A STUDY OF (7Li,t) REACTIONS ON LIGHT NUCLEI

MARTIN ELLIS COBERN

1974

A STUDY OF (?L i, t) REACTIONS

ON LIGHT NUCLEI

A Dissertation

Presented to the Faculty of the Graduate School

of

Yale University

in Candidacy fo r the Degree of

Doctor o f Philosophy

by

Martin ElUs Cobern

June 1974

ABSTRACT

A STUDY OF THE (?L i, t) REACTIONS

ON LIGHT NUCLEI

Martin E llis Cobern

Yale 1974

nTriton angular distributions have been measured from the ( L i, t)

reactions on targets o f carbon-12, nitrogen-14, oxygen-16 and neon-20

at lithium beam energies from 36-40 MeV in the laboratory. These reactions

selectively populate a small number of states at forward angles, and the

angular distributions of these states are prom arily forward-peaked.

An exact finite-range coupled-channels DWBA code has been used

to analyze the angular distributions of tritons populating the three lowest

members of the 4p-4h (K 7T= 0+) band in oxygen and the f irs t three members

TT +of the ground state (K = 0 ) band in neon-20. A lpha-particle reduced widths

w ere extracted fo r these states.

A Hauser-Feshbach analysis has indicated that fo r those states

se lectively excited at forward angles the reaction mechanism is prim arily

compound-nuclear in nature at angles > 9 0 °(c .m .). Angular distributions

have also been extracted fo r a number o f other states not strongly excited

at forward angles; these states appear to be populated almost entirely

v ia compound nucleus formation at a ll angles.

A qualitative analysis of the angular distributions for the *4N (7L i, t ) ^ F

reaction has shown that the two states at 9.472 and 11.074 MeV are probably

n +members of the K = 1 band of fluorine-18, which dominates the forw ard-

angle spectra.

Angular distributions for lithium elastic scattering from targets of

carbon-12, nitrogen-14 and neon-20 have been measured at a beam energy

o f 36 MeV. Optical potential parameters have been determined.

(c) Copyright by Martin E llis Cobern 1974

A L L RIGHTS RESERVED

TO DORIE

While this thesis bears a single name, there are many who have made

important contributions. It is a pleasure to acknowledge their e fforts.

I was fortunate in having in P ro fessor Peter D. Parker an advisor who

was never too busy to advise. His assistance, encouragement and unfailing

good spirits have lightened my labors considerably.

My friend and colleague, D r. Daniel J. Pisano has unselfishly given

untold hours throughout each phase of the research. In particular, he has

patiently explained, often several times, the intricacies o f the many computer

programs involved in the analysis. P ro fessor Robert J. Ascuitto is to be

thanked fo r tolerating, and even encouraging, the clumsy forays of this

experimentalist into the heady realm of reaction theory. The interest shown

by P ro fessor D. Allan Brom ley in this work is greatly appreciated.

This pro ject could not have been completed without the hospitality

o f two neighbor institutions. I would like to thank Dr. P eter Theiberger, M essrs.

Robert Lindgren and John Benjamin, and the staff o f the Brookhaven National

Laboratory Tandem Acce lera tor fo r their special efforts in my behalf. Their

willingness to g ive of their time o f the weekend was beyond what any guest could

rightfully expect. The staff o f the Courant Institute computer center were

most helpful, even in the face o f intransigent electronics.

The form factor calculation o f Dr. James P . Vary contributed much

to the validity o f this work. His efforts in our behalf are greatly appreciated.

The volume in your hands is the work of two talented people. M r.

Wayne Chomey expertly and rapidly drew the many figures. Ms. Carole

Lawson put in long, and often late, hours in typing this thesis. Through

ACKNOWLEDGEMENTS

their e fforts, this phase of the production was the only one to run entirely

smoothly.

The financial support of the National Science Foundation during the

f irs t three years o f my work, and of the Atohiic Energy Commission

thereafter has been most welcom e.

Finally and above a ll, there is my w ife, D orie, without whose

support, assistance and encouragement this work would have been long

abandoned. My debt to her w ill take a lifetim e to repay.

V

vi

Abstract

Chapter I: Introduction

A : The Cluster Structure o f Light Nuclei

B: Stellar Nucleosynthesis and Helium Burning

C: Lithium-Induced Reactions

Chapter II: Experimental Method and Data Reduction

A : Reactions

B: Ion Sources, A ccelerators and Scattering Chambers

C: Targets

D: Detectors

E: Electronics

F: Resolution

G: Data Reduction

Chapter n i: Theory o f the Reaction Mechanism

A : Plane-W ave Bom Approximation

B: Distorted-W ave Bom Approximation

C: Coupled-Channels Bom Approximation

1. Assumptions o f the DWBA

2. Assumptions o f the CCBA

3. Outline of the Source Term Method

Chapter IV: Experimental Results

A : 12C (7L i, t )160

B. 160 (7L i, t )20Ne

C: 14N (7L i, t )18F

D: 20N e(7L i, t)24Mg

E: E lastic Scattering

Chapter V : Analysis and Results

A : Analysis o f the Lithium-7 Scattering Data

B: Compound-Nuclear Reaction Calculations

TABLE OF CONTENTS

Acknowledgementsi

iv

1

1

4

8

13

13

13

14

15

18

20

22

24

24

28

29

30

31

32

35

35

36

37

38

38

40

40

42

Vll

C: Finite-Range Coupled-Channels DWBA Analysis:

12C(7Li, t)160 46

1. Program Testing 46

2. Reaction Calculations 47

3. Results of the Calculations 51

Ds Finite-Range Coupled-Channels DWBA Analysis:

160 (7L i, t)20Ne 56

Chapter V I: Conclusions 60

Appendix A : Target Energy Loss and Kinematic Broadening 63

Appendix B: Calculation o f the Source Term 68

Appendix C: Numerical Methods 77

1. Solution o f the D ifferential Equations 77

2. The Finite-Range Source Term 78

Bibliography 81

1

I f one were of poetic inclination, one might say that the motivation

fo r this thesis hails from the extreme dimensions on the scale of human

knowledge. The results of this research could, indeed, increase man's

comprehension of the structure of the atomic nucleus, and at the same tim e,

illuminate some facets o f the interiors of the stars. Such lofty aims can

never be fu lly satisfied, but, hopefully, each new datum advances the

endeavor.

This Introduction is an attempt to present the rationale fo r perform ing

these experiments. Part A concerns the m icroscopic and gives a brie f

survey o f one important aspect o f light nuclei — their tendency to form sub-

nuclear clusters. Part B pertains to the m acroscopic, the energy producing

mechanisms of stars. In Part C, we review the previous work on lithium-

induced reactions.

A . The Cluster Structure o f Light Nuclei

In any description of the structure of light nuclei, a perem ptory

consideration is the extrem e stability o f four nucleons, when coupled to zero

spin and isospin — an alpha particle. This fact enables one to describe many

features of these light nuclei, with reasonable accuracy, in terms o f "alpha-

particle clusters", o r "quartets".

This description was firs t formulated in 1958, as the "c luster m odel",

by Wildermuth and Kanellopoulos (Wi58), although the concept was, to some

extent, an expansion of the "resonating group" theory of W heeler (Wh37). In

I . IN T R O D U C T IO N

this model, the nuclear wave function is expanded in terms o f clusters of

nucleons, rather than in single-particle wave functions, as in the shell model.

The clusters may bo single nucleons, deuterons, alpha particles, or heavier

aggregations, depending on the nuclear state in question. If the choice of

clusters is physically rea listic , then such an expansion w ill contain only a

small number of term s.

The term "cluster model" has come to include a large variety of

disparate theories. These range from those which have a single alpha

particle o r other cluster orbiting an inert core (e .g . Wi62, H068), to the

extrem e "alpha-particle m odel", in which nuclei are built up so le ly as

pyramidal conglomerations o f alpha particles (e .g . Av72).

The f ir s t application of the cluster theory was to bery llium -8, where

it was quite successful in describing the low -lying states. This is under­

standable since the nucleus in its ground state is unstable against dissociation

into two alpha particles. That this clustering is p referred may be seen

by the fact that the ground state is bound against any other decay by 17 M eV.

The mass-7 nuclei have also been well-described by the cluster model

(e .g . Ta61, T 06I ; see Aj73 fo r additional references), as an alpha particle

coupled to a triton or helion, respectively. This model has been successful

in explaining the results of (a , y) capture experiments (Pa63, .To63 , Gr61).

From an analysis of elastic scattering experiments, Tom brello and Phillips

2(T 06I ) showed that the reduced widths for a "4 + 3" cluster structure, 0 4+^ ,

2are about 20 times the single-particle reduced widths, 0 . Brown and6+1

3Tang (B r68) have also been able to reproduce the a -t and H e-a scattering

phase shifts (Sp67, Iv 68), using a resonating group description of the mass-7

nuclei.

2

3

As the mass of the nucleus increases, the cluster structure becomes

less pronounced, but is s till significant. The even-even, N=Z nuclei, in

particular, have structures which can be visualized in terms of alpha-

clusters. Even oxygen-16, a spherical, doubly closed-shell nucleus in its

ground state, has many states which exhibit strong alpha-cluster character.

+In particular, the rotational band built upon the deformed 0 firs t excited

state at 6.05 MeV is apparently well described in terms of 4p-4h structure;

i .e . an alpha particle orbiting a carbon-12 core.

The cluster model also seems to present a valid description of the

16ground-state band of neon-20 (Ch69; see also references in Aj72) as an a - O

structure. A t least one of the excited bands (the one built on the 0 + state

at 7.196 MeVjcan be described as two alpha-particles outside and inert

carbon-12 core (v iz . , e .g . A172 and the references therein). The 0“ band,

built on the 5.785 MeV 1~ state in neon-20, can be thought of as an alpha

particle coupled to a lp - lh excitation in oxygen-16. Although the "nuclear-I

m olecular" description of magnesium-24 as two carbon-12 nuclei has been useful

in describing some highly-excited states (Go71),other cluster models fo r

magnesium have been less productive, and the extrem e alpha particle model

breaks down com pletely . (Ab71)

Even in the odd-odd nucleus, fluorine-18, the cluster model appears to

have some utility. Recently Rolfs and co-workers (Ro73) have shown that the

K 1 = 1+ rotational band, built upon the 1+ state at 1.701 MeV, is p rim arily

144p-2h in nature. Such states could be explained by an a- N description.

It should be noted that the cluster model is completely compatible

with other descriptions of light nuclei. Kanellopoulos (Ka60) has shown that the

cluster wave function can be expressed as an expansion of shell-model term s.

4

Abgrall (Ab71) has demonstrated the equivalence of the alpha particle

model with a deformed Hartree-Fock calculation. The advantage

of the cluster model is that, in cases fo r which it has physical validity, it

reduces the number of terms that need be considered in describing a state.

To study the structure of states which, display strong alpha-clustering

a mechanism is needed to transfer an alpha particle d irectly onto the core with

the requisite linear and angular momentum. It was fe lt that, due to the

strong cluster structure of the lithium-7 nucleus, the ( L i, t) reaction would

be an effective agent fo r this purpose. This has proven to be the case.

B. Stellar Nucleosynthesis and Helium Burning

Since 1939 (Be39) astrophysicists have recognized that most

stages of ste llar evolution have nuclear reactions as their energy source.

Beginning with the classic paper by Burbidge,Burbidge, Fow ler and Hoyle (Bu57)

attempts have been made to reproduce the observed galactic isotopic

abundances (Su56), using series o f known nuclear reactions that can ( and

hence w ill) take place in ste llar in teriors, under the varying temperature

and pressure conditions characteristic of the various stages of stellar

history. (F o r a thorough review of this area, see Clayton (C168). An ex­

tensive bibliography has been compiled by Fow ler and Stevens (Fo68), and

many o f the original papers have been reprinted by the A . I . P . (Am68) ).

A t each period in the life o f a star, there may be a wide range of

competitive processes that can take place simultaneously, but one o r two

reaction chains w ill dominate the others in a given region of the stellar

in terior. Most main sequence ste llar models begin with a homogeneous

5

star composed prim arily o f hydrogen, with up to about 30% helium (by mass)

and one or two per cent heavier elements. The firs t stage o f nuclear "burning"

involves the p-p chain or the C -N-O b i-cyc le (v iz . Bu57, CaG2, Pa64) by

which four protons are combined into an alpha particle.

When the hydrogen-burning reactions have nearly exhausted the

hydrogen in the stellar in terior, further nucleosynthesis is stymied by the lack

o f any stable nuclides with atomic mass A=5 or 8. Only through the mechanism

of a three-body collision can a heavier element, carbon-12, be created (Ho54,

Sa52, Se63). Thi.s reaction can take place only as a result of two favorable

factors:

—161) the beryllium -8 ground state has a lifetim e o f 2. 6 x 10" s e c . ,

which is long compared to the alpha-alpha collision time, and;

2) there is a 0+ energy leve l in carbon-12 at 7.6562 MeV (Au70) nearg

the energy of an a + Be combination, so the reaction is resonant

(Se63, Au70).

This reaction, the so-called "trip le-a lpha" reaction, is extrem ely

tem perature-sensitive and occurs extrem ely rapidly — instanteously, on

the cosm ic scale.

The triple-alpha reaction marks the beginning of the "helium'burning"

phase o f ste llar evolution (Sa57). Once carbon-12 comprises a significant

fraction of the local ste llar composition, a series of (a ,Y ) reactions converts

the carbon-12 into, sequentially, oxygen-16, neon-20, magnesium-24, and

silicon-28. A fte r such reactions deplete the helium content of the star, thefheavier elements produced interact, at higher temperatures, v ia chains

known as "carbon-burning" (Re59), "oxygen-burning" (Fo64), and "s ilicon -

burning" (Bo68).

6

In order to understand the coupled reaction chains at each evolutionary

phase, it is essential to determine which nuclear processes are responsible

fo r the energy generation, and which determine the rate o f nucleosynthesis.

This requires a quantitative knowledge of the relevant nuclear data. The

d irect measurement o f the reaction cross-sections at stellar thermal energies,

typically less than several hundred keV, is extrem ely difficult owing to the

vanishingly small probability o f penetrating the Coulomb b arrier at low energy.

By factoring out the barrier-penetration factor from the cross-section , the

astrophysicist obtains a quantity, S(E), which varies much more slowly with

the energy, and may therefore be more confidently extrapolated to the low

energies o f interest. Explicitly:

5 ( E ) - ( T ( E ) e * p p ^ el] * E «

However, caution must be used in perform ing this extrapolation since S(E)

param eterizes only the direct, non-resonant portion of the cross-section and

specifically omits the effects o f any low -lying resonances. Due to the narrow

range o f energies over which the reaction may take place (c. f. C168), and the

low density o f states in light nuclei, one does not, however, often find a

resonance in the area of interest. (A notable exception being the 7.6562 MeV

leve l in carbon-12, mentioned above.) What is more often the case is that

a reaction w ill be taking place in the "w ings" of a resonance, many widths

removed from the centroid. In these cases, although the resonant contri­

bution is extrem ely small, it can still be very important compared to the

non-resonant contribution. Thus, a small change in a reduced width for such

a resonance can have a significant effect on a reaction rate and thereby

7

A relevant example, cited by Couch et al. (Co73) is the measurement

2of the reduced alpha width (9^) of the 7.119 MeV state in oxygen-16. The

12 16current lim its on this parameter, critica l to the C (a , y ) O reaction,

range from 0.04 to 0. 64. Such values, when inserted into appropriate

stellar models, predict final carbon-12 mass fractions ranging from zero

to the observed values, and are consistent with the evolution of a star with a

core consisting entirely of oxygen-16. Such a divergence makes any pre­

dictions based on these solar models highly indeterminate. "Th is nagging

uncertainty plagues all calculations o f ste llar evolution and nucleosynthesis."

12(Co73) Since this state lies below the a + C threshold, (see F ig. 1), its

width cannot be measured directly in an alpha scattering or capture experi­

ment; the measurements which have been perform ed (Ba73) down to a lab­

oratory energy o f 1860 keV, have been unable to determine this parameter

any more accurately.

One possible way to resolve such difficulties would be to measure

the reduced alpha widths indirectly, through the use of an alpha-transfer

7reaction such as ( L i, t). I f this reaction proceeds via a direct process, one

2could determine 9 from the cross-section fo r the transfer.

a

In addition to the 7.119 MeV level mentioned above, other states

lying in the energy regions of astrophysical interest are:

the 9. 60 MeV ( l - ) level in oxygen-16

the levels between 4 and 5 MeV in fluorine-18

the 4.25 (4+), 5.63 (3 ) and 5.80 (1 ) levels in neon-20

Hence, in addition to a general study o f alpha particle clustering in

light nuclei, there was also a very specific motivation to study the properties

g re a t ly change the outcom e o f a s t e l la r m odel ca lcu lation .

Figure 1-1. The low -lying energy levels o f oxygen-16 (Aj71) and the

relevant reaction thresholds. The arrow indicates the thermal

12 8 energy o f the a - C system at a temperature o f 2 x 10 K,

typical of a ste lla r in terior.

4.694

l2C+7Li-t

10.353 4+

9.847

8.872

6 . 9 1 9 ^ — 2 ^

fi n s n 6 . 13 1 n »3 ~

16,0 +

7.161 \Tq=2

,2C + a

8

o f several levels in these nuclei, because o f their important role in the

helium-burning phase o f stellar evolution.

C. Lithium-Induced Reactions

With the development o f reliable sources o f negative lithium

ions (He68, Mi68), a new and valuable tool was made available to the nuclear

physicist. Tandem accelerators could then be employed to obtain lithium

beams at energies above the Coulomb barrier, fo r use in multinucleon

transfer reactions. The sudden interest in lithium-induced reactions that

followed these developments can be appreciated by observing that only two

years later one entire session at the 1969 Heidelberg conference (Bo70)

was devoted to them. Contributions from many laboratories in several

countries w ere presented, dealing with the use o f lithium beams to perform

scattering and transfer reactions. Reviews o f these and other reaction studie

are given by Bethge (Be70), Bassani (Ba71b), and Ogloblin (Og72).

The reason fo r the wide interest in lithium-induced reactions is that

these reactions have many of the advantages common to most heavy-ion

reactions, but without some o f the disadvantages. M ore specifically, these

qualities may be enumerated as follows:

1) The heavier lithium ions can carry more angular momentum into

the target nucleus than protons or alphas, and thus excite states of

higher spin,

2) M ore than a single nucleon may be transferred, thus enabling the

excitation o f multiparticle-multihole states. In addition, the w e ll-

documented clustering in the lithium ions favors certain cluster

transfers.

9

3) The small binding energy (of single nucleons or clusters) in the

lithium isotopes results in large positive Q-values fo r most transfer

reactions.

4) The light reaction products are usually the easily identifiable

isotopes of hydrogen or helium rather than heavier nuclides. In

addition, these particles do not have low -lying states that can com­

plicate a spectrum via mutual excitation, and w ill not decay in flight.

5) The incoming lithium ions w ill suffer less target energy loss

than heavier ion beams.

Much o f the research on lithium-induced reaction has been concerned

with the four-nucleon transfer reactions, although some work has been done

on one-, two-, and three-nucleon transfers (cf. Ba71b). Because o f the strong

binding of the alpha particle, a -clustering should be most pronounced, and

a-tran sfers easiest to observe. Both theoretical cluster model predictions

and experiments ve r ify this supposition (cf. part A above, and e .g . Ta61).

Some cf the experiments that show the strong alpha-cluster nature of

the lithium-isotopes are:

1) d irect ( a, Y) capture experiments (To61),

2) transfer reaction between pairs of lithium ions at low energies

(e .g . Mo61, Hu63, Ki66) and

3) observation o f the dissociation of lithium ions at high energy, into

( a+ d) and (a + t), respectively (0163, 0164).

The study o f alpha-transfer reactions on light nuclei, using tandem

accelerators has been carried out at several laboratories, principally: the

University o f Heidelberg (e .g . Be67, Me68, Pu70), the University of Penn­

sylvania (e .g . Mi68a, Sc69, Be70a, Fo71), C .E .N . S a c la y (e .g . Cu72> and

10

the University o f Rochester (e .g . Pa72, Go73). In addition, the cyclotron at

the Kurchatov Institute in Moscow has been able to produce lithium-ion

beams of comparable or higher energies since 1966 (Va69, Og72) that have

been used fo r a variety of transfer reaction studies (e .g . Ch67, Da68, Go69).

The qualitative results o f these experiments indicate that the ( L i, t)0

and ( L i, d) reactions are both highly selective, favoring the population of

natural-parity states with alpha-cluster structure. This seems to indicate

that these reactions proceed by a d irect mechanism which should allow their use

in extracting specific nuclear structure information about the final states. The

6 7( L i, d) reactions are somewhat less selective than the ( L i, t) reactions, as

measured by the relative population of unnatural-parity states, such as the

8.8717 MeV (2 ) level in oxygen-16. Such states cannot be populated

by a d irect one-step transfer. For this reason, we have concentrated our7

efforts on the ( L i, t) reactions.

The techniques fo r the analysis o f the results o f such reactions to

enable the extraction of spectroscopic information have improved greatly during

the last several years, and these w ill be treated in Chapter III. Several

authors, using techniques of varying sim plicity have used lithium-induced

alpha-transfer reaction to determine relative reduced widths (Ne70, Pu70).

There is, however, much insight into nuclear structure that can be gleaned from

these experiments, without resort to a particular model fo r the reaction

mechanism. Bassani (Ba71b) categorized two methods fo r doing so as " lo n g ­

itudinal" and "transverse" comparisons. In the firs t instance, one compares

the results of the ( L i, t) reaction,for example, on two neighboring nuclei

to extract parentage and structure information. Middleton (Mi69) has

successfully applied this technique to study states in fluorine-19 which are

11

based on the neon-20 ground-state rotational band, coupled to a P- /2 Pr°t°n

hole. In the second method, several d ifferent transfer reactions leading to

the same residual nucleus are compared. This allows discrimation among

the different m ultiparticle-hole configurations fo r excited states. Examples

o f this technique are the study of oxygen-16 via two-, three-, and four-

18 7 22nucleon transfers at Saclay (Ba71a), and the comparison of the 0 ( L i, t) Ne

21 22and Ne(d,p) Ne reactions at the University o f Pennsylvania (Sc69).

Recently, angular correlations from reactions induced by lithium ions

have been measured at several laboratories, with interesting results. A

12 6 16 12 group at the Kurchatov Institute has used the C( L i, d) O (a ) C reaction

to determine spins and parity assignments o f highly-excited states in oxygen-16

(Ar71). The state at 20. 8 MeV, previously thought to be the 8+ member o f the

6.05 MeV 0+ band (Co70), was given the assignment 7 . The 15.37 MeV

state in neon-20 has been given a 7~ assignment by a group at Saclay, using

the 2®0(^Li, t)29N e * (a )*80 (gs ) reaction (Vo73), and the 1972 Jahresbericht

o f the University o f Heidelberg reports several spin and parity determinations

using the lithium-induced reaction correlations ( Ca72).Balamuth (Ba71) has

measured the 180 (^ L i, t y )29Ne and the 480 (8L i, d y )29Ne correlations for

the y -ray transition 1.63(2+) -+■ g. s. in the Litherland-Ferguson Method II

geom etry (L i61). These correlations indicate m=0 substate populations of about

60% and 46% fo r the respective reactions. I f the reaction mechanism w ere

solely d irect alpha-transfer, these quantities should both be unity. Thus,

while the assumption of a d irect nature for the mechanism is substantially

correct there is evidence fo r compound nucleus formation, a two-step

process, a spin-orbit interaction, or some combination o f these.

Since it is believed that the compound nuclear contribution w ill diminish

12

with increasing beam energy, it was fe lt desirable to perform these reactions

with the lithium beam recently obtained at the Yale M P tandem accelerator.

While the experiments at Kurchatov have been perform ed at roughly comparable

energies, the poor resolution of those experiments makes them less useful

than those using a beam from an M P tandem accelerator. In the course of

these experiments we w ill attempt to elucidate the mechanism o f these

reactions, and, once having done so, to use the reaction model calculations

to extract spectroscopic information. Particu lar attention w ill be paid to

those states that are describable as an alpha particle cluster orbiting a core

nucleus. In addition, there are several reduced widths, essential to

astrophysical model calculations, that we hope to extract.

13

II. E X P E R IM E N T A L M E T H O D A N D D A T A R E D U C T IO N

A. Reactions

The reactions studied in this research were:

Q = 4.6939 MeV

Q = 1.9487 MeV

Q = 2.2623 MeV

Q = 6.8492 MeV

In addition, the elastic and inelastic scattering of lithium-7 ions

from targets of carbon-12, nitrogen-14 and neon-20 were also studied in

order to determine optical model parameters.

B . . Ion Sources, Accelerators and Scattering Chambers

the M P-1 Tandem van de Graaff accelerator of the A .W . Wright Nuclear

Structure Laboratory at Yale University. The negative ions fo r injection into the

accelerator w ere produced by a Penning-type direct extraction diode source

designed by Heinicke (He68). Using this source, a typical beam current of

source proved to be unreliable in the production of L i ions, due to contam­

ination of other elements which, made it difficult to control the rate of lithium

evaporation. The remainder o f the data were obtained using the M P-7 Tandem

at Brookhaven National Laboratory, which employs a duoplasmatron knockout

source (General Ionex Corporation) to produce negative lithium ions (M168).

The prelim inary phases of this research were carried out using

7 3+50-100 nA o f 38-40 MeV L i ions could be maintained on target. This

With this source, currents o f L i in excess o f 300 nA could be maintained

on target, at beam energies of 36-38 MeV.

A t both fac ilities, the reactions were measured in 30"-diam eter

Ortec scattering chambers that perm it two detector arrays to be moved in­

dependently and positioned to within to. 1°. The beam entering the chamber

was defined by a pair o f tantalum collim ators 1/16" in diameter, placed 50"

and 20" from the center of the chamber, defining a maximum beam diameter

o f 0.146" at a target placed at the center o f the chamber.

A fter passing through the chamber, the beam was stopped in a mag­

netically-shielded Faraday cup. The charge collected was measured by an

electronic beam current integrator (BCI). The accuracy o f the integration

has been checked and the results are consistent to within <1%.

C. Targets

The firs t targets studied were self-supporting carbon fo ils2

(Yissum Corporation, Jerusalem ), nominally 75 to 100 pg/cm thick. The

targets were mounted either normal or at 45° to the beam. The beam energy

7 3+loss in the target was between 45 and 85 keV fo r 38 MeV L i ions.

A gas ce ll was used fo r the nitrogen, oxygen and neon targets. This

20 7 24method was required fo r the Ne( L i, t) Mg experiment which was per­

formed to complement the 24M g(3He, ^Be)2GNe reaction studies o f Pisano

(Pi73). The gas ce ll also has advantages over fo il targets for oxygen and

nitrogen. One avoids the problems of target evaporation (as when melamine

targets are used) and contaminant peaks from other elements in the target.

In designing a gas cell, one must balance two opposing factors: a

high pressure is desirable to increase count rate, and thin windows are

14

7 3+

15

needed fo r good energy resolution. Since the lithium ions w ill experience

greater straggling than the tritons, 20 pin nickel foils (Chromium Corp­

oration of Am erica ) were used for the entrance window and 40y in foils

fo r the exit windows. To minim ize the area o f thin fo il necessary and thus

reduce the probability of leaks, a ce ll used previously at Yale (Fr72) was

modified by reducing the entrance window diameter to 1/4" (see F ig. 1 ). In

addition, one of the two posts near the beam exit point was removed to

facilitate measurements at small angles. Provision was also made to mount

a carbon fo il on top of the ce ll to provide an energy calibration.

The ce ll was successfully tested at pressures of up to 1/2 atmos­

phere. However, since the beam straggling and energy loss in the gas are

the dominant factors in the total resolution, (see Section F, below) the cell

was normally operated at pressures o f about 1/3 atmosphere.

The use of a windowless ce ll was considered and rejected as unnec­

essary since the 24 keV straggling in the entrance fo il was not a significant

contribution to the resolution (see Section F, below). The low er pressures

necessitated by differential pumping would also have given an unacceptably

low count rate.

D. Detectors

Totally depleted Si(SB) detectors (Ortec) were employed in A E + E

telescopes. The thicknesses o f these detectors were chosen as follows:

1) The tritons from low -lying states had energies of up to "40 MeV

ar forward angles. The telescope had to be thick enough to stop

these particles.

Figure I I—1. The s lit geometry employed to study reactions on a gas

target, illustrating some o f the quantities in equation (2). The

inset shows a cross-section of the actual gas cell.

" O

B E A MA

. - f ' '40/i in Ni

20/i in Ni

not to scale

4 0 /i in Ni

actual size

16

2) Since those particles which stop in the "A E " detector cannot be

identified, (see below) the " AE" detector must be thin enough to

allow passage of the lowest energy tritons of interest (~10 M eV).

3) The " AE" detector must, however, be thick enough to provide

a "A E " signal which can reasonably differentiate between deuterons

and tritons.

The detector thicknesses chosen by these criter ia were typically

150-200M fo r " AE" at forward angles and 90 y at back angles. The "E "

detectors were typically 4 mm (two 2mm detectors) at forward angles and

1.5-2 mm at back angles.

The "active volume" o f the gas target, seen by each telescope, is

determined by the intersection of two cones: one being the volume struck by the

beam, and the other defined by two rectangular slits placed between the target

and detector (see Fig. 1). For such a configuration, the yield, Y, is given

by S ilverstein (Si59) as:

Y = n J ± G r _ . _ d s _' S i n e 0 dJ-L ( )

where n = total number of bombarding particles

N = number of target nuclei per unit volume

= differential cross-section , and

0q = the (laboratory) angle o f observation.

The geom etric factor (G/sin0Q) as determined by the s lit geometry,

plays the rule o f the product o f target thickness and detector solid angle

(t x dft) fo r fo il targets. For a pair of rectangular slits of half-widths b^

17

and t>2, heights ^ and > Z2), and separation h, with the second s lit

a distance Rq from the target center, Silverstein shows that, to lowest

order, fo r an infinitely narrow beam:

4b b £1 2 2

G - Goo - I T <2>o.

A typical geom etry for these experiments had the following parameters:

b j = bg = 1/32", h = 5 .9 " , Rq = 7 .9" , and Z^ = Z2 = 1/4". (Th e s lit height

is lim ited to the diameter o f the detectors.) Thus:

G = 5. 33 x 10 ® cm -sr oo

This quantity is the leading term in a series expansion:

a ( l )"da~d fi

G - G (1 J A + A —-j—— + ...) (3)o o o l d 0 ' ' '

where A. is the i ^ order correction and a ^ i s the derivative o f ^ with

respect to 0q . The factors contributing to the A 's are a ll dependent on

increasing powers of quantities such as (b0/R ) which are much less than 1.z o

Calculation of the A ’ s also includes corrections fo r a finite beam size and

Kan (Ka73) gives a correction for a diverging beam. Pbr the geometry

described above:

a — J_ c o s Qo b j _ 2 _ ( K2" - b Z 1 (4)A o 3 S in1 © , **1 8 R*- ( )

f - 8.8 x 10"5 © 15°

V - 1 .54 x 10-4 @ 9 0 °

18

The lo w est o r d e r co rrection fo r fin ite beam s iz e is given by S ilv e rste in a s :

( A ° \ b = 1 6 ["sTvFei. H r ](5)

+2.79 x 10"4 @> 15°

-1.08 x lO -5 @ 90°

F or a diverging beam, Kan (Ka73) gives the correction:

G = G (Silverstein) oo oo ' 'o 2 ,2

4 JI

-1(6)

where c and d are the aperture radii and L is the distance between apertures.

For our system (see Section B, above) c = d = 1/32", L = 30 in, so the c o r -

—6rection is equal to -1 .6x10 . Thus, we can safely ignore all of these

corrections. The angular acceptance of our s lit configuration is to . 6°.

For the carbon fo il targets a single s lit, 1/16" o r 1/8" wide and 1/4"

high was placed 5 .9" from the target center; the w ider slit was used at

back angles to compensate fo r the sm aller cross-sections. These geom etries

“ 4 I ohave solid angles of 4.48 and 8.96 x 10 sr, and angular acceptances of t0 .3

and to. 6°, respectively.

E. Electronics

A typical configuration o f the electronics fo r this experiment is

shown in Figure 2. Partic le identification was achieved through the use of

a pulse m ultiplier designed by Radeka (Ra63). This design makes use o f the

Figure n-2. Partic le identification electronics. Key

BCI : beam current integrator

G+D : gate and delay generator

SCA : single channel analyzer

ADC : analog-to-digital converter

MCA : multi-channel analyzer

ELECTRONICS FOR PARTICLE ID E N T IF IC A T IO N

LINEARAMPAE.El

E l [ > - •

E2 0 -

LINEAR— • SUMAMP -•AMPLINEARAMP

IDENT10. BIASEDAMP SCA 6 + 0B2B3<B4«

B.C«D<

DELAY BIASEDEN LINEARAMP AMP GATEIN'

COINC— • G + D

2*3«SUMAMP

coTS?

AOC 4 ROUTEROUTELIN.

COIN,LINADC I LINEAR

MCA

I D E N T I F I E R O U T P U T E - AE ♦ E'10-AE ME0+E'+K*AE) E0lK ARE VARIABLES

- »0

19

Bethe-Block equation (Be32, B133):

(?)

where 0= v/c and C£ and are absorber constants. Thus, fo r a fixed

detector configuration:

dependent o f energy, fo r a particular species o f ions. A single-channel

analyzer (SCA) can then be easily set to select only those signals corresponding

to the particle species of interest. In this case, the output o f the SCA was

used to open a linear gate and allow the total energy signal, AE + E, to pass

into the multichannel analyzer (M CA).

As a check on the analyzer dead tim e, the BCI pulses were routed

into an unused quadrant o f the MCA. Comparison o f the counts in the

analyzer with those of a scalar indicated that such losses were always £ 3%,

and this factor was included in the calculation o f the cross-sections.

M Z2 = AE • E • f(E ) (8)

The pulse multiplier approximates this function with:

M Z2 a AE * (E + Eq + K . A E) (9)

where K and Eq are adjustable parameters. By careful adjustment of K and

Eq, the quantity on the right side of equation 9 can be made a constant, in -

20

The various contributions to the broadening o f the observed peaks

can best be understood by reference to a particular case. Consider, fo r

14 7 18example, the N( L i, t) F* reaction at 36 M eV, leading to the 1.701 MeV

state in 18F. For sim plicity, a laboratory angle o f 90° w ill be assumed.

The sources of line broadening are:

1) Intrinsic beam spread. For a model M P tandem accelerator, the

-4beam resolution AJ2/E has been measured as 1.6 x 10 fo r 14 MeV

protons. Using this figure the intrinsic beam spread is on the order of

6 keV which can be ignored since it is quite small compared to other

resolution effects, (c f . Le69).

2) Beam straggling in the entrance fo il. The Bohr straggling formula

(B ol5 , Co66) gives the full width at half-maximum in the cen ter-o f-

mass (c .m .) system,H, as:

q2 = 32 tt( to 2 )z2e4 NZ • A x (10)

where ze is the charge on pro jectile , N is the number o f nuclei per

cm o f the target, Z is the target atomic number and A x is the target

thickness in cm.

Applying this formula to the entrance fo il, 20 microinches of

nickel, one obtains n= 46 keV in the laboratory fram e, fo r the

straggling of the entrance beam. To obtain the e ffect of this energy

spread on the observed tritons, one differentiates the reaction kine­

matics equation (Ma68) to obtain:

F . Resolution

21

a t ' — '

d E j E ,

(m4 -M,)E, - (M tM 3EtE3) co&j ( M 3+ M ^ E 3- ( M l M 3E xE ^ kc o s vr

(11)

w h e r e t h e r e a c t i o n i s d e s i g n a t e d 2 ( 1 , 3 ) 4 ; ^ i s t h e l a b o r a t o r y a n g l e o f

p a r t i c l e 3 a n d i s t h e l a b o r a t o r y e n e r g y o f t h e i ^ p a r t i c l e . ( T h e

n o n r e l a t i v i s t i c f o r m u l a h a s b e e n u s e d f o r s i m p l i c i t y . ) A t 'P = 9 0 °

t h i s r e a c t i o n s i m p l i f i e s t o :

d E g M 4 - M x ( 1 2 )

dE1 Mg + M4

F o r t h i s r e a c t i o n t h e r a t i o i s . 5 2 4 .

T h u s t h e e n t r a n c e f o i l s t r a g g l i n g c o n t r i b u t e s 2 4 k e V t o t h e

t r i t o n r e s o l u t i o n .

3 ) B e a m s t r a g g l i n g i n t h e g a s . B y a n a n a l y s i s s i m i l a r t o t h e a b o v e ,

t h e b e a m s t r a g g l i n g I n t r a v e r s i n g 7 / 1 6 " o f n i t r o g e n - 1 4 a t . 2 4 1 a t m .

( a t y p i c a l v a l u e ) i s 7 5 k e V in t h e l a b o r a t o r y f r a m e . T h i s r e s u l t s i n a

3 9 k e V a d d i t i o n t o t h e t r i t o n r e s o l u t i o n .

4 ) T a r g e t t h i c k n e s s a n d k i n e m a t i c b r o a d e n i n g . A s t h e b e a m t r a v e r s e s

t h e a c t i v e t a r g e t - v o lu m e , i t l o s e s e n e r g y , a n d t h e a n g l e o f o b s e r v a t i o n

i n c r e a s e s ( s e e F i g . 1 ) . T h u s t h e s e t w o e f f e c t s , b o t h t e n d i n g t o

d e c r e a s e t r i t o n e n e r g y w i t h i n c r e a s i n g d i s t a n c e , w i l l a d d c o h e r e n t l y .

T h i s a d d i t i o n i s p e r f o r m e d i n A p p e n d ix A . T h e r e s u l t i n g c o n t r i b u t i o n ,

t h e d o m i n a n t o n e , i s 1 2 6 k e V .

5 ) T r i t o n s t r a g g l i n g i n t h e g a s a n d w in d o w . B y a n a n a l y s i s p a r a l l e l t o t h a t

i n ( 2 ) a n d ( 3 ) , a b o v e , t h e c o n t r i b u t i o n s f r o m t r i t o n s t r a g g l i n g i n t h e

g a s a n d e x i t w in d o w a r e 2 1 a n d 2 0 k e V , r e s p e c t i v e l y .

6 ) D e t e c t o r s a n d e l e c t r o n i c s . T h e r e s o l u t i o n o f t h e S i ( S B ) d e t e c t o r s

u s e d i n t h e 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 c a t i o n m o d e w e r e f o u n d b y

p u l s e r m e a s u r e m e n t s t o h a v e a r e s o l u t i o n o f a p p r o x i m a t e l y 5 0 k e V .

S i n c e a l l o f t h e s e c o n t r i b u t i o n s , e x c e p t t h o s e o f p a r a g r a p h ( 4 ) a r e

r a n d o m G a u s s i a n s p r e a d s , t h e y w i l l a d d i n q u a d r a t u r e , g i v i n g a c o m b i n e d

c o n t r i b u t i o n o f 7 5 k e V . T h e p e a k s h a p e r e s u l t i n g f r o m t h e t a r g e t e n e r g y

l o s s a n d k i n e m a t i c b r o a d e n i n g i s n o t G a u s s i a n ( s e e A p p e n d ix A ) a n d i t i s

n o t i m m e d i a t e l y c l e a r h o w t o c o m b i n e t h i s w i t h t h e o t h e r r e s u l t s . A d d in g

i n q u a d r a t u r e y i e l d s a p r e d i c t e d r e s o l u t i o n o f 1 4 7 k e V ; a l i n e a r c o m b i n a t i o n

p r e d i c t s 2 0 1 k e V . T h e o b s e r v e d p e a k w id t h o f 1 8 0 k e V l i e s b e t w e e n t h e s e

v a l u e s , i n d i c a t i n g t h a t s o m e i n t e r m e d i a t e m e t h o d o f c o m b i n a t i o n i s n e c e s s a r y .

A t o t h e r a n g l e s t h e s e v a l u e s w i l l v a r y s o m e w h a t , b u t t h e c h a n g e s

i n s o m e f a c t o r s c a n c e l c h a n g e s i n o t h e r s . T h e o b s e r v e d r e s o l u t i o n v a r i e d

f r o m 1 5 0 - 2 0 0 k e V , w i t h t h e l a r g e s t v a l u e s n e a r 4 5 ° i n t h e l a b o r a t o r y s y s t e m .

G . D a t a R e d u c t i o n

T h e t r i t o n e n e r g y s p e c t r a s t o r e d i n t h e M C A w e r e w r i t t e n o n

m a g n e t i c t a p e . P e a k a r e a s w e r e s u b s e q u e n t l y e x t r a c t e d f r o m t h e s e s p e c t r a

o n t h e W N S L I B M 3 6 0 / 4 4 c o m p u t e r , u s i n g t h e Y a l e / i B M n u c l e a r d a t a a c q u i s ­

i t i o n s y s t e m ( G e 6 7 ) . T h e R E P L A Y m o d e o f t h i s s y s t e m p r o v i d e d a n i n t e r a c t i v e

f r a m e w o r k f o r t h e s p e c t r a a n a l y s i s t h r o u g h t h e u s e o f a f u n c t i o n k e y b o a r d

a n d t h e C R T - l i g h t p e n c o m b i n a t i o n .

E a c h p e a k , o r g r o u p o f p e a k s , w a s f i t t o t h e s u m o f u p t o f i v e g a u s s i a n

c u r v e s a d d e d t o a p o l y n o m i a l b a c k g r o u n d o f u p t o f i f t h o r d e r . A n a d d i t i o n a l

o p t i o n a l l o w e d t h e u s e o f s u c h a p o l y n o m i a l f i t t e d t o t h e l o g a r i t h m o f t h e

22

23

b a c k g r o u n d . T h e l e a s t - s q u a r e s m i n i m i z a t i o n p r o g r a m , C U R F I T , b y

B e v i n g t o n ( B e 6 9 ) p e r f o r m e d t h e s e a r c h f r o m i n i t i a l p a r a m e t e r s c h o s e n w i t h

t h e l i g h t p e n . S i n c e s u c h a l e a s t - s q u a r e s m e t h o d u n d e r e s t i m a t e s t h e t o t a l

2p e a k a r e a b y t h e v a l u e X , t h i s q u a n t i t y w a s a d d e d t o o u r f i t t e d r e s u l t s i n

2p r o p o r t i o n t o i n d i v i d u a l p e a k a r e a s . T o m i n i m i z e X , t h e b a c k g r o u n d w a s

d e t e r m i n e d s e p a r a t e l y w h e n e v e r p o s s i b l e u s i n g a d j a c e n t r e g i o n s o n b o t h

s i d e s o f t h e p e a k s , t h u s r e d u c i n g t h e n u m b e r o f f r e e p a r a m e t e r s i n t h e f i t .

T h i s t e c h n i q u e w a s p a r t i c u l a r l y u s e f u l i n r e m o v i n g t h e b r o a d b a c k g r o u n d

a t h i g h e x c i t a t i o n d u e t o l i t h i u m b r e a k u p .

1 6I n s e v e r a l c a s e s , n o t a b l y t h e d o u b l e t s n e a r 6 M e V a n d 7 M e V i n O ,

t w o p e a k s w h i c h w e r e n o t c o m p l e t e l y r e s o l v e d h a d i n t r i n s i c w i d t h s m u c h

s m a l l e r t h a n t h e e x p e r i m e n t a l r e s o l u t i o n . I n s u c h c a s e s , i t w a s p o s s i b l e t o

i m p r o v e t h e f i t s , a n d t h e c o n s i s t e n c y o f t h e a r e a d e t e r m i n a t i o n , b y r e q u i r i n g

t h e p e a k w i d t h s t o b e e q u a l . T h i s t e c h n i q u e w a s n o t u s e f u l f o r t h e b r o a d s t a t e s

a t h i g h e r e x c i t a t i o n i n m o s t s p e c t r a , s i n c e t h e s e u n k n o w n w i d t h s a r e n o t

n e g l i g i b l e c o m p a r e d t o t h e e x p e r i m e n t a l w i d t h .

O n c e t h e y i e l d s w e r e e x t r a c t e d f r o m t h e s p e c t r a , t h e a n g u l a r d i s ­

t r i b u t i o n s w e r e c a l c u l a t e d b y f o l d i n g i n t h e a p p r o p r i a t e g e o m e t r i c , k i n e m a t i c

a n d t a r g e t t h i c k n e s s f a c t o r s .

24

7T h e s i m p l e s t m o d e l f o r t h e ( L i , t ) r e a c t i o n i s t h a t o f a d i r e c t " a l p h a

p a r t i c l e s t r i p p i n g " p r o c e s s . E v e n i f s u c h a s i m p l e m o d e l w e r e c o r r e c t , o n e

c o u l d n o t e m p l o y s u c h " s t a n d a r d " D W B A c o d e s a s D W U C K ( K u 6 9 ) o r J U L I E ( B a 6 2 )

t o a n a l y z e t h e t r i t o n a n g u l a r d i s t r i b u t i o n s , b e c a u s e o f t h e £ = 1 r e l a t i v e

m o t i o n o f t h e a l p h a a n d t r i t o n i n l i t h i u m - 7 , w h i c h n e g a t e s t h e a s s u m p t i o n

o f a z e r o - r a n g e s t r i p p i n g i n t e r a c t i o n e m p l o y e d i n s u c h p r o g r a m s .

S u c h c o d e s a r e a l s o u n s u i t a b l e f o r t h e f o l l o w i n g r e a s o n s :

1 ) T h e n o - r e c o i l a p p r o x i m a t i o n b e c o m e s a p o o r o n e w h e n a n a l p h a

p a r t i c l e i s t r a n s f e r r e d o n t o a l i g h t n u c l e u s . D o d d a n d G r e i d e r ( D o 6 9 )

h a v e s h o w n t h a t , a t h i g h e n e r g y , t h e s h a p e o f t h e a n g u l a r d i s t r i b u t i o n

i s d e t e r m i n e d e n t i r e l y b y r e c o i l e f f e c t s i n d e p e n d e n t o f a n y n u c l e a r

s p e c t r o s c o p y .

2 ) I t i s p o s s i b l e t h a t h i g h c r - o r d e r p r o c e s s e s m a y b e s i g n i f i c a n t . T h i s

i s s u g g e s t e d b y t h e f a c t t h a t s o m e s t a t e s , w i t h s m a l l r e d u c e d w i d t h s

f o r a d i r e c t t r a n s f e r , a r c p o p u la t e d w i t h a n o m a l o u s s t r e n g t h i n t h e

7( L i , t ) r e a c t i o n . ( F o 7 3 ) . S u c h e f f e c t s c a n b e p r e d i c t e d b y m a k i n g u s e

o f t h e c o u p l e d c h a n n e l s t e c h n i q u e ( A s 6 9 , T a 6 9 a n d r e f e r e n c e s t h e r e i n ) .

T h i s c h a p t e r d i s c u s s e s s e v e r a l m e t h o d s w h i c h c a n b e u s e d t o o v e r -

7c o m e t h e s e p r o b l e m s i n t h e a n a l y s i s o f t h e ( L i , t ) r e a c t i o n s .

A . P l a n e - W a v e B o m A p p r o x i m a t i o n

O n e o f t h e s i m p l e s t w a y s t o i n c l u d e t h e f i n i t e e x t e n t o f t h e

l i t h i u m - 7 i o n a n d t h e r e c o i l o f t h e r e s i d u a l n u c l e u s , i n t h e a n a l y s i s o f t h e

IE. THEORY OF THE REACTION MECHANISM

( L i , t ) r e a c t i o n s i n v o l v e s t h e u s e o f t h e p l a n e - w a v e B o m a p p r o x i m a t i o n

( P W B A ) . I t c a n b e s h o w n (G 1 G 3 ) t h a t , a s i d e f r o m a g e o m e t r i c c o e f f i c i e n t , t h e

s t r i p p i n g a m p l i t u d e , T , f o r a g i v e n v a l u e o f t h e a n g u l a r m o m e n t u m t r a n s f e r ,

L , i s t h e p r o d u c t o f tw o f a c t o r s . T h e f i r s t , 6 , r e p r e s e n t s t h e o v e r l a p o fXj

t h e f i n a l s t a t e w a v e f u n c t i o n w i t h t h a t o f a n a l p h a p a r t i c l e p l u s c o r e r e p r e s e n ­

t a t i o n . I t i s p r o p o r t i o n a l t o t h e r e d u c e d a l p h a w id t h o f t h e s t a t e . T h e s e c o n d ,

MB , i s t h e p r o b a b i l i t y o f t h e a b s o r p t i o n b y t h e t a r g e t o f a n a l p h a p a r t i c l e w i t h

s p e c i f i c l i n e a r a n d a n g u l a r m o m e n t a .

T h i s s e c o n d f a c t o r i s , i n g e n e r a l , a s i x - d i m e n s i o n a l i n t e g r a l o v e r t h e

e n t r a n c e a n d e x i t c h a n n e l s c a t t e r i n g r a d i i , a n d i s r a t h e r d i f f i c u l t t o e v a l u a t e i n

t h e f i n i t e r a n g e s c h e m e ( c f . A u 6 4 ) . T h e g e n i u s o f t h e P W B A l i e s i n t h e f a c t

Mt h a t , s i n c e t h e s c a t t e r i n g w a v e f u n c t i o n s a r e s i m p l e e x p o n e n t i a l s , B f a c t o r s1j

i n t o tw o t h r e e - d i m e n s i o n a l i n t e g r a l s . O n e o f t h e s e i n t e g r a l s , o r v e r t e x

f u n c t i o n s , r e p r e s e n t s t h e e n t r a n c e c h a n n e l a n d t h e o t h e r r e p r e s e n t s t h e e x i t

c h a n n e l ( G 1 6 3 ) .

7T h e f i r s t t o a p p l y t h i s m e t h o d to ( L i , t ) r e a c t i o n s , i n a s e m i - e m p i r i c a l

f a s h i o n , w e r e D a v y d o v a n d P a v l i c h e n k o v ( D a 6 9 ) w h o o b t a i n e d t h e e x p r e s s i o n :

4 ^ * - 4 s- g ( K ) (1)d-TL

25

7

w h e r e

F L W = i L W ' i L - l M i M W <2)

2 2I n t h i s e q u a t i o n R i s t h e m a t c h i n g r a d i u s , H k . / 2 m . = E ., a n d j- ^ ( x ) i s a

s p h e r i c a l B e s s e l f u n c t i o n o f o r d e r L . I n t h e i r a n a l y s i s , t h e l i t h i u m v e r t e x

f u n c t i o n g ( k ) w a s n o t c a l c u l a t e d a p r i o r i a s a f u n c t i o n o f t h e a - t r e l a t i v e

26

m o m e n t u m , k . I n s t e a d a n e m p i r i c a l v a l u e w a s o b t a i n e d b y d i v i d i n g t h e o b s e r v e d

c r o s s - s e c t i o n s b y t h e o t h e r f a c t o r s i n e q u a t i o n 1 . T h e f a c t t h a t n e a r l y a l l t h e

c a l c u l a t e d p o i n t s f r o m s e v e r a l r e a c t i o n s l a y o n a s i n g l e s m o o t h c u r v e f o r g ( k )

v s . k i n d i c a t e s t h a t t h e m o d e l m a k e s a v a l i d r e p r e s e n t a t i o n o f t h e a - t f o r m

f a c t o r i n l i t h i u m - 7 . S u c h w a s n o t t h e c a s e w h e n a s i m i l a r a n a l y s i s o f t h e

0( L i , d ) r e a c t i o n s w a s p e r f o r m e d .

A m o r e e x p l i c i t e x p r e s s i o n f o r t h e P W B A c r o s s - s e c t i o n w a s g i v e n

b y N e o g y e t a l . ( N e 7 0 ) a s :

i s t h e s i n g l e - p a r t i c l e r e d u c e d w i d t h . O t h e r t e r m s i n t h i s e x p r e s s i o n a r e m * ,

w h e r e 9 2 ( L ) E B l 9 o ( L ) (4)

- a n d02 T 1 d3 2 _ e0<L> “ 3 R “ l (K) (5)

w h i c h i s a r e d u c e d m a s s ; «T a n d J . t h e f i n a l - a n d i n i t i a l - s t a t e s p i n s . T h e tw of i

v e r t e x f u n c t i o n s a r e :

P ( K ) = 4 T r / ° ° jL ( K r ) <J> L i ( r ) r 2 d r (6)

a n d

WI > R) = R l R ' (7)

j a n d h ^ 1 * a r e a s p h e r i c a l B e s s e l f u n c t i o n a n d a s p h e r i c a l H a n k e l f u n c t i o n J-j L

o f t h e f i r s t k i n d , r e s p e c t i v e l y . T h e m o m e n t u m a n d r a d i u s v e c t o r s a r e

27

defined in Figure 1, and

h 2 t 2 / 2 m * = Ba ia (8)

w h e r e B ^ i s t h e a s e p a r a t i o n e n e r g y f o r a n a l p h a p a r t i c l e i n t h e f i n a l s t a t e .

T h e e x p r e s s i o n f o r P ( K ) w a s o b t a i n e d i n c l o s e d f o r m , u s i n g t h e f o l l o w i n g

e x p r e s s i o n f o r i n t e r n a l a - t w a v e f u n c t i o n o f l i t h i u m - 7 ( N o 7 0 ) :

o f t h e a - p a r t i c l e i n l i t h i u m - 7 . N i s a n o r m a l i z a t i o n c o n s t a n t , a n d t h e r a n g e ,

R o , i s a d ju s t e d t o r e p r o d u c e t h e l i t h i u m - 7 r m s r a d i u s . I n a d d i t i o n , t h e

d i s t o r t i o n d u e t o C o u l o m b e f f e c t s m a y b e i n c l u d e d t o f i r s t o r d e r b y u s i n g

t h e W K B c o u n t e r p a r t o f t h e m o m e n t a ( A u 6 4 ) , e . g . :

r a d i u s R . A s i m i l a r e x p r e s s i o n i s u s e d f o r K £ .

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

t r i b u t i o n s a n d h a v e m e t w i t h s o m e l i m i t e d s u c c e s s ( s e e F i g . 2 ) . H o w e v e r ,

t h e e x t r e m e s e n s i t i v i t y o f t h e s e r e s u l t s t o t h e p a r a m e t e r s c h o s e n , a n d , in

p a r t i c u l a r , t o t h e c u t o f f r a d i u s , R , m a k e t h i s m e t h o d h i g h l y u n r e l i a b l e f o r

t h e e x t r a c t i o n o f s p e c t r o s c o p i c i n f o r m a t i o n .

4

2 2w h e r e h a / 2 m * = B , m * a n d B a r e t h e r e d u c e d m a s s a n d b i n d i n g e n e r g y

(10)

w h e r e V . ( R ) i s t h e C o u l o m b e n e r g y i n t h e e n t r a n c e c h a n n e l , a t t h e m a t c h i n g

7F i g u r e i n —1 . V e c t o r s f o r P W R A a n a l y s i s o f t h e r e a c t i o n i ( L i , t ) f .

T h e r a d i i a r e s h o w n i n t h e u p p e r f i g u r e ; t h e m o m e n t a i n t h e

l o w e r .

r e a c t i o n ; T h e p a r a m e t e r s f o r t h i s f i t a r e : R = 8 . 0 f m

R = 0 . 5 5 f m , L = 2 a n d t h e n o r m a l i z a t i o n N = 2 9 7 . T l o

c h o s e n v a l u e f o r R r e s u l t s i n a n r . m . s . r a d i u s o

o f 2 . 6 0 f m f o r l i t h i u m - 7 .

Figure III-2. A Typical P W B A fit for the 12C(7Li, t)160* (6

l2C(7Li,t)l60(6.92,2+)

Ey=38MeV

a

28

T h e e x p r e s s i o n f o r t h e r e a c t i o n c r o s s - s e c t i o n i n t h e f i n i t e - r a n g e

d i s t o r t e d w a v e B o m a p p r o x i m a t i o n ( D W B A ) , i n c l u d i n g r e c o i l , w a s d e r i v e d

b y A u s t e m e t a l . in 1 9 6 4 (A u 6 4 ) . T h e p r o b l e m i s s o c o m p l e x , h o w e v e r ,

t h a t i t h a s t a k e n s o m e t i m e f o r t h e c o m p u t e r h a r d w a r e a n d s o f t w a r e t o

d e v e l o p t o t h e p o i n t t h a t o n e c a n u s e t h i s f o r m a l i s m f o r p r a c t i c a l c a l c u l a t i o n s

( e . g . D e 7 3 ) . I n t h e i n t e r i m , v a r i o u s s i m p l i f y i n g a p p r o x i m a t i o n s h a v e b e e n

t r i e d .

O n e s u c c e s s f u l c o m p r o m i s e h a s b e e n t h e f i x e d - r a n g e D W B A , e m p l o y e d b y

1 2 7P u h l h o f e r e t a l . ( P u 7 0 ) t o a n a l y z e t h e C ( L i , t ) r e a c t i o n . I n t h i s a p p r o a c h ,

tw o a s s u m p t i o n s a r e m a d e :

1 ) T h e r e a c t i o n o c c u r s o n l y o n t h e n u c l e a r s u r f a c e , a n d

122 ) T h e t h r e e p a r t i c l e s ( a , t , C ) r e m a i n c o l i n e a r d u r i n g t h e t r a n s f e r .

T h e f i r s t a p p r o x i m a t i o n i s t a k e n t o b e a c o n s e q u e n c e o f t h e s t r o n g a b s o r p t i o n o f

b o t h t h e l i t h i u m - 7 a n d t r i t o n i n t h e n u c l e u s . T h e s e c o n d a s s u m p t i o n i s a c r u d e

w a y o f i g n o r i n g r e c o i l e f f e c t s . ( F o r a n a n a l y s i s o f t h e i m p l i c a t i o n s o f

a s s u m p t i o n ( 2 ) s e e D e 7 3 ) .

O n t h e b a s i s o f t h e s e t w o p r e m i s e s , t h e p r o d u c t o f t h e i n t e r n a l w a v e

f u n c t i o n o f t h e l i t h i u m i o n a n d t h e s t r i p p i n g p o t e n t i a l i s r e p l a c e d b y a 6 -

f u n c t i o n a t f i x e d a - t s e p a r a t i o n , w i t h a c o l i n e a r s p a t i a l o r i e n t a t i o n . In t h i s

Mw a y , t h e s i x - d i m e n s i o n a l i n t e g r a l , B ^ , i s r e d u c e d t o o n e o f t h r e e d i m e n s i o n s

( a s i n t h e P W B A o r t h e z e r o - r a n g e D W B A ) . S i n c e t h e a - t r e l a t i v e m o t i o n

h a s £ = 1 , m o r e t h a n o n e L - t r a n s f e r c a n o c c u r .

U s i n g t h i s t e c h n i q u e P u h l h o f e r a n d h i s c o w o r k e r s ( P u 7 0 ) c o u l d r e ­

p r o d u c e m a n y o f t h e i r m e a s u r e d a n g u l a r d i s t r i b u t i o n s a t f o r w a r d a n g l e s , f o r

B. Distorted-Wave Born Approximation

1 2 7 1 0t h e r e a c t i o n C ( L i , t ) ’o . W h i l e t h e e x p e r i m e n t a l c r o s s - s e c t i o n s b e y o n d

9 0 ° w e r e m u c h l a r g e r t h a n t h e c a l c u l a t i o n s , m u o h o f t h i s s t r e n g t h w a s

a c c o u n t e d f o r b y u s e o f a H a u s e r - F c s h b a c h c o m p o u n d n u c l e a r c a l c u l a t i o n .

K u b o a n d H i r a t a ( K u 7 2 ) h a v e a t t e m p t e d t o e x p a n d u p o n t h e r e s u l t s

o f t h e f i x e d - r a n g e D W B A . W h i l e r e q u i r i n g t h e c o l i n e a r a s s u m p t i o n , a s

a b o v e , t h e y a l l o w t h e a - t s p e a r a t i o n t o v a r y a l o n g t h a t l i n e , w i t h a G a u s s i a n

p r o b a b i l i t y d i s t r i b u t i o n . T h i s h a s t h e e f f e c t ( a m o n g o t h e r s ) o f r e l a x i n g

t h e s e l e c t i o n r u l e s a n d a l l o w i n g ( f o r a 0+ t a r g e t n u c l e u s ) L - t r a n s f e r s o f

L = J j . o r L = t l , w h e n > 0 . F o r t h e f i x e d - r a n g e c a l c u l a t i o n , t h e L =

t r a n s f e r i s f o r b i d d e n . T h i s a d d e d c o m p l e x i t y a p p a r e n t l y c a u s e d K u b o a n d

H i r a t a t o e m p l o y i n i n s u f f i c i e n t n u m b e r o f p a r t i a l w a v e s , i n a n e f f o r t t o

r e d u c e c o m p u t a t i o n t i m e , t h u s c a u s i n g t h e d i f f r a c t i o n l i k e o s c i l l a t i o n s s e e n

i n t h e i r c a l c u l a t e d c u r v e s .

D e V r i e s ( D e 7 3 ) h a s r e c e n t l y d e v e l o p e d a c o d e ( L O L A ) w h i c h t r e a t s

t h e f i n i t e - r a n g e D W B A c a l c u l a t i o n f u l l y . W h e n t h e f u l l s i x - d i m e n s i o n a l

i n t e g r a l i s p e r f o r m e d , t h e t h r e e L - t r a n s f e r s m e n t i o n e d a b o v e a l l c o n t r i b u t e .

D e V r i e s h a s s h o w n t h a t i n s e v e r a l c a s e s t h e " n o n - n o r m a l " t r a n s f e r ( i . e .

L = J^ ) h a s s i g n i f i c a n t e f f e c t o n t h e a n g u l a r d i s t r i b u t i o n s . T h i s c o d e h a s n o t

7a s y e t b e e n a p p l i e d t o ( L i , t ) r e a c t i o n s .

C . C o u p l e d - C h a n n e l s B o m A p p r o x i m a t i o n

T h e u s e o f t h e c o u p l e d - c h a n n e l s f o r m a l i s m i n t h e a n a l y s i s o f t h e

7( L i , t ) d a t a w a s m o t i v a t e d b y t h e o b s e r v a t i o n t h a t c e r t a i n s t a t e s w e r e p o p ­

u l a t e d w h i c h c o u l d o n l y b e r e a c h e d v i a a t w o - s t e p i n t e r a c t i o n . I n a d d i t i o n ,

P i s a n o ( P i 7 3 ) h a d d e m o n s t r a t e d t h e i m p o r t a n c e o f i n e l a s t i c e x c i t a t i o n e f f e c t s

29

i n t h e " a l p h a - p i c k u p " r e a c t i o n ( H e , B e ) i n t h i s m a s s r e g i o n . T h e s a m e

f o r m a l i s m a n d c o m p u t e r c o d e w e r e m o d i f i e d t o t r e a t " a l p h a s t r i p p i n g " .

T h e a s s u m p t i o n s a n d f o r m u l a t i o n o f n u c l e a r r e a c t i o n s i n t h e c o u p l e d -

c h a n n e l s B o r n a p p r o x i m a t i o n ( C C B A ) h a v e b e e n p r e s e n t e d e l s e w h e r e ( A s 6 9 ,

G 1 6 9 , T a 6 9 ) s o a b r i e f o u t l i n e w i l l s u f f i c e h e r e , w i t h a d d i t i o n a l d e t a i l s o f

t h e d e r i v a t i o n s a n d p r o g r a m m i n g l e f t t o A p p e n d i c e s B a n d C .

1 . A s s u m p t i o n s o f t h e D W B A

T h e c o u p l e d - c h a n n e l s f o r m a l i s m w a s d e v e l o p e d t o r e m e d y s o m e

o f t h e i n a d e q u a c i e s o f t h e D W B A b y r e m o v i n g s o m e o f t h e s i m p l i f y i n g

a s s u m p t i o n s i m p l i c i t in t h e s t a n d a r d D W B A a n a l y s i s . T h e t h r e e p r i m a r y

a p p r o x i m a t i o n s t h a t a r e m a d e a r e ( c f . G 1 6 3 ) :

1 ) T h e r e a c t i o n p r o c e e d s v i a a d i r e c t p a r t i c l e t r a n s f e r f r o m t h e

g r o u n d - s t a t e e n t r a n c e c h a n n e l t o t h e e x i t c h a n n e l l e a v i n g t h e c o r e

( i . e . t h e t a r g e t i n a s t r i p p i n g r e a c t i o n ) i n e r t . T h i s i s i m p l i e d i n t h e

e x p a n s i o n o f t h e f i n a l s t a t e w a v e f u n c t i o n o n a b a s i s i n c l u d i n g o n l y

t h e u n e x c i t e d c o r e c o u p l e d t o t h e t r a n s f e r r e d p a r t i c l e ( o r c l u s t e r ) .

2 ) T h e o p t i c a l m o d e l w a v e f u n c t i o n s f o r e l a s t i c s c a t t e r i n g a r e g o o d

r e p r e s e n t a t i o n s o f e i g e n f u n c t i o n s o f t h e t o t a l H a m i l t o n i a n n e a r t h e

n u c l e a r s u r f a c e , w h e r e t h e t r a n s f e r i s a s s u m e d t o t a k e p l a c e . T h e s e

w a v e f u n c t i o n s a r e e m p l o y e d i n t h e D W B A .

3 ) T h e m a t r i x e l e m e n t c a u s i n g t h e t r a n s f e r i s s m a l l ( c o m p a r e d t o t h e

e l a s t i c s c a t t e r i n g , f o r e x a m p l e ) a n d m a y b e t r e a t e d t o f i r s t o r d e r .

T h i s i s t h e f i r s t - o r d e r B o r n a p p r o x i m a t i o n .

L e t u s c o n s i d e r t h e s e t h r e e a s s u m p t i o n s in t h e c a s e o f a h e a v y - i o n

30

3 7

31

r e a c t i o n . T h e t h i r d s h o u l d b e v a l i d , p a r t i c u l a r l y w h e n o n e c o n s i d e r s t h a t

a p a r t i c u l a r c h a n n e l w i l l h a v e a s m a l l a m p l i t u d e c o m p a r e d t o t h e s u m o f

a l l t h e o p e n c h a n n e l s i n s u c h a r e a c t i o n . T h e f i r s t t w o , h o w e v e r , a r e s u s p e c t .

T h e r e i s e v i d e n c e , ( F o 7 3 , S e 7 3 ) o f a n a n o m a l o u s l y s t r o n g p o p u l a t i o n o f t h e

+ 71 1 . 0 9 6 M e V ( 4 ) l e v e l i n o x y g e n - 1 6 i n t h e ( L i , t ) r e a c t i o n — m u c h s t r o n g e r

t h a n w o u ld b e p r e d i c t e d o n t h e b a s i s o f t h e r e d u c e d a l p h a - w i d t h o f t h i s

s t a t e . T h e s t a t e i s b e l i e v e d ( F o 7 3 ) t o h a v e a s t r u c t u r e b a s e d u p o n a n a l p h a

+ 12p a r t i c l e c o u p l e d t o t h e 4 . 4 3 M e V ( 2 ) l e v e l i n C , a n d h e n c e a m e c h a n i s m

i n w h i c h i n e l a s t i c s c a t t e r i n g i s f o l l o w e d b y a l p h a - t r a n s f e r c o u l d m a k e a

s i g n i f i c a n t c o n t r i b u t i o n t o t h e p o p u l a t i o n o f t h i s s t a t e .

A s c u i t t o ( A s 6 9 ) h a s a l s o s h o w n t h a t i n a d e f o r m e d n u c l e u s ( s u c h a s

+ +c a r b o n - 1 2 ) , t h e s t r o n g c o l l e c t i v e t r a n s i t i o n b e t w e e n t h e 0 a n d 2 l e v e l s

c a n c a u s e a v a r i a t i o n i n t h e w a v e f u n c t i o n i n s i d e t h e n u c l e a r s u r f a c e w h i c h

w i l l n o t m a n i f e s t i t s e l f i n e l a s t i c s c a t t e r i n g , s i n c e t h e l a t t e r i s s e n s i t i v e

o n l y t o t h e e x t e r n a l r e g i o n . T h u s , t h e s e c o n d a s s u m p t i o n m a y n o t b e

v a l i d e i t h e r .

T h u s , i t i s n e c e s s a r y t o u s e a t r e a t m e n t t h a t w i l l a l l o w i n e l a s t i c

e x c i t a t i o n s ( a n d d e - e x c i t a t i o n s ) i n b o t h t h e i n i t i a l a n d f i n a l n u c l e i , a s w e l l

a s t r a n s f e r b e t w e e n s t a t e s i n w h i c h t h e c o r e i s e x c i t e d . T h e s e v a r i o u s

e f f e c t s a r e i l l u s t r a t e d i n F i g u r e 3 . T h e p r o c e d u r e w e h a v e c h o s e n f o r

i n c l u d i n g t h e s e i n e l a s t i c p r o c e s s e s i s t h e " s o u r c e t e r m " m e t h o d o f A s c u i t t o

a n d G l e n d e n n i n g ( A s 6 9 ) .

2 . A s s u m p t i o n s o f t h e C C B A

T h e C C B A a n a l y s i s r e t a i n s t h e f i r s t - o r d e r B o r n a p p r o x i m a t i o n ,

w h i l e r e m o v i n g t h e t w o o t h e r s o f t h e p r e v i o u s s e c t i o n . T h e r e a r e s t i l l

F i g u r e n i - 3 . C o m p a r i s o n o f t h e a l l o w e d r o u t e s f o r t h e D W B A a n d

C C B A a n a l y s e s . T h e s e r p e n t i n e a r r o w s r e p r e s e n t i n e l a s t i c

e x c i t a t i o n s a n d d e - e x c i t a t i o n s , w h i c h a r e t r e a t e d e x a c t l y . T h e

s t r a i g h t a r r o w s r e p r e s e n t t r a n s f e r p r o c e s s e s , t r e a t e d t o f i r s t

o r d e r .

J3'

J r

TARGET (A ) RESIDUAL NUCLEUS (B)

DWBA

J3

J2

Ji

TARGET (A ) RESIDUAL NUCLEUS (B)CCBA

s e v e r a l i m p l i c i t a s s u m p t i o n s i n t h i s t r e a t m e n t .

7F i r s t , w e t r e a t t h e ( L i , t ) r e a c t i o n a s a s t r i p p i n g r e a c t i o n , a s s u m i n g

t h a t b o t h t h e l i t h i u m - 7 g r o u n d s t a t e a n d t h e f i n a l s t a t e s o f t h e r e s i d u a l n u c l e u s

h a v e a s t r u c t u r e w h i c h c a n b e r e p r e s e n t e d b y a f o u r - n u c l e o n c l u s t e r c o u p l e d

t o a t r i t o n o r t o s o m e s t a t e o f t h e t a r g e t n u c l e u s , r e s p e c t i v e l y . W e f u r t h e r

a s s u m e t h a t t h e t r a n s f e r r e d c l u s t e r h a s b o t h S = 0 a n d T = 0 ( i . e . i t i s a n

a l p h a p a r t i c l e ) a n d t h a t t h e c l u s t e r d o e s n o t c h a n g e i t s i n t e r n a l s t r u c t u r e

d u r i n g t h e r e a c t i o n . ( T h e i n t e r n a l s t r u c t u r e i s i g n o r e d . ) T h e s e c o n d i t i o n s

a r e i m p o s e d a p r i o r i ; t h e i r j u s t i f i c a t i o n l i e s i n t h e a b i l i t y o f t h e c a l c u l a t i o n

t o r e p r o d u c e t h e e x p e r i m e n t a l r e s u l t s .

3 . O u t l i n e o f t h e S o u r c e T e r m M e t h o d

T h e s o u r c e t e r m m e t h o d r e p r e s e n t s a c o n v e n i e n t m e a n s o f

d i v i d i n g a n e x t r e m e l y c o m p l e x a n d c u m b e r s o m e n u m e r i c a l p r o b l e m i n t o s e g ­

m e n t s t h a t a r e b o t h p h y s i c a l l y m e a n i n g f u l a n d m o r e e a s i l y m a n i p u l a t e d o n

a c o m p u t e r . S c h e m a t i c a l l y , t h e m e t h o d i n v o l v e s t h r e e s t e p s : 1 ) s o l u t i o n

o f t h e e n t r a n c e c h a n n e l s c a t t e r i n g p r o b l e m ; 2 ) c a l c u l a t i o n o f t h e t r a n s f e r

c o n t r i b u t i o n s t o e a c h e x i t c h a n n e l ( t h e s o u r c e t e r m s ) ; a n d 3 ) s o l u t i o n o f

t h e i n h o m o g e n e o u s e x i t c h a n n e l s c a t t e r i n g e q u a t i o n s .

I f w e u s e a s i n g l e s u b s c r i p t t o r e p r e s e n t t h e u n i q u e q u a n t u m n u m b e r s

o f a g i v e n c h a n n e l a n d d e s i g n a t e t h e r e a c t i o n A ( a , b ) B , t h e e n t r a n c e s y s t e m

e q u a t i o n s m a y b e e x p r e s s e d a s :

33

w h e r e t h e u n p r i m e d s u b s c r i p t , a , i s r e s e r v e d f o r t h e e l a s t i c c h a n n e l . ( U n l e s s

e x p l i c i t l y i n d i c a t e d , h o w e v e r , s u m s o v e r a ' i n c l u d e t h e t e r m a ) . N o t e t h a t

t h e s e e q u a t i o n s d i f f e r f r o m t h o s e i n a D W B A t r e a t m e n t o f i n e l a s t i c s c a t t e r i n g ,

i n w h i c h t h e i n t e r a c t i o n i s c o n s i d e r e d o n l y t o f i r s t o r d e r , r e d u c i n g t h e r i g h t

s i d e o f t h e e q u a t i o n t o ( 6 , - 1 ) V ,u . ( c f . G 1 6 9 ) . T h e b o u n d a r y c o n d i t i o n scl cl E E E '

r e q u i r e a n i n c o m i n g d i s t o r t e d w a v e i n t h e e l a s t i c c h a n n e l ( a ) a n d o u t g o i n g

w a v e s i n a l l c h a n n e l s .

T h e e x i t c h a n n e l e q u a t i o n s a r e i n h o m o g e n e o u s , a s e a c h s t a t e i s

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

f r o m t h e e n t r a n c e s t a t e s . E x p r e s s e d a l g e b r a i c a l l y , t h i s b e c o m e s :

<Tb + v bb - Eb > »b c r,£h v b ' b ' V - £1pb <12)b ’ = b a*

H e r e t h e r e i s n o d i s t i n c t i o n f o r t h e g r o u n d s t a t e ; a l l s t a t e s h a v e o n l y o u t ­

g o i n g w a v e s . A s i n t h e e n t r a n c e c h a n n e l p r o b l e m , i s a m a t r i x e l e m e n t ,

i . e . :

V b ’ b = <tf>b | V ^ B > ^ b ^ ( 1 3 )

H e r e <f>b r e p r e s e n t s t h e a n g u l a r p a r t o f t h e b - B r e l a t i v e w a v e f u n c t i o n , c o u p l e d

t o t h e i n t e r n a l w a v e f u n c t i o n o f B . T h e t o t a l a n g u l a r m o m e n t u m ( " c h a n n e l s p i n " ) ,

i t s z - p r o j e c t i o n , a n d t h e t o t a l p a r i t y ( I , M , a n d 71, r e s p e c t i v e l y ) h a v e b e e n

s u p p r e s s e d a s i n d i c e s , s i n c e i s d i a g o n a l i n t h e s e q u a n t i t i e s ( c f . A s 6 9 ) .

T h e i n t e r a c t i o n , V ( b , ~ fe), i s p a r a m e t e r i z e d a s a n o p t i c a l p o t e n t i a l a n d e x p a n d e d

i n s p h e r i c a l h a r m o n i c s a s a f u n c t i o n o f t h e n u c l e a r a n d C o u l o m b d e f o r m a t i o n s

( c f . G 1 6 9 ) .

atT h e s o u r c e t o r m , c o n t a i n s t h e 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

r e g a r d i n g t h e p a r e n t a g e o f t h e f i n a l s t a t e s , t h e n a t u r e o f t h e " s t r i p p i n g "

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

p r o j e c t i l e s i z e a n d t h e r e c o i l — i n s h o r t , a l l o f t h e r e l e v a n t n u c l e a r p h y s i c s

b e y o n d t h e s c a t t e r i n g p r o b l e m . A s c u i t t o ( A s 6 9 ) g i v e s t h i s t e r m a s :

34

N o t e t h a t i n a c c o r d a n c e w i t h t h e B o r n a p p r o x i m a t i o n , t h e m a t r i x e l e m e n t

o f t h e s t r i p p i n g i n t e r a c t i o n i s t r e a t e d o n l y t o f i r s t o r d e r .

T h e r a d i u s v e c t o r s i n t h i s e q u a t i o n a r e d e f i n e d i n F i g u r e 4 . + 0 ( r £ )

i s t h e i n t e r n a l w a v e f u n c t i o n o f p a r t i c l e a , a n d <j> , / r , A ’ ) u , / r ) / r i s t h e r e l a t i v e

w a v e f u n c t i o n o f a a n d A . T h e d e r i v a t i o n o f t h e e x p l i c i t f o r m o f p ^ i s g i v e n

i n A p p e n d ix B . A n o u t l i n e o f t h e n u m e r i c a l m e t h o d s u s e d t o s o l v e t h e e n t i r e

p r o b l e m i s p r e s e n t e d i n A p p e n d ix C .

F i g u r e H I - 4 . R a d i u s v e c t o r s o f t h e C C B A a n a l y s i s .

35

I V . E X P E R I M E N T A L R E S U L T S

A . 1 2 C ( 7 L i , t ) 1 6 0

T h e i n i t i a l m e a s u r e m e n t s o f ( L i , t ) r e a c t i o n c r o s s - s e c t i o n s w e r e

2 12m a d e o n 1 0 0 y g / c m C f o i l s . S p e c t r a w e r e t a k e n a t 3 6 a n d 4 0 M e V a t Y a l e ,

a n d a n a n g u l a r d i s t r i b u t i o n a t 3 8 M e V w a s o b t a i n e d a t t h e B r o o k h a v e n T a n d e m .

T y p i c a l s p e c t r a a t 3 6 a n d 3 8 M e V a r e s h o w n i n F i g s . 1 a n d 2 .

T h e m o s t s t r i k i n g f e a t u r e o f t h e s e s p e c t r a i s t h e e x t r e m e s e l e c t i v i t y

o f t h e r e a c t i o n . I n p a r t i c u l a r , t h e m o s t s t r o n g l y p o p u l a t e d s t a t e s b e l o n g

7T *f +t o t h e K = 0 b a n d , b u i l t o n t h e 0 l e v e l a t 6 . 0 5 M e V a n d i n c l u d i n g t h e

s t a t e s : 6 . 9 2 ( 2 + ) , 1 0 . 3 5 3 ( 4 + ) a n d 1 6 . 2 2 ( 6 + ) . T h e s t a t e a t 2 0 . 8 M e V w a s

o r i g i n a l l y f e l t t o b e t h e 8 + m e m b e r o f t h i s b a n d ( C o 7 0 ) , b u t h a s s i n c e b e e n

s h o w n , v i a a c o r r e l a t i o n m e a s u r e m e n t t o h a v e s p i n a n d p a r i t y o f 7 ~ ( A r 7 1 ) .

T h e s t a t e n e a r 2 1 . 8 M e V n o w s e e m s t h e m o s t l i k e l y c a n d i d a t e f o r t h e 8 +

m e m b e r .

T h e s t r o n g s t a t e s a l l d i s p l a y p r o n o u n c e d f o r w a r d - p e a k i n g i n t h e i r

a n g u l a r d i s t r i b u t i o n s , a s s e e n i n F i g u r e 3 . T h e a n g u l a r d i s t r i b u t i o n s l e a d i n g

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

a b o u t 9 0 ° . ( S e e F i g u r e s 4 a n d 5 . ) A l l o f t h e d i s t r i b u t i o n s f a l l t o a r e l a t i v e l y

c o n s t a n t l e v e l o f 1 0 - 3 0 y b / s r b e y o n d a b o u t 9 0 ° i n t h e c e n t e r - o f - m a s s s y s t e m .

T h e s e f a c t o r s i n d i c a t e t h a t w h i l e t h e d i r e c t r e a c t i o n m e c h a n i s m d o m i n a t e s t h e

r e a c t i o n f o r t h o s e s t a t e s w h i c h a r e s e e n t o b e s e l e c t i v e l y p o i u l a t e d a t f o r w a r d

a n g l e s ( e . g . 1 5 ° ) , t h e r e i s a n o n - n e g l i g i b l e c o m p o u n d - n u c l e a r c o n t r i b u t i o n t o

a l l s t a t e s . T h i s h a s b e e n b o r n e o u t b y t h e a n a l y s i s ( s e e C h a p t e r V ) .

7

F i g u r e s I V - 1 a n d I V - 2 . T r i t o n s p e c t r a a t 6 = 1 5 ° f o r t h e r e a c t i o n

1 2 C ( 7 L i , t ) 2 6 0 a t 3 6 a n d 3 8 M e V . E x c e p t f o r t h e s t a t e s a t 2 1 . 8

a n d 2 3 . 2 M e V i n F i g u r e 2 , t h e e n e r g i e s , s p i n s a n d p a r i t i e s

a r e f r o m r e f e r e n c e A j 7 1 .

CHANNEL

CO

UN

TS

EXCITAT ION ENERGY

TT +F i g u r e I V - 3 . T r i t o n a n g u l a r d i s t r i b u t i o n s f o r t h e 4 p - 4 h K = 0 b a n d

i n o x y g e n - 1 6 . E r r o r b a r s r e p r e s e n t s t a t i s t i c a l e r r o r s o n l y ;

w h e r e n o t s h o w n t h e y a r e s m a l l e r t h a n t h e d a t a p o i n t s .

10,000

l2C(7Li,t)60*

Eu = 38.0 MgV

1000

1000

100

XiJ .

5■osb■o

6.23(6+)

10.353(4+)

100

+100

0 $

10

r f

-O O-

6.9188(2+)

6.0502(0+)

?

0 20 40 60 80 100 120 140 160 180

♦%

I

9cm.

F i g u r e s I V - 4 a n d I V - 5 . T r i t o n a n g u l a r d i s t r i b u t i o n s t o o t h e r s t a t e s

i n o x y g e n - 1 6 . S e e c a p t i o n t o F i g u r e 3 .

100,2C(7Li,*)l6<f Eu * 38MeV

10• •

-9-

100

£ 3

cj"O

b ■o io

10

1.0

0.1

-O-n o

<f=.

8.8717 (2“ )

-9—<1

o-o-

-0-

7.11867(1")

i t 7 "*

T

6.13066 (3") =

g.s.(0+ )

$m

_d .

0 20 40 60 80 100 120 140 160 180ftcm.

(js/qt/)

up/x)p

1000 l2C(7U,t),60* EL| = 38 MeV i

1000 20.8 (7“ )

1000 14.82 (6+) =SS c p £

100 14.39(4+) _

100

13.869(4+) Z=

10011.096(4+) =

10SjE gE »3E

9.8469(1")

0 20 40 60 80 100 120 140 160 180%m.

36

1 6 7 2 0T h e 0 ( L i , t ) N e r e a c t i o n w a s s t u d i e d a t a l a b o r a t o r y b e a m

e n e r g y o f 3 8 M e V , u s i n g t h e g a s c e l l d e s c r i b e d a b o v e ( I I . C ) . A t y p i c a l

t r i t o n s p e c t r u m a t a f o r w a r d a n g l e i s s h o w n i n F i g u r e 6 .

7T h i s s p e c t r u m a g a i n d i s p l a y s t h e s e l e c t i v i t y c h a r a c t e r i s t i c o f ( L i , t )

r e a c t i o n s . W h i l e n e o n - 2 0 h a s m a n y l o w - l y i n g b a n d s ( c f . A j 7 2 , A 1 7 2 , F o 7 3 )

o n l y t w o a r e s t r o n g l y e x c i t e d . T h e f i r s t i s t h e g r o u n d - s t a t e b a n d w i t h

K 7T= 0 + , i n c l u d i n g t h e s t a t e s a t 1 . 6 3 3 8 ( 2 + ) , 4 . 2 4 7 3 ( 4 + ) a n d 8 . 7 7 5 ( 6 + ) M e V .

A n g u l a r d i s t r i b u t i o n s f o r t h e s e s t a t e s a r e s h o w n i n F i g . 7 . T h e 8 + l e v e l

a t 1 1 . 9 5 8 M e V , b e l i e v e d t o b e a m e m b e r o f t h e g r o u n d - s t a t e b a n d , ( M i 7 2 )

i s v e r y w e a k l y p o p u l a t e d . T h e o t h e r s t r o n g s t a t e s a r e t h e m e m b e r s o f t h e

K 71 = 0 ” b a n d , n a m e l y : 5 . 7 8 5 ( 1 " ) , 7 . 1 6 6 ( 3 ~ ) , 1 0 . 2 5 7 (5 ~ ) a n d 1 5 . 3 4 ( 7 - ) M e V .

A n g u l a r d i s t r i b u t i o n f o r t h i s b a n d a r e s h o w n i n F i g u r e 8 .

A b o v e 1 6 M e V , t h e r e a r e s e v e r a l s t r o n g s t a t e s , w h o s e s t r u c t u r e i s

n o t c u r r e n t l y k n o w n . T h e s e a r e l o c a t e d a t e n e r g i e s o f 1 6 . 6 3 ± . 0 3 , 1 7 . 2 7 ± . 0 2 ,

2 0 . 5 8 i 0 4 , 2 1 . 0 0 ± . 0 2 5 a n d 2 2 . 7 3 ± . 0 3 M e V . T h e e n e r g y a s s i g n m e n t s a r e t h e

r e s u l t o f a c a l i b r a t i o n p e r f o r m e d u s i n g t h e l o c a t i o n s o f t h e s t a t e s b e l o w

1 1 M e V i n e x c i t a t i o n , a t s e v e r a l a n g l e s .

I t i s a l s o i n t e r e s t i n g t o o b s e r v e t h a t c e r t a i n b a n d s a r e c o n s p i c u o u s l y

+ 4a b s e n t . T h r e e e x c i t e d 0 b a n d s a r e m i s s i n g , n a m e l y , t h e ( s d ) b a n d b e g i n n i n g

a t 6 . 7 2 2 M e V , t h e ( I p ) - 4 ^ ) 8 b a n d a t 7 . 1 9 6 M e V a n d t h e ( s d ) 2 ( f p ) 2 b a n d

a t 8 . 3 M e V ( c f . A 1 7 2 , F o 7 3 ) . I n a d d i t i o n , t h e o d d - J m e m b e r s o f t h e

5 - 1 t t 2 4 3 7( s d ) ( lp ) , K = 2 “ b a n d , w h i c h a r e s e e n i n t h e M g ( H e , B e ) r e a c t i o n

( P i 7 3 ) a r e a l s o a b s e n t .

T h e r e h a s b e e n c o n s i d e r a b l e d i s c u s s i o n o f t h e n a t u r e o f t h e 8 + s t a t e

B. 160(7Li, t)20Ne

1 6 0 ( 7 L i , t ) 2 ® N e * a t E = 3 8 M e V . W h e r e n o u n c e r t a i n t i e sLi

a r e g i v e n t o t h e e x c i t a t i o n e n e r g i e s , t h e n u m b e r s q u o t e d a r e

t a k e n f r o m A j 7 2 . W h e r e u n c e r t a i n t i e s a r e i n d i c a t e d , e n e r g i e s

h a v e b e e n d e t e r m i n e d i n t h i s w o r k a s d e s c r i b e d i n t h e t e x t .

Figure IV-6. Triton spectrum at 9 ^ = 15° for the reaction

CO

UN

TS

F i g u r e s I V - 7 a n d I V - 8 . T r i t o n a n g u l a r d i s t r i b u t i o n s t o s t a t e s i n

n e o n - 2 0 . F i g u r e 7 , o n t h e l e f t , s h o w s s t a t e s i n t h e g r o u n d - s t a t e

0 + b a n d . F i g u r e 8 , o n t h e r i g h t , s h o w s m e m b e r s o f t h e = 0 “

b a n d . S e e c a p t i o n t o F i g u r e 3 .

1000

100

l60 (7L i, t )20Ne*

Ey=38 MeV

100

-.100£XiJL3■osb K) •o

P.SCSfZ

10

0.1

k

:c;

5 7 7 3 0 (6 *)

H -rt:

-^— 4.2473 (4+ )

1.6330 12*)

g.s. (0 *) =

IQP00

1000

1000

-100

XiX

3

■o 100

W U .t )20^

Eli = 38 MeV

100

10

*5F

= 15.342 (7")

EE

10.237 (5~)

E 7.166 (3 “ )

5785 (D

0 20 40 60 80 100 120 0 20 40 60 80 100 1209,c.m. etc.m.

a t 1 1 . 9 4 8 M e V . W h i l e t h e s t a t e h a s b e e n u n a m b i g u o u s l y g i v e n a / = 8 +

a s s i g n m e n t ( K u 6 6 ) , i t s i n c l u s i o n i n t h e g r o u n d - s t a t e b a n d h a s b e e n q u e s t i o n e d

7( e . g . V o 7 2 ) . T h e e x t r e m e l y w e a k p o p u l a t i o n o f t h i s s t a t e i n t h e ( L i , t )

r e a c t i o n , r e l a t i v e t o t h e 6 + m e m b e r , f o r e x a m p l e , w o u ld t e n d t o i n d i c a t e t h a t

t h i s s t a t e i s n o t a m e m b e r o f t h i s b a n d . A r e c e n t c a l c u l a t i o n b y V a r y ,

B u c k a n d D o v e r ( V a 7 3 ) p r e d i c t s t h a t t h e 8 + m e m b e r o f t h e g r o u n d s t a t e

b a n d w o u ld b e d e g e n e r a t e w i t h t h e 7 " s t a t e a t 1 5 . 3 4 M e V . I f t h i s w e r e t r u e ,

i t c o u l d e x p l a i n b o t h t h e l a c k o f s t r e n g t h o f t h e 1 1 . 9 5 M e V l e v e l a n d t h e l a r g e

w i d t h ( - * 4 0 0 k e V f w h m ) o f t h e 1 5 . 3 4 M e V l e v e l c o m p a r e d w i t h , s a y , t h e ~ 2 0 0

k e V o f t h e 1 0 . 2 6 M e V l e v e l .

1 4 7 1 8C . N ( L i , t ) F

1 4 7 1 8T h e N ( L i , t ) F r e a c t i o n w a s s t u d i e d a t E ^ . * 3 6 M e V ,

u s i n g t h e g a s c e l l t a r g e t . A t y p i c a l f o r w a r d a n g l e s p e c t r u m i s s h o w n i n

F i g u r e 9 .

A s i n d i c a t e d i n t h e f i g u r e , a m o n g t h e p r o m i n e n t s t a t e s b e l o w 7 M e V

TT + +e x c i t a t i o n a r e f i v e m e m b e r s o f t h e K = 1 b a n d , n a m e l y : 1 . 7 0 1 ( 1 ) ,

2 . 5 2 3 ( 2 + ) , 3 . 3 5 8 ( 3 + ) , 5 . 2 9 8 ( 4 + ) a n d 6 . 5 6 7 ( 5 + ) . T h i s b a n d h a s b e e n

r e c e n t l y s t u d i e d a t t h e U n i v e r s i t y o f T o r o n t o ( R o 7 3 ) a n d h a s b e e n s h o w n

- 2 4t o h a v e a s t r u c t u r e t h a t i s ( lp ) ( s d ) i n n a t u r e . T h e a n g u l a r d i s t r i b u t i o n s

f o r t h e s e l e v e l s a r e s h o w n i n F i g u r e 1 0 .

W h i l e n o s p i n a s s i g n m e n t s c a n b e m a d e f r o m o u r a n g u l a r d i s t r i b u t i o n s ,

i t i s i n t e r e s t i n g t o s p e c u l a t e a b o u t p o s s i b l e e x t e n s i o n o f t h i s b a n d . F i g u r e 1 1

s h o w s t h e k n o w n e n e r g y l e v e l s i n t h i s b a n d p l o t t e d a s a f u n c t i o n o f J ( J + 1 ) .

A l s o i n d i c a t e d i n t h e f i g u r e a r e t h e t w o s t r o n g e s t s t a t e s a t h i g h e r e x c i t a t i o n

( 9 . 4 7 2 ± . 0 1 a n d 1 1 . 0 7 4 ± . 0 1 M e V ) , o n t h e u n f o u n d e d b u t n o t e n t i r e l y s p u r i o u s

37

F i g u r e I V - 9 . T r i t o n s p e c t r u m a t ® ^ = 1 5 ° f o r t h e * 4 N ( 7 L i , t ) 1 8 F *

r e a c t i o n a t E _ = 3 6 M e V . T h e f i v e s t a t e s w h o s e s p i n s a r e L i

i n d i c a t e d a r e m e m b e r s o f a = 1 + b a n d ; t h e i r e n e r g i e s a r e

t a k e n f r o m r e f e r e n c e R o 7 3 . T h e o t h e r e x c i t a t i o n e n e r g i e s

i n d i c a t e d w e r e d e t e r m i n e d f r o m t h e e n e r g y c a l i b r a t i o n .

EXCITATION

IIj07

4±Q

020 ,4 N(7L i, t ) ,8F

ELi=36MeV

0|ab=l5°

ENERGY (MeV)

F i g u r e I V - 1 0 . T r i t o n a n g u l a r d i s t r i b u t i o n s t o t h e k n o w n m e m b e r s o f t h e

71 +K = 1 b a n d i n f l u o r i n e - 1 8 . S e e c a p t i o n t o F i g u r e 3 .

(js/qr/) up/.op

1 0 0 0

100

100

,4N (7L i ,t ) l8F *

E y s 3 6 MeV

100

100

10

mi ±

a r

10

t t

t

- t - i

I f

T =

6—

6 .5 6 7 (5 " )

a- i -

5 .2 9 8 (4 " )

I e N e

3 .3 5 8 (3 ")

<j>= 2.525(2")E

’: r d i =

1.701(1'")

::d :

0 20 4 0 60 80 100 120 140

Qc.m.

+ 1 8F i g u r e I V - 1 1 . E n e r g i e s o f m e m b e r s o f t h e K = 1 b a n d i n F ,

p l o t t e d a s a f u n c t i o n o f J ( J + 1 ) . A l s o s h o w n a r e t w o o t h e r

s t r o n g s t a t e s p l o t t e d o n t h e a s s u m p t i o n t h a t t h e y a r e t h e 6 +

a n d 7 + m e m b e r s o f t h e b a n d .

Ex

(MeV

)

The K7 !+ Rotational Band in , 8 F

J=l 2 3 4 5 6 7 8

a s s u m p t i o n t h a t t h e s e s t a t e s a r e t h e 6 + a n d 7 + m e m b e r s o f t h e b a n d . I n

a d d i t i o n , a l l t h e s t a t e s In t h i s b a n d d i s p l a y m a r k e d l y s i m i l a r f o r w a r d - p e a k e d

s t r u c t u r e l e s s a n g u l a r d i s t r i b u t i o n s . T h i s m a y r e s u l t f r o m t h e f a c t t h a t , d u e

t o t h e J = 1 g r o u n d s t a t e o f n i t r o g e n - 1 4 a n d £ = 1 i n t e r v a l m o t i o n i n l i t h i u m - 7 ,

u p t o 5 A - t r a n s f e r s m a y c o n t r i b u t e t o t h e r e a c t i o n ( J - 2 ^ 1 $ J + 2 ) .

H e n c e t h e c h a n g e in d i s t r i b u t i o n f r o m o n e s t a t e t o t h e n e x t w i l l n o t b e

v e r y m a r k e d .

T h i s f a c t i s i l l u s t r a t e d i n F i g u r e 1 2 , w h i c h s h o w s a c o m p o s i t e o f t h e

a n g u l a r d i s t r i b u t i o n s t o t h e f i v e k n o w n m e m b e r s o f t h i s b a n d . A s c a n b e

s e e n i n F i g u r e 1 3 , t h e s t a t e s a t 9 . 4 7 2 a n d 1 1 . 0 7 4 M e V d i s p l a y v e r y s i m i l a r

a n g u l a r d i s t r i b u t i o n s , w h e r e a s s o m e o f t h e o t h e r o b s e r v e d s t a t e s d o n o t .

H e n c e t h e r e i s a n a d d i t i o n a l i n d i c a t i o n t h a t t h e s e s t a t e s a r e m e m b e r s o f t h e

K 77 = 1 + b a n d .

_ 2 0 XT .7 _ . . . 2 4 . ,D . N e ( L i , t ) M g

2 0 7 2 4T h e e x t r a c t i o n o f t r i t o n a n g u l a r d i s t r i b u t i o n s f o r t h e N e ( L i , t ) M g

r e a c t i o n w a s h a m p e r e d b y t h e a p p a r e n t d e c r e a s e i n t h e s e l e c t i v i t y o f t h e

r e a c t i o n m e c h a n i s m ( s e e F i g u r e 1 4 ) . T h i s f a c t , c o m b i n e d w i t h t h e h i g h

d e n s i t y o f s t a t e s i n m a g n e s i u m - 2 4 m a d e r e s o l u t i o n o f n e i g h b o r i n g s t a t e s

e x t r e m e l y d i f f i c u l t . A l s o , d u e i n p a r t t o t h e m o n a t o m i c s t r u c t u r e o f n e o n

g a s , t h e s t a t i s t i c s t o t h e l o w - l y i n g s t a t e s , w h i c h w e r e r e s o l v e d , w e r e

v e r y p o o r . ( T h e g r o u n d s t a t e t y p i c a l l y h a s £ 1 0 c o u n t s i n t h e p e a k . ) T h u s ,

n o r e l i a b l e a n g u l a r d i s t r i b u t i o n s c o u l d b e e x t r a c t e d .

E . E l a s t i c S c a t t e r i n g

38

A n g u l a r d i s t r i b u t i o n s o f l i t h i u m - 7 i o n s e l a s t i c a l l y s c a t t e r e d

F i g u r e I V - 1 2 . S u p e r p o s i t i o n o f t h e a n g u l a r d i s t r i b u t i o n s o f F i g u r e 1 0

w i t h a r b i t r a r y n o r m a l i z a t i o n . T h e s h a d e d a r e a r e p r e s e n t s t h e

g e n e r a l s h a p e o f a l l t h e d i s t r i b u t i o n s .

dcr/

dXl

(AR

BIT

RA

RY

U

NIT

S)

i

^c.m.

I V - 1 3 . T r i t o n a n g u l a r d i s t r i b u t i o n s t o f i v e s t a t e s i n f l u o r i n e - 1 8 .

T h e s h a d e d a r e a s r e p r o d u c e t h e o u t l i n e o f t h e c o m p o s i t e a n g u l a r

d i s t r i b u t i o n o f t h e p r e c e d i n g f i g u r e , s u i t a b l y n o r m a l i z e d .

dcr/

dflt(

/ib/s

r)

^c.m.

F i g u r e I V - 1 4 . T r i t o n s p e c t r u m a t 0^ ^ = 1 5 ° f r o m t h e r e a c t i o n

20 y 24N e ( L i , t ) M g * . E n e r g y l e v e l s i n d i c a t e d a r e t a k e n f r o m

r e f e r e n c e E n 6 7 .

COU

NTS

CHANNEL

39

f r o m t a r g e t s o f c a r b o n - 1 2 , n i t r o g e n - 1 4 a n d n e o n - 2 0 w e r e m e a s u r e d a t 3 6 M e V .

T h e r a t i o s o f t h e s e c r o s s - s e c t i o n s t o t h e R u t h e r f o r d c r o s s - s e c t i o n a r e

s h o w n i n F i g u r e 1 5 . I n a d d i t i o n , t h e a n g u l a r d i s t r i b u t i o n o f l i t h i u m - 7 i o n s

f r o m i n e l a s t i c s c a t t e r i n g t o t h e 4 . 4 3 3 ( 2 + ) M e V l e v e l o f c a r b o n - 1 2 i s s h o w n

i n F i g u r e 1 6 .

iE x t r a c t i o n o f t h e s e c r o s s - s e c t i o n s w a s h i n d e r e d b y t h e f a c t t h a t

e a c h p e a k w a s a d o u b l e t , t h e s e c o n d c o m p o n e n t r e s u l t i n g f r o m d e t e c t i o n o f

a l i t h i u m - 7 i o n i n i t s e x c i t e d s t a t e a t 4 7 8 k e V . I n a d d i t i o n , s i n c e n o p a r t i c l e

i d e n t i f i c a t i o n w a s p e r f o r m e d , t h e r e w e r e n o a d d i t i o n a l p e a k s r e s u l t i n g f r o m

o t h e r h e a v y i o n p r o d u c t s , i n c l u d i n g l i t h i u m - 6 , b e r y l l i u m - 7 , b o r o n - 1 1 , a n d

t a r g e t r e c o i l s , a t y p i c a l s p e c t r u m i s s h o w n i n F i g u r e 1 7 . T h e o p t i c a l

m o d e l a n a l y s i s o f t h e s e a n g u l a r d i s t r i b u t i o n s i s d i s c u s s e d i n C h a p t e r V ,

b e l o w .

3 6 M e V f r o m t a r g e t s o f c a r b o n - 1 2 , n i t r o g e n - 1 4 , a n d n e o n - 2 0 ,

d iv i d e d b y t h e R u t h e r f o r d c r o s s - s e c t i o n . S e e c a p t i o n t o

F i g u r e 3 .

Figure IV-15. Lithium-elastic scattering angular distributions at

0.1,2c

C5

1s 0Jc5

1■p

14N

20,

7Li-Elastic Scattering Eu =36MeV

OJNe • •

0.01

OjOOI

= t fc

*= u

t = f

0 0 2 0 3 0 4 0 5 0 6 0 70 8 0 9 0 1 0 0ftc.m.

d i s t r i b u t i o n s a t 3 6 M e V f r o m a c a r b o n - 1 2 t a r g e t . S e e c a p t i o n

t o F i g u r e 3 .

Figure IV-16, Elastic and inelastic lithium scattering angular

do7d

ft (m

b/sr

)

10

4.433 (2+)

l2C -7Li Scattering ~ E|_j=36 MeV -

100 u

g.s. (0*)io

o.i

0.010 10 20 30 40 50 60 70 80 90 100

0c.m.

1 2 7F i g u r e I V - l 7 . U n i d e n t i f i e d s p e c t r u m o f p r o d u c t s f r o m C + L i

s c a t t e r i n g . T h e s y m b o l L i * r e p r e s e n t s l i t h i u m - 7 o b s e r v e d

i n i t s f i r s t e x c i t e d ( 4 7 8 k e V ) s t a t e . T h e l a r g e u n l a b e l l e d

p e a k r e c o r d s t h e e l a s t i c a l l y s c a t t e r e d L i - i o n s .

4000

3000 -

in

3OO

2000-

1000 -

600 640 680 720 760 800CHANNEL

840 880 920

40

A . A n a l y s i s o f t h e L i t h i u m - 7 S c a t t e r i n g D a t a -

T h e l i t h i u m - n u c l e u s o p t i c a l m o d e l p o t e n t i a l s n e c e s s a r y f o r

t h e C C B A a n a l y s i s , h a d n o t p r e v i o u s l y b e e n d e t e r m i n e d f o r t h i s e n e r g y

r e g i o n . T h e r e f o r e , t h e l i t h i u m - 7 e l a s t i c s c a t t e r i n g a n g u l a r d i s t r i b u t i o n s ,

d e s c r i b e d i n C h a p t e r I V , w e r e f i t u s i n g t h e o p t i c a l m o d e l p a r a m e t e r s e a r c h

c o d e J I B 3 ( P e 6 3 ) .

T h e p o t e n t i a l s u s e d w e r e o f W o o d s - S a x o n f o r m , w i t h s u r f a c e

i m a g i n a r y t e r m s . T h e C o u l o m b i n t e r a c t i o n w a s t h a t o f a u n i f o r m l y - c h a r g e d

1 / 3s p h e r e o f r a d i u s r c A . T h e s t a r t i n g v a l u e s f o r t h e r e a l w e l l d e p t h a n d

r a d i u s w e r e s y s t e m a t i c a l l y v a r i e d o v e r a w i d e r a n g e , i n o r d e r t o i n c l u d e

p o t e n t i a l s f r o m d i f f e r e n t " f a m i l i e s " ( c f . B a 7 2 ) .

F o r e a c h t a r g e t , t h e b e s t f i t w a s o b t a i n e d u s i n g a w e l l a b o u t 1 6 0

M e V d e e p . T h e c a l c u l a t e d c u r v e s a r e c o m p a r e d w i t h t h e d a t a i n F i g u r e 1 ;

t h e b e s t f i t p a r a m e t e r s a r e l i s t e d i n T a b l e 1 . A l s o s h o w n i n t h e t a b l e a r e

7 1 6a s e t o f L i v- O p a r a m e t e r s w h i c h w e r e d e r i v e d f r o m t h e o t h e r s e t s b y

a n i n t e r p o l a t i o n .

A t t e m p t s t o f i t t h e d a t a u s i n g s t a r t i n g v a l u e s t a k e n f r o m e a r l i e r

s t u d i e s ( B e 6 9 a , W e 7 2 ) r e s u l t e d i n l a r g e , u n p h y s i c a l r e a l r a d i i . T h e s e

s h a l l o w e r w e l l s (U - 5 0 M e V ) w e r e u n a b l e t o g i v e f i t s o f c o m p a r a b l e

q u a l i t y t o t h o s e w i t h t h e d e e p e r w e l l s .

A f t e r t h e o p t i c a l m o d e l a n a l y s i s w a s c o m p l e t e , S c h u m a c h e r f i t J l L

( S c 7 3 ) p u b l i s h e d t h e r e s u l t s o f t h e i r c a l c u l a t i o n s f o r l i t h i u m - 7 s c a t t e r i n g

f r o m c a r b o n - 1 2 a n d o x y g e n - 1 6 a t 3 6 M e V . T h e " F a m i l y H I " ' p a r a m e t e r s

V. ANALYSIS A N D RESULTS

F i g u r e V - l . O p t i c a l m o d e l f i t s t o t h e l i t h i u m e l a s t i c s c a t t e r i n g a n g u l a r

d i s t r i b u t i o n s o f F i g u r e I V - 1 5 . T h e p a r a m e t e r s c a l c u l a t e d

b y t h e p r o g r a m J 1 B 3 ( P e 6 3 ) , a r e l i s t e d i n T a b l e 1 .

(d<7

/da)

/(d<

r/d&

)Rut

h

^c.m.

Table V-l: Optical Model Parameters for Lithium-7 Scattering at 36 MeV

T a r g e t V

( M e V )

ro

( f m )

a

( f m )

W D

( M e V )

r o I

( f m )

* 1

( f m )

r o C

( f m )

12c 1 6 0 . 1 . 0 8 6 0 . 8 2 9 8 . 7 1 0 1 . 5 5 0 1 . 0 5 6 2 . 5 0

1 4 n 1 6 0 . 1 . 0 3 1 0 . 9 0 6 1 2 . 7 4 7 1 . 7 5 4 0 . 6 8 5 2 . 2 4

* ° N e 1 9 7 . 5 1 . 0 0 9 0 . 8 7 8 2 2 . 5 1 7 1 . 4 4 8 0 . 8 2 2 2 . 1 3

1 6 0 1 6 0 . 1 . 0 2 0 . 9 1 6 . 1 . 6 0 . 8 2 . 2

W H E R E

U ( r ) = V c - V f(x) + 4i WD ^ f(Xj)

1 / r r > Rc

r = r a1 / 3

42

q u o t e d i n t h a t p a p e r a r e s i m i l a r t o o u r o w n b u t s l i g h t l y d e e p e r (U - 1 9 0 M e V )

a n d e m p l o y v o l u m e a b s o r p t i o n . I t s h o u l d b e n o t e d t h a t t h e r e g i o n b e y o n d

9 0 ° i n c e n t e r - o f - m a s s , w h i c h w a s n o t o b s e r v e d b y S c h u m a c h e r g l . w a s t h e

m o s t c r i t i c a l i n c h o o s i n g b e t w e e n t h e d i f f e r e n t p o t e n t i a l s .

B . C o m p o u n d - N u c l e a r R e a c t i o n C a l c u l a t i o n s

T h e p o p u l a t i o n o f s t a t e s f o r b i d d e n i n a d i r e c t o n e - s t e p r e a c t i o n .

a n d t h e c o n s i s t e n t s h a p e o f a l l t h e a n g u l a r d i s t r i b u t i o n s a t b a c k a n g l e s

1 2 7 1 6( i n t h e C ( L i , t ) O r e a c t i o n ) s u g g e s t t h e p r e s e n c e o f a c o m p o u n d -

n u c l e a r c o m p o n e n t i n t h e r e a c t i o n m e c h a n i s m . I n o r d e i s t o a s s e s s t h e

i m p o r t a n c e o f s u c h e f f e c t s , t h e H a u s e r - F e s h b a c h c o d e S T A T I S ( S t 7 3 ) w a s

e m p l o y e d t o c a l c u l a t e t h e s t a t i s t i c a l , c o m p o u n d - n u c l e a r c o n t r i b u t i o n s t o

t h e c r o s s - s e c t i o n s .

U p t o s i x c h a n n e l s , i n c l u d i n g t h e e n t r a n c e a n d e x i t c h a n n e l s a r e

a l l o w e d i n t h e c a l c u l a t i o n . O p t i c a l m o d e l p a r a m e t e r s f o r e a c h c h a n n e l

a r e u s e d b y t h e c o d e A B A C U S ( A u 6 2 ) t o c a l c u l a t e t r a n s m i s s i o n c o e f f i c i e n t s

T ^ ( E ) f o r a r a n g e o f e n e r g i e s u p t o t h e m a x i m u m a v a i l a b l e i n t h a t c h a n n e l .

T h e r e a c t i o n i s a s s u m e d t o p r o c e e d t h r o u g h a s t a t i s t i c a l l y - d i s t r i b u t e d g r o u p

o f l e v e l s i n t h e c o m p o u n d n u c l e u s , a b o v e t h e Y r a s t - l e v e l f o r t h e p a r t i c u l a r

J - v a l u e . T h e d i s t r i b u t i o n f u n c t i o n i s g i v e n b y :

i 2 a * (u * t Y u' t o ,f 2 J e4>

JLd Z ± )

201( i )

w h e r e : U = E - A , a2 = I t / h 2 , U = a t 2 - t

I = | M R 2 ( 1 . + . 3 1 0 + . 4 4 0 2 ) a n d R = r A 1 ^ 3u O

43

For our calculation:

a - . 14 A

r 1.2 fmo

3 = f 0.87 for fluorine-19 C.N.

L 0.64 for sodium-23 C.N.

In the low-excitation region, where this formula will be inapplicable

the levels of known energy, spin and parity may be explicitly included.

The level-density formula is then employed only above the energy of the

highest such level.

Using the above procedure, and the channel parameters listed in

Tables 2 and 3, compound-nuclear angular distributions were calculated

for the states of interest In oxygen-16 and neon-20. In each case the overall

normalization of these cross-sections was adjusted to correspond to the

amplitudes of the lowest Ju = 2 state, but the relative magnitudes were

preserved. The normalization factors were 0.35 for oxygen-16 and 3.0

for neon-20.

The calculated cross-sections for several of the less prominent

states in oxygen-16 are shown in Figures 2 and 3. Those for the 4p-4h

band, as well as those for several states in neon-20, are shown later in

this chapter. The experimental data show more structure than the

calculated curves, since the Hauser-Feshbach model predicts the shape

of energy averaged angular distributions.

It is of interest to note that this Hauser-Feshbach calculation

accounts for 50% of the forward-angle cross-section to the J71 = 4+ level

at 11.10 MeV. Several authors (e.g. Fo73, Se73) have noted "anomalously"

12 7 16Table V-2: Optical Model Parameters for Hauser-Feshbach Calculation of C( Li, t) O Reaction.

Compound Nucleus: Fluorine-19

Channel V

(MeV)

W„o " S D(fm) (fm) (MeV) (MeV)

ol(fm)

*1(fm)

oC(fm)

Reference

12C + 7 Li

16O + t

18F + n

18,O + p

17O + d

15N + a

160.

162.9

68.16

1.086 0.829

1.16

-0.851E 1.142 0.69

+0.00524E2

118.

154.1

1.0

1.43

0.6

0.56

0.

0.69 17.9

8.71

0.

8.72 0.

0. 4.9

4.74 0.

1.55

1.16

1.9

1.05

0.69

0.6

2.5

1.30

1.3

this work

Ga73

1.602 0.507 1.25 Le73 and Va69a+

Th69+

1.81 0.65 1.43 Jo70 +

U(r) = V - Vf(x) - i ^Wgf(Xj) - 4WD f(Xj) J ; see Table V - l.

+ quoted in reference Pe72.

Table V-3: Optical Model Parameters for Hauser-Feshbach Calculation of *G0 (7L if t)29Ne Reaction

Compound Nucleus: Sodium-23

Channel V ro a w s WD roI *1

Uou Reference(MeV) (fm) (fm) (MeV) (MeV) (fm) (fm) (fm)

160 + 7 Li 160. 1.02 0.9 0. 16. 1.6 0.8 2.2 this work

2°Ne + t 162.9 1.16 0.69 17.9 0. 1.50 0.82 1.30 Ga73

22Na + n

22Ne + p J41.7 1.25 0.67 0. 12.2 1.11 0.39 1.25 Hu69 +

21Ne + d 79.6 1.027 0.806 0. 15.43 1.583 0.613 1.3 De70 +

19F + « 140. 1.50 0.58 32. 0. 1.38 0.55 1.4 Ko69 +

u (r ) = vc " v f (x) ~ 1 \ y s n ^ ) ~ 4WD - ^ - f (x j)] (see Table V -l)

+ quoted in reference Pe72

Figures V-2 and V-3. Hauser-Feshbach calculated angular distributions

for states in oxygen-16. The normalization constant (0.35) was

determined by matching the magnitude to the observed angular

distribution of the 8.8717 MeV (2” ) level.

dcr/

dft

(/x

b/s

r)

46

7 6large cross-sections to this state in ( L i, t) and ( LI, d) reactions at

lithium energies of up to 24 MeV. The state had a strength of up to 50%

that of the 10.35 MeV (4+) level, although the ratio of reduced widths is

about 1/250 (Fo73). This was taken as an indication that a strong two-step

process, involving inelastic excitation of the 4.43 MeV(2+) level in carbon-12

was taking place.

If one crudely subtracts the compound-nuclear contributions from the

two 4+ levels In our data, (cf. Figure 10) the 11.10 MeV level has only about

6% of the strength of the 10.35 MeV level. Thus the two-step process

makes at most a 6% contribution even though the 11.10 MeV state has a

parentage based largely on the 2+ level in carbon-12. The effect on the

10.35 MeV level will be even smaller. Since the compound and direct

amplitudes are of the same magnitude, interference effects will largely

determine the shape of the angular distribution. Thus, any attempt to

employ a coupled-channels formalism in this case is undoubtedly futile.

This result seems to verify one of the assumptions underlying this

research — that the mechanism of the reaction would simplify as the

bombarding energies were increased from " 20 to " 40 MeV. Additional

discussion follows in Section C.

C. Finite Range Coupled-Channels DWBA Analysis: *2C(7Li, t)160

1. Program Testing

The angular distributions to the three lowest members

of the 4p-4h = 0+ band in oxygen-16, 6.05 (0+), 6.92 (2+) and 10.35 (4+),

were analyzed using the method described in Chapter III and in the Appendices.

47

One program check was performed by verifying that we could reproduce the

fixed-range DWBA calculations of Piihlhofer (Pu70) and Kubo (Ku72) by

suitable limitation of our code. In the colinear or "no-recoil", fixed-

range approximation, the vectors r& and rb(see Figure III-4) are required

to be parallel and the a-t radius, r^, is set equal to a constant R. These

restrictions combined with definition of the vectors result in a linear

relation between the lengths r and r. , and limit the allowed area ofa b

(r&, rb) space to a single line. The colinear requirement (r . ?b = 1)

also reduces the form factor (eq. B-18) to:

gK(rb* V = U1 (rl> U2 (r2> V <rl> <2)

eliminating both the integral and the K Independence.

For these tests the parameters of the harmonic potential were

varied in an attempt to duplicate the Woods-Saxon wave function of

Piihlhofer and of Kubo. We were able to closely reproduce Puhlhofer's

calculations but could also reproduce Kubo's highly oscillatory curves

by limiting the number of partial waves to a small, non-realistic value

(Amax ^ suggests a similar limitation in Kubo's calculation.

2. Reaction Calculations

The 6.05, 6.92 and 10.35 MeV states in oxygen-16

were first treated in one-channel finite-range DWBA calculations. These

calculations, because of their greater simplicity, compared to the CCBA,

facilitated an investigation of the effects of different form factors and

optical model parameters on the final angular distributions and normal-

48

The initial calculations were performed using harmonic oscillator

wave functions (HOWF). Since the 4+ level at 10.35 MeV is unbound, the

binding energy of an alpha-particle in oxygen-16 was arbitrarily increased

by 5 MeV, following the technique used by Puhlhofer e l al. (Pu70). This

allows all the states of immediate interest to be treated as bound.

The harmonic oscillator quantum numbers were determined by the

energy and angular momentum selection rules. The tails of the wave

functions were matched,at a sufficiently large radius, to Whittaker

functions which satisfy the extranuclear Schrodinger equation with the

appropriate (i. e modified) binding energy (cf. Pi73). The only remaining

variable, the range parameter v , was adjusted to reproduce the r. m. s.

radii measured via electron scattering (Co67). The values used for

-2oxygen-16 are: N = 8, v = 0 .75fm , 1= 0, 2, 4; and for lithium-7 are:

N = 3, v = 1.2fm"2,JI = 1.

At the time these calculations were under way, we were fortunate

enough to learn of some recent calculations of alpha-cluster wave functions

by Dr. James Vary and his coworkers (Va73, Bu73). These wave functions

are derived from a "folding" model in which the potential is a convolution

integral of the alpha-partlcle and core densities. These authors have

used these wave functions to successfully reproduce the locations andTT ±

alpha widths of states in the K = 0 bands in oxygen-16 and neon-20.

( Va73, Bu73). The two types of wave function are compared, for a

particular case, in Figure 4.

Finite-range DWBA calculations were performed using various

combinations of optical model parameters (see below) and both types of wave

izations.

Figure V-4. Comparison of the a - C form factors for the 6.92 MeV

(2+) level in oxygen-16. The dashed curve represents the

result of an harmonic oscillator potential; the solid curve

represents the result of a folded potential (Bu73). In each case,

u(r) is defined by X^(r) = u(r) Y^(r).

12

ALPHA-CLUSTER FORM FACTORS FOR l60(6.92,2+)

HarmonicOscillator

r(fm)

49

functions. A comparison between the results using the different form

factors is shown for one case in Figure 5. While there was no marked

improvement in the agreement with the data, the folded potentials were adopted

since it was felt they were better suited to these calculations, since they

successfully account for many of the properties of the states in question.

To test the sensitivity of these calculations to different optical

7 16model parameters six different combinations were tried. Two L i- O

potentials were used; that derived in this work and that used successfully

by Puhlhofer et al. (Pu70). Three different triton parameter sets were

used: that of Donau fit aL (Do67) (used by Piihlhofer) a potential of the

"average-geometry" type with rQ = 1.16 fm, obtained by Flynn et al. (F169)

to triton elastic scattering from somewhat heavier nuclei, and an average

triton potential for the s-d shell derived by Garrett et al. (Ga73) for

analysis of (t, p) and ( t , p) reactions. These potentials are listed in

Table 4. While little or no difference could be seen between the results

obtained using the Flynn and Garrett potentials, the use of the Donau

potential resulted in markedly inferior results. The difference between7

the deep and shallow Li potentials was not as distinct, but the deeper

well was thought better. A comparison of the results from two of the

six combinations of potential sets is shown in Figure 6. Although the

shapes of the curves vary considerably in some cases, the normalizations

do not. The deep lithium well (L I) and the Garrett triton parameters (T3)

were chosen for further use.

There remained the question of the "stripping" interaction. In a

(d, p) reaction one can argue, with reasonable validity, that is the7

appropriate potential (cf. Au70a). The analogous argument for the ( Li, t)

Figure V-5. Comparison of finite-range DWBA calculations using the

harmonic oscillator (dashed curve) and folded potential (solid

curve) form factors of Figure 4. Both curves are shown before

normalization.

cfoVd

ii (/ib

/sr)

Table V-4:7

Optical Model Parameters for Finite-Range DWBA and CCBA Analysis of ( Li, t) Reactions

Channel Potential V ro a w s WD

i—iou

h roC Ref.

12 7C + Li L I 160. 1.086 0.829 0. 8.710 1.550 1.056 2.50 this work

12C ♦ 7Li L2 50.0 1.47 0.83 0. 8.65 1.86 0.80 2.30 Pu70

160 + 7 L i L3 160. 1.02 0.9 0. 16. 1.6 0.8 2.2 this work

160 + 7 Li L4 33.1 1.73 0.85 0. 10.3 1.87 0.72 2.5 Be69a

16o + t T1 143. 1.42 0.54 0. 17.4 1.56 0.55 1.40Pu70Do67

16o + t T2 143. 1.16 0.752 15. 0. 1.50 0.817 1.25 F169

16o + t

20Ne + t j

T3 162.9 1.16 0.69 16.9 0. 1.50 0.82 1.30 . Ga73

cno

Figure V-6. Comparison of finite-range DWBA calculations using different optical potentials for the 7 Li + 22C and t + *80 systems. Both curves employ the folded potential form factors and are shown before normalization. The optical potential parameters are listed in Table 4.

dcr/

dii

(/xb

/sr)

reaction is somewhat less defensible. It rests on the somewhat naive assumptions:

i) that the cluster-model description of oxygen-16 is valid so that, hence V(t - l60) - V(t - a ) + V(t - 12C),

ii) that exchange effects can be entirely neglected, and12iii) that the (t - C) nuclear interaction is approximately

reproduced by the optical model potential derived from triton scattering on oxygen-16.Although none of these conditions may, in fact, be met, a (t - a )

potential was used for the stripping potential since no better one was available. A real Woods-Saxon potential with a radius of 2.05 fm, a diffuseness of 0.7 fm and depth of 91.2 MeV was employed (Ku72), to give the proper binding energy and the observed r. m. s. radius for lithium-7.

The area of integration is shown in Figure 7.

3. Results of the Calculation

Figures 8, 9, and 10 show the calculated cross-sections forthe 0+, 2+ and 4+ members of the 4p-4h band in oxygen-16. In thesecalculations, the folded potential (Va73) wave functions were employed,with optical model potentials LI and T3 (see Table 4).

The wave functions were constructed on the assumption that the states12are completely represented by a C-<* structure and the spectroscopic

factors, Yo in Equation B-6, were set to unity. Since only a single 2

value of y % was assumed for each state, the normalization constant is:2

51

ground state. The contours, labeled N, indicate the lines outside1 i -Nof which | gQ(r , r ) | < 10 . (The maximum value of g^ is26.48 x 10 ) The dashed lines indicate the limits of integration:

> 1.1 xr - 5.5 r, £ 0.62 x r +5.0 a b a0.11 ra <-10.0 ; 0. li. rbi 10.012 7 16For the C( Li, t) O reaction the lines limiting the reintegration

were

Figure V-7. Contour plot of the form-factor g^(r^, r ) for the neon-20

c o n t o u r p l o t o f g K <rb ,ra )

Figures V -8, V-9, andV-10: Finite-range DWBA calculations of the12C(?Li, t)160 reaction to the 6.05 (0+), 6.92 (2+) and 10.35 (4+) MeV levels of oxygen-16. The calculations used the folded potential form factors, and optical potentials LI and T3 (see Table 4). The normalizations for these curves is described in the text. Also shown are the results of a Hauser-Feshbach calculation normalized to the 8.87 MeV(2~) state.

dcr/

dfl(^

b/sr

)

1000 E ©

l2C(7Li,t)l60*(6.92,2+)

DWBA HF

20 40 60 80 100 120 140 160 180dt'em.

lOjOOO

IOOO

(O \£33.100 c?T>\bT3

10

\A*

V

£

l2C(7Li,t)l60*(l0.35,4+)

-\*\ •\— • \ \

-N-At

V

V

DWBA HF

J » V ,

20 40 60 80a

100 120

/ — v

140 160 180c.m.

52

Y represents the relative reduced width (or spectroscopic factor) for-finding the lithium-7 nucleus in the out configuration. This quantity isapproximately unity (T06I). Therefore, the normalization constant is a

2direct measure of Y^ or the a -particle spectroscopic factor in oxygen-16.2 2 This number is related to the reduced alpha-width, 0^ , by the equation:

(cf. e.g. Pu70)

0 2 = y2 — r 2 u (r ) (2)a 1 3 2 2 1 2} ( >

where is taken at the nuclear surface. In this expression u2(r2) is■f

defined by (r2) = u2(r2)Y&(r2), which differs from the expression2 2 2

in Appendix B by a factor of r0 . The reduced alpha-width, 0 , representsj u C l

the probability of the existence of an alpha-particle at the surface of2oxygen-16. Note that while Y is sensitive to the overall normalization

2 2 2 of the wavefunction, the0a is not. The calculated values of T and 9afor the three lowest members of the 4p-4h band are shown in Table 5. Theresults of a fixed-range DWBA analysis by Puhlhofer et al. are shownfor comparison.

2 +The anomalously small value of Y 0 for the 4 state results fromh + +the way in which its wave function was normalized. The bound (0 and 2 )

states were normalized to unity, but the unbound 4+ state was matched instead to a unit distorted wave (Bu73). Also, the tabulated wave functions extended only to 11 fm, and were extrapolated with an exponential tail

2

53

Table V-5: Spectroscopic Factors and Reduced Alpha-Widths for Oxygen-16

State Spectroscopic Factor ( Ya )This work Puhlhofer

et al. (Pu70)

Reduced a-width ( 6 )' a 'This work Puhlhofer

etal. (Pu70)

6.05 (0 )

6.92 (2 )

10.35 (4 )

3.0

0.17

8 x 10

0.2

0.7-5

.69

.033

.031

0.05

0.18

0.1 - 0.3 t

a„ 2 1 3 .2, . a 3 r2 U 2

r„ = 5.4 fm

+ C.M. Jones et_al. (Nucl. Phys. 37 (1962) 1) foimd a value of 20^ = 27% of the Wigner limit from the study of elastic a-scattering.

54

beyond that point. This extrapolation is probably valid in the case of the bound states where the tail region is approximately exponential. In the case of the 4+ state, the tail of the wave function at 11 fm is beginning the first of an infinite number of oscillations. Although ignoring these oscillations will probably not change the shape of the angular distribution, the absolute value of the cross-section is not well-defined since the method to be used to normalize the wave function is unclear.

In spite of this uncertainty, it was felt worthwhile to investigate the effect of inelastic processes on the angular distributions. While no effort was made to fit the data, a coupled-channels calculation was carried out to gauge the magnitude of the effect of inelastic excitations. The allowed routes for this calculation are shown in Figure 11. Excitation of the first excited (2+) level in carbon-12 was not included, because no reliablefactors of the type 12C(2+) x 4He (0+)

I -Is reasonable despite significant excitation of the 2 level In llthlum-

j were available. This omission

scattering since:TT +i) the K =0 band in oxygen-16 is felt to have a parentage

based almost entirely on the carbon-12 ground state (e.g. Fo73),ii) the 4+ level at 11.10 MeV, believed to be largely built

upon the 2+ level in carbon-12, is not very strongly excited beyond the calculated compound-nuclear strength (see Section B).

iii) in light of the above mentioned uncertainty in the normalization of the (dominant) 4+ state, it was not felt worthwhile to increase the number of channels six-fold (from 15 to 90) witha corresponding increase in computer time.In performing the CCBA calculations the parentage factors (

2

Figure V-ll Allowed routes in the CCBA calculations. See also Figure II-3.

A L L O W E D R O U T E S IN C C B A A N A L Y S I S

4 +

2 *

0 +

in Eq. B-6) for the 0+ and 2+ states were left at unity, while that for the 4+ state was set to .01. Since the cross-section varies as the square of this factor, the magnitude of the calculated cross-section for the 4+ state was within 10 or 20% of the experimental value. The effects of these inelastic processes can be seen in Figure 12, which compares the two types of calculations, with equal normalizations. As the figure demon­strates, the CCBA calculation differs only slightly from the DWBA in the shape of the angular distributions ;only the 0+ level is changed in magnitude.

Several factors should be mentioned with regard to the CCBA calculation. Since the K 77 = 0+ band in oxygen-16 is thought to be based on the ground-state band of neon-20 coupled to 4 holes in the lp shell, the quadrupole deformation parameter &2 was taken to be that of the neon-20 ground-state. This probably is an overestimate, but will give an upper limit to the effects. In addition, there Is an ambiguity in the signs of the parentage factors. All of these parameters were (arbitrarily) chosen to be positive, but a change in sign could reverse the effects of interference. Since the aim here was merely to estimate the magnitude of these changes, however, these signs were not critical.

It seems clear from this analysis that some improvement could be made in the CCBA calculation. Since a spectroscopic factor greater than unity is required to normalize the 0+ level cross-section, it may be that some additional process is contributing to the reaction. One likely possibility would be transfer following excitation of the 2+ level in carbon-12. Since such a process is known to interfere destructively with the direct transfer to J77 = 2+ states in deformed nuclei (cf.As72 ) this

55

Figure V-12. Comparison of the finite-range DWBA (dashed curve) and CCBA (solid curve) calculations for the 12C(7Li, t)160 reaction. The 0+ and 2+ states have unit normalization. The 4 state has been multiplied by a factor of Y2 = 10-4 in each curve as described in the text.

do

7d

&

{fxb/

sr)

56

additional route might additionally result in a larger reduced alpha-width for the 6.92 MeV level.

Finally, it should be noted that these calculations do not correctly predict the magnitude of the cross-sections in the region near 60°, particularly for the 2+ and 4+ levels. (This effect is also noted in neon-20; see below). Other routes, such as the one mentioned above, or a cascade down from the strong 6+ level may be the source of this strength or the discrepancy may be the result of the wave function used. The folded-potential form factors emphasize the nuclear surface region (where grazing collisions would result in a forward-peaked angular distribution) at the expense of the interior (where larger-angle deflections would originate).

1 fi 7 20D. Finite Range Coupled-Channels Analysis: 0( Li, t) Ne

12 7 16 In comparison with the C( Li, t) O reaction, the16 7 200( Li, t) Ne reaction offered relatively few problems. The principle states of interest in neon-20, the ground-state (0+), 1.63 MeV (2+) and4.25 MeV (4+) levels, are all bound, and thus the wave functions can be properly normalized. Since the band head is the ground state, the deformation parameter is known. The entrance channel is extremely simple as well. Oxygen-16 is spherical and has an exceedingly high

7T *4*first excited state (6.05 MeV). Since this state has J =0 there should be no inelastic excitation to it. The second excited state at 6.13 MeV is primarily lp-lh in nature and should therefore not couple strongly to the rotational states in neon-20. We are therefore well justified in treating only the elastic entrance channel.

57

Since the carbon-target analysis had shown the CCBA and DWBAcalculations to be relatively insensitive to optical model parameters, onlyone triton potential was used; the average sd-shell well of Garrett et al.(Ga73), listed as T3 in Table 4,which gave the best results in the 12 7 16C( Li, t) O analysis. Two lithium potentials were used for theDWBA analysis, an "average" potential (L3) for the 160-7Li systendeduced from the elastic scattering measurements made from neighboringtargets (see Section A), and the shallower potential derived by Bethge et al.(Be69a) from 7Li-*80 scattering at E = 20 MeV (L4). The results

LI

using the two potentials were virtually indistinguishable; only the deeperwell (L3) was used for the CCBA analysis. The folded-potential wavefunctions for neon-20 (Va73) were employed throughout.

The calculations for the 0+, 2+ and 4+ levels are shown inFigures 13, 14 and 15. The normalization factors for the DWBA curvesare, respectively: 0.17, 0.10 and 0,07. The CCBA calculation wasperformed with all the parentage coefficients set equal to unity, and thenan overall normalization of 0.1 was applied, thus not altering theinterference effects. As can be seen, there is virtually no differencebetween the DWBA and CCBA calculations, except in the case of the 0+state, which has had some of its structure dissipated by the CCBA.

The small effect of the inelastic transitions can be understoodsince the three states are comparable in magnitude. Therefore, thetwo-step processes, which are roughly a factor of 10 weaker will result

12 7 16in only a ~ 10% perturbation. This is in sharp contrast to the C( Li, t) O reaction in which the 10.35 MeV (4+) state was significantly stronger than the lower levels and hence could populate them via a cascade with

Figures V-13, V-14 and V-15. Finite range DWBA (dashed curve) and CCBA (solid curve) calculations leading to the g. s. (0+),1.63 (2+) and 4.25 (4+) MeV levels on neon-20. The normalization procedures are described in the text. Also shown are the results of a Hauser-Feshbach calculation (dot-dash curve) normalized to the 4.97(2”) MeV level.

do-/

df}(y

xb/s

r)

58

significant amplitude relative to the direct routes. It thus appears that12 7 lfialthough inelastic effects are significant in the C( Li, t) O reaction

1 fi 7 20they seem to have little bearing on the 0( Li, t) Ne reaction.As mentioned in Section C above, the folded potential wave functions

are normalized on the assumption of unit spectroscopic strength. Therefore,the normalization factors directly measure the relative width of the states.

2 2In Table 6, the extracted values of y and9a are compared with the predictions of Matsuse (Ma73) and of Hiura (Hi72).

Finally, mention should be made of the K 77 = 0~ band in neon-20.The strong population of this band tends to verify the theory of Horiuchi

7T ±and Ikeda (Ho68) that the K =0 bands are "twins" with the same intrinsic 16O - a structure in analogy to the motion of a heteropolar diatomic molecule. The splitting between the bands is attributed to the different reflection symmetry of the odd and even members, and the existence of a nuclear barrier to the transition of an alpha-particle through the core.

59

Table V-6: Spectroscopic Factors and Reduced Alpha-Widths for Neon-20

2 2 State Spectroscopic Factor ( Ya ) Reduced a -Width ( 0a )this work theory this work theory

(Ma73) (Hi72)

0.00 (0+) 0.17 0.295 0.039 0.118

1.63 (2+) 0.10 0.295 0.023 0.107

4.25 (4+) 0.07 0.280 0.012 0.083

a 2 2 1 2 .a Ya 3 r2U2(f2)

r2 = 5.6 fm

60

The alpha-transfer reaction ( Li, t) has been studied on four targets: carbon-12, nitrogen-14, oxygen-16 and neon-20. While the rather global aims of the Introduction have not been met, significant progress has been made.

12 7 16The investigation of the C( Li, t) O reaction has yielded several useful results. The transfer reaction to the 4p-4h band has been well described by means of an exact finite-range DWBA analysis, and spectroscopic factors have been extracted. A coupled-channels DWBA analysis has shown that exit-channel inelastic effects can appreciably alter the shapes of the predicted angular distributions when one state in the band is much stronger than the others. In examining the angular distribution of the 4+ level at 11.10 MeV (which should not be strongly populated by a direct one-step process) the Hauser-Feshbachanalysis has been successful in accounting for much of the strength, in contrast to the observed results at lower energies. This indicates that, at this energy, entrance channel inelastic effect make only an ~ 6% contri­bution relative to the one-step process.

16 7 20In the case of the 0( Li, t) N reaction, the results of the DWBA and CCBA calculations indicate that inelastic effects do not play any significant part in the reaction mechanism probably due to the nearly equal strength with which each of the three states is populated. Reduced widths have been extracted for the three lowest levels which are in reasonable agreement with theoretical calculations, considering the

VI. CONCLUSIONS

7

61

highly model-dependent nature of these quantities.While no detailed reaction calculations have been performed for

14 7 18the N( Li, t) F reaction, there is strong qualitative evidence tosuggest that the two states at 9.472 and 11.074 MeV are members of the77 +K =1 rotational band. The preferential population of this band in an

alpha-transfer reaction tends to confirm the findings of Rolfs et al. thatthis band has a 4p-2h nature.

20 7 24The analysis of the Ne( Li, t) Mg reaction presents some problems. An apparent decrease in the selectivity of the reaction coupled with the large density of states in magnesium-24 makes improved resolution imperative. Furthermore, the monatomic nature of neon-20 reduces the experimental yield in a gas target experiment by a factor of two, compared to diatomic gases such as C>2 and Ng. Therefore the use of narrower slits to reduce the dE/d6 kinematic broadening would reduce the count rate to an unacceptable level. The use of a magnetic spectrograph at these energies is precluded by the great magnetic rigidity of the high-energy tritons. The possibility of a compromise solution seems small.

This work has shown then that an exact finite-range analysis of these reactions is possible within the current limitations of computer size and speed. The technique is rather cumbersome and time-consuming, and can no doubt be greatly streamlined by making some astute approxi­mations or simplifications. These techniques lie more in the venue of the reaction theorists, and I will gladly leave the task to them.

Further calculations that are warranted for the present data are:71) A finite-range DWBA analysis of the ( Li, t) reaction to

TTthe oxygen-16 J =1 level at 7.12 MeV. The previous measure­ments of the reduced width of this state (Pu70) have been per­formed using a fixed-range code and assuming a pure lp-lh nature of the state. This state has significant (43%) 3p-3h com­ponent (cf. Zu69), however, which should be included. If an appropriate alpha-particle form factor can be computed, this calculation so vital to astrophycical theories (cf. Chapter I) can be easily performed.

TT -2) An analysis of the angular distributions of the K =0 band in neon-20. An improved measurement of the reduced widths for these states could test the validity of the assumption that this band is a "twin" of the ground-state band. (Ho68) Such an analysis is in progress.

This work also suggests several extensions in the experimental14 7 18area. One promising possibility is the N( Li, ty) F correlation

experiment to determine whether the two states mentioned above areTT +strongly coupled to the other members of the K =1 band. Measure­

ment of excitation functions for some of the strongly-excited states in oxygen-16, for example, could better determine the effects of compound- direct interference, or of two-step processes in rht reaction mechanism.

In summary, we can say that, with some exceptions, the experi­mental data for these reactions have been measured to a precision which the theories have not yet matched. If some of these nagging difficulties in the determination of form factors and in the normalization of unbound wave functions can be resolved, then the ( Li, t) reaction may become a highly powerful spectroscopic tool of the nuclear physicist.

62

63

As indicated in Section HD, the active target volume for a gas target is not a point, but rather a finite segment of the beam path through the gas. For the purposes of this Appendix, the beam will be taken as a line of zero width, and the detector will be assumed to be at 90° to the beam axis.

As the beam traverses the active volume, two effects will conspire to change the energy of a triton observed from a reaction at any particular point. First, due to the energy loss in the gas, the lithium energy will differ from the "nominal" energy, i.e. that at the exact center of the target. Second, the laboratory angle at which a triton must emerge to reach the detector will be greater or less than the nominal 90°.

It should be noted that these effects are not random dispersions, but systematic energy shifts, and thus the method of summation in quadrature does not apply. In addition, as reference to Figure la will verify, the two effects will add coherently. As the point of reaction is taken further "downstream" the beam energy will decrease and the observed angle of the emerging tritons will increase. Due to the reaction kinematics, this second effect also causes a shift to lower triton energy. We shall attempt to calculate the sum of these effects.

Since the number of reactions is much smaller than the total number of incoming ions, the beam intensity, and bence the number of reaction events per unit path length, dN/dz, is assumed to be a constant. The efficiency for observing events along the active target volume, however, is not a constant. Between points O and B, in Figure la, the fraction of the detector which can be "seen" by a triton is 1. Between points B and A, this

A P P E N D I X A. Turgot Energy Loss and Kinematic Broadening

Figure A-l. The upper figure (a) represents the slit system used in the measurements using gas target cells. The lower figure (b) shows how the kinematic shift and target energy loss combine in determining the peak shape. In the lower left quadrant (HI), the fraction of the detector area visible from various positions along the beam axis. The lower right quadrant (IV) shows the triton energy as a function of position (z). The dashed curve shows the effects of beam energy loss; the solid curve shows the combination of this effect and the kinematic shift dE/d0 . Finally, the upper right quadrant (I) shows the number of observed particles per unit energy:

dN/dE = dN/dzdE/dz

£ ------- hRo -

64

active fraction decreases linearly to zero. This is indicated in the third quadrant of Figure lb. (For clarity, only half of the symmetric figure is shown, the part of the active volume from A 1 to O Is ignored.)

The beam energy loss rate dE/dz is approximately constant over this small interval. Since, by equation n-12, the triton energy is proportional to the beam energy, the energy of a triton emerging at 90° will decrease linearly with increasing z. This is shown by the dashed cirve in the fourth quadrant of Figure lb. Since dN/dE =(dN/dz) /(dE/dz), the curve of dN/dE is the trapezoid shown in the first quadrant of the figure as the dashed line.

The observed triton angle is not 90°, however. The deviation from 90°, a, increases as the point of reaction moves from the center, O, to point A. From O to B the increase is linear, i.e. a - tana = z/^0» where z = 0 at point O. From B to A, the increase is again linear, but at a greater rate, since the center of the visible detector area moves linearly from OO to BB. If dE/d 0- constant, then the E vs. z, and the dN/dE vs. E curves take the shapes shown as the solid curves in quadrants 4 and 1, respectively.

The discontinuity in the dN/dE curve at EL, results from the non-r>physical assumption that the tritons from each point are observed at a single angle, at the center of the visible detector area. When the finite size of the detector is taken into account, this discontinuity will disappear.

The signal measured when an "ideal" distribution is viewed by an instrument with finite resolution ( and possibly aberration) has been cal­culated for a variety of cases in connection with magnetic spectrographs (e.g. Zi55). The equation for the observed distribution,I(x), is:

65

I(x) = L dy g(x - y) h(y) (1)

where h(y) is the ideal distribution and g(x - y) .represents the effect of the instrument. The two functions are normalized to unity.

In the region <; E *>Eq, the spread in accepted angles is a constant. If d /dft( 0) is also approximately constant over this small range (£1°), then the spreading is uniform this range, as is the "ideal" dis­tribution, i.e.:

h<f> = \ 2Wi 1 g(x-y)*/2w2 ,| x-y |<w2 (2)L.0 , otherwise \f) , otherwise

and w^ > w2« The integral is trivial, and the result is the trapezoid shown in Figure 2a. Note that, for this case, the full width at half-maximum, n , is unchanged.

In the region E' < E < E' , the spread in angle decreased linearly A D

from w 2 to zero, i.e.

h(y, = j r <c -«• 0<y<c g(x -y) = / W b > 1 x-yl<w« (3)

-0 , otherwise

w2where w(y) - — (c - y). Note that the point y = 0 is now taken as E = E' .C Ij

Performing the integral, equation (1), we find:

on the rectangular portion of the distribution in Figure 1. (b) The effect of the function g(x - y) of equation 3 on the triangular part of the distribution in Figure 1. (c) The sum of these two results. Note that the horizontal arrows represent the half-width at half maximum (n/2) for each distribution.

Figure A-2. (a) The effect of the broadening function g(x - y) of equation 2

0 w,-w2 w. W,+ W2

66

x+w2" V X< W 2w 2(c+w 2)

I(x) =<

(4)

2(c-x) w <x<c 2

0 otherwise

This curve is shown in Figure 2b. The combined effect of beam energy loss and the kinematic energy shift due to change in angle is to generate a peak shape as shown in Figure 2c.

After application of some elementary algebra and geometry, we find that (in the notation of Section IID):

In our problem, w^ represents the shift in energy of tritons from point B relative to those from O, due to energy loss and angular change; Wg represents the energy spread caused by a difference in angle equal to that subtended by half of the detector. Thus, w ^ ^ represents a half-width at half-maximum.

For our case dE (B) due to energy loss was 7 keV; dE (B) due tot Iangular change was 42 keV, so that w = 49 keV. wg is just equal to dEt(B), as can be seen by inspection, and !/h = 1/3. Therefore :

w = 49 + 14 = 63 keV total

Thus n ,the fwhm is 126 keV, from these causes.The final question concerns the way in which the width combines with

the other sources of line width to give the total. Zimmerman (Zi55) shows that, except in the simplest cases, the integral must be done numerically, which is highly impractical. However, some general conclusions are possible from observation of his results. The way in which the widths of two curves combine depends largely on the shape of the "tail". For example, two extreme cases are two rectangles, which have no area outside the fwhm; and two Cauchy curves which have "infinite" tails. In the first case, super­position of a second rectangle (of smaller width) has no effect on the fwhm; in the second case, the widths add linearly. The Gaussian curve, with an exponential tail, is an intermediate case with the widths adding in quad­rature.

Our curve has a linear tail and may be considered similar to a triangle. Thus, it may be expected to yield a width somewhat larger than that resulting from a gaussian of equal width. This is seen to be the case, since the observed resolution (180 keV) is greater than that predicted by adding 126 keV and 75 keV (from other effects) in quadrature (i.e. 147 keV).

67

68

As explained in Chapter III, most of the physical description of the reaction process is contained in the source term — that is, the inhomogeneity in the exit channel scattering equations. The derivation of this term, however, is mainly an exercise in angular momentum coupling, with the assumptions and approximations made lying buried under Clebsch-Gordan coefficients. We shall attempt to unearth them.

This presentation parallels that given by Pisano (P173) in his thesis, which in turn owes much to the seminal paper of Austern^t al. (Au64). The angular momentum algebra is taken from Messiah (Me63) and DeShalit and Talmi(De63).

We start with the expression (III-4):

A P P E N D I X B. Calculation of the Source Term

In this expression, the wave functions 0 represent the solutions to Schrodinger equations in the entrance and exit channels, with the radial part of the relative motion factored as u (r )/r . The bracket representsintegration over all coordinates, including r^ but not r b* The 9 's may be factored as follows:

(1)

(2)

a' a' a

[(Y J2b(?b) <lsb( b) }jb » IM (3)

69

$a £(Yi. / ra) <J>sa^ a ) )ja’ ♦ JA Ia ^J IM (4)

where <J>( t ) represents the internal wave function (and coordinates) of particle i.

At this point, we introduce a physical assumption: the projectile, a, and residual nucleus, B, may be well-represented on a cluster-model basis, as follows:

] v <=>M.sk

If the assumption is a good one, only a few Y’s will be large and these willaccount for nearly all of the wave function for a + B. In practice, a singlevalue of i is used, but this is not required. Notice also that no assumptionis made(yet) as to the states of the clusters in either channel— they neednot be the same. The wavefunctions for the relative motions of the clusters

llmay be expressed as = r. u.(r ) Yi 1 1 i

To express the matrix element compactly, we adopt the convention:f.("? ) = j» and thus:J j

70

To evaluate this matrix element, which we shall call <; V> , we must have compatible momentum coupling on both sides. After some gymnastics we find:

9r-%T £ c ^ A A A /s.

9iSa9aJt jk93j.959b

NJb i t 9, y; S‘b 5 a

I J® J IX *. jo , %

where j = (2j + 1) 2 and:

r

<

St 5 b

it k 94

j t Jb 9,

si sb 9s}(i, i „ 9t

ji 3z J <v»>

(8)

N = <C(fe^ 93 U AJ Sk>9, JaJI MWlK bteb U,«9b)VJH<9>

It is at this point that we make the distorted wave Bom approximation and treat V(r^) to first order. That is, we assume that V can change the relative motion of the clusters, but not their internal structure. This ass­umption is implicit at this and each successive decomposition. Thus, we write:

N = I M, (10)

The first inner product yields 6 6 and:A* A M A ’M A

71

TY rn

We have again kept the internal cluster structure constant.

C C v'vb.^5 (12)

^S,<i5 ‘Smjm,

c ; ^ 3 ^ «

piwhere:

3 < > b ) = < i x K A M w W \ X , M , t ^ (14>

* x /<**<>„ jd*?.

J is the Jacobean of the coordinate transformation from (r , r g) to (ra, r b). Following the method of Austern (Au64) we make the expansion:

z f ( O V ( 0 < ' ( ^ ( Y ‘ (15)

->■ 4* III*Since h^fr^, r ) transforms like Y (fi), where ft is the angle between

rjj and r , we may write, using the spherical harmonic addition rule:

72

l l 2

Following the method of Austem (Au64) we can write:

H , , L K , 0 = i Y _ ( s . O V X ‘ ( t , rI I

« E c-) i , i , K A* Aka qvc.L^ ^ Ab<L| q A,'A, Aq X, At® A«N L ( rO O O q Q O '"‘r o o o o o O

'W ( l2L, AaAk)tK.) ' n X t u k 1

iZA,l 2-xJ ’(17)

/m\_ m! Jion, \n/ n!(rr and r. = s.r + t.r, .In this expression, w n!(m-n)! “““ "i “La ’ “Lb

The two-dimensional form-factor g is defined by:is.

gK <rb> r a> E d/ Pk<1I> ul< rl )V r2)V <r l ) (18)

_ A / >where m = r& • r and Pk(P ) is a Legendre polynomial. The expressionof equation 17 differs from that given by Pisano (Pi73) for the pickup reaction3 7 ^+^l+ 2( He, Be) only by the phase (-)

73

Using the expansion of eq. 16, we write:

X m L.ty tV\,Mz

(19)

• \The first integral yields (-) fi 0 6 M ; the second becomesV l "H> 1

o 5m ,, • Call the third I 0 (r,).L2 Aa ^ a 1 2 b Then:

Working back, we find:

<9,nvW I W J C i L

H> Ha (21)x T / v

After converting these C-G coefficients to 3-j symbols and doing some rearrangement, we find:

< 9 ^ w \ 9^ - i r t ( o r « v ^ ^ +mX 'w °k p's

X (A h % \ ( i b l jtA U u \ ( A I * 1 W / V V v-/Jw-m |J«/ j (22)

Applying equation (c. 32) of Messiah (Me63), to the last summation, we find:

( w i v i v o • H i t * - r ‘ i ^ i ^

(23)

^ ^SbSb 939 5 949<o

C ) (24)

N J ^ W ; W i (S!>.V,W > 4 s / s fcs;"\MA

12 , Q -J2

^ ^9 jo.^)S2 Sbsi

75

Substituting this into <V> , and letting s E s. = s. , we find:a l 2

T 4 a. j \ j o . A /> a A j A A A2A9k

<rjkjtja ')rjisbSg"i fs^SbSj*Ja I j0 9 ^ ^ ^

Jz Jb-k

>S< Sb J / A 94 >

J I 9*. J a,

5 iV >v ’ U A ' f t f .? }(26)

After some rearrangement, we can write:

V ^ z^ 9 , U Ja S,

25* (A j|(27)

using a formula on p. 518 of DeShalit and Talmi (De 63). So finally:

M J O * J lA *3.5*

” I X j. sai jk 9, g4j, [ J_fc ]' fj> SA 1|9i a» j» j

$o< k 93( A A 9^ }

Jl Jlo J (X

t‘. .V- fE i co U a i/ 9*»U A i

(28)

76

In the special case of sa = 0, we may reduce the expression further

( A o j , 1U>b sa93J

U , J, o^ -'b 9j ~t‘

C-'j

(29)

f ° Sb 9a< L Jt* =

J z Jb

% A

J k Jcc Jz.

sb 9 , 0 U

h h (30)

M ba(rb ) s ^ o ) - j E ix y, (-)0 9 ‘

TA + I*ZSb+A-Jbb-94->tI M9,

4 A A A A 2* J, f V ' i r ^ b 9 4 ^ Z >i r Ja. 1

IJ, I J g A j * j, 5 ^ 1 sa i A i ,J

* 1 lM .(C A ^ A ,

l A A - d

(31)

A P P E N D I X C. Numerical Methods

1. Solution of the Differential Equations

In solving the reaction problem by the source term method,one must find solutions to the Schrodinger equation in the entrance and exit channels (equations ni-1 and HI-2, respectively). This part of the code was taken virtually intact from that used by Ascuitto, (As69), but modified to include spin.

During the initial pass, the program REAC solves for N linearly independent solutions to equation m-1 with the limiting values as r -*• 0 of

where the 1 is in the qth row. It then constructs a linear combination of

(1)

N(2)

At large values of r (greater than some value R outside the nucleus) the function U J r) must satisfy:

78

where I and O are incoming and outgoing distorted waves. These conditions

and their derivatives give 2N equations to be solved fo r the N values of a^

and the N S 's .q

In the exit channel, the equations are inhomogeneous. A particular

solution, W (r) is found by integration o f equation I II—2 from the origin with

the requirement that W(0) = W '(0 ) = 0. Then the homogeneous solutions

W ^(r) are found fo r the M exit channels, and again a solution to the 2M

linear equations:

M

H b q W o C r ) - W ( 0 =

(Sy O . W

v 5 m O mCO(4)

and the derivatives. This is done in a second pass theough REAC, after the

source term has been calculated.

The S-m atrix elements fo r the exit channel are passed to the program

CROSS, which calculates the differential cross-sections.

2. The Finite-Range Source Term

In order to solve the exit channel problem , one must calculate

the source term of equation (1) of Appendix B. The m atrix element M ^ r ^ )

is given, fo r s = 0 in equation (B-31). The crux o f the problem is the

evaluation o f the integral I ££ £ (r^ ) given in equation (B-19) as:

r°° 2I p p p = J r dr H. . . (r,, r )u . (r )/rb a 0 i i b V b a - V a

(5)

79

H££ £ (r b» r a) may be written as:b a

= Z. t h (x,,x1i k’) 9 K(YklO

< S 1r J V X ' ( t , r bf ' ( s 1r ^ ( t l r a V A l(6)

The coefficient TH is independent o f the radii, expressing only the angular

momentum coupling, i .e . (c f. eq. B-17):

The expression for r^, r ) is given by equation (B-18), and depends only

on the internal wave-functions o f the pro jectile and residual nucleus, and

the stripping potential. This quantity is calculated by the program GK and

stored on magnetic tape. The values of TH are calculated by a program of the

same name and stored on the disk. These results are combined, and the

summation of equation (C -6) is perform ed by the program HL and the values

o f H ^ (r^ , r^) are stored again on tape.

, v i r - A 2 / Z A . J A / S VTVA (X,,X; *4

8 0

A fter the firs t pass through REAC, the scattering wave functions of

the entrance channel are combined with the previous results by the program

ERS. This code perform s the integration of equation (5) and then the

summations o f equations (B-31) and then (B - l ) . The source terms are then

stored on disk to be used in the second pass through REAC. (See Section 1

o f this Appendix).

A schematic diagram of the program structure is given in Figure C - l.

Figure C - l . Flow chart o f the CCBA program . In the chart, the

osymbol represents card input; ( ) represents

intermediate tape storage, and the double arrows indicate

data stored on intermediate disk data sets.

Cross-Sections

81

AJ71Aj72Aj73

A172

Ab71

Am 68

Ar71

As69As72Au62Au64

Au70

Au70a

Av72Ba62Ba71Ba71a

Ba71bBa72

Ba73

Be32Be39Be67

Be69

Be69a

Be70Be70a

B133

B IB L IO G R A P H Y

Abgrall , Y. and E. Caurler, Suppl. J. de Phys. 32 (1971)C6-32.Ajzenberg-Selove, F .,N ucl. Phys. A166 (1971) L A jzenberg-Selove, F . , Nucl. Phys. A190(1972) 1.A jzenberg-Selove, F. and T . Lauritsen, Lemon A id Preprint LAP-116 (unpublished).Alexander, T .K . at Fifth Symposium on the Structure of Low - Medium Mass N u c le i, Lexington, K y . , Oct. 26-28, 1972, J. P . Davidson and B .D . Kern, eds. (The University P ress o f Kentucky, Lexington, 1973) p. 272.Am erican Institute of Physics, (ed) Synthesis and Abundance of the Elements ( A . I . P . , New York, 1968).Artem ov, K .P . , V .Z . Goldberg, I .P . Petrov, V .P . Rudakov,I .N . Serikov abd V. A . T im ofeev, Yad. F iz . 14 (1971) 292. ((tr .Sov. J. Nucl. Phys. 14 (1972) 165)). Supp. J. de Phys. 32 (1971) C6-125, 127. Phys. Eett. 37B (1971) 61.Ascuitto, R .J . and N .K . Glendenning, Phys. Rev. 181 (1969) 1396. Ascuitto, R .J . and B. Sorensen, Nucl. Phys. A190 (1972) 309. Auerbach, E .H . BNL 6562 (unpublished).Austem , N . , R .M . Drisko, E .C . Halbert and G .R . Satchler,Phys. Rev. 133B (1964) 3.Austin, S .M ., G .F . Trentelman and E. Kashy, Bull. Am . Phys.Soc. 15 (1970)1662.Austem , N . , D irect Nuclear Reaction Theories (W iley-Interscience, New York, 1970).Avishai, Y . , Phys. Rev. C 6 (1972) 677.Bassel, R .H ., R .M . Drisko and G .R . Satchler, ORNL 3240 Balamuth, D .P . , Phys. Rev. C (1971) 1565.Bassani, G . , N. Saunier , B. M. T raore, G. Pappalardo and A . Foti, Suppl. J. de Phys. 32^(1971) C6-135.Bassani, G ., Note - CEA - N - 1474 (unpublished).Bassani, G . , N. Saunier, B .M . Traore and J. Raynal, Nucl.Phys. A189 (1972) 353.Bam es, C .A . , P . D yer, M .R . Dwarakanath, D .C . W eiss ler and J. F. Morgan, reference (De73a) p. 683.Bethe, H .A ., Z. Phys. 7£ (1932) 293.Bethe, H .A ., Phys. Rev. 55 (1939) 434.Bethge, K . , K. M eier-Ew art, K. P fe iffe r , and R. Bock, Phys.Lett. 24B (1967) 663.Bevington, P .R . , Data Reduction and E rro r Analysis fo r the Physical Sciences (M cGraw-H ill Book C o ., New York, 1969).Bethge, K . , C .M . F ou an d R .W . Zurmuhle, Nucl Phys. A123 (1969) 521.Bethge, K . , Ann. Rev. Nucl. Sci. 20(1970) 255.Bethge, K . , D .J. Pullen and R. Middleton, Phys. Rev. C2 (1970) 395.Bloch, F . , Ann. Physik 16 (1933) 285.

82

Bo70

Bol5B068

B r68Bu57

Bu73Ca62Ca72

Ch67

Ch69C168

C066

Co67

Co70

Co73

Cu72

Da68

Da69

De63

De70De73De73a

Do67

Do69En67F169

Fo64F068

Bohr, N . , Phil. Mag. 30 (1915) 581.Bodansky, D . , D .D . Clayton and W. A . Fow ler, Phys. Rev. Lett.20 (1968) 161.Bock, R . , andW .R . Hering, eds., Proceedings of the International Conference on Nuclear Reactions Induced by Heavy Ions, Heidelberg July 15-18, 1969 (North-Holland Publishing C o ., Amsterdam, 1970). Brown, R. E. and Y . C. Tang, Phys. Rev. 176 (1968) 1235.Burbidge, E .M ., G .R . Burbidge, W .A . Fow ler and F. Hoyle, Rev.Mod. Phys. 29 (1957) 547.Buck, B ., C .B . Dover and J .P . V ary ,B u ll.A m .Ph ys .S oc. 18(1973)1414. Caughlan, G .R . and W .A . Fowler,Astrophys. J. 136 (1962) T53. Carpenter, R. T . , J .C . van Staden and K. Bethge, Jahresbericht (1972), II. Physikalisches Institut der Universitat Heidelberg (unpublished) p. 24.Chuev, V. I . , V .V . Davidov, A .A . Ogloblin andS .B . Sakuta,Ark . F is. 36 (1967) 263.Chang, F .C . , Phys. Rev. 178 (1969) 1725.Clayton, D .D ., Principles o f Stellar Evolution and Nucleosynthesis (M cGraw-H ill, New York, 1968).Comfort, J .R . , J .F . Decker, E .T . Lynk, M .O . Scully and A . R.Quinton, Phys. Rev. 150 (1966) 249.Collard, H .R . , L .R .B . Elton and R. Hofstadter, Numerical Data and Functional Relationships in Science and Technology (New Series)Vol . I I , K .H . Hellwege and H. Schopper, eds. (Springer-Verlag,Berlin, 1967).Comfort, J .R . , G .C . M orrison, B. Zeidman and H. T . Fortune,Phys. Lett. 32B (1970) 685.Couch, R .G ., W .D . Arnett and D .N . Schramm, at Austin Con­ference on Explosive Nucleosynthesis, A p ril 2-3, 1973. (unpublished). Cunsolo, A . , M .C .-L em a ire , M .C . M erm az, J .L . Qufebert and H. Sztark, Supp. J. de Phys. 33 (1972) C5-66.Davydov, V .V . , A .A . Ogloblin, S .B . Sakuta and V .I . Chuev,Yad. F iz . 7 (1968) 758. (tr. Sov. J. Nucl. Phys. 7 (1968) 463).Davydov, V. V. and I. M. Pavlichenkov, Phys. Lett. 29B (1969)551.de-Shalit, A . and I. Talm i, Nuclear Shell Theory (Academic P ress , New York, 1963).Dehnhard, D. and N .H . Hintz, Phys. Rev. C^ (1970) 460.DeVries, R .M ., Phys. Rev. C 8 (1973) 951. deBoer, J. and H. J. Mang, eds. Proceedings of the International Conference on Nuclear Physics, Munich, Aug. 27-Sept. 1, 1973 (North-Holland Publishing C o ., Amsterdam, 1973).Donau, F . , K. Hehl, C. R iedel, R .A . Broglia and P . Federman Nucl. Phys. A101 (1967) 495.Dodd, L . R. and K. R. G reider, Phys. Rev. 180 (1969) 1187.Endt, P .M . and C. Van der Leun, Nucl. Phys. A105 (1967) 1.Flynn, E .R . , D .D . Arm strong, J .G . Beery and A .G . B lair,Phys. Rev. 182 (1969) 1113.Fow ler, W .A . and F. Hoyle, Astrophys. J. Supp. 9^(1964) 201.Fow ler, W .A . an dW .E . Stephens, Am . J. Phys. 36 (1968) 289.

I

83

Fo71Fo73

Fr72Ga73Ge67

G163G169

Go 69

Go71

Go73

Gr61

He68

H172

Ho 54 Ho68 Hu63

Hu69

Iv68

Ka60Ka73K i66Ko69Ku66Ku69Ku72

Jo70Le69Le72Li61Ma68

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