o n of the

r e a c t i o n up to the point of fusion. F r om an e x t e n s i v e a n a ­

lysis of c o m p l e t e f u s i o n data it has been found that fu s i o n

will take place w h e n e v e r in c i d e n t e n e r g y and a n g u l a r m o m e n t u m

195

are such that d i s t a n c e b e t w e e n f r a g m e n t s Is less than or

equal to a c r i t i c a l d i s t a n c e R «* r (A, 17 A ,,17 3 -vCR CR 1 2 } ’

d e f i n e d as the c l o s e s t d i s t a n c e w h i c h the ions w i t h I = Z c r

can re ac h at e n e r g y E. T h is d i s t a n c e is gi ve n by the point

of i n t e r s e c t i o n b e t w e e n the k i n e t i c e n e r g y of the i n c i d e n t

n u c l e u s and the p o t e n t i a l e n e r g y curve b e t w e e n the two

nu cl ei for a gi ve n Z

V(r) = V (r) + V (r) + •n c 2yrz

The p a r a m e t e r r cp I s found to be a p p r o x i m a t e l y c o n s t a n t

over all of the p e r i o d i c table w i t h an a v e r a g e v a l u e

r ^ = 1.0 ± 0.07 fm, c o n s i s t e n t w i t h the a s s u m e d s e p a r a t i o n

in the Bass m o d e l c o r r e s p o n d i n g to the n u c l e a r s a t u r a t i o n

d e n s i t y in the r e g i o n of o v er la p. In this mo d e l , the

" s u d d e n a p p r o x i m a t i o n " is as s u m e d , in w h i c h the sh ap e and

i d e n t i t y of the n u c l e a r f r a g m e n t s are p r e s e r v e d up to the

point w h e r e the c r i t i c a l r a d i u s is r e ac he d. The n u c l e a r

p o t e n t i a l is d e r i v e d , in a s o m e w h a t d i f f e r e n t way, from the

n u c l e a r m a t t e r d e n s i t y of the i n t e r a c t i n g nuclei. The

n u c l e a r p o t e n t i a l at a given d i s t a n c e r is equal to the

d i f f e r e n c e b e t w e e n the n u c l e a r e n e r g y of the s y s t e m of two

i n t e r a c t i n g nuclei and the b i n d i n g e n e r g i e s of the two nu c l e i

at i n f i n i t e s e p a r a t i o n . The m e t h o d of B r u e c k n e r et al.

( Br 68) is used to c a l c u l a t e the b i n d i n g e n e r g y at a p a r t i c u l a r

po in t of the n u c l e u s as a f u n c t i o n of the n u c l e a r d e n s i t y p

196

at this point, w i t h the total b i n d i n g e n e r g y o b t a i n e d by

i n t e g r a t i n g o v er the e n t i r e n u c l e a r v o lu me . For two n u c l e i

s e p a r a t e d by a d i s t a n c e r, the d i f f e r e n c e b e t w e e n the

local binding, e n e r g y of the c o m p o s i t e s y s t e m of two n u c l e ar-

m a t t e r d e n s i t i e s and the i n d i v i d u a l n u c l e i at a p a r t i c u l a r

point P l o c a t e d a d i s t a n c e r^ from n u c l e u s 1 and r 2 from n u c l e u s 2 ma y be w r i t t e n

E[p1 (ri> + P 2 (r2)] - E[p1 (r1)]- E[p2 (r2)]

w h e r e p^(r^) and P 2 ^r 2 r e p r e s e n t the local n u c l e a r -

m a t t e r d e n s i t i e s of n u c l e u s 1 and 2 , r e s p e c t i v e l y , at

d i s t a n c e s r ^ and r 2 from the ce n t e r s . The n u c l e a r

p o t e n t i a l is then o b t a i n e d by i n t e g r a t i o n over the v o l u m e

of the two nucl ei :

V N U C I ( r = ■/‘(E (P1 (r l) + P 2 (r 2 1 “ M p j / r j . ) ]

- E [ p 2 (r2 ) ]} d r 3

The p r e d i c t i o n s of the Bass and " c r i t i c a l d i s t a n c e

of a p p r o a c h " m o d e l s are q u i t e s i mi la r, w h i l e the s i m p l e

a n a l y t i c a l form of the n u c l e a r p o t e n t i a l in the Bass m o d e l

m a k e s the e v a l u a t i o n of the l i m i t i n g a n g u l a r m o m e n t u m e l e ­

m e nt ar y. For this reas on , the d e t a i l e d p r e d i c t i o n s of the

Bass m o d e l will be tested in the next section.

197

E. C o m p l e x P a r t i c l e E m i s s i o n and L i m i t i n g A n g u l a r M o m e n t a

in H e a v y - i o n I n du ce d S t a t i s t i c a l C o m p o u n d R e a c t i o n s

In S e c t i o n C, we found that the light p a r t i c l e p r o ­

d u c t i o n in the ^ ZC + i n t e r a c t i o n at low e n e r g i e s

could be a c c o u n t e d for al mo st c o m p l e t e l y in terms of s t a t i s ­

tical c o m p o u n d n u c l e u s f o r m a t i o n and decay. In this s e c t i o n

we pres en t a s t a t i s t i c a l mo de l a n a l y s i s of the a v a i l a b l e

d ata on c o m p l e x p a r t i c l e e m i s s i o n for ^ ZC + r e a c t i o n s

w i th e m p h a s i s on b o th the v a r i a t i o n in the e n t r a n c e c h a n n e l

l i m i t i n g a n g u l a r m o m e n t u m and the c h a n g i n g na t u r e of the

r e a c t i o n m e c h a n i s m w i t h i n c r e a s i n g b o m b a r d i n g energy.

This s t a t i s t i c a l m o d e l a n a l y s i s of c o m p l e x p a r t i c l e

e m i s s i o n at i n t e r m e d i a t e and high e n e r g i e s was o r i g i n a l l y

m o t i v a t e d by c o n f l i c t i n g i n t e r p r e t a t i o n s of the m e c h a n i s m

for the ^ 2C (^ ,^ L i ) Z ^Ne re ac t i o n . The first d a ta on

this r e a c t i o n c o n s i s t e d of p a r t i a l a n g u l a r d i s t r i b u t i o n s

w h i c h were f o rw ar d pe ak ed and had s i z e a b l e m a g n i t u d e s s h o w i n g

a 2j + 1 d e p e n d e n c e . It was ar g u e d ( M a 7 1 , N a 7 1 ) t h a t

6 2 0the c o m p o u n d n u c l e u s a m p l i t u d e s for d e c a y into Li + Ne

w o ul d be e x p e c t e d to be small, and the d a t a t h e r e f o r e w e r e

ta ke n to c o n s t i t u t e e v i d e n c e for d i r e c t e i g h t - n u c l e o n t r an sf er .

H o w e v e r , the c o m p o u n d m e c h a n i s m can also p r o d u c e f o r w ar d

peaked cross s e c t i o n s h a v i n g a (2J+1) d e p e n d e n c e in

m a g n i t u d e . Th e i n i t i a l e v i d e n c e for the c o m p o u n d n a t u r e of

the r e a c t i o n was p r o v i d e d by B e l o t e et al. (Be73) who found

198

that when these m e a s u r e m e n t s are e x t e n d e d to b a c k w a r d an g l e s

by the i n t e r c h a n g e of target and p r o j e c t i l e , a n g u l a r

d i s t r i b u t i o n s for the * 2C (^^ N ,^ L i ) 2 ©Ne r e a c t i o n at

E cm = 36 M e V e x h i b i t an a p p r o x i m a t e s y m m e t r y ab ou t 9 0 ° (c.m.)

w h i c h is well r e p r o d u c e d by a (sin 0 ) - d e p e n d e n c e . The

a b i l i t y of s t a t i s t i c a l m o d e l c a l c u l a t i o n s to r e p r o d u c e both

the sh ap es and the r e l a t i v e and a b s o l u t e m a g n i t u d e s of these

cross se ct i o n s , as d e m o n s t r a t e d in this study, p r o v i d e s

c o n v i n c i n g e v i d e n c e for the c o m p o u n d n a t u r e of this r e a c t i o n

at both low and i n t e r m e d i a t e en er g i e s .

The i n i t i a l c o n f u s i o n c o n c e r n i n g the p r o b a b i l i t y of

c o m p l e x p a r t i c l e e m i s s i o n from a c o m p o u n d n u c l e u s is in

part a result of the e x t e n s i v e d i s c u s s i o n in the early

l i t e r a t u r e of " p r e f o r m a t i o n f a c t o r s " w h i c h w o u l d d i s c r i m i n a t e

a g a i n s t such h e a v y p a r t i c l e s . H o w e v e r , b e c a u s e of the

d e g e n e r a t e f e r m i o n n a t u r e of the n u c l e a r s y s t e m (Co60) ,

c o m p l e x p a r t i c l e e m i s s i o n ma y take pl ac e on an eq ua l ba si s

with si ng le p a r t i c l e e m i s s i o n , and w i ll be d i s c r i m i n a t e d

a g a i n s t o n ly on the ba si s of a v a i l a b l e p h a s e space w h i c h is

g o v e r n e d by the r e a c t i o n t h r e s h o l d , t r a n s m i s s i o n c o e f f i c i e n t s ,

and le ve l d e n s i t i e s .

H e a v y - i o n r e a c t i o n s l e a d i n g to the e m i s s i o n of c o m p l e x

p a r t i c l e s e n j o y c e r t a i n a d v a n t a g e s in the st u d y of the

e n t r a n c e c h a n n e l l i m i t i n g a n g u l a r m o m e n t u m . T h is s e c t i o n

w i ll b e g i n w i t h a br ie f d i s c u s s i o n of these f e a t u r e s of the

c o m p o u n d r e a c t i o n m e c h a n i s m .

199

Fig. V -7 i l l u s t r a t e s the d i f f e r e n c e in s e n s i t i v i t y

to the a n g u l a r m o m e n t u m c u t o f f in the e n t r a n c e c h a n n e l for

and 36 MeV. At Ec = 10 MeV, the a n g u l a r m o m e n t u m

cu t o f f is d e t e r m i n e d e s s e n t i a l l y by the g r a z i n g p a r t i a l

w a ve to be Yc ■ 11^» w h i l e at E__ = 36, the c u t o f f iscm ’

t a ke n to be the Bass mo d e l cu t o f f on the o r b i t a l a n g u l a r

m o m e n t u m , gi ve n by Yc ™ 18ft. Fig. V - 7a shows the e n t r a n c e

w h i c h is p r o p o r t i o n a l to the cr os s s e c t i o n for c o m p o u n d

n u c l e u s format i o n , as a f u n c t i o n of total a n g u l a r m o m e m t u m

J. It can be seen that, since the g r a z i n g p a r t i a l wa v e s

m a ke a n e g l i g i b l e c o n t r i b u t i o n , a c h a n g e in Yc by s e v e r a l

un it s at E c = 10 Me V wo u l d p r o d u c e a v e r y small c h a n g e

in the total cross s e c t i o n for c o m p o u n d n u c l e u s f o r m a t i o n ,

w h i l e a s i m i l a r c h a n g e in Yc at E = 36 Me V w o u l d have

a s i g n i f i c a n t effect. This s e n s i t i v i t y is also r e f l e c t e d

in the cross s e c t i o n s for light p a r t i c l e p r o d u c t i o n p o p u ­

la t i n g i n d i v i d u a l st at es in the r e s i d u a l nucl ei , i l l u s t r a t e d

in Fig. V-7c by the ^-^CCl^M ,d) re ac t i o n . It is u s e f u l

to note that w h i l e the total cr os s s e c t i o n for c o m p o u n d

n u c l e u s f o r m a t i o n i n c r e a s e s with i n c r e a s i n g b o m b a r d i n g ener gy ,

the cross s e c t i o n s for i n d i v i d u a l st at es d e c r e a s e b e c a u s e

the nu mb er of open c h a n n e l s is la r g e r and thus the f r a c t i o n

of a v a i l a b l e phase space for the d e c a y of the c o m p o u n d n u c l e u s

c o m p o u n d y i e l d s from the 1 2 C + r e a c t i o n at E cm 10

c h a n n e l factor

cm

12 14V-7. a) Hauser-Feshbach entrance channel factor for C+ Nreactions at E = 1 0 and 36 MeV as a function of total angular c.m .momentum J. Y represents the Bass model cutoff at these energies, cb) Number of open channels for ^2C+^4N induced reactions as afunction of total angular momentum at E = 1 0 and 36 MeV.c.m .

» , 24c) Angie-integrated cross sections to low-lying states in Mg for12 14the reaction C( N,d) at E = 10 and 36 MeV.c • m#

Fig

. V

-7

oj(mb) NUMBER OF OPEN CHANNELS ENTRANCE CHANNEL FACTOR

200

r e p r e s e n t e d by a si n g l e c h a n n e l is c o n s e q u e n t l y m u c h smaller.

T h i s i n c r e a s e in the n u m b e r of o p en c h a n n e l s for v a r i o u s

d e c a y p r o d u c t s in go i n g from low to h i g h i n c i d e n t e n e r g i e s

is sh ow n in Fig. V-7b.

Of even m o r e in t e r e s t for the p r o b l e m of e x p e r i m e n t a l l y

d e t e r m i n n i n g the e n t r a n c e c h an ne l a n g u l a r m o m e n t u m cu t o f f

is the i n c r e a s e in the r e l a t i v e n u m b e r of o p en c h a n n e l s for

he a v y p a r t i c l e s , e.g. ^ L i , ^ L i , c o m p a r e d to light r e a c t i o n

p r o d u c t s w i t h i n c r e a s i n g a n g u l a r m o m e n t u m . The r e a s o n for

this is that o n ly he a v y p a r t i c l e s are able to ca r r y off

e n o u g h a n g u l a r m o m e n t u m from the h i gh spin c o m p o u n d n u c l e u s

to p o p u l a t e the m o r e n u m e r o u s low spin st a t e s of their

r e s p e c t i v e r e s i d u a l nucl ei . As sh ow n in Fig. V - 8 , w h i l e the

p r o d u c t i o n cr os s s e c t i o n s for c o m p l e x p a r t i c l e s such as

f i 7°Li and Be are lower, they i n c r e a s e m u c h m o r e r a p i d l y

w i th i n c r e a s i n g a n g u l a r m o m e n t u m t h an the light p a r t i c l e or

total r e a c t i o n yields. In the r e g i o n w h e r e the d y n a m i c

r e a c t i o n m o d e l s p r e d i c t a c u t o f f for c o m p l e t e fusion, the

r e l a t i v e c h a n g e of ^Li yield per unit a n g u l a r m o m e n t u m

is four times that of the total yield. Thus, the m e a s u r e m e n t

of a b s o l u t e cross s e c t i o n s for c o m p l e x p a r t i c l e p r o d u c t i o n

r e p r e s e n t s a p o t e n t i a l l y s e n s i t i v e p r o b e for d e t e r m i n i n g

the e n t r a n c e c h a n n e l a n g u l a r m o m e n t u m c u t o ff , si nc e a s t a t i s ­

tical mo de l c a l c u l a t i o n u s i n g a "best" set of level d e n s i t y

and o p t i c a l m o d e l p a r o m e t e r s should be able to r e p r o d u c e

Fig. V-8. Total angle-integrated cross sections for various exit channels12 14of the reaction C + N at E = 3 6 MeV plotted as ac.m .

function of total angular momentum J.

( m

b)

J C fi)

Fig. V-8

201

a b s o l u t e cross s e c t i o n s to w i t h i n b e t t e r than a fa ct or of

two. An a d d i t i o n a l a d v a n t a g e of s t u d y i n g h e a v y p a r t i c l e

e m i s s i o n is that, b e c a u s e of the h i gh e n e r g y t h r e s h o l d for

he a v y p a r t i c l e de ca y, m u l t i p l e p a r t i c l e e m i s s i o n i n v o l v i n g

such p a r t i c l e s should not be as la rg e as for light pa r t i c l e s .

T h e r e now exist e x t e n s i v e e x p e r i m e n t a l d a t a on h e a v y

p a r t i c l e p r o d u c t i o n , e s p e c i a l l y 8Li p r o d u c t i o n , by the

12c + r e a c t i o n over an e n e r g y ra ng e up to E ™ 72.5cm

MeV. T h e s e d a ta are of i n t e r e s t b o t h for a s t u d y of

e n t r a n c e c h a n n e l l i m i t i n g a n g u l a r m o m e n t a and also for the

light they shed on the q u e s t i o n of w h e t h e r the c o m p o u n d

n u c l e u s r e a c h e s s t a t i s t i c a l e q u i l i b r i u m f o l l o w i n g c o m p l e t e

fusion, p a r t i c u l a r l y at h i g h en er g i e s . It sh o u l d be no te d

that ^ 2C + is the l i g h t e s t s y s t e m for w h i c h a d e t e r ­

m i n a t i o n of a c r i t i c a l a n g u l a r m o m e n t u m has b e en a t te mp te d.

L i g h t e r s y s t e m s p r e s e n t the d i f f i c u l t y that the m a s s e s of

the h e a v y e v a p o r a t e d p a r t i c l e s are c l o s e to the m a s s e s of

the target and p r o j e c t i l e and thus it is d i f f i c u l t to

d i s t i n g u i s h c o m p o u n d d e c a y from p r o j e c t i l e or t a r g e t f r a g m e n ­

t a t i o n or m u l t i n u c l e o n transfer.

In this s e c t i o n we p r es en t a c o m p l e t e s t a t i s t i c a l

m o d e l a n a l y s i s of all a v a i l a b l e d a ta for ^ 2 C + IAjj i n du ce d

r e a c t i o n s l e a d i n g to c o m p l e x p a r t i c l e e m i s s i o n at e n e r g i e s

up to E cm “ 72 MeV. O p t i c a l m o d e l , level d e n s i t y , and

202

p a i r i n g p a r a m e t e r s used in the s t a t i s t i c a l m o d e l a n a l y s i s

are th os e listed in Sec. C, w h i c h have been sh ow n to yield

good a g r e e m e n t b e t w e e n c a l c u l a t e d and m e a s u r e d cr o s s s e c t i o n s

19 1 Afor C + N i n d u ce d r e a c t i o n s l e a d i n g to li gh t p a r t i c l e

p r o d u c t i o n at low e n e r g i e s (E cm 10 MeV) w h e r e c o m p o u n d

n u c l e u s f o r m a t i o n is kn o w n to be p r e d o m i n a n t .

The cross s e c t i o n s e v a l u a t e d w i t h the H a u s e r - F e s h b a c h

e x p r e s s i o n and the c o n c l u s i o n s d r a w n from them r e g a r d i n g the

l i m i t i n g a n g u l a r m o m e n t u m are s u b j e c t to u n c e r t a i n ­

ties in the o p t i c a l - m o d e l t r a n s m i s s i o n c o e f f i c i e n t s and level

d e n s i t i e s of the r e s i d u a l nucl ei . Th e trartsmisslon c o e f f i c i e n t s

for the exit c h a n n e l s 8Li + 29Ne and 2Be + 4 9 F w e r e d e r i v e d

from the s u r f a c e a b s o r p t i o n p a r a m e t e r s of Beth ge , F o u , and

Z u r m u h l e (Be69). If their v o l u m e a b s o r p t i o n p a r a m e t r i z a t i o n

of the ^Li + o p t i c a l p o t e n t i a l had b e e n used, the

c a l c u l a t e d ^Li c r o s s s e c t i o n s w o u l d be i n c r e a s e d by ab o u t

15% for E = 2 4 M e V and r e d u c e d by ab ou t 25% forc.m. y

E c.m. * 38 MeV. A f u r t h e r u n c e r t a i n t y in the t r a n s m i s s i o n

c o e f f i c i e n t s c o n c e r n s the effect of ^Li d i s s o c i a t i o n and

di r e c t t r a n s f e r r e a c t i o n s on the i m a g i n a r y p o t e n t i a l d e r i v e d

from a n a l y s e s of e l a s t i c s c a t t e r i n g data. Since b r e a k u p and

d i r e c t t r a n s f e r r e a c t i o n s r e m o v e flux from the ^ L i e l a s t i c

s c a t t e r i n g c h a n n e l w i t h o u t c o m p o u n d - n u c l e u s f o r m a t i o n , the

use of o p t i c a l - m o d e 1 t r a n s m i s s i o n c o e f f i c i e n t s o b t a i n e d from

fits to e l a s t i c s c a t t e r i n g r e s u l t s in an o v e r e s t i m a t e of the

203

c o m p o u n d - n u c l e u s f o r m a t i o n cr os s section. If, for e x a m p l e

the ^Li i m a g i n a r y p o t e n t i a l d e p t h is r e d u c e d by 50%

in o r d e r to s i m u l a t e the e f f e c t s of d e p l e t i o n of a b s o r b e d flux

by b r e a k u p and d i r e c t r e a c t i o n s , the c a l c u l a t e d c r o s s s e c t i o n s

for the r e a c t i o n ^ 2^ (14 jj 16^ ) 20jje at ^ 0 3$ d e c r e a s e by ab ou t

R e a s o n a b l e v a r i a t i o n s in the a - p a r t i c l e o p t i c a l - m o d e l p a r a ­

m e t e r s ma y be e x p e c t e d to lead to a n c e r t a i n t l e s in the ALi

cr os s s e c t i o n s of a s i m i l a r m a g n i t u d e (i.e., 'v 30%), since

the a - p a r t i c l e yield is the d o m i n a n t c o n t r i b u t i o n to the

d e n o m i n a t o r of the H a u s e r - F e s h b a c h e x p r e s s i o n .

At h i gh e n e r g i e s , the c h o i c e of the ^ 2C + ^4^ o p t i c a l -

m o d e l p o t e n t i a l has p r a c t i c a l l y no ef fe ct on the p r e d i c t e d

cr os s s e c t i o n s b e c a u s e the e n t r a n c e c h a n n e l t r a n s m i s s i o n c o e f ­

f i c i e n t s are l i mi te d by the am o u n t of a n g u l a r m o m e n t u m w h i c h/2 ftthe °A1 c o m p o u n d n u c l e u s can s u p p o r t or by the d y n a m i c a l

an g u l a r m o m e n t u m c u t o f f in the e n t r a n c e c h a n n e l to those

w h o s e £ v a l u e s c o r r e s p o n d to c o m p l e t e a b s o r p t i o n .

The level d e n s i t y p a r a m e t e r s a gi v e n in Ref. ( F a68) w e re o b t a i n e d from an a n a l y s i s in w h i c h a r i g i d - b o d y m o m e n t

of i n e r t i a given by r “ 2 / 5 m A R 2 and R 0 1.4 A ^ / ^ f m was

used to p a r a m e t r i z e the spin d i s t r i b u t i o n of the level

d e ns it y. We have e m p l o y e d a s o m e w h a t la r g e r m o m e n t of iner ti a,

w i t h ra d i u s R = l . S A ^/3 fm in c a l c u l a t i n g the s p ib cu t o f f

fa c t o r a 2 . T h is s e r v e s to i n c r e a s e the d e n s i t y of h i gh spin

204

for c a l c u l a t i o n s at E ° 36 M e V is about a fa ct or ofc.m.

s t a t e s in the r e s i d u a l n u c l e i and t h e r e b y r e d u c e the c o m p o u n d

n u c l e u s cr o s s s e c t i o n for a p a r t i c u l a r state. Th e r e d u c t i o n

2 .

Since the v a l u e of R = 1.4 fm u s ed in Ref. (F a68)

was o b t a i n e d from a 9 h e l l - m o d e l c a l c u l a t i o n of the a v e r a g e

va l u e of (the p r o j e c t i o n of the total a n g u l a r

m o m e n t u m for st a t e s ar ou nd the Fermi leve l) , it is not

u n r e a s o n a b l e that a la r g e r v a l u e of the m o m e n t of iner ti a

sh ou ld be a p p r o p r i a t e in a s t a t i s t i c a l c a l c u l a t i o n w h e r e v e r y

high e x c i t a t i o n e n e r g i e s and a n g u l a r m o m e n t a are in vo lved.

At low b o m b a r d i n g e n e r g i e s (E„ 10 MeV) , this ch a n g ec • Tn •

in the r a d i u s has l i t t l e effect.

An e x p l i c i t c o n s i d e r a t i o n of l s o s p l n c o n s e r v a t i o n hlas

not b e en i n c l u d e d in the p r e s e n t c a l c u l a t i o n . T h is is e q u i ­

v a l e n t to a s s u m i n g that all le v e l s p o p u l a t e d in this r e a c t i o n

in the c o m p o u n d n u c l e u s and in the r e s i d u a l n u c l e i have

T = 0 or T « 1/2 a c c o r d i n g to w h e t h e r the r e s i d u a l n u c l e u s

is even or odd A. Un de r this a s s u m p t i o n , the T = 1 state

at 3.562 M e V in ^Li should not be p o p u l a t e d p r o v i d e d

9 nthat the r e s i d u a l Ne n u c l e u s is left in a T = 0 state.

It is i n t e r e s t i n g to note, in this rega rd , that in the ^Li

s p e c t r a of Refs. (Ma71, Na71, Be73), no g r o u p a p p e a r s w h i c h

w o u l d c o r r e s p o n d to bLi(T * 1, E x -3.56) + 2^N e( T = 0, E x ** 0)

W i t h i n the s t a t i s t i c a l p r e c i s i o n of these sp e c t r a , i s o s p i n

9 6a p p a r e n t l y r e m a i n s a good q u a n t u m n u m b e r in Al even to

205

e x c i t a t i o n e n e r g i e s as h i gh as 50 MeV.

A c h e c k on the v a l i d i t y of the s t a t i s t i c a l m o d e l for

e n e r g i e s up to E cm ■ 27 M e V is p r o v i d e d by a c o m p a r i s o n

of s t a t i s t i c a l m o d e l c a l c u l a t i o n w i t h m e a s u r e m e n t s by

H o l u b et al. (Ho73) of 7B e , 22Na, and ^ N a r a d i o a c t i v i t y

p r o d u c e d in the b o m b a r d m e n t of a th i c k c a r b o n t a r g e t by

16-58 Me V ions, shown in Fig. V-9. The 2^Na yi el d

has b e en c a l c u l a t e d a s s u m i n g the s u c c e s s i v e e m i s s i o n of

dEtwo prot on s. T h i c k ta rg et y i e l d s Y * * /

f e (dE/dx)B E A M

w e r e c a l c u l a t e d u s i n g the s t o p p i n g po w e r s gi ve n by N o r t h c l i f f e

and S c h i l l i n g (No70). A n g u l a r m o m e n t u m li m i t s in the e n t r a n c e

c h a n n e l w e r e d e t e r m i n e d at low e n e r g i e s (E < 20 MeV)C • TT1 *

e s s e n t i a l l y by the g r a z i n g p a r t i a l w a ve d e f i n e d by T^ ** 0.5

and at h i g h e r e n e r g i e s by n e g l e c t i n g the X 7 C + e n t r a n c e

c h a n n e l spin of l<tf and s e t t i n g the cu t o f f on total a n g u l a r

m o m e n t u m eq ua l to the Bass m o d e l p r e d i c t i o n for the o r b i t a l

a n g u l a r m o m e n t u m c u t o f f ^ c . If the c o m p o u n d n u c l e u s is

indeed formed and r e a c h e s e q u i l i b r i u m , and if c o m p l e x par-

I Ft i d e s such as Li are e m i t t e d t h r o u g h s t a t i s t i c a l p r o c e s s e s ,

then the e m i s s i o n of 7Be should also o c c u r in a p r o p o r ­

tion g o v e r n e d p r i m a r i l y by the r e a c t i o n t h r e s h o l d t r a n s m i s ­

sion c o e f f i c i e n t s , and level d e n s i t i e s . The e x c e l l e n t

a g r e e m e n t sh ow n h e re i n d i c a t e s the i m p o r t a n c e of c o m p o u n d

12 14p r o c e s s e s for c o m p l e x p a r t i c l e e m i s s i o n in C + N

Fig. V-9. Absolute statistical model calculations compared with12 14experimental integrated yields for various C + N

24reaction products (Ho73). The Na yield has been calculated assuming the successive emission of two protons. The stopping powers given by Northcliffe and Schilling (No70) were used in the calculation of thick target yields.

INTE

GRAT

ED

YIELD

(m

bxmg

/cm2)

1 2 ^ 1 4

I03C+ N Delayed Activit ies

(Holub, et al.)4f

2 2 m _ —9

- • ---------------

©

O O -Na III —" *L A A-O V

¥A / f\ f A - .w /

: -4/ :

/ inrNa... 7/

■»

— /—f B e /

/ I .

/ / / t

I -

— // k!1 * V 1

10 '

10

10 -i

10-2' 20 40 60ELAB <MeV)

Fig. V-9

206

I n du ce d r e a c t i o n s in the i n t e r m e d i a t e e n e r g y range. In

ad di ti on , we note that 7Be has about the same t h r e s h o l d

for b r e a k u p as 8Li, and the good a g r e e m e n t found for

7Be e m i t te d with b o th low and h i g h e n e r g i e s from the c o m ­

pound n u c l e u s s u g g e s t s that b r e a k u p e f f e c t s for 7Be and

for ^Li may not be a s e r i o u s p r o b l e m for the p r e s e n t

c a l c u l a t i o n s .

A d d i t i o n a l e v i d e n c e for the s t a t i s t i c a l c o m p o u n d

n a t u r e of the 12C ( i ) 20Ne r e a c t i o n in this i n t e r ­

m e d i a t e e n e r g y ra ng e is p r o v i d e d by an a n a l y s i s of the data

of Vo l a n t et al. (Vo75) who have m e a s u r e d cr os s s e c t i o n s

to i n d i v i d u a l st a t e s at s e v e r a l an g l e s o v er an e n e r g y ra ng e

E £ m = 9 - 2 7 MeV, and of that of M a r q u a r d t et al. (Ma71)

who have m e a s u r e d c o m p l e t e a n g u l a r d i s t r i b u t i o n s to low

lying st at es in 2 9 Ne at E = 2 4 MeV, (Fig. V-10).c.m.

E x c e l l e n t a g r e e m e n t is o b t a i n e d w i t h the sh a p e s and r e l a ­

tive m a g n i t u d e s of all the e x p e r i m e n t a l data, and good

a g r e e m e n t w i t h the a b s o l u t e m a g n i t u d e of the V o l a n t d a ta

r e s u l t s w h e n an a n g u l a r momenturn c u t o f f p r e d i c t e d by the

B ass mo d e l is i m p o s e d in the e n t r a n c e chan ne l. In g e ne ra l,

we r e qu ir e a cu t o f f one unit h i g h e r in a n g u l a r m o m e n t u m

than that used by Vo l a n t et a l . in th ei r s t a t i s t i c a l m o d e l

a n a l y s i s in o r d e r to o b t a i n the best a g r e e m e n t w i t h the

a b s o l u t e m a g n i t u d e of their data. V o l a n t et al. have also

m e a s u r e d s e ve ra l p o i n t s at E = 24 M e V and find the

Fig. V-10. Angular distributions for the 42C(*4N, 6Li)2°Ne reaction atE = 2 4 MeV from (Ma71). Statistical model cross c.m .sections, shown here by dashed lines, were calculated withan angular momentum cutoff of J = 14ft and normalized down-cward by a factor of 2. Subsequent measurements by Volant et al. (Vo75) showed the magnitude of these experimental cross sections to be too low by a factor of three and thus a cutoff of Jq= 15ft is required to reproduce the absolute cross sections at this energy.

100

l2C(l4N ,6 L i)2 0 Ne Ec.m.=24 MeV (Marquardt.et al.)

10

(/)

j Q

Cj* 0

10

b“O

100

10

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m XII ) . 0 0 M «

)+ ■

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100

10

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I

t

- K 2

. 9 7 M e

> ~

>V „/

. //— z —

/s'

5 2 , 5 . 7 *

" 1 "

3^ vO-

___ 3 “

- . V - J

0 / //i

//s . ...

> r Ts’

r

I --------= r

S ' / ^ . 0 0 - 7 . 2 0S* " \ /

/if* -/

• ✓ u'^ 1 —

0 ° 60 ° 120° 180° 0 ° 60 ° 120° 180°

f trc.m.

Fig. V-10

207

a b s o l u t e n o r m a l i z a t i o n of the d a t a of M a r q u a r d t et al. to

be in error. If the m a g n i t u d e of the e x p e r i m e n t a l cross

s e c t i o n s is I n c r e a s e d by the i n d i c a t e d fa c t o r of three,

then an a n g u l a r m o m e n t u m c u t o f f of J c ■ 15fi is r e q u i r e d

to r e p r o d u c e these cr os s sectio ns .

At h i g h e r b o m b a r d i n g e n e r g i e s , the c a l c u l a t e d a b s o ­

lute cross s e c t i o n s b e c o m e i n c r e a s i n g l y s e n s i t i v e to the

ch o i c e of l i m i t i n g a n g u l a r m o m e n t u m J c in the c o m p o u n d

nucl eu s. At E = 3 6 MeV, the cr os s s e c t i o n s I n c r e a s ec.m.

(decrease) by r o u g h l y a fa c t o r of 2 for an I n c r e a s e

(decrease) of one unit in the v a l u e of J c . T h is s e n s i ­

t i v i t y is r e d u c e d at lower e n e r g i e s and c o r r e s p o n d s to a

fa ct or of 1.75 at E = 2 4 MeV.O • ID •

An a n g u l a r m o m e n t u m c u t o f f of J £ = 184i is r e q u i r e d

to o b t a i n a g r e e m e n t w i th the a n g u l a r d i s t r i b u t i o n s for

st at es in 2 0 Ne b e l o w 10 M e V for the 12C (14N ,6L i ) 2 0 Ne

r e a c t i o n m e a s u r e d at E„ __ = 36 M e V by B e t o t e et al.c.m. J

(Be73) at O x f o r d (Fig. V-ll). The o v e r a l l a g r e e m e n t for the

s h a p e s and r e l a t i v e and a b s o l u t e v a l u e s of the cr os s

s e c t i o n s is e x c e l l e n t . Since the g r a c i n g a n g u l a r m o m e n t u m

in the e n t r a n c e c h a n n e l is 'v. 21/K, this c o m p a r i s o n c l e a r l y

shows the e x i s t e n c e of a l i m i t i n g a n g u l a r m o m e n t u m .

The e n e r g y s p e c t r u m of ^Li p a r t i c l e s at E„ = 36C • TTi •

MeV and © i ab = (Fig. V-12) i n d i c a t e s a n u m b e r of

Fig. V - ll . Absolute Hauser-Feshbach statistical model calculationscompared with, experimental angular distributions from

20(Be73) for low-lying states in Ne populated by the*2C(*4N ,8Li)29Ne reaction at E = 3 6 MeV. An angularc .m .momentum cutoff of J^= 18ft was applied in the entrance channel.

dcr/dil

(fjLb

/sr)

l2C(l4N .6 L i)2 0 Ne Ec.m. = 36 MeV(Belote.et al.)

Fig. V-ll

208

s e l e c t i v e l y p o p u l a t e d s t a t e s w h i c h a p p e a r as r e s o l v e d or

p a r t i a l l y r e s o l v e d gr o u p s s u p e r i m p o s e d on a s m o o t h l y

v a r y i n g b a c k g r o u n d . T h e s e gr o u p s c o r r e s p o n d m o st li k e l y

to h i g h - s p i n states. The o r i g i n of the b a c k g r o u n d ab ov e

10 Me V e x c i t a t i o n is p r e s u m a b l y the m a n y c l o s e l y spaced

le ve ls of lower s p in w h i c h are not r e s o l v e d in the e x p e r i ­

ment. At R c .m . “ 55.4 M e V and = 2 °, h o w e v e r the

^Li s p e c t r u m (Fig. V-13) w i t h the same e x p e r i m e n t a l e n e r g y

r e s o l u t i o n shows li t t l e e v i d e n c e of the s e l e c t i v e p o p u ­

l a t i o n of h i gh spin states.

The shape of the ^Li " b a c k g r o u n d " yield m a y be

c a l c u l a t e d w i t h i n the s t a t i s t i c a l m o d e l by u s i n g a level

d e n s i t y f o r m u l a to a p p r o x i m a t e the s p e c t r u m of e x c i t a t i o n

in 2 r Ne. The r e s u l t s of such a c a l c u l a t i o n at E„ _ ■ 36c.m.

and 55.4 M e V for s e v e r a l v a l u e s of the a n g u l a r m o m e n t u m

cu t o f f J £ are c o m p a r e d to the e x p e r i m e n t a l d a ta in Figs.

V-12 and V-13, r e s p e c t i v e l y . The c o n t r i b u t i o n from the

d o m i n a n t t w o - s t e p d e c a y p r o c e s s *-2C ( ,ot) 2Na (^Li) ^ 0

has been e v a l u a t e d for s e v e ra l J c v a l u e s u s i n g the

s e q u e n t i a l t h r e e - b o d y k i n e m a t i c s p r o g r a m S F O U E L w h i c h is

part of the m u l t i - s t e p d e c a y code B R E A K U P (Ha75).

A p p r o p r i a t e k i n e m a t i c c o r r e c t i o n s have b e en m a d e to e x p r e s s

the d i f f e r e n t i a l cross s e c t i o n at a p a r t i c u l a r an gl e in the

l a b o r a t o r y frame. Si nc e the a b s o l u t e n o r m a l i z a t i o n for the

e x p e r i m e n t a l s p e c t r a are not given in Ref. (Be73), these

Fig. V-12. Statistical model calculations of the shape of the Li"evaporation" spectrum at E = 3 6 MeV and 0 , . = 10°.c .m . LABThe data are from (Be73) and have been normalized to thestatistical model calculation corresponding to the Bass modelcutoff Jq= 1877 . The solid lines correspond to statisticalmodel absolute cross sections at 10° in the lab frame for

0"first chance" Li emission for various values of the angular momentum cutoff. The dashed lines represent similar calcu­lations for the dominant two-step process of alpha-particle

0emission followed by Li emission. A sequential three-body decay kinematics program was used in calculating the

t

two-step cross sections.

0

doVd

cu

dE

||ab(

mb/

sr

MeV

)

20 30 40 6,

50 60 70

E L (MeV)

Fig. V-12

209

s p e c t r a have b e en d i s p l a y e d In a wa y to f a c i l i t a t e c o m p a r i s o n

w i t h the c a l c u l a t e d ^Li yi el d c o r r e s p o n d i n g to the Bass

m o d e l cutoff. W h i l e the p r e d i c t e d and m e a s u r e d sh a p e s of

the 8Li s p e c t r a a g r e e fa i r l y w e ll at E = 3 6 MeV, atc.m.

the h i g h e r e n e r g y p c m ” 55.4 MeV, o n l y the h i g h e n e r g y

part of the s p e c t r u m is r e p r o d u c e d . The low e n e r g y p o r t i o n

of the c a l c u l a t e d second ch a n c e ^Li s p e c t r u m is e n h a n c e d ,

but the o v e r a l l m a g n i t u d e is down by a fa ct or of 10 and

thus ca nn ot a c c o u n t for the d i s c r e p a n c y . An a n g u l a r

m o m e n t u m c u t o f f of J c < 164 w o u l d be r e q u i r e d to c o r r e c t l y

r e p r o d u c e the shape of the sp ec tr um . H o w e v e r such a c o m p a r i ­

son is m e a n i n g l e s s in the a b s e n c e of an a b s o l u t e n o r m a l i ­

zation, a l t h o u g h K l a p d o r et al. (K175) w o u l d m a k e such a

shape c o m p a r i s o n the c r i t e r i o n for d e t e r m i n i n g the l i m i t i n g

a n g u l a r m o m e n t u m . Th e p o s s i b i l i t y of c o n t r i b u t i o n s from

n o n - e q u i l i b r i u m p r o c e s s e s w i ll be d i s c u s s e d s h o r t l y in

light of n o r m a l i z e d high e n e r g y data.

The a b s e n c e of s e l e c t i v e l y p o p u l a t e d gr o u p s in the

8li s p e c t r u m at E *\> 55 Me V (Fig. V-13) can bec.m*

e x p l a i n e d by the s t a t i s t i c a l mo de l. At lower b o m b a r d i n g

e n e r g i e s (E < 36 M e V ) , a s t r o n g a n g u l a r m o m e n t u mc.m. — ^

m i s m a t c h b e t w e e n the m a x i m u m a n g u l a r m o m e n t u m In the c o m p o u n d

n u c l e u s and the m a x i m u m o r b i t a l a n g u l a r m o m e n t u m a v a i l a b l e

C O Ato the Li + Ne s y s t e m r e q u i r e s that the r e s i d u a l st at es

O Ain ‘ 'Ne have h i gh spin. In going to h i g h e r b o m b a r d i n g

gFig. V-l 3. Statistical model calculations of the shape of the Li"evaporation" spectrum at E = 55.4 MeV and © TA = 7°.

c.m. L A B

See caption for Fig. V-12 for further details.

dcr/d

cudE

jab

(m

b/sr

M

eV)

E » ( M e V )

Fig. V-13

2 1 0

en er g i e s , the m a x i m u m a n g u l a r m o m e n t u m of the c o m p o u n d

n u c l e u s J £ i n c r e a s e s slowly, w h e r e a s the i n c r e a s e d

k i n e t i c e n e r g y a v a i l a b l e in the Li + 2 Ne c h a n n e l r e s u l t s

in an en h a n c e d cr os s s e c t i o n for the l o w e r - s p i n st a t e s in

20Ne. Thus, it is the r e d u c t i o n in this a n g u l a r m o m e n t u m

m i s m a t c h for a r e a c t i o n p r o c e e d i n g t h r o u g h the c o m p o u n d

n u c l e u s w h i c h is r e s p o n s i b l e for the a b s e n c e of s e l e c t i v e l y

p o p u l a t e d gr o u p s at v e r y h i gh b o m b a r d i n g energi es .

In o r d e r to m e a s u r e the c r i t i c a l a n g u l a r m o m e n t u m

at yet h i g h e r e n e r g i e s and to test the a s s u m p t i o n of

s t a t i s t i c a l e q u i l i b r i u m , m e a s u r e m e n t s at T e x a s A & M w e r e

14p e r f o r m e d at N b e a m e n e r g i e s of 86.7 and 156 MeV

(E => 40.1 and 72.5MeV). In this case the total y i e l d sc.m. J

of ^ H e , bLi , 7 Li , ^Be, and 9 Be w e re d e t e r m i n e d . S t a t i s ­

tical m o d e l c a l c u l a t i o n s for the y i e l d s of these p a r t i c l e s ,

i n c l u d i n g c o n t r i b u t i o n s from se c o n d c h a n c e e m i s s i o n , are

c o m p a r e d in Fig. V-14 to the e x p e r i m e n t a l d a ta at 40.1 Me V

6 7 9c.m. The a g r e e m e n t for Li , Be, and Be y i e l d s is

q u i t e good w h e n the total a n g u l a r m o m e n t u m limit of 19

p r e d i c t e d by the Bass mo d e l is e m p l o y e d . The m e a s u r e d ^Li

y i el d is not well r e p r o d u c e d , h o w e v e r , and wo u l d r e q u i r e a

cu t o f f equal to the g r a z i n g a n g u l a r m o m e n t u m to yield the

m e a s u r e d cross section. H o w e v e r , it may be c o n c l u d e d

from the o v e r a l l a g r e e m e n t for the three i s o t o p e s ^Li , 7Be,

and 9 Be that s t a t i s t i c a l e q u a l i b r i u m is a c h i e v e d and that

Fig. v-: . Total yields for 6Li, 7Li, 7Be, and 9Be for the *2C + *4Nreaction at E = 4 0 .1 MeV from (Ch75), and corresponding

c •in*

statistical model predictions including first and second chance emission.

<

l2c + '4n

cm

i EXPERIMENT:CHULICK, NAMB00DIRI,

et al.

nd i STATISTICAL2 chance ■ CALCULATIONS:

HANSON et al.

* 40.1 MeV

^max" ^ ^

J = * > ± i d

l ( T * ± ) = 2 2 1)

V I chance

211

the factor of two d i s c r e p a n c y for L i r e p r e s e n t s e v i d e n c e

for n o n - s t a t i s t i c a l p r o c e s s e s in the 7 L i + ^9Ne channel.

The cross s e c t i o n for a l p h a p a r t i c l e p r o d u c t i o n is

^1200 mb, about half of w h i c h is a c c o u n t e d for by the

p r e d i c t e d first and se co nd ch a n c e oi-emission. H o w e v e r

it m a y he p o s s i b l e to a c c o u n t for a large f r a c t i o n of the

1 2d i f f e r e n c e as yi el d from the d i s s o c i a t i o n of the C target

into three a l p h a p a r t i c l e s .

The n o r m a l i z e d ^Li e n e r g y s p e c t r u m for E = 40.1cm

Me V has also been m e a s u r e d at T e x a s A& M and is s h o w n in

Fig. V-15. The s t a t i s t i c a l mo de l cu r v e for J = 19di, thec

p r e d i c t e d Bass m o d e l cutoff, a g r e e s f a i r l y w e ll in sl op e and

a b s o l u t e m a g n i t u d e w i t h the d a ta in the r e g i o n of e x c i t a t i o n

e n e r g y b e t w e e n the po in t at low E^ w h e r e the le ve l d e n s i t y

f o r m u l a may b r e a k d o wn and the r e g i o n at h i g h E x wh e r e

n o n - s t a t i s t i c a l p r o c e s s e s m i g h t be e x p e c t e d to b e c o m e i m p o r ­

tant. The s t a t i s t i c a l e r r o r s on the d a ta are not k n o w n so

it is d i f f i c u l t to ju dg e the q u a l i t y of the fit at low

e x c i t a t i o n energi es .

A c o m p a r i s o n of the s t a t i s t i c a l m o d e l to the i n t e g r a t e d

6 ? 7 qe x p e r i m e n t a l y i e l d s of Li, L i , 'Be, and Be at

E c m = 72.5 Me V s u g g e s t s that there m a y be s u b s t a n t i a l

d e p a r t u r e s from e q u i l i b r i u m at this e n e r g y a n d / o r di r e c t

r e a c t i o n c o n t r i b u t i o n s . A s i g n i f i c a n t d i f f e r e n c e is o b s e r v e d

in the shape of the ALi e n e r g y s p e c t r u m (Fig. V - 1 6 ) , w h i c h

Fig. V-0model calculations of the angle-integrated Li "evaporation"

12 14spectrum for the C + N reaction at E = 40.1 MeVc.m .

. Comparison of shape and absolute magnitude of statistical

with normalized data of Chulick et al. (Ch75).

dcr/d

E (m

b/M

eV)

Ecin(MeV)

Fig. V-15

gmodel calculations of the angle-integrated Li "evaporation" 12 14spectrum for the C + N reaction at E = 7 2 .5 Mevc.m .

Fig. V-16. Comparison of shape and absolute magnitude of statistical

with normalized data of Chulick et al. (Ch75).

LI ENERGY SPECTRUM

l2C (l4N, 6L i)20Ne

= 7 2 .5 MeV

NORMALIZED TEXAS A SM DATA

HF STATISTICAL MODEL CALCULATIONS

Jc ■ ENTRANCE CHANNEL ANGULAR MOMENTUM

CUTOFFJc 8 26

BASS MODEL CUTOFF

23 BASS MODEL CRITICAL ANGULAR MOMENTUM

50 60

Ec.L.(MeV)

Fig. V-16

212

shows an e n h a n c e d yi e l d in the low e n e r g y p o r t i o n of the

s p ec trum. It is i n t e r e s t i n g to note that the h i gh e n e r g y

p o r t i o n of the s p e c t r u m is well r e p o r d u c e d in b o t h s h a p e and

a b s o l u t e m a g n i t u d e by the s t a t i s t i c a l m o d e l c a l c u l a t i o n

u s in g an a n g u l a r m o m e n t u m cu t o f f Jc ** 23Jf, w h i c h c o r r e s p o n d s

to the c r i t i c a l a n g u l a r m o m e n t u m p r e d i c t e d by the Bass model.

H o w e v e r , no c r i t i c a l a n g u l a r m o m e n t u m has b e en o b s e r v e d even

1 A 9 7at v e ry high e n e r g i e s for the N + ‘‘'Al s y s t e m (Na75),

4

and f u r t h e r d a t a at h i g h e r e n e r g i e s will be r e q u i r e d b e f o r e

any c o n c l u s i o n can be r e a c h e d r e g a r d i n g a c r i t i c a l a n g u l a r

m o m e n t u m for the ^ 2 C + system.

The r e s u l t s of these a n a l y s e s of l i m i t i n g a n g u l a r

m o m e n t u m for I-2C + i n d u ce d r e a c t i o n s are s u m m a r i z e d

in T a b l e V-4 and Fig. V-17. The s q u a r e s in Fig. V-17

r e p r e s e n t the l i m i t i n g total a n g u l a r m o m e n t a J £ d e t e r m i n e d

from s t a t i s t i c a l m o d e l a n a l y s e s of the a v a i l a b l e e x p e r i m e n t a l

data, as d i s c u s s e d in this section. An e r r o r of ± Hi is

a s s i g n e d to each e x p e r i m e n t a l point to a c c o u n t for the

u n c e r t a i n t i e s in the s t a t i s t i c a l c a l c u l a t i o n s . The solid line

r e p r e s e n t s the Bass m o d e l p r e d i c t i o n for the e n t r a n c e c h a n n e l

o r b i t a l a n g u l a r m o m e n t u m c u t o f f £ . Th e Bass m o d e l p r e d i c t sc

a c r i t i c a l a n g u l a r m o m e n t u m of £ ■ 23tf, w h i l e thec

W i l c z y n s k i mo de l (v er ti ca l d a s h e d line) p r e d i c t s an e n e r g y -

i n d e p e n d e n t c u t o f f at £ = 17*i. The p o s i t i o n of thec

g r a z i n g a n g u l a r m o m e n t u m £ ^ y (defined by ^ £ gr = 0*5) is

12 14 26Fig. V-17. Limiting angular momenta for C + N and Al based on statistical model analyses described in the text. The data are from Volant et al. (Vo75), Marquardt et al. (Ma71), Belote et al. (Be73), and Chulick et al. (Ch75).

Ecm(M®V)

i2C + ,4N

100 200 300 400 SOO 600

X(M«V)In

26a i

Fig. V-17

213

is i n d i c a t e d by the d o t t e d line. Y r a s t lines for s e v e r a l

2 cd i f f e r e n t r i g i d - b o d y m o m e n t s of i n e r t i a for the Al

c o m p o u n d sy s t e m are s h o w n by the d a s h e d lines.

The a g r e e m e n t b e t w e e n the Bass m o d e l p r e d i c t i o n s and

the data, as d e t e r m i n e d u s i n g i n d e p e n d e n t l y d e r i v e d t r a n s ­

m i s s i o n c o e f f i c i e n t s and level d e n s i t y p a r a m e t e r s in a

s t a t i s t i c a l m o d e l a n a l y s i s , is e x c e l l e n t and e x t e n d s the

range of a p p l i c a t i o n of this mo de l to the lower 2s~ld shell.

TABLE V-4. 12C + *4N Limiting Angular Momenta

Ec.m .(MeV)

^GRAZING(ft )

Bass^CUT(ft)

Type of a Normalized

DataExperimental

JCUT(ft)

Ref.

16.2 12 12 A 12 Vo7518.5 13 13 A 13 Vo7520.8 14 14 A 14 Vo7523.1 15 15 A 15 Vo7524.0 15 15 A 15 Ma7125.4 16 16 A 16 Vo7527.7 17 17 A 17 Vo7536.0 20 18 A 18 Be7340.1 21 19 E,T 19 Ch7572.5 29 24b E 23 Ch75

3, 20 A = angular distributions to low lying levels in Ne; E = angle-integrated energy spectra ; T = total cross6 7 7 9sections for Li, Li, Be, and Be emission.

b Bass model predicts net force at is not attractive at ^ ^ = 2 4 , and that critical or absolute angularmomentum is I ____ = 23h

v U 1

215

CHAPTER VI CONCLUSIONS AND PROJECTIONS

A. I n t r o d u c t i o n

T h r o u g h o u t the p r e v i o u s d i s c u s s i o n , the u n i q u e role

of h e a v y - i o n s in both d i r e c t and c o m p o u n d r e a c t i o n s has

been e m p h a s i z e d . E x p e r i m e n t a l d a t a and c a l c u l a t i o n s h a ve

be en p r e s e n t e d as e v i d e n c e for n e w p h e n o m e n a and u n u s u a l

r e a c t i o n m e c h a n i s m s in h e a v y - i o n r e a c t i o n s w h i c h have no

p r o m i n e n t c o u n t e r p a r t in l i g h t - i o n induced, r e a c t i o n s . In

this c h a p t e r , the m a j o r r e s u l t s and c o n c l u s i o n s of this

th es is based on the study of m u l t i s t e p p r o c e s s e s in two-

n u c l e o n t r a n s f e r r e a c t i o n s and the i n v e s t i g a t i o n of the

m e c h a n i s m for m u l t i - n u c l e o n t r a n s f e r and c o m p l e x p a r t i c l e

e m i s s i o n in *^C + ^4 jj r e a c t i o n s w i l l be p r e s e n t e d in

turn, and some p o s s i b l e e x t e n s i o n s of this w o r k w i l l be

s u g g e s t e d .

216

B. T r a n s f e r R e a c t i o n s

A n u m b e r of n e w and i m p o r t a n t r e s u l t s h a ve e m e r g e d

from this st ud y of m u l t i s t e p p r o c e s s e s in h e a v y - i o n

i n du ce d t w o - n u c l e o n t r a n s f e r r e a c t i o n s . We have d e m o n s t ­

rated that w i t h the p r o p e r i n s t r u m e n t a t i o n it is p o s s i b l e

to o b t a i n a d e q u a t e r e s o l u t i o n to ca rr y out h e a v y - i o n

r e a c t i o n s on d e f o r m e d n u c l e i in the rare e a r t h region.

In the first such e x p e r i m e n t s , we have o b s e r v e d s e v e r a l

new p h e n o m e n a i n c l u d i n g s t r o n g i n t e r f e r e n c e m i n i m a r e s u l ­

ting from d e s t r u c t i v e i n t e r f e r e n c e b e t w e e n d i r e c t and two-

step t r a n s f e r and the first e v i d e n c e for the s t r o n g i n f l u e n c e

of C o u l o m b e x c i t a t i o n on a t r a n s f e r r e a c t i o n . The s e n s i t i ­

v i t y of these r e a c t i o n s to the u n d e r l y i n g s t r u c t u r e of

the o r b i t a l s p a r t i c i p a t i n g in the t r a n s f e r has b e e n i n v e s t i ­

gated t h r o u g h the e f f e c t s of a s y s t e m a t i c d e c r e a s e in

q u a d r u p o l e t r a n s f e r s t r e n g t h w i t h i n c r e a s i n g n e u t r o n n u m b er .

Th e role of m u l t i s t e p p r o c e s s e s no w a p p e a r s to be

s u f f i c i e n t l y well u n d e r s t o o d that it is p o s s i b l e to test in

d e t a i l the p r e d i c t i o n s of n u c l e a r s t r u c t u r e m o d e l s . The

i n f o r m a t i o n d e r i v e d from such s t u d i e s m u s t n e c e s s a r i l y he

i n d i r e c t , si nc e the n u c l e a r s t r u c t u r e i n f o r m a t i o n en t e r s

the c a l c u l a t i o n in the form of input p a r a m e t e r s for a

p a r t i c u l a r m o d e l , and the C C B A c a l c u l a t i o n s then e x p l o r e

the c o n s i s t e n c y of the m o d e l d e s c r i p t i o n w i t h the e x p e r i m e n t a l

217

data. Howe ve r, b e c a u s e of the s e n s i t i v i t y to d e t a i l s of

the n u c l e a r w a v e f u n c t i o n s i n t r o d u c e d by n u c l e a r - C o u l o m b

i n t e r f e r e n c e , C C BA a n a l y s i s of e x p e r i m e n t a l d a t a p r o v i d e s

a p o w e r f u l test of the p r e d i c t i o n s of n u c l e a r s t r u c t u r e

models.

W h i l e the a g r e e m e n t b e t w e e n C C B A c a l c u l a t i o n s and

the d a ta is in g e n e r a l good, there is st il l need for i m p r o v e ­

ment in the t h e o r e t i c a l an al y s i s . In p a r t i c u l a r , there is

the need for a m o r e r e a l i s t i c p a i r i n g m o d e l to i m p r o v e

the c o n v e r g e n c e of the U V fa ct or d e s c r i b i n g o c c u p a t i o n

p r o b a b i l i t i e s of o r b i t s ar o u n d the Fermi level. L i k e w i s e ,

m o r e c a r e f u l c o n s i d e r a t i o n sh ou ld be gi ve n to the type of

o p t i c a l m o d e l p o t e n t i a l most s u i t a b l e for a d e s c r i p t i o n of

h e a v y - i o n s c a t t e r i n g . Also, w h i l e r e c o i l e f f e c t s are

e x p e c t e d to be r e l a t i v e l y sm al l for th e s e h e a v y n u c l e i , a

C C B A c a l c u l a t i o n s h o u l d be r e p e a t e d w i t h full re c o i l

c o r r e c t i o n s i n c l u d e d to v e r i f y that this is the case.

The e f f e c t s of i n e l a s t i c e x c i t a t i o n s of the light

p a r t i c l e in the init ia l and final f r a g m e n t a t i o n ma y be

I m p o r t a n t and sh ou ld be studied. A s c u i t t o (As75c) has

i n v e s t i g a t e d this ef fe ct for the ground state t r a n s i t i o n

in the *8^ W ( *2C ,*^C) r e a c t i o n by r e v e r s i n g the r o l e of

target and p r o j e c t i l e in the c o u p l e d - c h a n n e l s code and p e r ­

m i t t i n g v i r t u a l e x c i t a t i o n s of the p r o j e c t i l e w i t h the

218

target in its ground state. The effect of this renormali­

zation in the entrance channel is to raise the 0+ cross

section at forward angles, thus bringing it into better

agreement with the data. A similar enhancement of the

0+ angular distribution has been reported recently for

calculations on the X ^ S n ( X8o 7 80) reaction by Glendenning

and Wolschin (G175).

IFinally, there is a need to test the sensitivity of

these CCBA results to variations in the deformation

parameters entering the inelastic form factors. It would

he interesting to know whether, for example, the shapes

of the calculated angular distributions are sensitive to

changes by a factor of two in the nuclear and Coulomb

hexadecapole deformations.

It should be noted that one additional and important

test of the reaction model is possible by extending the

present data. When the intrinsic states of the heavy

system are described in terms of a simple pairing model for

identical rigid rotors of the type used in this analysis,

the angular distributions for stripping and pickup between

the same nuclei and with the same kinematic conditions

should be identical in shape. This prediction may be

tested in the most sensitive way by doing the two-neutron

stripping reaction on X84W using either a XAC or X80 beam.

219

Possible direct extensions of the present work

include the investigation of transitions to higher rotat­

ional bands for nuclei in the tungsten region and contin­

uation of the present two-neutron transfer measurements

through the osmium region. These latter measurements are

of interest for observing the transition in the transfer

pairing field from prolate to oblate to spherical. Since

the collectivity of these transition nuclei is no longer

simply described, a more elaborate model will be required

to adequately describe the structure of the intrinsic

states.

Another possibility is to extend these measurements

to the actinide region where the deformed nuclei again

are good rigid rotors but where relatively less is known

about the deformations and intrinsic nuclear structure.

This offers an interesting opportunity to apply these

reactions as a spectroscopic tool in relatively unexplored

territory. Because of the reduced energy spacing of the

rotational band members in the transuranlc region, the

resolution of the lowest members of the ground state

rotational band for these nuclei represents an experimental

challenge, which is not, however, beyond the capability of

the QDDD magnet if proper attention is given to target

construction and beam transport. This close energy spacing

increases the probability of nuclear and Coulomb excitation

220

to higher members of the ground state rotational band and

opens the possibility of observing interference minima in

the angular distributions for transitions to these higher

states. Two proton transfer on the actinides is not easily

explored with light ion reactions and offers another

possibility for future work.

While light-ion reactions have been used in the past

to study more exotic multistep processes (As71a), the use

of heavy-ions to investigate such higher-order processes

offers exciting possibilities for future work because of the

greater freedom from ambiguity in the interpretation of

heavy-ion experimental angular distributions. The most

immediate examples are to be found in inelastic and sequen­

tial transfer reactions, which populate states that are

inaccessible through any simple direct or inelastic-plus-

transfer route by going through a virtual intermediate

fragmentation. While the model description of such

processes becomes increasingly more complicated, the en­

hanced L-space localization and sensitivity to underlying

nuclear structure inherent in such multistep processes may

give access to details of nuclear structure unattainable

with light-ion reactions. In any case, it is essential to

determine the role of "simple" and more exotic heavy-ion

induced multistep transfer processes in all mass regions

if heavy-ion transfer reactions are to realize their poten­

tial as a serious spectroscopic tool.

221

C. Compound Reactions

The extensive data on light and complex particle19 1 /production in the C + N-induced reactions, taken at

low energies as part of the work of this thesis, and at

higher energies by many other researchers, appear to he

well understood, on a quantitative basis, at least at

energies of Rc ,m . < 40 MeV, in terms of a simple

statistical model for compound nucleus formation and decay.

There is definite evidence at energies E > 20 MeVc . m . —for a restriction on the maximum allowed angular momentum

of the compound nucleus resulting from dynamical conditions

in the entrance channel. Using the prescription of Bass

to determine this angular momentum cutoff, we are able to

reproduce within the statistical model most features of

the available data, including the slope and absolute ma g ­

nitude of the high energy part of the ^Li evaporation

spectra, at energies up to E ■ 72.5 MeV.C « TTt •

Of particular interest for further study is the

systematic increase in the yield of low kinetic energy

heavy particles of non-statistica 1-compound origin. Such

yield makes up a significant fraction of the total cross

section for BLi emission at high energies, but its origin

in terms of direct or deep-inelastic transfer processes

remains an open question of some interest. In addition,

while the available data on ^ 2C + reactions is

222

consistent with a critical, or absolute limiting angular

momentum of ^CRIT = predicted by the Bass model,

higher energy data are needed before any definite conclu­

sion can be drawn.

While a large amount of data on heavier target and

projectile combinations has been amassed in recent years

and found to be consistent with predictions of the Bass

model (Ba73,Ba74), the system is the lightest

compound system to be examined extensively in terms of

dynamical limitations on the complete fusion cross section.

This.study takes on added interest because it serves to

extend the range of application of this dynamical model

to the lightest mass range where complete fusion remains

a meaningful concept.

The most exciting applications of this rapidly

accumulating knowledge on fusion and many-particle transfer

processes lie in future explorations far from the line of

nuclear stability in both excitation energy and particle

number .

223

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