lllHHlHlllHlWWWWWHIHWHIHHINIHIIHIHI

   

traces!

 

This is to certify that the

thesis entitled

Elastic and Inelastic Scattering of 6Li from 58Ni

presented by

Cecil L. Williamson

has been accepted towards fulfillment
of the requirements for

Masters degree in Physics

flow

fl

Major professor

 

0-7 639

 

 

 

OVERDUE FINES ARE 25¢ PER DAY
PER ITEM

Return to book drop to remove
this checkout from your record.

 

 

 

ELASTIC AND INELASTIC SCATTERING OF 6L1 FROM 58Ni

BY

Cecil L. Williamson

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Physics
1979

ACKNOWLEDGMENTS

I would like to thank Dr. A. Galonsky and Dr. R.
Ronningen for their careful reading of this thesis and for
their discussions and suggestions concerning proper presenta-
tion of the material. Also, Dr. Galonsky has been most
helpful and encouraging in his assistance and guidance in
analyzing the data and interpreting the results.

The assistance of Dr. P. Miller in preparing, and some-
times repairing, the cyclotron is gratefully acknowledged.
Setup of the detector and wiring of electronics was skill-
fully accomplished by Dr. R. Markham, who was also quite
helpful in gathering the data. The willing assistance of
Mr. R. Huffman in running the experiment during the long
hours of early morning will be long remembered.

Timely and courteous explanations of the coupled
channels code ECIS was furnished by several people. They
were Mr. P. Deason of Michigan State University and Dr. S.E.
Darden and Dr. L. Foster of the University of Notre Dame.

The perserverance of Mrs. Carol Cole in typing both the
rough and final drafts of this thesis is greatly appreciated.

Finally, I would like to thank my friends and family for
their unflagging support throughout my years of graduate
study. Most especially I would like to thank my wife, Mary,

for her devotion and patience during this final year of

research.

ii

ABSTRACT

ELASTIC AND INELASTIC SCATTERING OF 6L1 FROM 58Ni

BY
Cecil L. Williamson

Elastic scattering of 6Li from 58Ni at 73.4 MeV was
studied, and five sets of optical model parameters were
obtained which gave equivalently good fits to the data.
These data were gathered using surface-barrier detectors in a
forty-inch diameter scattering chamber.

Nine inelastic states of 58Ni were studied via the same
reaction at 71.2 MeV. These data were gathered using a
position-sensitive proportional counter placed in the focal
plane of a split-pole spectrograph. Two methods were
employed to analyze the data. They were the distorted wave
Born approximation (DWBA), and the method of coupled
channels. A comparison of these methods and the information
they provide about the nuclear deformation lengths is pre-
sented. It was anticipated that comparison of the experi-
mental inelastic angular distributions with theoretical
predictions would produce an unambiguous choice of optical
model parameters to describe the elastic scattering process.
This was not possible, however, because each set of param-

eters gave equally good fits to the data.

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . .
LIST OF FIGURES o o o o o o o- o o o o o o o o

1. INTRODUCTION . . . . . . . . . . . . . . . .

2. INELASTIC SCATTERING OF 6Li FROM 58Ni AT 71.
2.1 Introduction . . . . . . . . . . . .
2.2 Experimental Procedures . . . . . . .
2.3 DWBA Analysis . . . . . . . . .
2.4 DWBA Results . . . . . . .
2.5 Coupled Channels Analysis and Results .
2.6 Conclusions . . . . . . . . . . . .

APPENDIX A - Elastic Scattering of 6Li from
73.4 Mev O O I O O O O O I O O O O O O O O 0

APPENDIX B - The Theory of Inelastic Scattering

Direct Transitions . . . . . . . . . . . . .

8.1 The Distorted Wave Born Approximation .
B.2 Calculation of Form Factors . . . . .

APPENDIX C - The Method of Coupled Equations . .

LIST OF REFERENCES . . . . . . . . . . . . . . .

iii

58

55

55
58

63
70

LIST OF TABLES

Table

1.

A1.

Data from the 71.2 MeV experiment. The cross
sections and uncertainties are in :mb/sr unless
otherwise specified . . . . . . . . . . . . . . .

Optical model parameter sets which give the best
fits to the elastic scattering data . . . . . . .

Nuclear deformations and deformation lengths of
this analysis with deformation lengths of previous
analyses. The values reported for this analysis
are the averages of the values obtained inde-
pendently from the five optical. model sets lof
Table 2. The numbers in parentheses under these
values are the standard deviations associated with
the averaging processes. Previously reported
values include the standard deviations in % and
number of observations, respectively, in paren-
theses under their average. For (p,p'), (a,a'),
and other reactions, 6 is the average of values
reported in references obtained from 20 and in
References 17-19. GAy’is the simple average of all
references incorporated in the six other columns of
previously reported values. Methods of analysis
used in extraction of 6 V include DWBA, Austern-
Blair diffraction mode , coupled channels Born
approximation, and measurement of electromagnetic
transition strengths . . . . . . . . . . . . . . .

Final values of the nuclear deformations,
deformation lengths, and one- plus two-phonon
mixing parameter BT obtained from coupled channels
analysis . . . . . . . . . . . . . . . . . . . . .

Elastic scattering data from the 73.4 MeV experi—
ment. The cross sections are in mb/sr . . . - - .

iv

Page

13

28

36

4S

LIST OF FIGURES

Figure

1.

Al.

A2.

Block diagram of the apparatus for the inelastic
scattering experiment. F +'Front, B + Back,
DYN + Dynode, AN + Anode, FC + Faraday Cup,
TC + Timer-Counter, CD + Current Digitizer,
PA + Preamp, DISC + Discriminator, PS + Pulse
Shaper, TSCA z'Timing Single Channel Analyzer,
TAC + Time-Analog Converter, G&D + Gate and
Delay, SC + Scalar . . . . . . . . . . . . . . .

Typical spectra for the inelastic scattering
experiment. The elastic peak is cut off for
clarity in the 20 spectrum . . . . . . . . . . .

Fits to the data for each optical model set using
8 scaling (solid line) and BR scaling (dashed
line). Data for which error bars are not shown
have uncertainties less than the size of the
points . . . . . . . . . . . . . . . . . . . . .

Fits to data using V = 60 MeV (solid line) and
V = 295 MeV (dashed line) optical model sets of
Table 2 when using 8 scaling. Data for which
error bars are not shown have uncertainties less
than the size of the points - - . - . . - . . . .

Nuclear deformations of this analysis along with
the ranges and averages of previous analyses. 0
for BRI (DWBA, B scaling), A for BR (DWBA, BR
scaling) ,+ and I for 8 (coupled channels, first
2 and 4 only). See ables 3 and 4 for related
values . . . . . . . . . . . . . . . . . . . . .

Comparison of ECIS (solid line) and DWUCX 72
(dashed line) for elastic (ratio-to-Rutherford)
and first two excited states - - - - - - - - - -

Block diagram of apparatus for the elastic
scattering experiment. For explanation of
abbreviations, see Figure l - - - - - - - - - - -

Typical spectrum for the elastic scattering
experiment 0 O O O O O O O O O O O O O O O O O O

Page

20

26

31

38

43

44

Figure

A3.

A4.

A5.

C1.

Three dimensional plot of x2 vs. V and W for a
grid on V a W using the geometry parameters of

Chua et al. (viz. V = 232 MeV, r = 1.3 fm,
a = .70 fm, W = 20 MeV, rI = 1.7 fm,
a1 = .90 fm). . . . . . . . . . . . . . . . . . .

Results of searches on optical model parameters
performed in fitting the elastic scattering data

Fits to the elastic scattering data provided by
each of the five sets of best fit optical model
parameters of Table 2 . . . . . . . . . . . . . .

Schematic illustration of the various single and
multistep excitation modes included within the
coupled channels formalism. The dashed arrows of
case (b) illustrate three alternatifi ways of
depopulating the 2 state (from Hillis ) -

vi

Page

46

52

53

67

1 . INTRODUCTION

The complex optical model potential has been used for
many years in the successful reproduction of elastic scatter—
ing angular distributions. Usually, several sets of optical
model (OM) parameters may be found to fit experimental angu-
lar distributions over the range for which data are gathered.
It is not frequently possible, however, to make a conclusive
statement indicating that a particular set of OM parameters
gives the most accurate description of an interaction. This
may be due to the interdependent nature of the parameters or
it may simply be a property of the potential.

Optical model analysis has also been logically extended
to the treatment of the inelastic scattering process and has
been employed to provide information on nuclear shapes. This
process is usually treated as a direct transition from the
ground state involving only the first derivative of the OM.
Also, it is most common to assume that the wavefunction
describing the projectile has the same form following an
interaction as it had prior to the interaction. This last
assumption is a part of the distorted wave Born approximation
(DWBA). With the advent of high speed computers in the mid-
1950's, implementation of these assumptions into a machine
code has led to generally successful studies of the inelastic
scattering of many projectiles with a wide variety of

energies from most known stable nuclei.

Advances in speed and efficiency of computers have made
possible more comprehensive approximations of the scattering
processes. In particular, indirect transitions which may
occur via multistep processes may now be treated directly in
the codes by use of the coupled channels method of analysis.
In this type of analysis, it is possible for any transition
in) a final state (or channel) to affect transitions to any
other state through the coupled system of equations which
describes the interaction. Coupled channels (CC) calcula-
tions of this type for the higher excited states of low lying
collective bands have proven quite successful in providing
improved fits to data when the DWBA was deemed inadequate.

DWBA for nine inelastic states and .CC for the first
vibrational band have been herein employed to describe 6Li

58Ni at a lab energy of 71.2 MeV. The

scattering from
nuclear deformation lengths extracted are compared to pre-
viously determined values and the fits to the data provided

by each method are compared and contrasted.

2. INELASTIC SCATTERING OF 6Li FROM 58

Ni AT 71.2 MeV
2.1. Introduction
a 1-5 . 0 3'4 . 6 o 0
Elastic and inelastic scattering of Li by medium
weight nuclei have been studied recently, but none of the
58

studies has investigated inelastic transitions in Ni with

projectile energies above 34 MeV. In the present work nine

58Ni at 71 MeV. Each

inelastic transitions were observed in
transition was treated as a one-phonon collective transition
in the DWBA using complex coupling. Since the deformations
of separate portions of the interaction potential (real,
imaginary, Coulomb) need not necessarily be equal, the calcu-
lations were performed twice with each set of optical model
parameters which fit the elastic scattering data.5 In one set
of calculations, the deformations were held equal, and in the
other set of calculations, the deformation lengths were held
equal.

Preliminary coupled-channels calculations were also
performed for the lowest energy, 0+-2+-4+, vibrational band
which coupled the ground state and first two excited states.
Investigations into multiple-plus-direct with admixtures of
one- and two-phonon transitions for the first 4+ state at
2.45 MeV were performed withcl-particle scattering as early
as 1962 by Buck6 and as recently as 1972 by Horen et al.7
These studies were prompted by the fact that this state does

not follow the Blair phase rule. According to this rule,

angular distributions corresponding to even values of angular
momentum transfer L should be out of phase with angular
distributions corresponding to odd L transfer. Also, angular
distributions with odd L should be in phase with the elastic
distributions.

90 6

For the Zr + Li reaction at 34 MeV 8

3 and 75 MeV,

sizable differences have been noted between extracted defor-
mation lengths and previously reported values. Anticipation
of these observations was one motivation in performing this

investigation.

2.2. Experimental Procedures
The MSU sector-focused cyclotron was used to produce an

6Li++

extracted beam of 200-300 nA of + ions. An arc-type ion

source9 produced the beam through the sputtering action of Ne

6L1. Electrodes of tantalum

on LiFl pellets, enriched in
(source life, approximately three hours) were used, but
hafnium was briefly employed in an attempt to increase source

life.lo

(Hafnium did increase source lifetime by approxi-
mately a factor of two, but on-target current was reduced by
a factor of approximately four.) Two analyzing magnets and‘
several quadrupole focusing magnets were used to give a
rectangular beam spot of approximately 2 mm x 4 mm on a foil
target of 1.02 mg/cmz, 99% enriched 58Ni. On-target beam
intensity of 10-50 nA at 71.2 MeV was monitored by stopping
the beam in a Faraday cup and sending the current to an Ortec

charge digitizer for charge measurement.

A detector with two resistive-wire, proportional
counters11 in sequence backed by a scintillator, placed in
the focal plane of an Enge split-pole spectrograph was used
to gather the data. Two-dimensional gating techniques (AE
vs. TOF (time—of—flight), AE vs. position, TOF vs. position,
and light vs. position) were used for multiple identification
of the scattered lithium ions. A PDP-11/45 on-line computer
was used for gating, display, and collection of data. A
block diagram of the apparatus is shown in Figure 1.

The low beam intensity restricted the range over which
it was possible to gather data to lab angles less than
approximately 45°. A FWHM 2 90 keV made it possible to
resolve nine inelastic states. Typical spectra are displayed
in Figure 2. The data were normalized using the 74 MeV
elastic scattering results of Huffman et a1.5 gathered at the
MSU Heavy Ion Lab using a 40-inch scattering chamber equipped
with two telescope-mounted AE—E silicon surface-barrier
detector pairs. The optical model parameter set of Table 2
with V = 160 MeV was used to calculate the elastic scattering
at E = 71 MeV. The data of this experiment for which

LAB
elastic scattering was observed (9 2 18°) were then

cm
normalized to this angular distribution. For angles below
18°, the elastic peak was not recorded (see upper spectrum of
Figure 2) in order to decrease dead-time in the detector.

The magnitude of the correction required to obtain normaliza-

tion of the data was significant mostly for angles greater

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ADC ’s

 

 

 

 

 

 

 

 

 

 

 

 

Strobe i
<———J
S C S C
Busy lnflbit
D

Figure 1. Block diagram of the apparatus for the inelastic
scattering experiment. F +-Front, B + Back,
DYN + Dynode, AN + Anode, FC + Faraday Cup,
TC + Timer-Counter, CD + Current Digitizer,

PA + Preamp, DISC + Discriminator, PS + Pulse
Shaper, TSCA + Timing Single Channel Analyzer,
TAC + Time-Analog Converter, G&D + Gate and
Delay, SC + Scalar.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

400 I T
_ 58Ni(6Li, 6Li’) 9° 72 MeV
EXCITATION ENERGY (MM ‘
g 300- 8 7 6 5 3 ' 3 r z I _
:4: - - +
U 3 2
{5 200- -
O.
U) ..
g .
8 IOO- -
o
250 500 750 IOOO
CHANNEL NUMBER
200 I r
58M (6U, 6Li’) 20° 72 MeV
EXCITATION ENERGY (MeV)
d _ 7 6 5 4 3 2 I o .
é
- +
5 3 2 (EELS.
5: I00- u
D.
(I)
g...
4' 4+
4“ 244‘ 2+
0
250 500 750 I000

CHANNEL NLMBER

Figure 2. Typical spectra for the inelastic scattering
experiment. The elastic peak is cut off for
clarity in the 20 spectrum.

 

 

 

 

 

moumhao.v monmmho.~ mmm.aq

monumma.~ Hansemm.~ mH¢.mm

moumfleo.a Houmamm.~ H¢~.hm

mcammmo.m moumomm.m ~mo.mm

moummaa.e co memc.a omm.~m

mcumeom.v co moma.a mom.om

«oumamm.H Houmwoo.¢ qom.m~

moummna.~ oo mmmo.a oce.-

Noumhm~.~ co mosa.m Ham.m~

HoImHNm.H co wooc.m mH~.m~ m.m mmq.o wem.a¢
HoImmoH.H co mm¢~.v mHH.q~ m.p mn~.o msm.mm
Noummsm.m oo mamv.~ mac.m~ ~.m mH.H mo~.pm
moumqam.m co mmm~.~ oam.a~ p.a mn.~ omo.mm
HoImmho.H co weem.m oam.o~ m.~ mm.~ mem.~m
Hoqmmmp.a Ho umm~.H mHA.mH H.H mm.» www.cm
Housmm~.~ Ho mamm.H «Hm.ma h.o ~.wH 45¢.mm
Manuema.~ Ho mesa.a «Hm.ha m.o m.a~ ahm.-
acumomp.a cc m~m¢.o oaq.ma m.c c.m~ mm~.m~
Hoummno.m ac mwma.a som.mfl m.o m.a¢ sma.m~
Hoummm~.m Ho u~H¢.~ mo~.ea o.o a.mp omo.q~
oo mov~.H Ho ummm.¢ ooH.ma «.9 m.pp~ ~am.-
Hoummmm.q Ho mamm.m mmm.aa m.o m.mma mmm.a~
Hoummam.e Ho mmhm.a Hmm.oa m.o m.~c~ emp.o~
Monmomo.m Ho mcvm.a om~.m m.o ~.om~ mmo.¢H
HoImHHm.m Ho mmaq.m Hmo.m m.o «.ome «mm.ma

ca ce\oc soc Amvoa cc\oc sum
+~ >0: ovme.a ofiummflm
.pofiuwoomm mmfizumcuo mmoacs um\ne
5 mum mmfiucfimuumocs can mcowuomm mucuo one .ucoewummxm 3.5 ~.:. 05 50: 3mm .H manna.

 

mo-flmmv.a monumwo.m mmm.a¢ moummm~.~ noanHm.m mao.av
moImamm.~ mausohm.m mmq.mm moummsm.m «cummop.m «sq.mm
«cummam.m mcumflsm.~ mp~.~m melam~¢.m «cumomo.~ mo~.sm
moummmm.~ mo-fl-¢.p mac.mm moummmm.m Noumhm¢.m mmo.mm
ma-fl~pm.~ moummhm.~ HH¢.~M woumosa.h «cumaeo.a qom.~m
moumcmm.~ mo-Em~a.m m~p.om mcumpmo.o monmhmm.m was.om
moumqm¢.e Noum¢m¢.~ «mm.m~ mcummHN.m Heuuhqo.a mmm.m~
monumem.q monumsm.~ mme.- moumham.m «annomm.q m~¢.s~
mcImmme.q mcummm~.H hmm.m~ mouuomo.m Noumflmo.q cmm.o~
melamma.a «clumco.~ mmm.m~ Noumpmm.~ Honmmhm.a ~m~.m~
moumsam.h moumsmo.~ ama.¢~ Noumwom.m acumoeq.m mma.qm
moummhm.s NOImemm.m mmo.m~ Noummmm.~ Hoummpo.v mmo.m~
mOImmhm.a Noummmm.v mma.a~ Noumhom.~ HOImoma.~ mma.H~
Noumsme.a Helmssm.a hmm.c~ Noumama.H Hoummmm.a ~mm.c~
NcImmm~.« Houmpam.~ mms.m~ nonme¢~.~ acummvc.~ cm~.md
noumapq.a moumh¢5.m ~mo.wa NOImmoo.m co mama.H www.ma
mo-@som.a Noumomm.m m~m.ka «cummmo.e co ammo.a mum.ha
Noumeqm.a «cumwom.o m~¢.ma «oumv¢~.h cc momH.H -q.oa
moummam.~ «cummma.m -m.mH «cumaao.m Hoummmm.m mam.ma
moummmo.m acummom.a mH~.¢H Noummmh.¢ acumomm.e ¢H~.qa
moumsom.~ Houmme~.~ MHH.MH Noummmm.m cc mmqo.d OHH.MH
Houmoso.a co ammo.~ moo.-
acummoo.~ acumsmp.m ~cm.oH Honmsch.a co mmoq.m mam.o~
Noummoa.s acummem.m mmp.m chmcom.~ co unsa.m qms.m
HoImmm¢.H Houuaaa.~ omo.m Houmomm.~ co unam.m mam.m
ca eE\oe Eco ca cexoe soc

N >m= mmhh.~

q >mz Hmmv.m

 

 

.A.w.ucoov H OHQMB

10

 

melmmma.m Nclmmhm.a omm.Hv mclmmmm.~ melamaa.m wa.ae
nonmamm.m NOIMmmm.N www.mm molmmqa.m Nonmmaa.~ omv.mm
mclmham.m mclmmhv.m cm~.hm mcIMNmH.m melummm.w mam.hm
McIm°~m.m NOImoom.m moa.mm moummmw.¢ «clacmm.m moa.mm
NoImHmm.H HoIMQNm.H mmm.~m NoIM¢m¢.H Nolmoom.m mam.mm
moImwmm.m Nolmhaa.w wmn.cm mclm~h¢.h NOImmvh.m ¢~>.om
melmbvm.a Helmmmo.a ~¢m.m~ mouflmmm.m Nolmmmh.m 5mm.mm
melmom~.H Helmamo.m mvv.hm «clmamc.a HOIMHQ¢.H o¢¢.hm
NOImcom.~ HOImNHm.v mvm.mm NonmoHN.N Hotmmmc.m Nem.mm
moImmvm.m Helmmvm.v www.mm Nclmmom.m Holmoaa.v m¢~.mm
Noumhom.m HOIMFom.N m¢H.¢N Nelmmna.m HOImMoh.~ vea.¢m
molmmmm.H Helmmvc.m hvo.m~ Nelmmcb.a HOIMmOh.H mqo.m~
Nclth~.m HOIMvmm.m mqm.H~ Nolmmmm.~ Holuhmm.m mwm.am
Nelmwmm.m HOIMQae.m mvm.c~ melammm.b Hqummm.m Hem.c~
Nelmhwm.w co m~m¢.H «wh.ma NoImhm¢.m cc mth.H mmn.ma
«elucmb.m co mm¢¢.H mmm.ma NoIMwm¢.w ca m¢HM.H mmo.ma
N¢IM¢Oh.m Holmm¢H.m wmm.hH «clummm.m Helmamm.m mmm.ha
NOImmmo.h co Hmmc.a ~m¢.ma Nclmmvh.m Holmoma.h mmv.ma
doImHmm.H co Maha.a mmm.ma Helmmca.a co mhmq.a mmm.ma
Helmvwm.a co MHHm.m m-.va HcImeoH.H co Meam.~ o~N.¢H
Melamho.a co mmhh.e QHH.MH Helumhm.a co mach.m maa.ma.
HoIM¢mm.H co M¢~N.m NHc.~H HoImemH.H co mmcv.~ oHc.NH
Holmmem.a co mmmh.m mom.oH HcIMQNm.H co mamo.m <om.ca
HcIMmhm.H co memo.m ocm.m HOIM¢o~.H co mmhm.a mah.m

HOImM¢¢.N oc mawm.m moo.m HoImHmH.N oo mmqm.m Noo.w

o< co\oo sec ea cc\oo soc

N >02 mmom.m

N >O= mhmo.m

 

 

.A.v.ucoov H manna

11

 

eouMmm¢.m ma-meH~.H mmo.H¢ Mennomc.a nonmamm.a ome.ae
moumaem.~ moummsm.m mm¢.mm moumcma.e «cuuaac.~ ssq.am

mcummma.m Nauumom.a oom.nm
mcumqoa.~ moImnmm.m mua.mm monmmmm.H mcummoa.m mHH.mm
mcumbmm.m Nouma~q.a mmm.~m moumoom.~ moummcm.~ ~ma.~m
moumomm.e «cumms~.~ m¢>.om «snakes.» Naqumm.o mqs.cm

moumhom.o monmcam.m cmm.m~
mcumeso.m «oummma.m mmv.s~ moumsam.p mouuhmd.p ~m¢.>~
mo-Mm¢a.m «cunnam.q amm.o~ NquHmH.H Noummwm.m «mm.u~
Noumamm.H ~cIqum.» oo~.m~ Noumom«.~ Houuomp.a qm~.m~
«cumsma.a HcImHm~.H cma.¢~ moumcmp.~ acumqmo.~ em_.e~
NoIMmH¢.H acum~ma.a amo.n~ Noumcmm.a acummoa.~ emc.m~
Noumm-.H Noum~m¢.m pma.a~ Noumomv.a Hoummqm.a ~ma.a~
~o-Em~e.a Heammqm.a mmo.o~ Noummmm.a Houuoo~.H omm.o~
«cumm~¢.~ Hauumsq.~ ~m>.ma Noumscm.~ Hoummmm.~ m¢5.aa
moummm~.m Houuose.m asm.ma NoumH-.« Nounsee.m mew.ma
Noumcc¢.m Houummm.~ mem.p~ Noummao.m Houmaem.o Hem.sa
Noummem.m Hoummms.~ ce¢.oa ~oumamm.¢ HcImmm~.m pm¢.ma
Nouma-.m Hoummwm.m mmm.ma Noumame.m Houumpm.m mmm.ma
Noum~m~.m Houmm-.o cm~.ea Noumaom.e Houmaom.¢ -~.qa
mcummam.o co unoH.H «ma.ma Noumamm.e cc mmmc.H Hma.ma
Noumsms.s cc mmec.a oHo.~H NcImHmm.m cc mamm.a mao.~H
NOImHHm.m cc M¢HH.H Ham.ca acumaoo.a cc mums.a mom.oa
moummoh.» Honmmmo.o mom.m acummmo.a co u~mm.a ~cm.m
Ho-mm>¢.a Hoummma.m hmo.¢ Hoummmn.a co mmme.a omo.o

o4 ce\oe Eco ca ce\oe sum

N >02 mmmm.m

w >w2 vomo.m

 

 

.A.©.ucoo. H OHQME

12

 

 

 

mcum¢mm.a noanmc.m ome.a¢ moummom.m «clavmo.m hso.aq
moumqmm.m Noummoo.H Ham.mm monumoo.a Noummmm.m mom.mm
mo-u~mq.m «cumaoo.m ~mm.pm mcumemm.m «anamh~.m «mm.sm
ma-Mm~¢.m Noumoom.H mea.mm moumom~.H Houmocc.~ H¢H.mm
~oIMmm~.H mcImwwm.m Hom.~m «cummmm.~ Houumam.~ omm.~m
monmmsm.m monumpm.m ohn.cm acumma¢.H HoImmm«.~ mo>.cm
mcumvmo.m moumwso.q mpm.o~ «cummom.~ Houmomh.e mmm.m~
moummas.o moImmmH.m o>¢.>~ NcImmHm.~ Houmooa.m c~¢.s~
«cummmp.a HoIm~co.H hem.o~ «cummmo.e oc msflo.a Hem.o~
moummma.m Hoummma.m www.mm «cumohn.m co usao.a Hh~.m~
moumnmo.~ Houmhmm.~ ona.«~ moImoom.m cc mmmc.H HAH.¢~
mcnmsoo.~ Hoummom.~ spo.n~ Noumqqe.m co macs.a meo.m~
moumaom.a Houummm.H ~>m.a~ «enumam.m co usem.~ som.a~
mcnmawo.~ Hoummmo.~ aeo.o~ «cummmfl.m co mmqa.m «om.o~
Noumem~.m Holmoho.¢ mop.ma woumcmm.m co ampa.m Hen.ma
moummoo.m Hoummdm.p Hmo.ma moummao.m co umHa.~ hmo.ma
Noum~pa.m flanneem.a hmm.na Houmvea.a co mam~.m mmm.ha
Noumohn.m Hoummcm.p qu.ma Houmomm.a co mooc.o meq.ea
«cummmm.s Houmama.s mem.ma HOImmmm.~ co ummm.> ~qm.ma
«cummom.m Hennamm.> o¢~.¢H Housemm.a oo mema.m hmm.¢H
moummm~.h co mamm.H mma.ma Hoummms.fl co meqH.m cma.ma
Houmomo.a oc mnmc.~ omo.ma HoummHm.H oo whoo.m «mo.~H
Houmoom.H co momm.~ aHm.oH Houmqmm.~ co mmea.m oas.ca
HoumomH.H oo node.~ Haw.a HoImMA¢.~ Ho mmom.H mom.m
HoImmme.H co unma.~ mo~.m Houmvmo.q He mmmm.H Hon.»
oq ee\oe soc o< ce\oe 20o
q >6: emh.q m >0: ms¢.«

+

 

 

.A.e.u:oov a manna

l3

 

 

 

Table 2. Optical model parameter sets which give the best
fits to the elastic scattering data.
V rR aR W rI aI
60.00 1.4315 .8351 17.5852 1.7828 .7790
110.00 1.2643 .8642 17.9915 1.7720 .7767
160.00 1.1727 .8672 18.7939 1.7577 .7837
225.44 1.0743 .8870 19.9029 1.7444 .7755
295.77 1.0063 .8899 21.5315 1.7210 .7802

 

than 30°. Therefore, it is assumed that corrections to the
small angle data would have been negligible and no further
investigations into normalization of the data were performed.
The accuracy of the data, unless otherwise specified, is t 5%

relative with an additional 1 5% absolute.

2.3 DWBA Analysis
Optical potentials with volume real and volume imaginary
terms of the standard Woods-Saxon form were used in DWBA

analysis of the data:

U(r) = -V f(r) - i Wg(r)

r-R
[l + exp (-3-B)]-lp

R

where f(r)

r-R
[1 + exp (—3—l)]‘1

I

9(r)

_ 1/3 _ 1/3
and RR - rRA , RI - rIA .

Added to this was the potential due to a spherically sym-
metric, uniform charge distribution of radius Rc'

We used a first-derivative, collective-model form
factor, with Coulomb excitation included, to describe the

interaction:

F(r) = FD(r) + Fc(r)

= _ dfgr) . dggr)
FD(r) [v RR dr + 1 w RI dr 1

15

L
BZZ'e2 Rc

Fc(r) ‘ (2L+1)r(L+l)'

 

r > R
C!

0 , r < Rc'

1/3

where Rc = 1.40 A fm and L is the angular momentum trans-

ferred. Coulomb excitation was significant mostly at small
angles.

Calculations of the theoretical differential cross
sections were performed on an XDS 2—7 computer using the code

DWUCK 72.12 In the collective model,

do
L - 2 do
(36‘)Exp ‘ BL (5§)Th .

Implicit in this model is the assumption that the deformation
parameter BL applies equally to the real, absorptive, and

Coulomb terms in the potential, i.e. SLR = BLI = BLC'.

However, by appropriate scaling of V, W, and the Coulomb

excitation scale factor, it is possible to set the deforma—

tion lengths 6 = BLR equal for each term in the interaction

L

potential, i.e. BLRRR = BLIRI = BLCRc'

to the DWBA analysis will henceforth be referred to as B

These two approaches

scaling and BR scaling, respectively.

When discussing deformations measured by different
experimental techniques, it is more common to compare defor-
mation lengths 5L = BLR. The nuclear matter deformation may
then be obtained from the potential deformation through the

relation BLMRM = BLPRP' where RM' RP, BLM' and BLP are the

mass and potential radii and deformation parameters

respectively. The choice of RR, RI, or Rc as R will be

P
discussed in the next section.

Investigations were performed to facilitate a propitious
choice for the matching radius and integration step size to
be utilized in the distorted wave calculations. These
investigations were performed on the elastic as well as the
inelastic cross sections. In the latter case, angular
distributions were obtained for integration step sizes
between .035 and .30 fm, for a matching radius of 14 Em and
between .05 and .30 fm for a matching radius of 20 fm. As
step size was decreased, the calculated differential cross
sections for the two matching radii asymptotically approached
the values obtained for a step size of 0.04 fm and .05 fm,
respectively. These observations were based on an angular
range of 00-1800. However, it was found that in the angular
range of our experiments, 50-500, the calculated angular
distributions were quite similar for integration step sizes
between 0.08 and 0.12 fm. Therefore an integration step size
of 0.10 fm was used in subsequent analyses. Using this
integration step size, angular distributions were then
obtained for matching radii between 15 and 25 fm. When
matching radii between 17 and 23 fm were used, the calculated
angular distributions were nearly identical. Therefore, in

subsequent analyses, we used 20 fm for the matching radius.

17

2.4 DWBA Results

The optical model parameter sets of Table 2 give the
best fits to the elastic scattering of Huffman et al.5
Results of the searches performed in fitting their data
displayed signs of both continuous and discrete ambiguities.
Each optical model (OM) set of Table 2 corresponds to the
best fit for each of the five "families” of parameters ob-
served. It was anticipated that our inelastic scattering
data would remove or reduce the observed ambiguities when
compared to DWBA inelastic angular distributions generated
using these OM sets.

The theoretical inelastic angular distributions jpre-
dicted by these five OM sets were very similar over the
angular range for which data were obtained. In particular,
the only observable differences were slight continuous
changes in the depths of the minima with increasing value of
the real potential depth V; These changes were most easily
observable in the 2+ states, but did occur to a much lesser
extent in the 3- and 4+ states. Also, these changes, even for
the 2+ states, were less than the uncertainty in the data.

The above observations hold for both 3 and BR scaling;
the difference between the two types of analyses being that
BR scaling gives slightly deeper minima and a slightly'
steeper overall slope for all states. Also, there is little
change in the quality of the fits to the data when switching
from one type of analysis to the other for a particular set of

OM parameters.

18

To illustrate the high degree of similarity between the
theoretical angular distributions calculated using different
OM sets the extremes of these calculations are shown in
Figures 3 and 4. Figure 4 shows the V = 60 MeV (solid line)
and V = 295 MeV (dashed line) fits to the data when the
B-scaling analysis technique is employed. In Figure BC the
V = 160 MeV fits to the data for B scaling (solid line) and
BR scaling (dashed line) are shown together. Five of the
states are fitted well by the DWBA but the predicted phase of
the oscillations does not agree with the observed phase for
the other four (viz. the 2.45 MeV (4*), 2.73 MeV (2*),
3.90 MeV (2+), and 4.48 MeV (3‘) states). For the 2.45 MeV
(4+) state, this problem is given more consideration in the
next section. .Alao, small-angle scattering (9cm 5 15°) is
not predicted well for any of the 2+ states.

Investigations were performed to find which element, if
any, of the interaction potential (real, imaginary, Coulomb)
was the major contributor to the cross sections. This
information could lead to a proper choice of RR, RI’ or RC to
be used as RP in calculating the deformation lengths of the
potential. Results showed that the imaginary term in the
interaction potential gave contributions to the cross
sections a factor of approximately three greater than the
contributions due to the real term for each of the three
angular momentum states observed. Also, Coulomb contribu-

tions to the cross sections were important only for the 2+

Figure 3.

19

Fits to the data for each optical model set using
B scaling (solid line) and BR scaling (dashed
line). Data for which error bars are not shown
have uncertainties less than the size of the
points.

20

 

 

IOO A V=60 MeV
.0 0
IO .. ’
o o
I
”\‘ O. “
’1 \ . O
I - f\\"“ . l.45(2"’)
1' MeV
rx‘i .\ _. ' O I \-
IO \,-. 5‘ ’ 2.45 (4*)
.. ’i " I "
.o’ . i - I
'0 ‘1- O f I‘ i
00 " 9 '
A 0.. .. 'i
{, ' O 9. ‘ 2.78 (2+)
.0 . -
E .
V O.
c, I o I 3.04 (2")
.0 -
\ I
B , . -; i
. \
1’ , _ f c 3.26 (2+)
. v C
I .Q. , a
I” i I - 3.62 (4*)
” .9
l0,’ 0’ f
I... d ‘9’
I \~/ \ O. \ 1 3.90 (2+)
0 \ ‘\
s’ \
0 o
\ "\
| \l \
O. \\ I“\
1‘ 4.4 -
.O'r ‘ 5(3 )
4.75 (4*)
I0'3 1 ' ‘ J
0 IO 20 3O 40
Handbeg)

Figure 3A

21

 

   
 
   

 

IOO , V=IIO MeV
’0 o
.0
I0 °
9
. o
l
o 9 \ o
I ~- ! ' ., , _l.45(2"')
!. 29" MeV
. 9 I " 9\
e "
IO _ f . - 2.45 (4+)
. i r , i -’
O... 0 if i ‘
IO 0 ‘ , f
.. O O u, ’
... . I
’2 J - +
€ I O o. ’ 2.73 (2 )
a I”
E
C: l ° «I _-‘ 3.04 (2+)
2 .-
e . ‘
I g 3.26 (2*)
.I
3.62 (4+)
00
IO I
. O
o , 3.90 (2+)
I o. 1‘“ "
3’ .
A
I V
0. ‘ \\’4“
’ ' 4.45 (3-)
.0l ‘
I 4.75 (4+)
-3 1 I l l
'0 0 IO 20 30 40
Gemideg)

Figure 3B

22

  

V=I60 MeV

  
  

  
 
 
 

 

' L45 (2")
f !0 MeV
- 0 °
IO ‘_ «‘0 ; 2.45 (4+)
.9 i \ i '\ i \"
0°. 5' i
'0 ‘ O. \J’ If ’\‘ I
00... . . \ “i, ’ sf
2 | , , , I,’ .- 2.78 (2*)
g o ,6 " _
E ‘5’ I’ \I
V I O ’ _ 3.04 ‘2...)
3 E I 3 ,2
b I \
'0 l - 3.26 (2*)
.I
3.62 (4*)
00
IO
. I
0° 1
I o O: 8 V
; o
I \”
o. \f"
’ 4.45 (3‘)
.0l
V 4.75 (4")
-3 4 l l L
IO 0 '0 20 30 40
Gemideg)

Figure 3C

23

 

   
  

 

r I ' I
IOO ~ V=225 MeV
'0 o
.0
l0 °
. O
u o ' -
O ’
r ”\ O . l .
IE-’, I 9 o ,7 - L45 (2*)
o ‘ I
14 , MeV
! . o i 9.
0
l0 . o 7 2.45 (4+)
i! 1 ,. f I“
O... O f‘ !
l0 ~’ \ o O \ i A I
00 ° I
A 0.. . ’ " . \i
a I 7 ‘. ”I 2.78 (2*)
\ I I
.a V
E S ' II
I ° ' +
g 4;} A 304 (2 )
3 ~- . ’
.0 l _ 3.26 (2")
.l
3.62 (4*)
’1 .0
IO VII
. I
o 4 f 3.90 (2")
I o 00 V -»
. \O’IO
' ..
O. \ IO.~
4.45 (3")
.0! . K V
’ 4.75 (4")
IO‘3 ‘ l 1 L
0 IO 20 30 4O
9c.m.(deg)

Figure 3D

24

 

 

I r T I
IOO V=295 MeV
.0 o
lo. .. °
0 o
I . ’
. -
o 0 4 O
I - 9 ', ; _l.45(2"’)
I i !.d if ‘I MeV
o . ’ .
O 4
'0 . o 2.45 (4*)
y 5 , i .
o . i I ~
.0 o I
l0 s’-\ O i ’ i
o. " °
0.. . ’ J . ’ i
t +
Q I t .. I," ; 2.78 (2 )
a I \’
E I
v O. . I +
c: ' ”I 3.04 (2 )
u U
s . . ’ '
| 5° . . ’ “f ,. 3.26 (2*)
. ‘1
.l ' 0°. -’
i , i _ , 3.62 (4*)
co ’
I0 0 o i ‘ ' I
0.. .o I
. V , 3.90 (2+)
| 00
\Ii“ ‘
o
4.
I \ I
.0 \t’.‘
4.45 (3-)
.0l ‘
’ 4.75 (4*)
-3 L l L l
9c.m.(d99)

Figure 3E

Figure 4.

25

Fits to data using V = 60 MeV (solid line) and
V = 295 MeV (dashed line) optical model sets of
Table 2 when using 8 scaling. Data for which
error bars are not shown have uncertainties less
than the size of the points.

26

 

‘ I r I
I"\
I \
IOO ~’ ,’\
.0 0
l0 .° °
0 r\
l
O -\
I"\ O . O
| ~ ’ , I l.45(2"’)
! a I. ‘1’ MGV
,\ o i
’l \ O ' O
"3%" o - 2.45 (4+)
v‘ 1 1 ~ '
- ., i f 1
. i

 

,5 , ’ . ,. 2.78 (2*)
\ o O ‘ ‘ ;
.o -‘ ‘
é ' 00 304 (2*)
g I- \i \ .3 .
\
8 0 fl. u i i 326 (2*)
l i. o I ‘ ‘
I ’..
i f 3.62 (4*)
O. i
I0 0 o '
0
4" O. 4’ U 1 \ 3.90 (2")
I O. »
- ' O
O O
.l
o. '
‘ 4.45 (3")
.0l
4.75 (4")
.3 1 l l l
GCOmIdeg)

Figure 4

27

.mcumcouum cowufim:Muu ofiuocmmEOHuomHo mo ucoEwusmmoE can .cofiumefixOummm zoom
maoccmco cwamooo .Hoooa cowuomuuufic ufimamIcuoumsd .amza moodocfi ><c mo cofiuomuuxo
5 com: mamaamcm mo mcofioz .mmsag wouuomou moofroum no masoaoo “.930 in
on» :H monouomuoocfi mmocououwu Ham mo mmm~m>m mamefim on» m“ .24c .malha moocwumumm
cfl now am EOuu cmcfimuno mmocouOMou cw cmuuom0u mmoam> mo manum>m on»
mg a .mcofluomou nonuo can .A.o.av ...m.mv mom .mmm~m>m uwozu Hows: mononucoumm
cfi .>Ho>fluoommou .mcofiuo>uomoo no umnasc can a so mcofiumfi>ov cumocmum on» mcoaocfi
mosam> couuommH >Hm90fi>mumw smomm000um mcwmem>m on» cufiz cmuMMUOmmm mcofiu
Imfi>mo cumccmum on» can mooam> ommnu Hows: mmmosucmumm ca mucosa: was .N manna
mo muom Hocoe Hmofiumo m>fiu can scum hauconcommccfi cocwmuoo mosam> on» no mova~m>m
on» EOum mum mwmaamcm was» uOu nouuomou mooam> one .momzamcm mzofi>mum mo mnumcoa
cofiumEHOMoo no“: mfimhamcm was» no mnumcoa cofiumsuOch can mcofiumEHOch umoaoaz

.m 032.

 

28

.m.mHV

“my

Amy

Amy

 

«4. mm. em. omo.o A+qvmh.e
Am~.hav Amy Acv .m.
am. no. «a. ~H.° Aum.m4.¢
A~.mv AHV Amy Amv
ma. ma. mm. ~mo.o A+mvom.m
Am.m~v Amy Amy Amy
mm. «N. am. meo.o A+4V~m.m
Av.mav Aqv Amy Ah.
mm. mm. m4. omo.o A+~vw~.m
Am.oav Amy Amy Amy
am. «m. as. omo.o .+~Voo.m
Amy Amy Amy
Hm. HH. 4H. Hmo.o A+mcmh.~
.4.4Hv any 24. A4.
Hm. Hm. hm. «mo.o A+4.wv.m
.mq.~av AH. .hv Am.
ma. FH.H H¢.H H~.o A+~cm4.a
w30w>wum ucwmwum ucwmmum
><o mm Hmn an

 

.m manna

29

 

A~.m

 

as. mm .+4.m5.4
Aa.4H. A~.4 Am.m~c AOH.~H.
A mp. . A mw.. . cm.a . ow. no. Aumvm4.4
.0: .03 .0 U .0 O
m n ma ma 5H. mH.I .+~Voa.m
mm. mm. 2+4V~m.m
«N.
a;
pm. as. am. A+mvmm.m
..0mH o3 2.0 we
A~.ma
aw. mm. A+quo.m
A.m.0v
Hm A+~Vms N
A~.mav
hm. o4. mm. .+4vmq.~
A.m.mv
A~.o~. AoH.NH.
oH.H mc.a pm.
..0ma.0ma. A.c.:c .xm .Hsoo
A~.4~c .4.m. 25.5v A~.mac .oa.m. .NH.~HV
NH.H Hw.a mm. cm. as. oc.H A+~om4.a
. . o o
A OmH Cody A. mm any A.U 0v A.w WV
muosuo m ..o.ovo A.m.mvo

 

 

.A.c.ucoo. m wanna

Figure 5.

30

Nuclear deformations of this analysis along with
the ranges and averages of previous analyses. 0
for 8R1 (DWBA, B scaling), A for BR (DWBA, 8R
scaling)“. and I for 8R1 (coupled channels, first
2

and 4 only). See Tables 3 and 4 for related
values.

31

 

'5 I l l r r T I I 1
L5"

I.4I- .

i;
I

 

I
__l
I

b
I

1

l

0.9 '-

 

 

0.8 '-

0.7 '-

0.5 '-
I.
0.4 '-

0.3 *-

DEFORMATION L ENGTH (fm)

I“.

DO
l-HDo
I—m

— — .
0.2 i

0.. - 1

01) l 1 l l l 5, l _1 I

 

 

 

I.4I ' 2.78 ' 3.26 3.90 ' 4.75

2.45 3.04 3.62 4.48
EXCITATION ENERGY (MeV)

Figure 5

32

states and then only for angles less than approximately 15°.
Thus, RI was chosen to be RP and subsequently used to
calculate the nuclear deformation lengths of the potential.
The imaginary radius has also been used to calculate deforma-
tion lengths for other targets at different 6L1 beam
energies.3’4

In Table 3, values of B, BR (from B scaling analysis

I
technique) and BR (from BR scaling) are listed along with the
6 values of previous analyses. The values of 8, BR, and BR.I
listed in Table 3 for this analysis were obtained by
averaging the five values independently obtained from each OM
set for a given state. The standard deviations associated
with these averaging processes are also listed in the table.

From Table 3 we see that the BR values are always about

I
20% greater than the BR values. In comparison to previous
determinations of 6, BRI is closer to GAV for five of the
states and BR is closer for four of the states. Also,BRI is
within the range of previous determinations of 6 for only two
states, whereas BR is within the recorded range for five
states. This is shown more clearly in Figure 5, where high,
low, and average values of 6 are shown with BRI and BR.

In addition, the standard deviations associated with the
BRI are larger than those associated with BR by a factor of
approximately three. Also noticeable for any given state is

a trend in the individual values of BR. (before averaging) to

I
increase with increasing depth of the real potential when

33

going from one OM set to another. No such trend was noticed
in the BR values.

The BR values differ from the previously reported 5
values, most noticeably for the 4+ states, but the difference
is less than with B scaling. BR values for the other states
do not agree well with the 6 values either, except for the 2+
states at 3.26 and 3.90 MeV. For the 4+ states, the BRI
values agree with the 6 values better than do the BR values,
but the differences are still quite significant.
Interestingly, note that BRI and 6 agree best (5% difference)
for the 3- state. Overall, though, the values of BR and BRI
of this analysis do not agree very well with the previously
reported values.

Significant also is the fact that the ratios of
BZRI/B4RI (8 1.67) and 82R/84R (8 1.75) are quite similar and
differ from the ratio of (52/64 (a 1.23) by about 40%. A
similar observation may be made from recent analysis of
lithium inelastic scattering from 90Zr.3 Thus, it would seem
inaccurate to scale experimental. lithium inelastic cross
sections for all observed states by the amount required to

obtain agreement with previously reported nuclear deformation

lengths for 3' states.

2.5 Coupled Channels Analysis and Results

Coupled channels calculations were performed using the

13

code ECIS. The optical potential used was that of Table 2

34

with V = 160 MeV, and the computational parameters were set
equal to those used in DWUCK 72: Rcoul = 1.40Aé/3, 85
partial waves, and integration to a maximum radius of 20 fm
in steps of 0.10 fm. The deformation parameters for each
portion of the potential were set equal (i.e. BR a BI a BC)
and the transition matrix elements were calculated
internally, in ECIS.

Only the ground state, 1.45 MeV (2+), and 2.46 MeV’(4+)
states were coupled in the present analysis. These states
may be effectively coupled in the first-order vibrational
model when the 4+ state is assumed to be an admixture of one-

6'7’15 This allows for direct

and two-phonon_ components.
transitions to the 4+ states, which are not usually included
in first order. However, the second-order vibrational model
was used because it was believed that reorientation matrix
elements would have a sizable effect on the calculated
angular distributions. In fact, this effect was subsequently
found to be negligible.

Initially, B2 and B1 were set to the values obtained
from the previous DWUCK 72 analyses. The mixing parameter
(BT) describing the mixing of one- and two-phonon components
for the 4+ state was initialized at 18.40 (10% 2 phonon and
90% l phonon). Searches were then performed on 82, B4, and
BT, independently, forcing a simultaneous fit to the 2+ and

4+ states. Then a simultaneous search on B2 and B4 was

performed and finally a simultaneous search on all three was

35

performed to optimize the fit (minimize X2) to the data.
These results are given in Table 4. It is seen that the
coupling has decreased 82 and increased 84.

Results of the coupled-channels calculations are shown
in Figure 6. For elastic scattering, the differential cross
sections for angles 15250 were identical to those predicted
by DWUCK 72, and for angles 2 25°, the differential cross
sections were slightly lower and somewhat out of phase. This
difficulty might be overcome by using the ECIS code to search
on the OM parameters while fitting the elastic and inelastic
scattering simultaneously. The fit to the 2+ data was good
except that the amplitude of the oscillations was too small.
Therefore, the fit to the data was not quite as good as that
predicted by DWUCK 72. The encouraging aspect of this
analysis for the 2+ state is that the small angle data
(9cm 5 15°) were fitted much better than the DWUCK 72 case.

Finally, the fit to the 4+ state was vastly improved.
The coupled channels analysis reproduced the phase of the
data extremely well, and the amplitude of oscillation sug-
gested a very good fit to the data. These were difficulties
which the DWUCK 72 analysis could not overcome.

Listed with the final ECIS-related values of 82’ B4 and
ET in Table 4 are the calculated values of BRI' The final
value of BT represents a mixture of 53% 2 phonon and 47%
1 phonon coupling for the 4+ state. The 82 value from the

coupled channels analysis does not agree with the previously

Table 4 .

36

Final values of the nuclear deformations, deforma-
tion lengths, and one- plus two—phonon mixing

 

 

 

parameter, BT, obtained from coupled channels
analysis.
2+ 4"
BL 0.161 0.0686
BLR 1.09 0.467

BT 43.75

 

37

Figure 6. Comparison of ECIS (solid line) and DWUCK 72
(dashed line) for elastic (ratio-to-Rutherford)
and first two excited states.

 

38

 

      

 

 

 

_ ' I ' I E
a: E E
b .. —— ECIS -
b I _ --- DWUCK 72 _=
S I“ -
$4 I X j
’ \
IOO a": / \ ‘E
I 4 2:.
A — ' '1
g. ELASTIC d
(D (0:. \ , _
\ : (I! \ E
E I ,5"‘o( J \ * I
v _ I" U 9 \_
I \ /\ \
SE; | :- l' \ ‘‘‘‘ o I ‘ ’t’i
E x \ I \ ..
\ :‘l \ ‘ 3e ‘l.45/Me¥_
. \ ‘
b " ‘ / ‘ an ‘
.0 \ I \ d
“x ,
: J \ 1 ’—
'.'.' §¢ \ ‘I
_ ‘\l \ _
0 \
IO‘Zg- 2.45 MeV ‘.
:. ‘4fl’ I xi:
: ‘1
-3 1 L 1 1
IO
0 20 40

Figure 6

39

reported value quite as well as that found with the DWUCK 72
analysis but the B4 value from ECIS does agree better.
Outstanding is the fact that the BRI values determined here
agree with previously reported values much better than does
either the B or BR scaling analysis techniques used in
DWUCK 72. Also, both BRI values are within the range of
previous values.

Results of the present work confirm the observation of

7 that the 4+ state at 2.45 MeV is an

Buck6 and of Horen et al.
admixture of one- and two-phonon transitions. Buck investi-
gated only the effects of multiple-plus-direct transitions
and obtained good agreement with his 40-MeV alpha-scattering
data when the direct two-phonon transition was enhanced by a
factor of 1.5 over theoretical predictions. With the
diffraction model of Austern and Blair, Horen et al. were
unable to simultaneously fit the magnitude and slope of their
alpha-scattering data at any of their experimental energies.
However, they were able to reproduce the phase of the
oscillations of their data” Coupled-channels calculations of
the present work reproduced the magnitude, slope, and phase

6Li data. Also, coupled-channels effects for the

of our
remaining 4+ states of this analysis at 3.62 and 4.75 MeV
appear to be much smaller because DWBA analyses reproduce

these distributions well.

40

2.6 Conclusions

No resolution or reduction of the OM ambiguities was
possible through .this DWBA analysis. Each of the five OM
sets used to describe the interaction gave virtually identi-
cal fits to the elastic data over the angular range for which
data were obtained and gave very similar fits to the

16'22 that more backward angle

inelastic data. It is believed
data for the elastic scattering could remove the observed
ambiguities.

Coupled-channels calculations were performed only as a
preliminary study to explore their possible effect on
inelastic states poorly fitted by the DWBA. It was found
that these coupled-channels calculations were able to more
accurately predict the phase of the oscillations for the
2.46 MeV’ (4+) state but only at the expense of reduced
quality in the fit to the 1.45 MeV (2+) inelastic scattering.
Finally, the deformation lengths obtained with the B values
extracted from the coupled-channels calculations agreed with

previously reported values better than did those produced by

the DWUCK 72 calculations.

APPENDICES

APPENDIX A

Elastic Scattering

Elastic scattering data were gathered using the 40"
scattering chamber at the Michigan State University Cyclotron
Laboratory. Cyclotron, source, and beam parameters were the
same as those for the inelastic scattering experiment except
that a beam energy of 73.4 MeV was employed. Initially, a
movable AE-E detector pair was used along with a monitor fixed
at 20°. In order to reduce energy dispersion with this
arrangement, the target was rotated with respect to the beam
by an angle half as large as the detector angle. Periodic
checks on the position of the beam axis relative to the
scattering chamber were performed by measuring the count rate
at equal angles on opposite sides of the beam. 'This system was
awkward because of the changes in effective target thickness
with angle, and because the checks on the beam axis consumed a
portion of the small (approximately three hour) source
lifetime.

Subsequently a new detector system was used with two‘AE-E
detector pairs situated at opposite but equal scattering
angles. Thus, the target was kept at right angles to the beam

at all times. Also, the orientation of the beam axis was

41

42

constantly monitored by observing spectra from each detector
as they were gathered through the use of a PDP 11/45 on-line
computer. Most importantly, this doubled the effective count
rate at each angle and provided a more efficient usage of the
beam time. All data presented in this study were gathered
using this latter technique. A block diagram of the apparatus
is presented in Figure Al and a typical spectrum is presented
in Figure A2. Accuracy of the data, unless otherwise
specified, is i 4% relative with an additional i 5% absolute.
The data are presented in Table A1.

Analysis of the data was performed using an optical
potential with volume real and volume imaginary terms of the

Wbods-Saxon form with a different geometry for each term:

U(r) = —Vf(r) - in(r)

r-R

 

 

where f(r) = [l + exp( RXl-l I
a
R
r-R
g(r) = [l + exp( a I)]-1
I
3 1/3 = 1/3
and RR rRA , RI RIA ,

Added to this was the Coulomb potential due to a spherically

1/3 fm.

symmetric charge distribution of radius Rc = 1.40 A

Calculations of the theoretical angular distributions and
searches on the parameters of the optical potential were per-
formed on an XDS Sigma-7 computer using the code SGIBELUMP.23

As for the inelastic scattering, a study of the matching

43

 

 

 

GE
III

TC

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ADC’s

 

 

1 Strobe
>

Busy

 

 

 

 

 

Figure Al. Block diagram of apparatus for the elastic scat-
tering experiment. For explanation of abbrevia-
tions, see Figure l.

 

44

 

 

 

 

 

 

 

.ucmefiummxw mowumuumom ofiummam ozu no“ Esuuoomm Hmon>B .~< magmas
. mum—>52 IEZdeo
0mm 00m Ooh cox. O_mm
I _ i m , u_ . _ _ y _ __ ‘ r , . l“
w , _ e l _ E h o.
3
I l O
.. n
T 1 N
n .. 1
n m S
n. l 00. mm.
1 L H
l L
I I O
m m V
"I ll m%U_ WW
3
u 1 1.
m m
m _ _ _ p m «o.

45

Table A1. Elastic scattering data from the 73.4 MeV experi-
ment. The cross sections are in mb/sr.

 

 

 

9cm do/dfl Aa(%)
7.72 8.83453 04 4.8
8.83 4.05413 04 3.1
9.93 2.40633 04 3.1
11.03 1.38763 04 3.1
12.14 7.23153 03 2.3
13.24 3.64503 03 2.4
14.34 2.22583 03 2.3
15.44 1.55923 03 2.2
16.54 1.04523 03 2.3
17.64 5.82043 02 2.5
18.74 3.01163 02 2.8
19.84 1.98943 02 2.6
20.94 1.68333 02 2.7
22.03 1.33493 02 2.7
23.13 7.80283 01 3.2
24.23 4.38163 01 3.2
25.32 2.32233 01 3.5
26.42 2.03163 01 3.3
27.51 1.95773 01 3.2
28.61 1.54253 01 3.5
29.70 9.59663 00 3.4
30.79 4.43273 00 3.5
31.89 2.40793 00 3.9
32.98 2.54533 00 2.6
34.07 2.76073 00 3.1
35.15 2.33023 00 3.3
36.24 1.46863 00 3.7
37.33 5.49953-01 3.8
38.41 2.76113-01 4.5
40.58 4.49803-01 5.0
42.75 2.62663-01 6.6
43.83 1.29973-01 6.4
44.91 3.06143-02 7.9
45.98 2 66313-02 6.6
48.14 5.45173-02 8.2
50.28 2.98153-02 8.8
52.43 4.79073-03 30.0
54.56 1.13593-02 11.0

 

46

. x’m =240

“ (V=232,W=20)
i
\0

O o‘ e ;
.‘.“‘ “‘.‘“““ \ xzm =27

2 vs. V and W for a

Figure A3. Three dimensional plot of X
grid on V and W using the geometry parameters of
Chua et al.1 (viz. V = 232 MeV, rR = 1.3 fm,
aR .70 fm, W = 20 MeV, r = 1.7 fm,

aI .90 fm). I

 

47

radius and integration step size was performed to find values
of these parameters for which the theoretical angular distri-
butions were approximately constant. First, matching radii
between 10 and 25 fm were examined with a resulting choice of
17 fm for future calculations. Using this value, the
integration step size was then varied between 0.05 and
0.15 fm with a resulting choice of 0.10 fm for future
calculations.

To become familiar with the optical model (OM) parameter
space which could fit the elastic scattering data, a grid
over V and W was completed using fixed geometry parameters

from the 50.6 MeV elastic study of Chua et al.1 The criteria

for a good fit was a minimum in the value of x2:
x2 . E’ {[c (e , -o (6 )J/Ac <9 n2
i=1 ex i TH i ex i

where N is the number of data points. In a three dimensional
plot of x2 versus V and W, it was very difficult to see the
minima due to their extreme depth compared to the surrounding
area of high x2 values. Therefore a suitably scaled plot of
l/x2 versus V and W was deemed the most appropriate for lucid
visualization of the results. This plot is given in
Figure A3. The 'zero" level of the z—axis in Figure A3

5 and the

corresponds to a x2 per point (xz/N) of 26 x 10
pinnacle of the highest peak corresponds to a xg/N of 6.8. It
is seen from the plot that the best fits to the data occur for
values of V between 130 and 220 MeV, and for values of the W

around 20 MeV} These results are borne out by the best fit OM

48

sets presented in Table 2. In fact, the peak which

2 (v = 170, w = 20 MeV)

represents the deepest minimum in x
corresponds closely with V and W for one of the best fit
parameter sets of Table 2. Also evident from Figure A3 is an
indication of the energy dependence of V and W for the
6Li + 58Ni reaction. The best fit values of V = 232 and
W = 20 MeV for E = 50.6 MeV are found to shift to V = 170 and
W = 20 MeV for 73.4 MeV scattering when the geometries are
held fixed.

To find the OM sets which produced the best fits to the
data, SGIBELUMP was used in a search mode which minimized x2.
Three distinct sets of preliminary searches were performed
using the 50.6 MeV parameters of Chua et al.1 as a starting
point. For each set of searches, a grid was performed on the
real well depth V between 10 and 300 MeV in 10 MeV incre-
ments. In the first set of searches, the OM parameters were
varied in pairs (r

rR, W; a aR, W) followed by a

R' 31’
as a group and finally varying

RI r1;
combination varying W, r1, aI

r aR, W, rI, aI as a group. This sequence was initially

RI
completed for the V = 160 MeV data point, the results of
which were then used as starting parameters in the V = 150
and 170 MeV search sequences. The results of the V = 150 MeV

search sequence were then used as starting parameters for the

V 140 MeV grid point. This algorithm was continued down to

V 10 MeV and similarly from V = 160 MeV up to 300 MeV.

49

The second set of searches employed the same search
sequence as the first set, the difference between the two
sets of searches being that in the second set the 50.6 MeV
parameters were used as starting parameters for each grid
point, The values of the OM. geometry parameters were
slightly different between the two sets of searches with a
noticeable difference between the values found for the imagi-
nary well depth W. For V = 200 MeV the difference was most
notable in that the second set of searches gave values of W
approximately 50-60% lower than the first set of searches.
Values of x2 were comparable.

The third set of searches was the'most lengthy. The
50.6 MeV parameters (except rR) were used as starting param-
eters for each grid point. The value of rR was held fixed
during the majority of this set of searches at 1.26 fm in an
n ambiguity. This value of r is the

R R
average obtained in the second search sequence. The

attempt to avoid the Vr

remaining 0M parameters were searched upon in pairs (aR, r1;

aR, W; aR, aI) followed by a simultaneous search on W, r , a

I I
as a group. The OM parameters which gave the best fit to the

data from this search sequence were then used in the same
search sequence for each data point. The results of this
second run-through of the sequence were then used as starting

parameters for a search on r followed by a simultaneous

R

search on r , aR, W, r a for each data point.

R I'I

50

Finally, the results of all three sets of searches were
examined and combined to form a composite grid representing
the best fits to the data. The OM parameter sets for each
grid point were used as starting parameters for a search on V
followed by a search on all six OM parameters. Most of the
grid points remained approximately constant in x2 and
retained OM parameters very near their starting values. How-
ever, there were five distinct areas in which x2 was
considerably reduced. For each of these minima, a five
parameter search was then performed while holding V fixed for
grid values in steps of five MeV for a total of 30 MeV’in each
direction from the minima. Starting parameters for each of
these grid points were the OM parameters (except V) for the
nearest value of V representing one of the five minima in x2.
It should be noted that in all of the above searches, the
uncertainty for the data point at 9cm = 52.40 was artifically
increased from 18% to 30%.

The results of all the searches performed are shown in
Figure A4. From the plot of xz/N versus V, we observe the
familiar continuous and discrete ambiguities. The values of
the OM parameters for the five minima in x2 are given in
Table 2 and their respective fits to the data are displayed
in Figure A5. From the plots of Figure A5, we see that there
is no possible resolution of the OM ambiguities represented

in Figure A4 over the limited angular range for which data

were gathered. It was anticipated that comparison of the

51

Figure A4. Results of searches on optical model parameters
performed in fitting the elastic scattering data-

52

 

Ogr'thr'VirVT'ViTVTIfiUrV'rTTTi'
o

 

 

 

 

 

 

 

 

oon ooooo
.- 0.8 00 o 0“,. no 009 009° .
O a, 000000 “000°0 000 “no" 0°90
0.7
l.9 .0, -
l. “o .
L—H '3 00000 0 0000900 09000 00000000009 cocooooooooo‘
. o o
l.6- . ° ”.0 °o .
o
)- o o J
30 P 000 0 0 j
3 20- mono ”00°00 0900“ ooooooooo°°°°°°°°°°°°°°°-
IO 0°
0'9 P 0900000 90 °°°°°oa°°o°° a°°o ('00 ooo°°°oor
O 0.8 P0000 0 000 0000 ..
0'71.
r “o +
L4. 0 a. .
. "coon
- 0 -
L.“ '2? °°°°°°°° o ””009“ a “on
IO- 0 °°°°oo
0 o
,4. o o o no ..
_ a a a a _
E : o 0 0° 0 no °o on
N " 0 0 0 -I
>< | i°i°9030 .‘ao: LL1°202 Li. . :“g {or LL03°209

lIOO I50 200 250 300
V (MeV)

Figure A4

 

53

 

T
d
q
-
-—|
d

I I In"

V860

L\
nu I In"

VSHO

“Trina

\\

\FHSO

 

V8225

 
   
  
   

':

m

\

DI ‘
35

of E
E” V

'U

U I lllllll

.Ol

I l VIIIIII

 

———‘

.II'li—n— I-IIIII—-

———-_

8

'0'50 :0 20 so 40 50
9c.m.(deg)

Figure A5. Fits to the elastic scattering data provided by
each of the five sets of best fit optical model
parameters of Table 2.

 

54

experimental inelastic angular distributions with those pre-
dicted by the above OM sets would remove or reduce the OM
ambiguities. This was found not to be the case and it is

believele'22

that the OM ambiguities may be resolved by
obtaining elastic scattering data at more backward angles

and/or higher energies.

APPENDIX B

The Theory of Inelastic Scattering

via Direct Transitions

8.1 The Distorted Wave Born Approximation

Consider the Hamiltonian:

452 2 +

a = -75 + U(r) + Hg - V(rf,§) (l)
where r represents the position of the projectile and i;
represents the internal degrees of freedom of the target
nucleus. The optical potential is U and Hg is the Hamil—-
tonian which describes the internal motion of the target.
The kinetic energy is expressed in terms of the reduced mass
of the system and V is the interaction potential involving
the final state channel :f. This total. Hamiltonian has

solutions W to the Schroedinger Equation

RV = 3? (2)
Also, for the target Hamiltonian Hg, we have

Hgvv(€) = evv(E) (3)

where the subscript v on the solutions may refer to either

55

56

the initial state i or the final state f. In the asymptotic

region, the solution to (2) looks like

.‘l: .‘* 'k
1 . r. 1 fr

f
- z e (4)
f Afvf(€) -—FE—‘

(+) +
T vi(E)e

where ki and RE refer to the relative momenta of the system in
its initial and final states respectively.

The differential scattering cross section for inelastic

 

scattering may be derived24’25 from (2) and is given by
9%=<“ >212 211-12 <5)
d Zflhz ki Av fl
2

where the T are amplitudes of the scattering matrix and

fi Av
indicates a sum over final spin states and an average over

initial spin states. Specifically, Tfi is given by

Tfi = <vf<€)xé"(fif.?f)IV(?f. £)|¥‘+’> (6)

This is an exact solution. The distorted waves x(-) in (6)
describe the elastic scattering of the projectile as

prescribed by

2 2n

2 (‘) -
[Vf - 4&7 U(rf) + kaX - 0 (7)

f

If we knew the solution ‘1'”) to (2), we would be able to
calculate Tfi exactly. However, this is not the case and we
must therefore make an ansatz of the scattered wave function
that will produce a good approximation of the transition
matrix.

A standard method is to assume the distorted wave Born

(+)

approximation, for which Y is approximated by

57

(+) _ (+) + +
The transition matrix thus becomes
- + + + + +-
Tfi = <vf(E)xé )(kf.rf)lV(rf.€)|vi(€)x§ )(Ii.ri)> (9)

Assuming that the wave functions do not change appreciably
over distances on the order of ranges of the nuclear poten-
tials which make up V, we may make the zero range approxima-

tion: rf = r1 = r. The transition matrix thus becomes

Tfi=Id3r xé') (if.?)<vf(€) IV(?.£)|vi(§)>xi‘+’ (121.?) (10)

We may now go about finding the matrix elements of the inter-
action <vflVlvi>. Expanded into multipoles, V becomes:

V(?,€) = Z .2 m . *
z'm Vlmtl Y2(r)] (11)

must behave under rotations of coordinates like

the spherical harmonics 32(3): and have parity (—).2'. The

where the Vim
factor 12’ is included to insure the reality of the nuclear
reduced matrix elements. Also, we must now make formal use
of the spins J and projections Mj of the initial and final
state. Employing (11) in (10) and making use of the Wigner-

Eckart theorem, we find:

<vf|VIvi> <JfoIVIJiMi>

Z 2 m A *
£'m<JfM' JiiMim><JfI|V£mllJi>[j_ Y2 (1')] (12)

 

58

The reduced matrix element is now a function of radius only
and is usually referred to as the product of a strength A2 and

a form factor F2:
A£F£(r) = (JfllvimllJi> . (13)

Then Tfi becomes:

- z
Tfi — £m<Jfo|JimMim> Bzm , (14)
where
__ 3 (-)*+ + .9. ma (4») ‘* +
32m = Id r Xi (ki,r)A2F£(r)[l Y2(r)]*xf (kf,r) (15)

Summing over final and averaging over initial spin states in
Tfi gives a term of (2Jf + l)/(2Ji + l)(22 + 1), so the

differential cross section (5) becomes:

2
kf 2Jf+1) 2 leml (16)

d0=( u )2 __ (______ ______
an 2Wfi2 ki 2Ji+l) £,m (22+l)

8.2 Calculation of Form Factors

All that now remains in order to get an accurate expres-
sion of the differential cross section is to calculate the
form factor. Ihl the collective model used, the rotational
and vibrational model form factors are equivalent to first
order in the deformations. Therefore only the derivation of
the rotational model form factor is herein presented for
even-even nuclei.

It is reasonable to assume that the total optical poten-
tial strength depends only on the distance of the projectile

from the nuclear surface

59

U E Ufr — R(G',¢‘)] (17)

where R is a nonspherical surface with body fixed coordinates

0', 4' and is usually given by:

3(0',¢') = ROEl + ggoqug(e',o')3 . (18)

The strengths “kg for a quadrupole deformation, k = 2, are
related to the familiar deformation parameter B and asymmetry

parameter Y'by

0120 = Bcos y; 02:1 = 0; 012:2 = Bsin Y/ 2 (19)

Assuming axially symmetric nuclei requires q = 0, Y = 0 and

(18) becomes:
I .. o I
R(e ,0) - ROEI + EB£Y£(9 ,0)] (20)

A Taylor-series expansion of the total optical potential

about R = R0 yields:

 

8U(r-Ro) 1 2 82U(r-Ro)
UopU: " R0) = U”: ' R0) ' 5R——a-r'—-+ 55R 3 r2 +..
(21)
where
6R = (R - R0) = R0 §B£Y§(e',0) (22)

The first term of the expansion (21) is identified with the
spherical optical potential used to describe elastic scatter-
ing. The other terms are identified with inelastic
scattering. Assuming the term of order 32. to be the major

contributor to the inelastic scattering, the interaction

potential (11) becomes

60

= z .2 d0 0 .
v 2 1 R082 3;1Y£ (0 ,0) . (23)

Converting to space-fixed coordinates via the spherical

harmonic addition theorem,

2 A A
Yim'm) filggmiz Y’funfi‘m . (24)

(where E are the polar angles of the nuclear symmetry axis)

 

yields
a g .2 gg 4H m* A m
V 2,m1 R082 dr V2111 Y2 (r)Y2(§) ' (25)
Comparing (25) with (11), we see that the Vim values are
given by
2 A
v = i R 82 9-9- 4" Yfi‘ug) (26)

2m 0 dr 45;:1‘

Using separation of coordinates for the initial and final

wave functions, we can say

a O ‘
”1(5) Y°(§) 0
Mf A
vf(€) = YJ (5) ° ¢ (27)
f
For rotational excitation, the radial components 4) of the
nucleus remain constant and therefore do not contribute to
the form factor. Thus, using (26) and (27), the reduced

matrix element of the potential becomes:

_ .1 4n dU Mf* A m A o A
AgFg(r) - 17m ROB: Egi YJf (Snzunoumg.
Or,
_ .2 -1/2 dU

f f

61

For the present analysis, we use an optical potential of the

 

form
Uop = -[Vf(r) + in(r)] , (29)
where
r-R
f(r) = [l + exp(-——B)]-l ,
a
R
and
r-R
g(r) = [1 + exp( a 13.1
I
. 3 1/3 _ 1/3
with R rRA , RI - rIA . Thus (28) would have two

terms in the derivative and should look like:

_ _.£ -l/2 dfgr) . dggr)
AzF2(r) ‘ 1 (22 + 1) [BBRRV dr + lslew dr Joafzamfm
(30)
Using (30) for the form factor and the expansion of Xé+) into
partial waves:
(+) + + a 4n 2 .1. M A M* 31
xi (“r”) kfr LM 1 XL (krr’ YLmYL (Rf) ( )
along with the time reversal relation:
x“’*(E.‘r*) = x”) (42.?) (32)

the 82m values may now be determined and the cross sections
calculated.

As stated earlier, derivation of the first order vibra-
tional model form factor for even-even nuclei would give a

result equivalent to (28). The difference being that 82

62

would be interpreted as the root mean square deformation

in the ground state due to zero-point oscillations:

2 z
82 <mlalmlz>

(29. + l) (ml/cw

where aim are the phonon operators of (18),“hw2 is
the energy of each phonon, and CR is the restoring-force

parameter.

APPENDIX C

The Method of Coupled Equations

26

Following the notation of Hillis, the Hamiltonian for

the reaction A(a,a')A' is
H =tr4-H(€)-+\n?,€) (34)

where T is the kinetic energy operator of the incident pro-
jectile, H(g) is the Hamiltonian which describes the internal
motion of the target and projectile, and V(?,€) is the total
potential describing the interaction between the target and
projectile. The coordinates E are vector quantities
describing the nuclear surfaces of the target and projectile.

The Schroedinger equation H? = E? may then be written as
[T + 0(a) + v(?.s)lv(?.£) = EW(?.€) (35)

where the energy E is the total energy of the system. As in

the solution to the Schroedinger equation presented for the

 

DWBA
3(E)Wd(€) = ana‘g’ = (Ea + EA)wa(ap)wA(gT) . (36)
Thus, the total energy of the system is
fizkz
E = 3 + 2 a (37)
a "a

63

64

Introduced here is the notation for the entrance channel
a = a + A. All possible exit channels will be represented as

a' = a' + A'. Thus Ea = Ea + E is the reduced mass of

A’ ua
the system, and kc is the wavenumber of relative motion.
The solution to the total Hamiltonian may then be written

as
W(?.£) = gwa(£)xa(?) . (38)

where x ('1') describes the relative motion of the target and

a
projectile. Multiplying the Schroedinger equation for the
total Hamiltonian from the left by ¢;,(€) and integrating over

5, one obtains

N + +
(E - Ea - T)xa(?) = X V§.a(r)xa(r) . (39)
a=1
*
where Vd,a(?) 5 f¢a.(€)V(?:€)Wa(§)d€ , (40)
+ - < '| I >
or Vo'c(r) - a V a - (41)
Now, Vfi.a(?) may be separated into diagonal-plus-off-diagonal

elements so that the Schroedinger equation looks like
E N I 7 42
E - Ea - T - Vda(r)]xa(r) = z Vd.a(r)xa(r) . ( )
afa'
Using separation of variables in a partial wave expansion of

X0 gives

 

U (I) .
xa(r) = fig at Y§(9) . (43)

Implicit in this expansion are the assumptions that the

incoming projectile may be represented as a plane wave and

65

that no exchange of nucleons occurs between the target and
projectile.
Using (43) in (42), multiplying (42) from the left by

I
Y%.(0') and integrating over 9' gives

U (r)
' . - + E .2... M .
(5:240 )[E - Ea - T vaa(r)] w r YL(n)d0
. N _, U .(r)
= Iy’fiun') a'ic vamm “r fifmww . (44)
LM

The result of this integration gives a set of coupled equa-
tions for the radial functions U(r) of the scattered particle
for each total angular momentum and parity of the system. For
each exit channel 0' they have the form

2 La(La+1)

[L________+ 1‘2 “V (r)]U (r)

drz r2 0. 00. a

- 2

- “If“ Vaughn (Ua:(r) I (45)
where

va'a(r) = §§ EM I Y§:(Q)Va:a(;)Y%(Q)dQ . (46)

This equation specifically shows the infinite system of
coupled equations which must be solved to determine the
wavefunction xa. In practice, the number of off-diagonal
elements in the coupling potential Vd,a is limited to a few of
the low lying bound states which compose the majority of the
inelastic cross sections, or which are mainly excited by
multistep processes. The off-diagonal elements of Vd.a

correspond roughly with the interaction potential V(?f,g) in

66

the Hamiltonian of equation (1) used to derive the dif-
ferential cross section in the DWBA analysis of Appendix B.

The diagonal potential Vac is the usual complex optical
model (OM) potential describing elastic scattering. However,
the elastic scattering is not independent of the off-diagonal
matrix elements of Va'a due to the coupling action of the
system of equations represented by (47). As shown by (b) in
Figure C1, a target nucleus may be excited and undergo a
subsequent de-excitation to the ground state while the
projectile is still within range of the interaction potential,
thus contributing to the elastic cross section. For this
reason, elastic and inelastic scattering should be considered
simultaneously when searching for an optical model (OM) to
describe the elastic scattering process.

The mode of elastic scattering represented by (b) in
Figure Cl is not possible in the direct interaction theory
used in first order DWBA calculations. To account for this
possibility in DWBA, second and possibly higher order terms in
the perturbation theory must be employed. The calculations
and necessary computer programming, however, become
increasingly difficult for additional orders in the Born
approximation and this alternative is thus discarded. The
coupled channels (CC) method of calculation, which includes
these effects, is the best alternative to the standard DWBA
method.

The off-diagonal matrix elements of Va'a contain all the

information on the inelastic scattering process and therefore

67

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

:11
A M
II
II
II
II
2* )1)
/"\ HMI'F A) M
II ||
II II
II ||
" n
J: 4) II
ll 4)
II
II
o i)
(a) (b) (c) (d) (0)
Figure C1. Schematic illustration of the various single and

multistep excitation modes included within the
coupled channels formalism. The dashed arrows of
case (b) illustrate three alternathfi ways of
depopulating the 2 state (from Hillis ).

68

must be chosen very carefully. A mathematical derivation of
the matrix elements will not be herein presented, but an in-
depth solution is presented by Tamura.27 Tamura concludes

that

Voua = télvit)(r)<Illoft)!II'>A(£jI,£'j'I',AJS)

where v£t)(r) contains all the optical model dependence and A
is a geometrical factor. The reduced matrix element of Q{t)
between initial I and final 1' spin states contains all the
information on nuclear structure. Specifically, Ogt) is
either the phonon operator of the vibrational model or the
multipole moment operator of the rotational model. For
example, in the first order vibrational model Q{t) is a sum of
creation and destruction operators.

As in elastic scattering, the system of coupled equations
represented by (46) gives rise to inelastic transitions which
occur via multistep processes and cannot be calculated with
standard DWBA techniques. Examples of this type of transition
are represented schematically in (b)-(e) of Figure C1.
Multistep processes of this type are most important in nuclei
which exhibit strong collective natures in the low lying
levels. Specifically, the higher phonon states of the
vibrational nuclei and the higher angular momentum states of
rotational nuclei are the most likely to be affected.

Another contribution which is more often important in

heavy ion scattering is the "reorientation” effect. The

69

incoming projectile excites a given state in the target
nucleus. Then, while still within range of the interaction
potential, it causes a reorientation or change in the rate of
;precession of the nuclear spin vector. 'This process is
represented schematically by (e) of Figure C1.

The method of coupled channels thus gives.a more accurate
representation of the interaction mechanisms between target
and projectile for nuclear bands *with strongly' collective
natures. The standard DWBA is still a good description when
used as intended: to explain transitions to excited states

that occur primarily through single step processes.

LIST OF REFERENCES

 

10.

11.

12.
13.

14.

LIST 03 REFERENCES

L.T. Chua, F.D. Becchetti, J. Janecke and F.L. Milder,
Nucl. Phys. A273, 243 (1976).

R.I. Cutler M.J. Nadworny, and K.W. Kemper, Phys. Rev.
C15, 1318 (1977).

R.J. Puigh and K.W. Kemper, Nucl. Phys. A313, 363
(1979).

K.W. Kemper, A.F. Zeller, T.R. Ophel, D.F. Hebbard, A.
Johnston, and D.E. Weisser, Florida State University, to
be published.

R. Huffman, A. Galonsky, C. Williamson, and R. Markham,
Michigan State University, to be published.

B. Buck, Phys. Rev. 127, 940 (1962).

Daniel J. Horen, J.R. Meriweather, B.G. Harvey, A.B.
de Nercy, and J. Mahoney, Nucl. Phys. 12, 97 (1965).

R. Pardo, R. Markham, W. Benenson, A. Galonsky, and E.
Kashy, Phys. Lett. 718, 301 (1977).

E.D. Hudson and M.L. Mallory, Nucl. Sci. NS-23, 1065
(1976).

P.S. Miller, H. Laumer, M.L. Mallory, and J.A. Nolen,
Jr., Proceedings of Particle Accelerator Conference of
March 1979 in San Francisco, CA, to be published in
Proceedings of IEEE Transactions in Nuclear Science June
1979.

L. Robinson, Annual Report of the Michigan State Uni-
versity Cyclotron Laboratory, 1977-78.

P.D. Kunz, University of Colorado, (unpublished).

J. Raynal, Centre d'Etudes Nucleaires de Saclay,
(unpublished).

D.L. Hillis, E.E. Gross, D.C. Hensley, C.R. Bingham,
F.T. Baker, and A. Scott, Phys. Rev. C15, 1467 (1977).

70

 

15.

16.

17.

18.

19.
20.

21.

22.

23.

24.

25.

26.

27.

71

N. Lingappa and G.W. Greenlees, Phys. Rev. C_2_, 1329
(1970).

D.A. Goldberg, S.M. Smith, H.G. Pugh, P.G. R003, and
N.S. Wall, Phys. Rev. C1, 1938 (1973).

R.H. Bassel, C.R. Satchler, R.M. Drisko, and E. Rost,
Phys. Rev. 128, 2693 (1962).

H.W. Broek, J.L. Yntema, B. Buck, and C.R. Satchler,
Nucl. Phys. 64, 259 (1965).

G.R. Satchler, Nucl. Phys. 10, 177 (1965).

S.A. Fulling and C.R. Satchler, Nucl. Phys. A111, 81
(1968).

C. Michael Lederer and Virginia S. Shirley, Table of
Isotopes, Seventh Edition, John Wiley & Sons, New York,
1 7 , pp 170-171.

D.A. Goldberg and S.M. Smith, Phys. Rev. Lett. 22, 500
(1972).

R.R. Doering, A.I. Galonsky, R.A. Hinrichs, J. of Comp.
Phys. 12, 498 (1973).

N. Austern in Selected Topics in Nuclear Theory, IAEA,
Austria, 1964.

C.R. Satchler in Lectures in Theoretical Physics,
V01. 8C, University of Colorado Press, Boulder, 1966.

D.L. Hillis, Ph.D. Thesis, University of Tennessee, 1976
(unpublished).

T. Tamura, Rev. of Mod. Phys. 31, 679 (1965).