E-NEUXSTIC PROTON SCATTERING FROM 333 BA AND .144

SM AND rrs MICROSCOPIC INTERPRETATION f,   ,, .3." 195,1;

fh‘esi-s for the Degree of Ph. D; f  
woman STATE UNIVERSE?!
DUANE CLARK LARSON

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ABSTRACT

INELASTIC PROTON SCATTERING FROM 138BA and 1445M
AND ITs MICROSCOPIC INTERPRETATION

BY

Duane Clark Larson

Differential cross sections for elastic and inelastic

138 1“Sm have been

scattering of 30 MeV protons by Ba and
measured. The total resolution for the inelastic peaks was
7-10 keV FWHM, which permitted the observation of 20 excited

133 144Sm below Ex=3.4 nev. and

states in Ba and 18 states in
measurement of excitation energies accurate to 2 keV or less
for these states. Based on characteristic shapes derived
from angular distributions to states of known J", spin-parity
assignments are made for the majority of the observed states.
Collective model DWBA calculations were performed and deform-
ation parameters extracted for all states of assigned J".
Microscopic DWBA calculations which included the exchange
amplitude were performed for the 2;,2' 41,2 and 61,2 states
in both nuclei, using large-basis shell model wave functions
to describe the nuclear states. These wave functions also
provide an excellent description of the excited states in

138 144

Ba, and a good description of the energy levels in Sm,

as measured in our experiments. The two-body force used in

Duane Clark Larson

The inelastic scattering calculations was obtained from a
recent survey of inelastic scattering analyses. Polarization
charges for the nucleons were extracted, and found to be
essentially state and multipOle independent. Two sets of
shell model wave functions were employed for the 1383a
calculation, and it was found that inelastic proton scatter—
ing clearly distinguished between the two sets, thus providing
a sensitive test of the wave functions. Careful consid-
eration of the transition densities derived from the wave

functions enable one to directly study the pr0perties of

the wave functions.

INELASTIC PROTON SCATTERING FROM 1383A AND 1448M

AND ITS MICROSCOPIC INTERPRETATION

BY

Duane Clark Larson

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Physics

1972

®\ ACKNOWLEDGEMENTS

I would like to thank the entire Cyclotron staff
for their many kindnesses which made these experiments
possible. Specific thanks are due the following peOple:

Dr. Stan Fox for his cool head during
difficult experimental times;

Drs. Helmut Laumer and Lola Panaggabean for
standby help during data taking;

Richard Au for immeasurable computer assistance;

Norval Mercer and the machine shop staff for
various favors;

Andy Kaye for procuring materials and photo-
graphs with utmost efficiency;

Professor Hugh McManus, Dr. Fred Petrovich
and the rest of the theoretical group for
many valuable discussions;

Julie Perkins for typing this thesis;

Dr. Barry Preedom for taking time to relate
the story of DWBA theory, and many useful
discussions;

My thesis advisor, Professor Sam Austin for
his help in all aspects of this work,
especially during the data taking and the
writing of this paper;

Professor Hobson Wildenthal, whose influences
are present in all aspects of this work, for
introducing me to many facets of nuclear
physics, especially the shell model, and for
providing a home for my tractor;

ii

My wife, Dr. Nancy Larson, for help in data
taking, and frequent discussions of theoretical
problems associated with this work, as well as
cooking my meals;

My parents, for their continuous encouragement
of my educational pursuits;

and finally, I acknowledge the financial support

of the NSF, and a three year NASA Traineeship
during my graduate studies.

iii

TABLE OF CONTENTS

Page
ACKNOWLEDGEMENTS . . . . . '. . . . . . . . ii
LIST OF TABLES O O O O O O O O O O O O O 0 Vi

LIST OF FIGURES . . . . . . . . . . . . . vii

I 0 INTRODUCTION 0 O O I O O I O O O O 1
II 0 EDERIMENTAL O 0 O O O O O O O O O 6
A. Proton Beam and Particle Detection. . . 6

B. Preparation of Targets. . . . . . . 8

C. High Resolution System. . . . . . . 9

III. EXPERIMENTAL RESULTS. . . . . . . . . 11
A. Energies of States in 1383a and 144Sm. 11

B. Angular Distributions . . . . . . . 12

C. Discussion of States in 138Ba . . . . 14

1. 2+ States. . . . . . . . . . 15

‘ 2 O 3- States 0 O O O O O O O O O 16

3 0 4+ States 0 O O I O O O O O O 16

4. 6+ States. . . . . . . . . . l9

5. Other States. . . . . . . . . 20

D. Discussion of States in 144Sm . . . . 21

l C 2+ States 0 O O I O I O O O O 21

2. 3’ States. . . . . . . . . . 22

3. 4* States. . . . . . . . . . 22

4 C 6+ States 0 I O O O . O O O O O 23

5. Other States. . . . . . . . . 24

IV. =82 WAVE FUNCTIONS . . . . . . . . . 26

A. Conventional Shell-Model . . . . . . 26
B. Other Structure Calculations. . . . . 29

V. OPTICAL MODEL . . . . . . . . . . . 31

iv

VI.

VII.

VIII.

’ DWBA THEORY. . . . . . . . . . .

A. ~Collective Model. . . . . .

B. MicroscOpic Model . . . . .

. Distorted Wave Calculations.

. Spectrosc0pic Amplitudes. .

. Elements of the DWBA Formalism
(Direct Amplitudes). . . . .

Transition Density. . . . . .

Discussion of Selection Rules and
Non-Normal Parity States . . .

Ul-b NNH
I O O O
O O O 0

DISCUSSION OF RESULTS . . . . . . .

A. Comparison of Shell Model Calculations
to Experiment . . . . . . .

B. Collective Model Results . . . . .
C. Microscopic Model Analysis . . . .
1. MicroscOpic DWBA Calculations for
138Ba 0 O O O O I O O O

2. Calculations with Other Forces. .

3. Microscopic DWBA Calculations for
1448!“ O O O O I I O O I

CONCLU S I ON 0 O O O O O I O O O 0

REFERENCES 0 O O O O O O O O O O O O 0

APPENDIX

A.

B.

C.

TABES O O O O O O O O O O O O
FIGURES O O O O O O O O O O O O

COMPILATION OF EXPERIMENTAL ANGULAR
DISTRIBUTIONS . . . . . . . . .

Page
34
34
36
36
37

38
42

46
50
50
52
54

56
62

64
67
70

74
81

105

LIST OF TABLES

Table Page
1. Energy Levels of 1383a and 144Sm. . . . . 75
2. Optical Model Parameters for 138Ba(1448m). . 76
3. Deformation Lengths and Transition Strengths

for 1338a and 144Sm . . . . . . . . 77

4. Polarization Charges for Transitions in .
Ba I I I I I I I I I I I I I 78

5. Major Components of 0+ and 62 Wave Functions
Calculated with BotE Interactions, and the
Resulting Transition Densities. . . . . 79

6. Comparison of Direct and Direct + Exchange

Cross Sections for Various Interactions
in 1383a I I I I I I I I I I I I 80

vi

Figure

1.

LIST OF FIGURES

Page

Slit and counter arrangement of high

resolution measurement-Optimization-
stabilization system located in the focal

plane of the spectrograph. The experimental
parameters are adjusted until the maximum

amount of elastically scattered beam passes
between the brass jaws, thus insuring

minimum line widths of the groups of

scattered particles along the focal plane. . 82

Spectrum of 138Ba(p,p') at 35 degrees, with
resolution of about 10 keV, FWHM. The broad
bumps correspond to protons which scatter

from Mg and Si impurities and because of
kinematic differences‘have planes of focus

in the spectrograph. The yields of the 2+

state at 1436 keV and of the 3' state at

2881 keV were too intense to be counted on

this plate I I I I I I I I I I I I 83

Spectrum from 144Sm(p,p') at 40 degrees,

with resolution of about 7 keV, FWHM. The

broad bumps under certain of the peaks

correspond to protons which scatter from Mg

and Si impurities, as in the 138Ba spectrum.

The yield of the 3' state at 1811 keV was

too intense to be counted on this plate . . 84

Elastic scattering.angular distributions

measured for 1388a and 144Sm. The curves

are results of Optical model calculations

made with the Optical model parameters of
Becchetti and Greenlees (Ref. 34) . . . . 85

Characteristic curves obtained by averaging

our measured angular distributions from

groups of states in 1383a and 1 48m which .
had previously assigned J1T values . . . . 86

vii

Figure

6.

10.

11.

12.

States in 138Ba which have angular distri-
butions in agreement with the characteristic
2+ shape, which is the line drawn, through
the data. we assign all of these states
J"=2+ with the exception of the state at

3050 keV. . . . . . . . . . . . .

States in 144Sm which have angular distri-
butions in agreement with the characteristic
2+ shape, which is the line drawn through the
data. We assign all of these states J"=2+
States in 138Ba and 144Sm which have angular
distributions in agreement with the charact-
eristic 3' shape, which is the line drawn
through the data. We assign these states

Jfl=3.- I I I I I I I I I I I I I I

States in 138Ba which have angular distri-

butions in agreement with the characteristic
4+ shape, which is the line drawn through the
data. We assign all of these states J‘"=4+ .

States in 144Sm which have angular distri—
butions in agreement with the characteristic
4+ shape, which is the line drawn through
the data. We assign all of these states
J"=4 I I I I I I I I I I I I I I

Angular distributions for the previously
assigned 6+ state at 2090 keV, and the weak
2201 keV member of a doublet. Both the 4+
and 6+ characteristic curves are drawn
through this anguli£4distribution. The

2324 keV state in Sm has been assigned

6+, but due to the rise of the data at for-
ward angles, this state may be a close lying
doublet. We assign J =6+ to the 3308 keV ’
state in 144Sm. . . . . . . . . . .
Angular distributions of states in 1383a for
which no J" assignment could be made from
this work. Spins of 5+ and 3+ have been
suggested for the states-at 2415 and 2445
keV respectively (Ref. 37). . . . . . .

viii

Page

87

88

89

90

91

92

93

Figure

13.

14.

15.

16.

17.

18.

19.

Page
Angular distributions of states in 144Sm
for which no J1r assignment could be made
from this work. The state at 3123 keV
has been assigned J“=(7i) in Ref. 11 . . . 94

Comparison of results of shell model cal-
culations discussed in Sec. IVA with
experimentally known energy levels in 1”Be . 95

Comparison of results of shell model cal-
culations discussed in Sec. IVA with
experimentally known energy levels in 144Sm . 96

Results of collective model calculations for
inelastic scattering on 1333a. The Optical
model parameters given in Table 2, Set SII

were used, and both the real and imaginary

parts of the Optical model were deformed.

The calculations shown are for the lowest

2*, 3’, 4+ and 6+ states . . . . . . . 97

Results of collective model calculations for
44Sm. The Optical model parameters given

in Table 2, Set SI were used, and both the

real and imaginary parts of the Optical

model were deformed. The calculations

shown are for the lowest 2*, 3‘, 4+ and

6+ states . . . . . . . . . . . . 98

MicrosCOpic model DWBA calculations f r

the inelastic scattering to the 2 , 41 and

6: states in 1388a. The calculations are

i entical except for the wave functions;

the left hand column employed the A=136-l40

set while the A=136-145 set was used for

the right hand column. Both sets of wave
functions predict very similar angular
distributions . . . . . . . . . . . 99

Microsc0pic model DWBA calculations for the
inelastic scattering to the 25, 45 and 63

states in l33Ba. The calculations are.

identical except for the wave functions;

the left hand column employed the A=136-140

set while the A=136-l45 set was used for

the right hand column. These calculations
indicate that the A=l36-l40 set of wave
functions provides the better description

of the low lying states in 388a. . . . . 100

ix

Figure

20.

21.

22.

23.

Page

Transition densities for the 2+ 2, 4; 2 and

SI 2 states in 1338a calculate 'with both

sets of wave functions. The 2:, 4i and

GI densities are very similar, and predict

very similar cross sections. The differences

in the 23, 42 and 65 cross sections are

directly related to the differences in the
transition densities for these states . . . 101

C mposition of the 6; transition density in

1 Ba, calculated with both sets of wave
functions. The left hand figure is cal-

culated with the A=l36-140 set while the

right hand column is calculated with the
A=136"145 Set 0 o o o e o e e o o o 102

MicroscOpic model DWBA calculations for I
inelastic scattering to the 3i and 5+ states

in 1333a calculated with the A=l36-1 0 set

of wave functions. The two-body force

included central, tensor and LS terms,

but no collective enhancement was included.

Only the Direct + Exchange calculations

are shown . . . . . . . . . . . . 103

MicrosCOpic model DWBA calculations for the
inelastic scattering to the 21 2, 4i,2 and

61 states in Sm calculated with the
A=136-145 set of wave functions. The

inadequacy of the basis space is demonstrated

in the calculated angular distributions for

the 25, 4g and 6; states . . . . . . . 104

I. INTRODUCTION

The recent interest in the N=82 nuclei stems from
a number of sources.‘ Foremost among these is the observa-
tion that in a shell model picture, low-lying states in
these nuclei are expected to be formed predominantly from
proton configurations, the neutron shells being closed with

(1-2)

82 neutrons. Experimental evidence from proton and

neutron(3-5)

transfer experiments confirm this expectation.
Both proton stripping and pickup reactions on the N=82
nuclei populate only the lg7/2, 2d5/2' 2d3/2, 331/2, and
lhll/Z single particle orbitals, thus indicating that
proton orbitals other than these do not play a significant
role in the wave functions for states of these nuclei.
Neutron pickup reactions indicate that the orbitals above
the last filled neutron major shell are empty, and con-
versely, neutron stripping reactions populate only the
orbitals above the closed major shell, thus indicating that
neutron shells below this are filled. The picture of the
N=82 nuclei which emerges from these experiments is that
of a closed 82 neutron, 50 proton core, with low-lying

states in these nuclei being formed by couplings of various

numbers of valence protons in the aforementioned orbitals.

The single particle nature of these nuclei has
been thoroughly studied with the proton and neutron transfer
reactions described previously. Isobaric analog resonance
experiments‘s) have been used to determine, among other
things, the positions of the low-lying neutron particle-
hole states, which signal the breakdown of our model.
Electromagnetic decay aspects of the N=82 nuclei have been

(9) (n,n'y)(lo) and

measured through (8,7),(7-8) (n,y),
(a,an)(;1) studies. These studies have been useful in
making precise energy level assignments, as well as often
limiting the possible J"r assignments to a few values. The
(a,xny) studies have led to the observation of a series of
isomeric 6+ states in the even-even N=82 isotones.

Charged particle inelastic scattering studies have
been limited mainly to the observation of the strongly
excited states. Early inelastic scattering experiments‘lz)
determined the positions of the first collective 2+ and 3-

(13)

states. The (p,p'y) and (d,d'y) reactions on the even-

even isotones were performed in an attempt to lOcate the
positions of excited 0+ states in these nuclei by observing

the E0 conversion eleCtrons emitted in the transition to

139 ,)'(l4)

the ground state. More recently, the reactions La(a,q

140 (14) 141Pr(a’a.)’(15) 138 a,)'(16)

SmiPIP'):

Ba(a,
l4

Ce(a,a'),

144 (17) 144 (17)

Smis,s'), and 4Sm(h,h')(18) have
been used to study the collective nature of the strongly

excited states in these nuclei. The angular distributions

obtained from the latter five reactions were analyzed

(19)

with the standard collective model formalism and from

that work J1r assignments were made, and deformation

139 140Ce experiments

parameters BL extracted. The La and
have also been analyzed with the collective model approach,
and in addition, microscopic calculations have been per-
formed using a Gaussian two-body interaction and zero-order
pseudo-spin orbit wave functions to describe the nuclear
states. However, since alpha particles are strongly absorbed
near the surface of the nucleus, only the tails of the wave
functions are important and this reaction is thus rather
insensitive to the details of the wave functions.

In this paper we present results from inelastic
proton scattering experiments performed at a bombarding

138 144

energy of 30 MeV on Ba and Sm. Use of the high

resolution system developed at Michigan State by Blosser,

(20) resulted in a total energy resolution for the

9.2.2;-
inelastic peaks of typically 7-10 keV FWHM. Excitation
energies accurate to 2 keV were extracted and found to
be in good agreement with those obtained by other methods.
Using empirical characteristic shapes derived from angular
distributions to known states, we assign spins and parities
to the majority of the observed states.

Most previous microscopic DWBA calculations for
inelastic scattering have been restricted to regions of

(21,22)

the periodic table where particle-hole or simple

(23,24)

shell-model wave functions were adequate for the

desoription of the nuclear states. A second aspect of
this work, in addition to spectroscopy, was to attempt to
determine if large basis shell model wave functions can
account for (p,p') measurementszon nuclei with several
valence nucleons. To this end, we have modified the Oak
Ridge-Rochester shell-model codes(25) to calculate the
necessary structure amplitudes in a form convenient for
DWBA calculations. By using two sets of shell-model wave

(26'27) in the DWBA calculations for 13888: we

functions
show that inelastic proton scattering provides a sensitive
test for these wave functions, and allows us to choose one
set as preferable to the other. Information concerning the
structure of the wave functions can be extracted by a
detailed consideration of the resulting transition densities.

Use of a model proposed by Atkinson and Madsen,(28)
and McManus(29) for including effects due to excitations
of nucleons out of the core enables us to extract information
concerning the polarization charge of the nucleons in this
mass region.

In Section II we describe the experimental details.
of this work, followed in Section III with an individual
discussion of all states observed up to Ex=3’3 Mev, and
assignments of excitation energies and spin-parities,

where possible. Section IV is a discussion of the shell

model calculation which we use in this work, and Section V

presents thecptical model parameters we obtained from our
elastic scattering. Section VI outlines the relevant
aspects of the DWBA theory used to analyze our angular
distributions, and lastly Section VII is a discussion of

the results of this experiment.

II. EXPERIMENTAL

A. Proton Beam and Particle Detection

The measurements were made with a 30 MeV beam of
protons from the Michigan State University sector focussed
cyclotron. An Enge split-pole spectrograph was used to
detect the scattered particles. The amount of beam.on
target was monitored both with a current integrator in
conjunction with a Faraday cup, and with a 5 mm thick
silicon detector placed at 60° with respect to the incident
beam. A set of removable slits located immediately prior
to the spectrograph scattering chamber was used periodically
to check the position of the beam on target. The typical
size of the beam spot was 2 mm.high by 4-5 mm wide. ”The
entrance aperture of the spectrograph was 2° wide by 1.60
high, corresponding to a solid angle of 0.98 msr. During
data accumulation this entrance aperture was the Only slit
between the cyclotron and the focal plane of the spectra-3
graph. The absolute energies of the proton beams were
obtained from NMR calibrations of the transport system
magnets. The uncertainty in this absolute scale was 10.1%.
The absolute beam energies far this experiment were 29.8 MeV

138 144

for Ba and 29.9 MeV for Sm.

Angular distributions for elastic scattering and
for scattering from the strong first 2+ and 3- states in
both nuclei were measured using a 300 micron thick solid
state position sensitive detector mounted at the focal plane

30 Signals proportional to E, the total

of the spectrograph.
energy loss of the_particles passing through the detector,
and xE, where x is the position along the detector, were
analyzed for the type of particle and its location along
the detector using a two dimensional data taking program.31
The xE signal was divided by the E signal, and energy vs.
position spectra were displayed on a storage scope, where
suitable lines were drawn defining the proton band. The
computer then identified all events falling between the
lines as proton events and stored them in a position
spectrum. The resolution obtained with the position sensi-
tive detector was typically 30 keV FWHM. Its charge
collection efficiency was mapped bygridding the peak from
the elastically scattered particles across its surface in
3 mm steps. During data accumulation, the particles to be
detected were positioned in a region which had been deter-
mined to have uniform efficiency.

The remainder of the inelastic scattering data
were measured with Kodak NTB 25 or 50 micron nuclear
emulsions placed in the focal plane of the spectrograph.

Aluminum.absorbers were used to stop all particles of greater

stopping power than protons. Emulsions were exposed every

138

5° between 20° and 80° for Ba and from 12° to 95° for

144Sm. Two exposures were made at each angle, a short
exposure to obtain data from the first 2+ and 3- states
(for normalization purposes), and a sufficiently long
exposure to obtain data with good statistics for most of
the remaining states. The agreement between the position
sensitive detector data and the emulsion data for the 2+

and 3- states was within statistics, so the data were

averaged to obtain angular distributions for these states.

B. Preparation of Targets

Isotopically enriched compounds of Ba(N03)2
(99.8%) and Sm203 (95.1%) obtained from Oak Ridge National
Laboratory were used in the fabrication of the targets.
The desired compound was placed in a Zr boat and heated
in a vacuum, causing reduction of the compound to the
enriched metal and simultaneous evaporation of the metal
onto the target backing, which consisted of a 20 ug/cm2
carbon foil plus a 3-5 ug/cm2 layer of formvar supporting
the carbon. The target material was evaporated over a
surface 5/8" in diameter and appeared to be quite uniform.
Typical target thicknesses ranged between 50 and 300 ug/cmz.
The targets were stored and transferred under vacuum to
reduce oxidation. Since complete oxidation occurred in

138

only a few seconds for a thin Ba target, and a few?

144

minutes for a Sm target, thickness was therefore estimated

by comparing the measured elastic scattering to optical
model predictions for the scattering.

The observed contaminants in the targets were
carbon, oxygen, magnesium, and silicon, determined from

analysis of the inelastic scattering spectrum.

C. High Resolution System
The emulsion data were all taken using the high
resolution system develOped at Michigan State by Blosser,

(20) This system relies on dispersion matching,(32)

seei-
kinematic compensation, and a feedback systemiwhich
compensates for possible drift of any magnets in the
cyclotron-beam transport-spectrograph system. In a
dispersion matched system the line width of the scattered
particles at the focal plane of the spectrograph is nearly
independentof the energy spread of the incident beam.

This is accomplished by using the focussing and dispersive
elements of the beam transport system to adjust the disper-
sion of the beam on target to match the dispersion of the
spectrograph. Kinematic compensation corrects for the
change in energy of the scattered particles across the
finite entrance slit width, which arises from recoil of

the target nucleus. This is accomplished by shifting the
kinematic focal plane from its zero order position (position

for scattering from an infinitely heavy target nucleus).

The approximate beam transport quadrupole settings for

10

dispersion matching and correct position of the focal plane
for kinematic compensation are calculated for a given

(33) Final minimization of

reaction via a computer code.
the line width of the scattered protons is accomplished by
using a stepped slit and detector device located at the

focal plane of the spectrograph and illustrated in Figure 1.
Fractional transmission of the elastically scattered

protons through the 4 mil slit is maximized (thus minimizing
the line width) by adjusting the dispersive elements of

the beam transport system and other parameters of the
experimental setup. After minimdzation of the line width,
this device serves as a feedback system, controlling the
spectrograph magnet to keep the elastically scattered protons
centered on the transmission slit. This insures that the
scattered protons remain at fixed points on the focal plane,
independent of drifts in the system. Using this high
resolution system we routinely obtained resolutions of '

7-10 keV FWHM at 30 MeV incident proton energy for the
inelastically scattered particles. Figure 2 shows a spectrum

l383a at a laboratory angle of 35°, and

144

obtained from

Sm obtained at 40°.

Typical beam currents on target were 100 na for 1388a

(target limited) and 900 na for 144Sm.

Figure 3 shows a spectrum from

III. EXPERIMENTAL RESULTS

138 144

A. Energies of States in Sm

Ba and
The high resolution system described in the previous
section, in conjunction with nuclear emulsions, was used

to obtain precise energies for 20 excited states in 138Ba

and 18 states in 144

Sm below Ex=3.4 MeV.

Peak centroids and intensities were extracted from
the spectra obtained from the scanned emulsions at each
angle. This was done with an automatic peak-fitting
program, which aided in removing ambiguities in the back-
ground subtraction and afforded a consistent method of
separating members of close lying doublets. The final
adjustments to the basic energy calibration of the spectro-
graph were determined by fitting certain strong, isolated
peaks in our (p,p') spectra to excitation energies
previously determined for these levels by Ge(Li) spectro-
meter studies of gamma rays emitted in (8,7) experiments.
These calibration energies, along with their errors, are
noted in Table 1. This calibration of the spectrograph
was then used for interpolatiOn and extrapolation to other
excitation energies. These results enabled us to assign
an excitation energy to each observed peak at each angle,

provided the peak was not obscured by a contaminant. The
11

12

extracted energies were averaged over all angles of observa-

tion, and the mean error in the centroid was calculated.

138 144

The energies we have assigned to levels in Ba and Sm,

along with the combined random and systematic errors,

are listed in Table 1. These energies are in excellent

agreement with energies from (8,7) work on 1388a and

144

(a,2ny) work on Sm.

B. Angular Distributions

Angular distributions have been measured for 18

of the 20 states observed in 1388a and 15 of the 18 observed

144

states in Sm, and for elastic scattering from both

nuclei.

The elastic scattering angular distributions from

138 144

Ba and Sm are shown in Figure 4. The curves through

the data are optical model calculations using parameters

(34) The elastic angular distrie

of Becchetti and Greenlees.
bution data were normalized to the optical model calculations
to Obtain an absolute normalization.) A comparison with

calculations using other sets of Optical model parametersi35)
results in an estimate of 10% uncertainty in the experimental
absolute cross sections. Relative uncertainties in our

(p,p') cross sections arising from scanning errors, monitor-

ing errors and statistical errors are typically 7%.

13

To facilitate a systematic analysis of our (p,p')
data we derived from our data empirical characteristic

4.

shapes for 2+, 3‘, 4 , and 6+ angular distributions in the

following way. Examination of angular distributions for

138 144Sm reveals that

the known 2+ states in both Ba and
they all have essentially the same shape. Using this

fact, one average 2+ shape was obtained from all of the
known 2+ distributions in both nuclei. This characteristic
2+ shape was then used as a standard and compared to angular
distributions for states of unknown J“. Identical
techniques were applied to the angular distributions of

all assigned 3-, 4+, and 6+ states in these nuclei to

obtain 3-, 4+, and 6+ characteristic. shapes. The resulting

+, and 6+

empirical characteristic shapes for 2+, 3-, 4
angular distributions are compared in Figure 5. We emphasize
that the characteristic shape for a given J"T is an average

of angular distribution data for all known states of that

J1r from ngh_nuclei. The states used in the determination

of the empirical shapes are noted in Table l. The angular
distribution data, along with the characteristic shape of

the appropriate J1T which best approximates the data, are
shown in Figures 6through 11. The remaining states, whose
angular distributions are not similar to any of the charact-

eristic shapes, are shown in Figures 12 and 13. They may

be grouped into two classes; either they are very weakly

l4

excited in this reaction, or they peak farther out in angle
than the 6+ states, implying they may be high spin states.

As explained in Sec. VIBS, direct DWBA theory
would predict L-transfers of 2, 3, 4, and 6 for transitions
from a 0+ ground state to states with Jfi=2+, 3-, 4+, and
6+, respectively. There are sufficient differences between
the characteristic shapes, as seen in Figure 5, to allow us
to uniquely assign an L-transfer of 2, 3, 4, or 6 to the
majority of the Observed transitions. This in turn leads
to assignments of J"=L, with parity (-)L, if it is assumed
that non-normal parity states are weakly excited. In Sec.
VIBS we discuss the evidence which indicates that this
assumption is valid. For example, an angular distribution
for a state of unknown J1r which matches the 4+ character-
istic shape has an L-transfer of 4. This in turn leads to
a J1T assignment of 4+, and not the non-normal parity 3+
or 5+, as direct'DWBA theory would also allow.‘

Our assignments for the spin and parity of excited

138 144Sm, based on the agreement

states we observe in Ba and
of the measured angular distributions with the empirical

characteristic shapes, are given in Table 1.

C. DiscusSion of States in 1388a

To organize the discussion of our experimental
results, it is convenient to divide the levels which we
observe into groups, each group being identified by its

appropriate J1r assignment.

15

1. 2+ States (Figure 6) .

The states at 1436, 2218, 2639, 3339, and
3368 keV are all in good agreement with the 2+
characteristic shape; we thus assign Jfi=2+ to

these states. The state at 1436 keV is firmly

established as 2+ from Coulomb excitation,(36)

(37)

conversion coefficient measurements, and

(16)

(a,a') inelastic scattering. . The state at

2218 keV is observed to have a strong decay branch

(9)

to the ground state in neutron capture and

(B.Y)(7-8’37) decay experiments. This limits the

spin to l or 2. Achterberg, et al.(37)

assign
positive parity to this state from conversion
coefficient measurements. Thus our assignment of
J"=2+ is consistent with previous data. The level
at 2639 keV, observed in (n,y)(9) and (8,y)(7-8)
studies, also has a strong decay branch to the
‘ground state, thus limiting its spin to J=l,2.
However, a log ft value of 7.4 for 8 decay from

138Cs given by Carraz, et al.(8)

the Jn=3- state of
eliminates J=1. The two levels at 3339 and 3368 keV
also decay directly to the ground state, limiting
their spins to J=l,2. Angular distributions from

(16) also suggest a spin of J=2 for

(a.a') studies
these states. The angular distribution for the

state we observe at 3050 keV is not in agreement

16

with the 2+ characteristic shape at forward angles,
thus we leave it unassigned. Hill and Fuller(7)

assign J=l,2 to this state.

2. 3’ States (Figure 8)

138

The only 3- state we observe in Ba is the

one previously assigned‘12'16) at 2881 keV. It is
the strongest state in the (p,p') spectrum, and is

)(13) and (a,a')(16)

strongly populated in (d,d'
experiments and in (n,y)(9) studies, where it decays

strongly to the 2+ state at 1436 keV.

3. 4+ States (Figure 9)

The states at 1898, 2308, 2584, 2779, and
3156 keV have angular distributions which are in-
good agreement with the characteristic 4+ shape;
we assign J'"=4+ to these states. The level at
1898 keV is eStablished as 4+ from (O,a')(16)
angular distributions, (d,3He)(2) measurements
which suggest an assignment of 4+ or 6+, and con-

(37)

version coefficient studies which, when combined

with the (d,3He) work, limit the spin to 4+.
The state at 2308 keV is assigned Jfl=3+, 4+
from conversion coefficient studies of Achterberg,

et al.(37) A J=3,4 assignment has also been

 

suggested by (n,y)(9) and (B:Y)<7) studies based

17

on the strong decay to both the first 2+ and 4+
states. Since non-normal parity states such as

3+ are very weakly excited in inelastic scattering

on heavy nuclei, we assign Jfl=4+ to this state,

which is also the suggested assignment from (a,a')(16)
studies.

7) and Mariscotti, et al.(9)

Hill and Fuller<
observe a state in the vicinity of the state we
see at 2584 keV. Hill and Fuller limit the spin
of this state to J=l,2 on the basis of a gamma
ray branch to the ground state. However, the angular
distribution we measure has a 4+ shape. Thus, we
believe there are two distinct levels in this
vicinity since the state seen by Hill and Fuller
is clearly not a 4+ by virtue of the gamma ray to
the ground state, and ours is not a J=l or 2 by
virtue of the angular distribution. It isinforma-
tive to consider ratios of intensities of two gamma
rays depopulating this level, as measured by Hill
and Fuller in their (8,y) work and Mariscotti,
gt_gl, in their (n,y) studies.

I(2583+1436)

I + = “708 (nIY)

I(2583+l436) _ ‘
1(258310) ‘ ~2°3 (5'7)

 

(n, ) _
Thus Tag—)- — ~3.3

Similar ratios for the 2583-2218 keV transition
relative to the ground state transition are also
in the ratio of ~3:1. The non-equality of these
ratios can be taken as an indication that the (n,y)
and (B,y) experiments are populating twolevels
with different intensities in the vicinity of

2583 keV. In addition, Hill and Fuller see a
broadening of the gamma ray line connecting their
state at 2583 keV with the 2445 keV state. They
attribute this to the decay of the 2445 keV to the
2307 keV state, but we suggest this could also

be due to a state at 2584 keV decaying to the

2445 keV state. The broadened decay line they
observe would then correspond not to two but to
three different transitions. We will find in

Sec. VIIA that the shell model predicts a l+-4+
doublet near this energy, which would be in excellent
agreement with experimental observation. A l+ state
would decay to ground via an M1 transition, but
since it is a non-normal parity state we would not
expect to populate it strongly in (p,p'). The
angular distribution we measure for this doublet
would then assume the shape characteristic of the

4+ member of the doublet.

19

The state at 2779 keV is observed in (BIY)I(7)

)(9) work and decays to both the first

but not (n,y
excited 2+ and 4+ levels, thus limiting its spin

to J=2, 3, 4. It may also correspond to the state

at 2.79 MeV in (d,3He)(2) experiments.

The state at 3156 keV which we observe is pro-
bably not the same state reported by Hill and Fuller(7)
at 3164 keV, since the 8 keV difference in energies
is well outside the combined errors. However, they
assign spin limits of J=2, 3, 4 to their state based
on its decay to the first excited 2+ and 4+ states.

These two experiments are the only ones which report

a state near this energy.

4. 6+ States (Figure 11).

We assign Jfl=6+ to the state at 2090 keV, and
a tentative (6+) to the level at 2201 keV. Angular
distributions for known 6+ states are scarce in the
literature. We have obtained an angular distribu-
tion for the state at 2090 keV, which has been

assigned 6+ by Carraz, et a1.(8)

on the basis of
its measured half life of 0.8 usec. This assign-

ment is consistent with systematics of 6+ states in

N=32 DUCIGi; isomeric 6+ states have been identified‘38)

in all even-even isotones from 134Te to 146Gd. A

+ . . a . .
6 a331gnment is also consxstent Wlth the absence

20

of this state in the (n,Y work, and its

negligible feeding in the B decay of the 3- ground
state of 138Cs.
The state at 2201 keV is the subject of some

controversy. Carraz, et al.(8)

propose that this
state has J"=(5-), while Achterberg, gt_§l.(37)
propose J"=(4+,5+), based on the measurement of

log ft values from the decay of l38““ng and the
assumption of Jn=3-, (6-) for the ground and

isomeric states respectively of 138Cs. This state

is quite weak in our spectra, and not well resolved
from the strong 2+ at 2218 keV. Our angular
distribution for this state is not inconsistent

with any of these tentative assignments. Both the

4+ and 6+ characteristic shapes are drawn through
the angular distribution for this state in Figure 11.
We would favor a (6+) assignment for this state,
based on the predictions of shell model calculations,

and to some extent, on the shape of the angular

distribution.

5. Other States (Figure 12)

We make no Jfl'assignments for the states-we.
observe at 2415, 2445, 3254, and 3285 keV. The
transitions to these states are all very weak, the

largest cross section at any angle being less than

21

10 ub/sr. Achterberg, gt_gl.(37) have assigned
J =3+ to the state at 2445 keV, based on angular
correlation and conversion-coefficient measure-
ments. They also suggest a JTT=5+ assignment for
the state at 2415, based on log ft values of the

l3Bm'ng. Our angular

B decay feeding it from
distributions for these states, by virtue of their
magnitude, support non-normal parity assignments.

The two remaining states at 3254 and 3285 keV are

not seen in previous work on this nucleus, and our
angular distributions do not shed any light on
possible J1T assignments for them. The 2929 and

2990 keV states which were observed only at one angle
are very weak and are therefore not likely to be

low spin normal parity states. Hill and Fuller(7)

suggest spins of J=l,2 and l,2,3,4 respectively for

these states.

D. Discussion of States in 144Sm

1. 2+ States (Figure 7)
We assign Jfl=2+ to the states at 1661, 2423,
and 2800 keV. The 1661 keV level is the first

144Sm; its J1T is firmly established

(39) :Y) (40)

excited state in

studies and

from Coulomb excitation, (B

(a,a') and (p,p')(l7) experiments. Barker and

(17)

Hiebert observe a 2+ state at 2.45:0.02 MeV

22

via (p,p') and (a,a') reactions which we assume
corresponds to our level at 2423 keV. The state

at 2800 keV has been assigned Jn=2+, from comparison
of its angular distribution with the 2+ character-
istic shape. It has also been observed weakly in

(40) 144

the 8+ decay of the 1+ ground state of Eu.

2. 3_ States (Figure 8)
We assign J"=3_ to the states at 1811 and 3227
keV. The state at 1811 keV is well established to

have J =3- from (ara') and (p,p') experiments,(l7)

and is the strongest state observed in our 144Sm
spectra. The angular distribution of the previously
unobserved state at 3227 keV is very similar to that
of the 1811 keV state, as well as the angular distri-
bution for the collective 3_ state in 1388a. On

this basis we assign the 3227 keV state Jfl=3-.

3. 4+ States (Figure 10)

We assign J"=4+ to the states at 2191, 2588,
2883, and 3020 keV on the basis of agreement of
their angular distributions with the 4+ character-
istic shape. The level at 2191 keV has previously
been assigned Jfl=4+ on the basis of gamma ray

(38)

systematics, and recent (a,a') and (p,p')

(17)

experiments verify this assignment. We asSume

that this is the level at 2.21:.02 uev observed in

23

(13) The states at

(d,d') and (p,p') studies.
2588, 2883, and 3020 keV have not been previously
reported. They are in good agreement with the
characteristic 4+ shape, with the possible exception
of the 3020 keV state, which falls off somewhat too
rapidly for angles larger than 50°. These 4+
assignments are consistent with the fact that
Kownacki, gt_al,(ll) have not observed these states

14 144

in the 2Nd(a,2n) Sm reaction which preferentially

populates states with Ji8. Studies of the B decay

144 (40) alSo show no

of the Eu 1+ ground state
evidence for levels at these energies, which is in
agreement with our 4+ assignments since population
of 4+ states via this decay would be unique second

forbidden.

4. 6+ States (Figure 11)

The Stockholm group(38) has assigned the state
at 2324 keV a spin-parity of 6+, based on its life-
time of 0.88 usec. This agrees with systematics
of 6+ states in the N=82 nuclei. However, our
angular distribution for this state is only in
qualitative agreement with the characteristic 6+
shape due to the rise of the data at forward angles.
This could be due to a very close-lying low spin.

state which would cause the experimental angular

24

distribution to rise at forward angles. This
possibility is weakly supported by the fact that
the FWHM of this peak is consistently 10-15% larger
than that of other nearby peaks. We also assign
Jfl=6+ to thegreviously unobserved state at 3308 keV
on the basis of the shape of its angular distribu-

tion.

5. Other States (Figure 13)

We make no J1T assignments for the states we
observe at 2826, 3123, 3196, and 3266 keV. The
state at 3123 keV has been assigned 7(1) by the

(11)

Stockholm group in their search for high spin

142Nd(a,2n )144Sm reaction and

states using the
coincidence techniques. Our angular distribution

for this state is consistent with a high spin

state; if this is the same state seen by the StOckholm
group, we would favor a 7- rather than 7+ assign-

ment. The angular distributions for the levels at

2826 and 3196 keV peak far out in angle, suggestive

of a high spin state, but the Stockholm group does

not place any levels at these energies. The

2800 (2+) and 2826 keV levels are seen as a doublet

(17)

in the (d,d') and (PIP') work of Barker and Hiebert.

We also note the similarity of the 3123 and 3196 keV

25

angular distributions, which leads one to doubt
that they are both doublets. The state at 3266 keV
has not been observed previously, and is excited
very weakly in the present experiment. The angular
distribution for this state does not allow us to

make any suggestions for its spin and parity.

IV. N=82 WAVE FUNCTIONS

A. Conventional Shell Model

As discussed in the Introduction, we have used
large basis shell-model wave functions to describe the
nuclear states involved in the present experiment. This
section describes the details of the conventional shell

(ZS-27) which was done with the Oak Ridge-

(25)

model calculation
Rochester shell model code.

The basis space for the shell-model wave functions
consists of the lg7/2 and st/2 orbits, plus one-proton
excitations from this subspace into the 331/2 or 2d3/2
orbits. The two-body interaction between the valence
nucleons was parameterized in terms of a modified surface
delta interaction (MSDI), with the four single particle
energies and the two MSDI parameters fixed by fitting to
energy levels of known J1T in the N=82 nuclei.

Two Hamiltonians were calculated. The MSDI para-
meters for the first Hamiltonian were obtained by fitting

136

to levels of known J1T in N=82 nuclei from Xe through

145Eu(A=l36-l45), while those for the second Hamiltonian

136

were obtained by fitting to levels from Xe through

l4OCe(A=136-140). The A=136-l45 interaction was applicable

26

27

138 144

. to both Ba and Sm, while A=l36-140 was designed for

the lower mass N=82 isotOnes, and hence was used only for

138Ba. Thus two sets of wave functions were calculated

for 1388a, but only one set for 144Sm. The basis space
was the same for all calculations; only the parameters of
the Hamiltonian differed.

The basic difference between the two Hamiltonians
is the 97/2-d5/2 single particle energy splitting, which
increased from 500 keV for the A=l36-l45 interaction to
900 keV for the A=136-l40 interaction. It is found that
the eigenvalues and eigenvectors calculated with the
A=136-140 interaction yield better agreement with experiment-
ally known spectra, pickup and stripping Spectroscopic
factors and electromagnetic data for the lower N=82 isotones
than do those calculated with the A=136-l45 interaction.
In particular, the A=l36-l40 calculations are in excellent

(27) 1335b and 134Te, the

agreement with recent data on
one and two proton N=82 isotones, although levels from these
nuclei were not included in the search procedure which fixed
the parameters of the Hamiltonian. It is reasonable that

the A=l36-l45 Hamiltonian might give poorer results for

the lighter N=82 isotones. For.the upper N=82 isotones,

the effects of the limited basis space appear to be important.

144

For example Sm, which has 12 valence protons, effectively

has a basis space consisting of two holes in theg7/2-d5/2

28

orbits and one-particle excitations out of these orbits.

144

The physical low-lying states of Sm presumably have

substantial amplitudes in their wave functions for con-

figurations outside of the allowed basis space, such as

2 2

2 .
(1h , (2d3/2) and (331/2) . Including these states

11/2)
in the searching procedure which determines the parameters

of the A=136-l45 Hamiltonian could well distort the para-
meters to compensate for components outside of the basis
space. This would in turn decrease the accuracy of the

calculations in the lower isotones. In light of this,

138

the superiority of results calculated for Ba with the

A=136-l40 interaction over those calculated with the

A=136-l45 interaction is expected. Moreover, it is expected

that the best results calculated for 138Ba should be

superior to the best for 144Sm, since the basis space is

more complete for A3140 . We have used both the A=136-140

and A=l36-145 sets of wave functions in the DWBA calcula-

138

tions for Ba, to see if inelastic proton scattering can

identify one set of wave functions as definitively better

than the other. Only the A=136-l45 interaction was used

in the 144Sm calculation, as previously noted. With the

138

basis space used in this calculation, states in Ba have

between 50 and 220 components in their wave functions,
while states in 144Sm have between 10 and 45 components.
The notation used to discuss the different states will be

J1, where i refers to the first or second excited state of

spin and parity J".

29

B. Other Structure Calculations

Properties of the N=82 nuclei have also been
calculated by methods other than the conventional shell
model approach of Wildenthal.(26) Rho‘41) performed a
two-quasiparticle calculation for the even N=82 nuclei,
employing a Gaussian form as the residual nucleon-nucleon
interaction. At the time of that work, the single particle
energies needed in the calculation were not experimentally
known. The results of this calculation are in qualitative
agreement with experiment, but no 0+ or 6+ levels were
calculated. In a later calculation, Waroquier and Hyde‘42)
used an approach similar to Rho, but employed the inverse

(43) to obtain the single particle

gap equation technique
energies for their calculation. In addition theycalculated
the 0+ and 6+ states and in general obtained good agreement
with existing data. Comparisons of their calculated energy
levels and those of the conventional shell model which we

use in this work are given in Ref. 42.

A new coupling scheme, developed by Hecht and

Adler,(44) and employed by Baker and Tickle,(l4) Baer,
et al.(15) and Jones, et al.(2) in their work on N=82

 

nuclei, predicts energy levels in good agreement with our

138B

experimental results for a. This pseudo spin-orbit

coupling scheme takes advantage of the fact, observed in

(26)

the shell-model calculation, that for the N=82 nuclei

each group of levels with seniority v has some of these

30

levels depressed in energy. Calculations performed<2> with
this coupling scheme and using a basis space very similar
to that of Wildenthal result in wave functions with at most
23 components. It is encouraging that this scheme, with
its apparent simplicity, is able to reproduce the salient
features of the conventional shell-model which we have.
employed in this work. A comparison of the energy levels
predicted by the pseudo spin-orbit scheme and the con-
ventional shell model are given for $38Ba in Ref. 2. This
coupling scheme also predicts approximate selection rules
for relative strengths of excited states observed in

inelastic scattering. This will be discussed further in

Sec. VIB5.

V. OPTICAL MODEL

Essential ingredients in any BWBA calculation are
the optical model parameters which describe the elastic
scattering in the entrance and exit channels. A number of
optical model parameter studies(34_35’45) have been made
for 30 MeV protons incident on nuclei with A between 40 and
208. From these parameterization studies, formulae result
which allow one to interpolate the "best fit" parameters
to nuclei and energies other than those specifically used
in the studies. However, there is a large gap from tin
(A~120) to lead (A:208) for which very little precise
elastic scattering data exists, and thus no information
for the region was included in the optical model studies.
It was not clear, therefore, that the parameters which
these studies predict would properly account for the elastic

138 144

scattering from Ba and Sm.

Fortunately, parameters predicted by two previous

(34-35)

studies yield results which are in very_good agree-

ment with our elastic scattering data. The optical model

parameters of Becchetti and Greenlees<34)

138

predict an elastic

l4

scattering angular distribution for Ba( 4Sm) which

results in a chi square per point between theory and

31

32

experiment of 3.4 (5.3), while Set II from Satchler's

(35) yields a fit with xz/N=4.0 (6.0). As discussed

analysis
previously in Sec. IIIB, these excellent theoretical fits
allowed us to normalize our elastic angular distribution
data to theory in order to obtain absolute cross sections
for the inelastic scattering data. After this normalization
was determined, the predicted parameters were allowed to
vary in a search to obtain the optimum parameters for use

in the DWBA calculations.

The optical model potential used in our analysis

has the usual form

V(r) = -VR(l+exR) - i(Wvol 4Wsurf d:IM)(1+ IM)-l
+<I-5’1-:-5)2 VLs §§E<1+eXLSf1 (Eq. 1)
where XR = (r-rRAl/3)/aR
xIM = (r-rIMAl/3)/aIM

st = (r-rLSAl/3)/aLS -

The Coulomb potential is that of a uniformly charged sphere
with radius rcA1/3.

To obtain an optimum set of optical model parameters,
three different sets were utilized as starting parameters

for the searching procedure. They are l) the set predicted

33

from the global optical model analysis by Becchetti and

Greenlees,(34) 2) Set I, and 3) Set II from the work of

(35) on 30 MeV proton elastic scattering.

Satchler
The final best fit parameters for each set are
presented in Table 2. The final values are all within 5%
of the starting values, with the exception of the surface
absorption strength in Satchler, Set I, which decreases

by 15% for 138Ba. The total DWBA cross sections for the

+ 138

first 2+, 4 , and 6+ states in Ba calculated with each
set of parameters are also given in Table 2. These give

an indication of the sensitivity of the DWBA calculations

to changes in optical model parameters. The cross section
ratios for J=2:4:6 are essentially constant; hence the main
difference between the various sets is an overall normaliza-
tion which is at most 10%. For the calculations presented
in this paper, Set SII in Table 2 was chosen for 138Ba as

it gave the smallest Xz/N, and resulted in total cross
sections intermediate in value between BG and SI. The

144Sm calculations were done with Set SI.

VI. DISTORTED'WAVE BORN APPROXIMATION

(DWBA) THEORY

A. Collective Model DWBA

We have performed conventional collective model(19)
DWBA calculations, deforming both the real and imaginary
terms of the optical model potential. This type of analysis
has been thoroughly studied for 30 MeV (p,p') reactions.(46)
We have extracted deformation parameters 3' for all states
which have been assigned a J1T value by normalizing the
calculated angular distribution to the measured distribution

(47) and emphasizing

using a chi-squared fitting program,
the forward angle data. The resulting 8' values may be
used to estimate the strength of the corresponding B(EL)
transition between the ground state and the state of ’
interest. The formalism of Bernstein‘48) has been designed
to describe the extraction of isoscalar transition rates
from inelastic a scattering. However, the method can be
applied to our results if one assumes 1) that spin and iso-
spin flip amplitudes are small, which is reasonable for this
mass region, and 2) the contribution from the interior of

the nucleus, which contributes to proton but not alpha

inelastic scattering, can be neglected. There is some

34

35

evidence that these are reasonable assumptions, since
deformation lengths 6'=B'R' obtained from inelastic proton
and alpha scattering are often in good agreement.(l7’48)
The most accurate comparisons can be obtained for the
relative strengths for a fixed L transfer in a limited mass
region, since theoretical and experimental errors will tend'
to cancel out. In this case, the B(EL) values which this
method of analysis yields will be useful as a guideline for
comparison to B(EL) values predicted by nuclear structure
models, such as the shell model. The isoscalar transition
rates are calculated (in single particle units) from the
(48)

equation

(zsm) 2 ( am 2
L = 4fl(2L+l)

 

(Eq. 2)

where 8m, the mass deformation parameter, is obtained

from the prescription of Bernstein<48)

BIRI = BmRm ' _ (ch 3)

o
where 8' is obtained from the relation EEEEE-= (8')2.
thy
R' is the imaginary radius obtained from optical

model fits to elastic scattering, and Rm§1.2 Al/B, the
cutoff radius for a uniform charge density model of the
nucleus. The results for GL for a more realistic Fermi
charge distribution can be obtained by multiplying the

result from Eq. 2 by a tabulated correction factor given

36

in Ref. 48. The results of this analysis will be presented

in Sec. VIIB.

B. Microscopic DWBA
l. Distorted Wave Calculations
The microscopic DWBA calculations were per-
formed with the code DWBA7O written by Raynal and

(49) This code is based on the helicitY

Schaeffer.
formalism of Raynal(50) which automatically accounts
for all values of orbital angular momentum L and
spin angular momentum S that can be transferred in

a given transition from a J"=0+ ground state. The
knock-on exchange term, which describes exchange
between the projectile and target nucleons, is
included in the scattering amplitude. Central,
tensor and L-S interactions can be included in the
two-body force between the projectile and target
nucleon.

The cross section is defined such that

do 1 )3

= _ f* f (Eq- 4)
35 2 OlOf ch,ol ofM,ci

.M '
where
_ m 2 JT ( 1 - . (+)
- Z. X (j 3 )Julle > (Eq. 5)
UfM’Oi 21m 3h3 k C P h 1‘- °i

37

l. ci, of are the helicities of the incoming and

outgoing particles, respectively.
2. M is the helicity of the residual nucleus.
3. k1, kf are the momenta of the incoming and
outgoing particles, respectively.
A target nucleus with spin zero ground state
is assumed, and the final state of angular momentum
J is described by a particle in orbital jp and a
hole in orbital jh. v is the two-body interaction
between the projectile and target nucleons.
The quantity ZgTj
h P

for the transition and contains all of the nuclear

is the spectroscopic amplitude(51)

structure information needed to describe the initial
and final states of the target nucleus. A full
description of the helicity formalism is given in

Ref. 50.

2. Spectroscopic Amplitudes

The spectroscopic amplitudes qu

th
P
major interest in this paper. The procedures used

are of

to calculate these quantities depend upon the model
used to describe the states of the target nucleus.
In this work the target states are described by the
large basis shell model wave functions described

in Sec. IVA. We have modified the Oak Ridge—

(25)

Rochester shell model codes to calculate the

38

spectrosc0pic amplitudes ngj
h p

for DWBA calculations. The j-j coupling formalism,

in a form convenient

used by most workers performing DWBA calculations,
is more familiar than the helicity formalism pre-
viously discussed. The following discussion will

therefore be based on notation very similar to that

(23) (52)

of Satchler and Petrovich. For the direct

term of the scattering amplitude, the spectroscopic
amplitudes are related to the single particle
matrix elements defined by Satchler.(23'24)

JIJ

H!)

J?
p

T Q . T O U : T U . _
[ML(3p3h)5s,o + NLlJ(3p3h)53,1]‘MLSJ(3p3h) 3
. F
l 1 JT . 1 . l
* .. _. *

(ETTa’Tb Talzrb>tszjh <£pjp§J[TLSJ(8,¢,g)1j|£hjh§>

(Eq. 6)

where T is the spin angle tensor, and 1 is the

LSJ

isospin Operator.

3. Elements of the DWBA Formalism (Direct Amplitude)

The transition density can now be defined

LSJ,T _ f

T I I
F - . . (j j )u (r )u (r ) (Eq. 7)
(r1) Jpjh MLSJ p h nplp l n 2 l

h h

where un£(r) are the bound state wave functions

of the active valence particles. We use harmonic~

39

oscillator wave functions for the bound state, with

the oscillator parameter given by

45 25
hm = —I7§-- -§7§- (Eq. 3)
A A
LSJ,T

The form factor G for the direct term of the

(r)
0
scattering amplitude is related to the transition

density through the relation

Gfiing = (FiiifT LT(r0’rl) Ii drl (Eq' 9)

th

where ViT(ro,rl) is the L multipole in the

decomposition of the two-body force between the

bound nucleon and the incident projectile. The
LSJ,T
(r0)

information which describes the states of the target

form factor G thus contains all of the structure
nucleus, as well as the information concerning the
form of the interaction between the projectile

and target nucleon.

The form factor G?::)T

the incident and exit channel optical model wave

is then folded in with

functions, and squared to get the direct contribution
to the DWBA cross section. It is clear from the
previous discussion that the magnitude and the

shape of the angular distribution is dependent on

1) the Optical model parameters which describe the

4O

elastic scattering from the target nucleus, 2) the
form.of the two-body force between the projectile
and target nucleons, and 3) the transition density.
We now consider these elements of the cross section
in turn.

The procedure for determining the optical
model parameters is well defined. The various
parameters of the optical model are adjusted until
one obtains the best fit to the measured elastic
scattering from the target nucleus, as was discussed
in Sec. V.

The form of the two-body force is, however,
not well defined. The most desirable two-body force
to use would be one which describes two-nucleon
scattering, such as the Ramada-Johnston potential.(53)
However, due to its hard core, this potential cannot
be used in its original form. A common technique
is to apply the Scott-Moszkowski separation method(54)
to the attractive-even state components of the
Hamada-Johnston potential, and neglect the odd state
parts on the basis that they are much weaker than
the even state components. This results in an
even state force similar to the Serber force. This

(24)

approach, used by Love and Satchler, results in

a potential which retains the basic features of the

41

original potential at low energies and is usable

in inelastic nucleon-nucleus scattering calculations
as well as in bound state calculations. Petrovich,
et al.(55)

w

Kallio-Kolltveit

have used a similar approach with the
(56) interaction as the effective
interaction and they find that this realistic force

gives a good account of proton scattering from 12C

and 40Ca. Other often-used interactions have Yukawa
or Gaussian radial shapes with strengths and ranges
chosen to reproduce low energy nucleon-nucleon
scattering data. Comparisons between these various
interactions are given in Ref. 24.

Many different exchange mixtures other than the
Serber mixture have been used in nuclear structure
calculations. Often they have large odd-state
components, thus differing quite sharply with the
forces predicted by realistic interactions. We
have tried seven of the more commonly used structure
forces(57) to see how their predictions for inelastic
scattering compare with those for the central force
which we use in this work. This force is an even
state Yukawa force with a range of 1.4 F, and a
strength chosen to be consistent with a recent

(58) The

survey of inelastic scattering analyses.
results of this study will be presented in Sec.

VIICZ.

42

For a zero range force, it is clear from Eq. 9
LSJ,T
(r0)

the transition density. As the range of the two-

that the form factor G is given by r2 times

body force increases, the form factor continues to

reflect the radial shape of the transition density
LSJ,T
(r1)

defined element in the cross section, namely, the

F .' This brings us back to the least well

transition density.

4. Transition Density

Referring back to Eqs. 6 and 7, we see that
this quantity is determined by the wave functions
which describe the states cf the target nucleus.
At this point it is useful to present a short review
of the properties of single particle wave functions

u which occur in the transition density. In this

n1
work we use harmonic oscillator wave functions,
hence the single particle wave functions are
completely specified by the principal quantum number
n and the orbital angular momentum 2. The quantum
number n specifies the number of nodes in the wave
function; we use the convention that n starts from
1. We recall from Sec. IVA that the single particle

orbitals which form the shell model basis are the

1g7/2, ZdS/Z’ 2d3/2, and 331/2. The single particle

43

wave functions of interest in this work are

therefore u14, u22, and “30’ which have 1, 2, and

3 nodes respectively. The transition density will

be constructed from a sum of products of single

particle wave functions, each product being weighted
. T . . . . .

by the appropriate MLSJ(jpjh), which in turn is

dependent upon the spectroscopic amplitude ng. .

3
It is clear that a term such as u14ul4 in theh p
sum will contribute an unstructured shape, while
a term such as uzzu3O will have a very structured
contribution. An example of a typical transition

density is that for the transition to the 2: state
138

in Ba. It is given by
F202'° = -o 339u u +o l35u u -o 089u u +0 031u u
(r1) ° l4 l4 ' 14 22 '.. 22 22 ° 22 30

The coefficients Mg02(jpjh) are determined by the

wave functions for the 03.3. and 2: states.
Different.wave functions will give different
coefficients, and thus a transition density of
different shape. This will in turn modify the

shape of the form factor, and also the cross section,
as discussed earlier. The magnitude of each MESJ(jpjh)

is related to the coherence properties of the amplitudes

of the components of the wave functions; in general

44

k

a larger MESJ(jpjh) will imply constructive
interference between the amplitudes while a small
number will imply destructive interference. However,
in our case the smallness of the MESJ(jpjh) involving
the 2d3/2 or 381/2 orbitals is also in part due to
the restriction on the occupation of these levels,

as discussed in Sec. IVA. Hence, there are two
aspects of the transition density which affect

the cross section; the first being the magnitude

of the individual components, and the second being
the interference among the terms of the sum.
Specific examples of transition densities will be
illustrated in Sec. VIICl, where we compare the

138Ba calculated with both

transition densities for
the A=l36-140 and Ael36-145 sets of wave functions.
we will find that one can easily distinguish the
two sets of wave functions by virtue of the resulting
transition densities and the angular distributions
which they predict.

we note at this point that other methods exist
for extracting the necessary transition density.
One promising approach is through inelastic electron

(59) It can be shown_that the transition

scattering.
density is simply related to the measured form
factor in (e,e'), and high quality (e,e') data over

a wide range of momentum transfer can fully determine

45

the proton transition density, since electrons
are only sensitive to protons in a nucleus. This
can then lead to a determination of the neutron
transition density if proper account is taken of
core polarization effects.

Since 30 MeV prdtons are not strongly absorbed
the cross section is sensitive not only to the tail
of the transition density and its value in the
vicinity of the nuclear surface as in (d,d'), but
also to the transition density inside of the
nucleus. For this reason, medium energy inelastic
proton scattering provides a very sensitive test
of the wave functions involved in a transition.

The previous discussion has been concerned only
with the direct DWBA contribution to the scattering
amplitude. The transition density is not defined
as such for the exchange amplitude, since for
exchange, the bound state wave functions have differ-
ent radial co-ordinates. However, a corresponding
quantity. to MESJUPjh) for the direct term exists

(24) and can be used to obtain

for the exchange term,
an estimate of the magnitude of the exchange contri-
bution to a given transition. In the case of the
exchange amplitude, the form factor is different

for each pair of partial waves, and involves many

46

multipoles of the two-body force for an orbital

angular momentum transfer L.
However, it has been found by a number of

(24'60-62) that the direct and exchange

authors
amplitudes are constructively coherent in general,
and identically so for a zero-range even state
force. Atkinson and.Madsen(28) have done a
particularly complete study of the properties of
the direct and exchange amplitudes for transitions
in single closed shell nuclei (such as the N=82
nuclei). They find that with a Serber force the
shapes of the exchange contributions to the angular
distributions are very similar to those for the
direct term. Since we have also used a Serber
force, the previous transition density discussion
appears to remain valid when the exchange amplitude
is included in the cross section; the main effect
of the exchange amplitude being a renormalization
of the magnitude of the cross section.
5. Discussion of Selection Rules and Non-Normal

Parity States

In terms of the transferred orbital angular
momentum L, spin S, total angular momentum J and
isospin T, the selection rules for the direct

amplitude in the DWBA are

47

i f
An = (-)L
where Aw denotes the change in parity. Both 138Ba
and 144Sm have JE=O+, so J=Jf.

For the exchange term in the DWBA amplitude, the
selection rule A1r=(-)L no longer holds. Thus for
exchange, all four triads (LSJ)=(JOJ), (JlJ),

(J-l lJ), and (J+l lJ) can contribute to the

cross section, while for the direct term, only

the first or second pair can contribute. The terms
which the second pair cf triads give rise to are
commonly referred to as non-normal parity terms

and are found in general to be small except in the
case of transitions to high spin states, where they-
can become non-negligible.(24) .

The only experimental angular distributions for
non-normal parity states (states for which nf(-)J)
in masses A>80 of which we are aware arerthose for

138

the 3+ and (5+) stateszin Ba, which we observe,

and an angular distribution for a 4- state in

48

2oapb.(63) In all cases, the cross sections

are less than 20 ub/sr at all angles. The fact
that so few non-normal parity states are observed
in inelastic scattering is evidence in itself that
cross sections to such states must be small. In
conclusion, both theory and experiment indicate
that non-normal parity states (also often referred
to as spin-flip states) are very weakly excited in
medium energy inelastic proton scattering on heavy
nu01ei.

As mentioned in Sec. IVB, the pseudo spin-

(44)

orbit coupling scheme of HeCht and Adler predicts

a selection rule for inelastic scattering. This
selection rule is based on the assumption that the
transition densities are independent of the quantum

numbers j jh and depend only on the orbital angular

P
momentum transfer L. For (d,d') this is a reason-
able assumption, since the alpha particles are
strongly absorbed by the nucleus and are sensitive

to the transition density only at the nuclear sur-
face where for the case of N=82 nuclei, the transition
densities are quite similar as illustrated in

Ref. 14. However, this may not be a good assumption

for protons, which are not strongly absorbed. The .

selection rule states that AB=O where B is the total

49

pseudospin. For an even number of protons, B ranges

from v/2 to 0, where v is the seniority of the

138Ba which we have

state. The wave functions for
used have mixed seniority, but projecting out wave
functions of good seniority reveals that the low-
lying J#0 states are more than 80% seniority two
states. In the Hecht-Adler scheme, the lowest

2+, 4+

, and 6+ states have v=2, B=0 and transitions
to these states from the v=0, B=O ground state are
allowed. Our data show them to be relatively
strongly excited. The next group of states have

v=2, B=l so transitions to these states would
violate the AB=0 rule. With two exceptions,

Figure 2 shows that the positive parity states
between 2.2-3.2 MeV are weakly excited. The third
group of states predicted in the pseudo spin scheme
are v=4, B=0 states. The relatively strong 2+ states

at 3339 and 3368 keV in 138

Ba are perhaps members~
of this group, since the shell model also predicts
these levels to be mostly seniority four states.

we thus find that the pseudo spin orbit selection
rules predict results which are in qualitative

agreement with our data.

VII. DISCUSSION OF RESULTS

In this section we present results of the calcula-
tions described in Sections IV and VI, and compare the
theoretical predictions from those sections with the

experimental results from Section III.

A. Comparison of Shell-Model Calculations to Experiment
we begin by contrasting the results of the shell

138 144

model calculations fOr Ba and Sm described in

Sec. IVA with the experimental energy level spectra obtained

138Ba resulting

in Sec. IIIA. The energy level spectra for
from shell model calculations with both the A8136-l45 and
A=136-l40 Hamiltonians, together with the experimental

level spectrum as determined from our measurements, are
shown in Figure 14. The calculated energy level spectrum,
using the A=136-140 interaction, is in excellent agreement
with experiment. There is oneeto-one correspondence between
theory and experiment up to 2.9 MeV of excitation, with the
exception of the experimentally missing excited 0+ state.
Each of the states predicted by theory up to 2.85 MeV are

in agreement with their experimental counterparts to within

200 keV, and for most states the agreement is better than

50

51

100 keV. In Sec. IIIC4 we noted that the angular distri-
bution for the state at 2201 keV did not enable us to
assign this state J"=6+, although a 6+ assignment is not
inconsistent with the data, as seen in Figure 11. Compar-
ing experiment to theory, one clearly finds additional
support for a 6+ assignment to this state. The states at
2415 and 2445 keV, which have been assigned (5+) and 3+
respectively,(37) have structureless angular distributions
and are excited very weakly in our (p,p') measurements,
thus leading us to concur with the non-normal parity
assignments. These JTr assignments are also in excellent
agreement with the shell model predictions. Recalling from
Sec. IIIC3 the discussion of a l+n4+doublet at 2583-2584 keV,
we see such a preposal is strongly supported by predictions
of the shell model.

In the discussion of Sec. IVA, we indicated that
the parameters in the A=136-l45 interaction may reflect
deficiencies in the basis space for the upper N=82 isotones,
since states of these isotones used in the determination
of the interaction parameters may have configurations lying
outside the present basis space. We observe from Figure 14
that the A=136-l45 interaction does not reproduce the.
138Ba energy levels with the accuracy of the Asl36-l40
Hamiltonian. The major differences are in the first 4+-6+
splitting, and in the spacing of the levels from 2.3 to

2.6 MeV. A more sensitive test of the relative quality of

52

the wave functions resulting from these interactions is
found in the microsc0pic DWBA calculations, which will be

discussed shortly.

The experimental energy level spectrum for 144Sm

along with the predicted energy level spectrum calculated
from the Ae136-l45 interaction is shown in Figure 15.. The
general characteristics of the experimental spectrum are

reproduced; however the specific state-by-state agreement

is not as impressive as the agreement between theory and

138

experiment for Ba. Allowing more than one proton

excitations into the 351/2 and 2d3/2 orbits, and inclusion
of pairs of particles in the lhll/Z orbit would be expected

to improve the agreement between theory and experiment.

B. Collective Model Results

In this section we present the results of a

138 144

collective model analysis of all states in Ba and Sm

for which J1r assignments have been made. Figures 16 and 17

show our measured angular distributions for the lowest—lying

138 144

2+, 3-, 4+, and 6+ states in Ba and Sm, along with

the collective model predictions for these states. The

optical model parameters labeled Set SII(Set SI) in

138 14

Table 2 have been used in the calculations for Ba( 4Sm),

but any of the three sets give fits of similar quality. As

seen in Figures 16 and 17, the predicted angular distributions

53

are in good agreement with the data for the 2+ and 3-
states, but this agreement deteriorates as one goes to the
higher spin states. Results of calculations for the
remaining states obserVed in our measurements are not
shown, since our use of characteristic shapes indicates
that all experimental angular distributions for a given
J1T are very similar in shape. The empirical observation
that the shapes of the angular distributions appear to be
independent of excitation energy in the energy range from
l-3.5 MeV has been verified through our collective model
DWBA calculations.

Deformation parameters B£,as discussed in Sec. VIA,
have been extracted by normalizing the eXperimental and
theoretical cross sections over the angular range of the
data. The results of this analysis are given in Table 3.
The smallness of these parameters, except for the lowest
3- states, is another indication that the states are not
strongly collective. Barker and Hiebert‘I7) have also
studied 144Sm(p,p') at 30 MeV, and the results of their
collective model analysis are in good agreement with ours
for the four states which they observe (see Table 3).

In Table 3 we also list the isoscalar transition
strengths GL, in single particle units, calculated from
Eqs. 2 and 3. we note they are only about one-half the

value obtained from Coulomb excitation measurements for the

54

+
21

this mass region,

states. This phenomenon persists for other nuclei in
(16) which seems to be the only region

which does not exhibit equality between the isoscalar and
electromagnetic transition rates when they are extracted

using this model.

C. Microsc0pic Model Analysis

As discussed in the introduction, one of the main
purposes in undertaking this work was to determine if large
basis shell-model wave functions could accurately account
for both the shape and magnitude of measured angular
distributions obtained from inelastic proton scattering.
To this end, we have applied the microsc0pic DWBA theory

discussed in Sec. VIB to calculate angular distributions

for the 2+, 2:, 4:, 4;, and 6:, 6; states observed in
1388a and 144Sm.

The pertinent details of the inelastic scattering
calculation are as follows. The Optical model parameters

labeled Set SII(Set SI) in Table 2 are used to describe

138 144

the incident and exit channels of Ba( Sm). The bound

states are described by harmonic oscillator wave functions,
with an harmonic oscillator constant obtained from Eq. 8.
The two-body interaction between the projectile and target
nucleons was central only, with a Serber exchange mixture

and a Yukawa radial dependence. The range of the force
S=0

=-804 MeV,
PP

was taken to be 1.4 F and its strength (V

55

S=O

where Vpp is the S=0 part of the proton-proton interaction)

was chosen to be the mean of the strengths obtained in a
recent survey of inelastic scattering analyses.(58)
Core polarization, i.e. the effect due to contri-
butions to the cross section from nucleons outside the
explicit shell-model basis space, must be included for a

(64) In the

prOper description of inelastic scattering.
case of electromagnetic transitions these effects also
appear and are accounted for by renormalizing the charge
on the nucleons, i.e., by introducing a "polarization

(28) (29) have shown that one

charge". Madsen and McManus

can similarly correct for finite basis-space effects in

inelastic scattering by renormalizing the strength of the

two-body force which mediates the transition. Thus One

has an "effective force" for (p,p') which is analogous

to the "effective charge" for electromagnetic transitions.(65)
One can get an idea of the amount of core partici-

138Ba by noting that for

pation in the low-lying states of
the wave functions used here, the calculated B(E2; 2I+OI)
is a factor of 3.2 too small(66) if no polarization charge
6e is used. This implies that (1+6e)2=3.2, or de=0.8. To
account for the contribution to the (p,p') reaction of
protons excited from the core, one therefore renormalizes

the interaction strength V to (1+6e)Vpp. However, neutron

PP

56

core excitations also contribute to (p,p') cross sections
and are, in fact, more important than those for protons,

since the proton-neutron two-body interaction Vpn is stronger

than V . If it is assumed, as has been found by Bernstein(67)

PP
(68)

and Astner, et al. that contributions from neutron and

proton core excitations are approximately in the ratio of
N/Z (the ratio expected in a collective model picture),
we need an additional term, N/Z 6e Vpn' For the Serber
exchange mixture we use, V =2V . Thus one obtains a

Pn PP

total ff ct’ f rc of 1+6 V +2N Z6 V = 1+6 1+2N Z V
e e ive o e ( e) pp / e pp [ e‘ / )J p

i.e. the strength is increased by a factor of (1+6e(1+2N/Z)).

I

P

We use this result in all calculations in this paper. We
have also inverted this process and used the measured
enhancement factors to extract polarization charges for other

138

transitions in Ba, for which electromagnetic transition

data are not available. Calculations of cross sections

were performed for the 2:, 2;, 4:, 4;, 6:, and 6; states
in 138Ba. The enhancement factors were extracted by

normalizing the experimental and theoretical integrated
cross sections over the angular range of the data. The
results are shown in Column 3 of Table 4 and we see that
the 6e are constant within the probable overall uncertainty

in the analysis.

138

l. Microsc0pic DWBA Calculations for Ba

For 138Ba, we have calculated angular distributions

using both sets (A=136-l40 and A=136-l45 Hamiltonians) of

57

wave functions for the aforementioned 6 states. Figures 18
and 19 show the angular distributions predicted by each
set of wave functions, together with the apprOpriate data.
We see that the magnitudes and shapes of the calculated
angular distributions are in good agreement with the data
for the 2:, 4:, and 6; states for both sets of wave
functions. However, the agreement between theory and
experiment for the 2+, 4:, and 6; states is much better
for the A=l36-l40 set of wave functions than for the
A=l36-145 set, which predicts cross sections which are an
order of magnitude low, and also exhibit poorer agreement
in shape. From this analysis it appears that the A=136-l40
set of wave functions gives the better description of low-
lying states of 138Ba.

To find the reason for this behavior, recall from
Sec. VIB4 that all of the nuclear structure information
entering the DWBA calculation is contained in the transition
densities. Thus it is informative to compare their
differences in structure as calculated from each set of
wave functions. Figure 20 shows such a comparison. The
magnitude of the transition density is plotted versus the
radial coordinate in units of roA1/3; thus unity corresponds
to the nuclear surface. Inspection shows that the transition

+

densities for the 2:, 41, and 6: states are quite similar

for both sets of wave functions. They peak slightly inside

58

the nuclear surface, which is a general characteristic(52)

of these quantities. We recall that the cross sections
calculated for these states were very similar for both sets
of wave functions. However, looking at the transition

4;, and 6; transitions, we note

densities for the 2:,
distinct differences. The A=l36-l45 wave functions yield
transition densities with a pronounced decrease of the
surface peaking, and an increase in magnitude of the peak
near 0.6 of the nuclear surface, relative to the A=l36-l40
results. For the 6; state, this effect is so great that the
transition density resembles that for a much lighter
nucleus. Since the transition density (and therefore the
form factor) peak too far inside the nuclear surface for
this transition, the diffraction pattern predicted by the
DWBA is pushed out, as if the target nucleus were indeed

much lighter. This is exactly what we observe for the

calculated 6; cross section for the A=l36-l45 set of wave

functions.

We can infer specifically what is wrong with this
set of wave functions by considering the individual contri-
butions to the total transition density for the 6; state,
for example. Figure 21 shows this transition density
broken down into its component parts un£=u14 ul4 and
ul4u22. The A=136-l40 set of wave functions enhances
the ul4u22 term relative to the ul4ul4 term, which results

59

in the surface peaking. However, this enhancement is not
present for the A=136-145 wave functions, which results in
the transition density being very small at the surface due
to a cancellation effect. Tracing back one step farther,

we can look at the spectroscopic amplitudes which contribute
to ul4ul4 and u14u22 and thus directly study the wave
functions. Only the <g7/2|a+a|g7/2> matrix element
contributes to ul4ul4, while both the <g7/zla+a]d5/2> and
<d5/2|a+a|g7/2> matrix elements contribute to the ul4u22
term of the sum. Table 5 shows the structure of the largest
components of the pertinent wave functions, as well as the
individual contributions to the 6: transition density

calculated with both sets of wave functions. The two

transition densities are thus

F606,0

(r1) = 0.1292 ul4ul4 + 0.3223 u14u22 A=136-140
F606'0

(r1) = 0.1765 ul4u14 + 0.1978 ul4u22 A=l36-l45

The quantity of interest is the ratio u14u22/u14u14 which
has the value 2.50(l.12) for the A=136-l40 (A=136-145)
interaction. The major reason for the smaller ratio from
the A=136—145 wave functions is the smallness of the u14u22
term. Even though this set of wave functions has the
smaller g7/2-d5/2 single particle splitting and hence would
be eXpected to have the larger u14u22 term, a detailed

comparison of the wave functions in Table 5 show this is

60

not true. The largest contributions to uuu22 for the
A8136-l40 set come from the (g7)5+(g7)5(d5)l and

4 2+ 4 2
(97) (d5) (97) 5)

wave functions are much smaller. This arises from the

(d amplitudes, which for the A=136-l45
observation that the smaller 9.7/2-d5/2 splitting leads to
a fragmentation of the strength of the 6; wave function
over many components. Many of these components cannot be
connected by the one body (p,p') Operator (a+a) to the
strong components of the much less fragmented ground state,
and hence do not contribute significantly to the transition
density. It thus appears that this decreased mixing
between the 97/2 and d5/2 orbitals is the required
ingredient in obtaining a reasonable fit to the angular
distribution for the 6; state. A similar analysis applied
to the transition densities for the 4; states arrives at
the same conclusion-namely that there appears to be too
much mixing of the 9.7/2-d5/2 orbitals in the higher lying
states as calculated with the A=136-l45 interaction. Specifimx
information concerning the structure of the wave functions
can thus be obtained by analyzing the composition of the
transition densities in this manner.

Comparing the DWBA calculations result’ A“rom the
collective model (Fig. 16) and microscopic moo. .18)
for the 2+ state, we see that the collective modeJNa

l
somewhat better job of fitting the data. This is due shinly

61

to the 2; microsCOpic calculation slipping out of phase

with the maxima and minima of the data. We have performed
a collective model calculation using the form factor
obtained by deforming only the real part of the Optical
model potential, and obtained an angular distribution nearly
identical to the angular distribution obtained from the
microsc0pic model. This implies that if one were to
combine the form factor obtained by deforming the imaginary
part of the Optical model with the real form factor used

in the microscopic analysis, the resulting angular distri-
bution would be in as goOd agreement with the data as that
of the full deformed collective model. Unfortunately, the
computercode we use for the microsc0pic DWBA calculations
does not at present have the Option of using a complex'form
factor; thus the above conclusion is rather speculative.
However, it has recently been found that using a complex
microsc0pic form factor results in appreciably better fits

(69) and asymmetry(70) data. It

to angular distribution
would also be interesting to determine if the addition of

an imaginary term to the form factor, which appears to
improve the fit to the 2:, would destroy the relatively good
fits to the 4: and 6: states obtained with the present

micrOSCOpic approach.

62

It must be noted that our calculations use a
multiplicative constant in the form factor to account for
core polarization and the angular distributions take the
shape predicted by the purely microsc0pic calculation.
Other microsc0pic calculations(64) frequently employ an
additive term in the form factor based on the collective
model to simulate the effects of core polarization. Often
times, using this latter procedure, the collective term
is the dominant contributor to the cross section, so the
predicted shape for_the angular distribution is in reality
due to the collective model contribution. Thus, these
"microscopic" calculations achieve good agreement with
the data as regards the shape of the angular distribution,
due to the fact that collective model predictions for the
shape are in general better than those calculated from a

purely microsc0pic model such as we have used.

2. Calculations with Other Forces

A Serber exchange mixture, which is an even state
force, is generally employed in microscopic inelastic
scattering calculations. It has been found to contain
about the correct exchange strength to give the required
enhancement for different L transfers. However, in nuclear
structure calculations, many other types of forces have
been employed, and we have tried a number of these to see

if they give as good a description of the inelastic scattering

63

process as does the Serber mixture. We have made calculations
for each of the forces "Cal", "Cop", Clark Elliot I,
Ferrell-Visscher, Rosenfeld and SOPer summarized

in Ref. 57, all of which contain non-negligible odd-state
components. The triplet-even parts of the various forces
l=-40 MeV. The range of the

TE
two-body force, which had a Gaussian radial shape, was

have all been normalized to V

1.67 F; this translates into 1.346 F for a corresponding
Yukawa radial dependence which we use. The strength of the
Serber fOrce is very similar to the strength we have used
in the calculations discussed previously. The results

are shown in Table 6, along with the results obtained with
the Serber mixture.

We observe that only the Clark-Elliot I and Soper
interactions (which have weak odd state components) yield
results similar to the Serber mixture, which reproduced
the data. The forces with strong odd-state components cause
the direct and exchange cross sections to exhibit the wrong
dependence upon L—transfer and in addition, often the
theoretical angular distributions acquire shapes which do
not resemble the data. We conclude that while such forces
may be acceptable for nuclear structure calculations, the
strong odd-state components make them inadequate to describe

inelastic proton scattering.

64

we have also performed calculations including tensor
forces, consistent with OPEP(58) and find that the tensor
force contributions to the cross section are at most 7%,
and this is for the 6+ states, where the non-normal exchange
amplitudes are becoming non-negligible.(24) The spin-orbit
force may be important‘7l) for the 6+ states, and this
possibility is being investigated further.

In addition, we have calculated angular distributions
for the non-normal parity 3+ and 5+ states, using only the
A=136-l40 set of wave functions. Such states are not
expected to be enhanced due to collective effects, but
should be sensitive to the tensor and LS forces. The pre-
dicted cross sections calculated with central,and central
plus tensor plus LS forces obtained from the Hamada Johnston
potential and given in Ref. 58, are presented in Figure 22,
along with the data. The enhancement due to core polariza-
tion is not included, and the direct plus exchange calcula-

tions underestimate the data by a factor of 3-5 for the

central plus tensor plus LS force.

144

3. MicrosCOpic DWBA Calculations for Sm

Only the A=136-145 Hamiltonian was applicable to
144Sm, so only the set of wave functions corresponding
to this interaction was employed in the DWBA calculations.
The results are shown in Fig. 23. The D+E angular distribu-

tions for the 2: and 4: states are a factor of two smaller

65

than the data at forward angles, while the 6: calculation
is not in good agreement with the experimental distribution.
However, we recall from Sec. IIID4 that this state is
possibly a doublet, which would make any theoretical
conclusion based on this state tenuous at best. The
underestimation of the magnitude is undoubtedly due in

part to the restricted basis space in which the shell model
calculation was performed. This inadequacy of the basis
space is clearly pointed out when one considers the
calculation for the 2; state. All of the strength has

been concentrated in_21, causing the angular distribution
for the 2; state to fall a factor of 25 below the data.

In addition? the shape is not qualitatively correct; the
maxima at 40° is larger than the first at 20°. The
calculated 4: angular distribution is a factor of73 lower
than the data, while the calculation for the 6; bears no
resemblance to the data. This can easily be predicted from
observation of the transition density for the 6; state; it
is negligible at the surface of the nucleus. All of its
strength is concentrated around 0.6 of the nuclear radius,
as in the similar case for the 6; transition density in

138Ba calculated with this set of wave functions. Most

144Sm microsc0pic calculation is the

disturbing about this
fact that a constant polarization charge does not reproduce

the magnitude of the angular distribution for different

66

L-transfers. One is required to use a state dependent

polarization charge, whereas a state independent charge

138 . (65)

was found to be adequate for states in Ba However,

we feel this also is a manifestation of the limited basis

144

space for the Sm shell-model calculation.

VIII. CONCLUSION

High resolution medium energy inelastic proton
scattering has been shown to be an effective method for
«:btaining precise information concerning the excited states
(of a nucleus. We have obtained excitation energies for

levels in 138

‘wwr‘““*+—flm-r

Ba which are in excellent agreement with
previous gamma ray work on this nucleus, and in addition.
‘we.are able to reduce the uncertainty on previous spin-
parity assignments, and in most cases suggest an absolute
assignment. We have increased the number of known levels

up to Ex=3.4 in 144

Sm from 8 to 18 and suggested spin-
parity assignments for the majority of these states. The
experimental information concerning the levels up to

138 144Sm has been reviewed, and found

Ex=3.4 MeV in Ba and
to be in good agreement with recent shell model calculations
for this region.

Collective model DWBA calculations have been carried
out for both nuclei, using a form factor obtained by
deforming the real and imaginary parts of the optical
model potential. The results were analyzed using the

formalism of Bernstein to obtain estimates of the strengths

of the corresponding electromagnetic transition rates.

67

68

MicroscOpic DWBA calculations including the exchange

amplitude were performed for the 2; 2, 4: 2, and 6; 2 states
I I I
in 1383a and 144Sm, using shell model wave functions(19-20)

and a realistic two-nucleon force. The necessary structure
amplitudes were calculated with a modified version of the
Oak Ridge-Rochester shell model code. In the case of 138Ba,
two sets of shell model wave functions were calculated,

and it was determined from the DWBA calculations that
inelastic proton scattering clearly distinguished one set
as superior to the other. With this result the transition
densities, which provide the link between the wave functions
and the DWBA reaction model, were then analyzed to see

if one could determine the undesirable properties of the
poorer set of wave functions. The results of this analysis
indicated that the large mixing between the 97/2 and d5/2
orbitals was the problem in the set of wave functions which
yielded the poorer result. We find that a careful con-
sideration of the transition density between two states

can provide useful information concerning the structure

of the wave functions for these states.

The use of large basis shell model wave functions
such as we have used in this work appears to yield quite
satisfactory results and this technique should be useful
in performing consistent studies of inelastic nucleon

scattering. Such an analysis will hopefully enable one

69

to study the effects of core polarization and the nature

of the polarization charge of the core nucleons.

wssi“‘“‘”"’m"'

10.

11.

12.
13.

14.
15.

16.

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i

 

APPENDIX A

TABLES

74

 

;,|

 

75

 

 

 

 

Lble 1: Energy Levels of 138Ba and lem.
1388a _ . .. ‘ w 1““5111
resent Work Previous Work Present Work Previous Work
BXEJflla’d EXEJ’H" EXEJMaéd ._ Excmc
i838g 11.0 2* 1835.7 2* 18813*11.0 2* 1880.811.0 2*
1898g*11.0 8* 1898.810.3 8* 1811g 11.2 3' 1810.1 3‘
2090g*1l.0 * 2090.110.8 <8)* 2191g*11.0 8* 2190.811.0 8*
2201 12.0 2203.2f 2328e 11.0 2323.2 8*
2218 11.0 2* 2217.9 2 2823* 11.0 2* 2823.811.0
2308* 11.0 8* 2307.810.3 (3,8) 2878 11.9 2878.3 0*
2815 11.2 2818.9 (5*)h 2588 11.0 8+
2885 11.2 2885.8 3* h 2881 11.8
2582.8 1,2 2800 11.8 2*
2588 11.0 8* 2828 11.8 2830
2839 11.2 2* 2839.3 2 2883 11.9 *
2779* 11.0 8* 2779.210.5 2,3,8 3020 12.0 8*
2881g 11.2 3' 2880.5 3' 3080 12.0
(2929) 12.0 2931.1 1,2 3123 11.8 3123.8
(2990) 12.0 2990.8 1,2,3,8 3198 11.9
3050* 11.0 3089.911.0 1,2' 3227 12.0 3‘
3158 11.2 8+ 3266 12.3
3183.5 2,3,8 3308 12.1 8*
3258 11.2 ‘
'3285 11.8
3339 11.8 2* 3339.5 1,2
3352.2 1,2
3368 11.8 2*” ‘ 3385.9 1,2
a ‘

Excitation energies in keV.

From Ref. 7, except see (f,h) below.

From Ref. 17 and references cited therein and Ref. '40.
Subset of levels used in energy calibration is marked
with an asterisk (*) .

Unresolved doublet whose angular distribution is con-
sistent with a spin 6* state plus a lower spin state.
From Ref. 8.

gUsed in determination of characteristic shapes.

From Ref. 37.

b

 

 

76

 

 

Table 2. Optical Model Parameters for 138Ba(luuSm).
Adjusted ' Satchler Satchler
Becchetti-Greenlees Set SI Set SII
Rc 1.25 (1.25) -1.2 (1.2) 1.2 (1.2)
VR 53.26 (53.20) 56.80 (56.10) 58.80 (55.16)
PR 1.162 (1.172) 1.122 (1.13) 1.1“ (1.139)
aR 0.75 (0.75) 0.75 (0.75) 0.75 (0.75)
WVOL 3.78 (3.50) 3.0 (2.70) 3.0 (2.65)
rVOL 1.32 (1.32) 1.33 (1.33) 1.33 (1.33)
aVOL 0.66 (0.615) 0.696 (0.667) 0.672 (0.65)
WSURF 6.27 (6.11) 6.49 (6.72) 6.86 (7.09)
rSURF 1.32 (1.32) 1.33 (1.33) 1.33 (1.33)
aSURF 0.66 (0.615) 0.696 (0.667) 0.672 (0.65)
VLS 6.20 (6.20) 6.“ (6.4) 6.1 (6.1)
PLS 1.01 (1.01) 1.12 (1.12) 1.19 (1.125)
aLS 0.75 (0.75) 0.75 (0.75) 0.75 (0.75)
X2/N 1.8 (2.8) 1.8 (1.6) 1.1 (1.8)
+
OD+E(21) 0.215 ( .212) 0.235 ( .208) 0.229 ( .203)
OD+E(N:) 0.0678( .0629) 0.0727( .0613) 0.0717( .0599)
OD+E(6:) 0.0831( .0258) 0.0858( .0281) 0.0855( .0237)

 

5ww~wrr‘fi-v
-J
1

not

77

 

 

 

 

Table 3. Deformation Lengths and Transition Strengths for
138 188
Ba and Sm.
138Ba 188Sm
Ex Ex
Energy J BLR' G(L) Energy J BLR' G(L)
1838 2 0.83 8.1 1881 2 0.88(0.88)a 8.7(8.8)a
2218 2 0.23 1.7 2823 2 0.29(0.29) 3.5(2.9)
2639 2 0.07 0.2 2800 2 0.18 0.8
3339 2 0.15 0.7 1811 3 0.87(0.82) 38.0(28.8)
3368 2 0.17 1.0 3227 3 0.07 0.3
2881 3 0.75 18.8 2191 8 0.33(0.32) 5.8(8.3)
1898 8 0.31 8.1 2588 8 0.21 2.3
2308 8 0.21 1.9 2883 8 0.25 3.2
2588 8 0.08 0.2 3020 8 p 0.23 2.8
b b
0.11 1.0
2779 8 0.12 0.6 2328 (6 0.18 2.8
3156 8 0.07 0.2 3308 6 0.15 1.8
2090 6 0.26 5.1
2201 (6 ) 0.15 1.7
aNumbers in parenthesis are from Ref. 13.
bThese numbers represent the extreme credible fits to the data.

sew-cw
f

-u‘}

78

 

 

Table 8. Effective Charges for Transitions in 138Ba.
Transition ' Lb) 68¢)
0* 2: 2 0.82 .2
0* 8: 8 0.88 .2
0* 6: 8 0.81 .2
0* 2; 2 0.98 .2
0* 8; 8 1.07 .2
0* (8;)a) 8 0.73 .2

 

aThis state has not been unambiguously assigned 6+, but its
angular distribution, together with the shell-model predictions,
suggest this assignment.

This is the L-transfer for the dominant amplitude. Non—normal
parity amplitudes also contribute to the cross section, and are
included in the calculations. See Ref. 28.

c . . 2_
Calculated from the relationship [(1+6e(1+2N/Z)] ’Oexp/Otheory

is calculated using

b

where the theoretical cross section otheory

the shell model wave functions described in the text.

ersrsrrrrmra-Iv
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79

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APPENDIX B

FIGURES

81

82

 

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.I is o 10
8% O 0 d
. moo .w i a:
my. a
3 o + 9
8 .I a 10 3
o 3 0 H
w o i H
98 mm .m w
-a A s \ 8
f >02 0&0 m 3 3 3 o "I «I 1m .414
. mu r. r. 8 .h
. Ah avommn. .w m m m m i
8 h 3 .h 3
b b c b h‘ . b § 5 O

 

 

 

OOOI

.mumHm was» :0 counaoo on
on wmnmucw oou mm3 >mx HHmH um oumum m may m0 meH5 was .Esuuommm mm on» :H mm .meuwunm
IEH Hm can at Scum Hmuumom 50H£3 mcoumum on Ugommmuuoo mxmmm on» mo :Hmmmmo Hound mmsbn omoun

one .zmzm .>mx 5 usonm mo GOHuaHommH nqu .mmmumop ov um A.m.mvam¢vH Scum Eauuoomm .m musmHm

mmmzaz szzmzu
com com oor com com o3 o

 

 

 

 

 

 

84

 

 

 

 

 

 

 

 

$8 mm fig
1 mm a .0 m
m . a c. N
.0. 8.8 e 8 w l
T “a w m we. 8 w M S
+ m. h . R afl nu .0
.h... .f O 3
m. a H
a 8
n 8 W JnU fiu
meL 8. w 0 w
.3 ow a 8 N
.. >22 81$ .8 w .m 2L
3.3.53. w. w.
c . Q
h P P p p 0
0

85

m0>H50

0.2.

cm"

L .:L .J
Era-C1112. ..- filiaiwln ..

va

can

UGM MmmmH

3.3.5.6
om 9w Hz. om

 

8...—

ddqddddddd‘dqdddd

.2

.2

.2

.2

.2

cm—

382 .5
can 00 Am.

.3. cm

.A¢m .momv mmchmme cam
Huumnooom «0 muouoamumm Hmwos HMOHumo ms» nu«3 move mGOHuMH50Hmo Hoooa Hmoaumo mo muHsmmu mum
Cam

 

qua—dqd—duq_jfi—dqq—uqq‘

 

lllLL l L ihlllll l l

.2

.2

.2

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om

(IS/aw) tsp/DP

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86

 

 

 

:- a
I- "3 P
h:
o 3
10 .. .. 3’
Z . ,3
p d
. '3
b d
b- u:
I
10' ..
r :
h d
- 4
10-2 lJJllJllllllllLlLJlJlJl

 

o 20 80 60 80 100 120
90m. (deg)

Figure 5. Characteristic curves obtained by averaging 0&58
measurfid angular distributions from groups of states in Ba
and Sm which had previously assigned J values.

87

     
 

 

 

 

 

 

 

3. - r- .
-: 104: -:
-3 .- ..
-3 10-8 1
‘7 F a
3 4 17
d y. 3
a p 1
g 10"1 .- 2 10", 3.339. 2 ~ -.
'0 Z : t 1
L- 1 Z, l
3- 1 3‘ «4
3 4 . 3
10-8 2&339.22* 10-1
C I C I
r- q r- ‘3
10-311111L11L141U4111LJA11 -2AlllLlllejlllllJIJJLLJ
0 20 ‘00 60 80 100 130 0 30 ‘30 60 80 100 120

Figure 6. States in 138Ba which have ngular distributions
in agreement with the characteristic 2 shape, which is the
line+drawn through the data. We assign all of these states
J =2 with the exception of the state at 3050 keV.

Lens,»
1
.‘
.x

(A_'.‘“ s- "-

“n“

‘
...
J — ..

g“

Figure 7.

(niu80

davdn

144

States in

10°

1 VIVFUU'

F

10°

1 T 'V'II'

 

j

10'1

10'1

10'2

I I V'U'U'

V

 

10'3 ...1.ii1...12..1...1...

88

 

 

1

d

1 1111111

1

1 1111111

 

1

1 411 1111

 

 

2010

etunfideg)

in agreement with the characteristic 2
line+drawn through the data.

J =2 .

8080

100 120

Sm which have ngular distributions

shape, which is the

We assign all of these states

‘15...- W..- moon-L‘s JAW
4.

89

1.1.."
.... . .1. he
1‘ I 1.0.... ug‘la‘. 1

, . . mu b
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on» nuH3 usufimmumm 2H mcoHuanHuume umHamcc m>mn 50Hn3 EmevH can mmmmH 2H mmumum .m ouanm

332$ ‘ 3.3.5.6

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I 1 .2 .. 1 .2 b
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4 -m E3 3. 0
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a U H . w
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n. a h n. u. I\
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I
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90

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3.3 us 3.2 e 8m
cm” 8a DO 00 0... ON 0 .ON« cod 00 O“ or ON 0
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n i n
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. Lu 69.” . .
f 1 I p
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m . + m m m u
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a . iv mom N w
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n ‘ . m. . .m
- .8 .25 . 1 .2 . . 72
T A I 1
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3.25.. 3.25...
cm" 8" cm 8 or cm 9 cm" 2: cm 8 2.. cm a

4.....*_.:_..._..._...NIOH §NIOH
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1

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11111 1 1
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92

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. o vmcmHmmm coma mm: 2m
+mum mo>uso owumwuouomummw

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m can « 02» spam

0”“

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59—. 9m 00 .3. em a

4dJ1dJ—4qd—‘dd3-ddd-ddd

om“

 

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omHu+on

n :mHumc 03
cu mac pan
.20HuanauumHv AMHsmcm mHnu cmsoucu :3muv

2H uoHndoc a mo Honsme >82 Homw xmw3

cosmHMmm mHmsOH>on ommHuOH unawuanwuumHn .HH shaman

 

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add—.1d—«qqqqdddqdq_#qd Ib—
n U
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mama 0n UHnoo ucmesmammm :n 0: nng3 new on 2H moumum mm chHUfinHuumHu AMHsmcd .NH shaman

 

 

 

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303.88% 3ch E 0%
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1 I I + H
E...
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H o+++ . . . U
H mmm.m + U n H
m. WIS .: “:3
by (
kw o + 1 . + L
. . . . . + .
. .3 + . . H
m . + m m 5.. + u
w 2mm m hung v QWN 8.033

 

 

 

 

94

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coma on uHsoo usoficmammm 2n oc+noH 3 you 2m 2H noun». mo nsowusnwuuch uanmsd .mH musmwh

. . va . .
3.28.8 3.258
cm" .8" cm cm or am am I Om emu c3 on 8 o: . 0.0. o IOH
. i . . .
... + + H H U
I A + :m-2 : . . .. -2
wow... . + . a my
. . . . . W
m H H on on. H U
m . . m m 8... .. l m )
._.I II hulOH n: h IOH W
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I I .. I I 1
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m 83 m n m
n. h cod v mm..m . mane“

 

 

 

 

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j? a:

:1 LL 3.33 3.35 3,36 3‘38 ’ 29 3.3.. 3.37
-,L¢__-{1—-, .31:._; 327 3.29 326 _--"-’———0 .5;__: 3.29 3.26 3,23
‘ * j I a. 1 3.22 3.2“ #553; 3.23 3.22 3'25
.2:- _ 3 ,7 3.19 .320 3.19 h
3.1% . LLb—fi 3’16

3,32 3.13 _ ’ !l* 3,13 3.13
39 m :j—D: _“1_—* 3.05 3.06 3'08 3.05
- . 9.. .0:— 2 98 .a__+ 2.99 3-01 2.99
D. o
- 3,92 2.93 - 2+ ~3— 233 3. 2.93
_:r+__ 2+ 7 5. 33° 2.29
- + g; 2083 - q‘ 2.83 2.85 [1*
2.78
Ho 29
2.6“ 25"
L 2‘ 2,30 2.60 - 1+ 2+ 2.37 2.58 3’. “O.
- 5+ ‘3:— 2953 a 51
2'“ ' 3+ - 15+ 1 L— 2 w
Iii—h a go 2..“ 5+ 232 2.91 '
- 3+___ ' 2.35 2.37
M ‘19
2.31 El 2.3!
5+ 2 £ 2.20
b
2.99
L— u" 5’ 192 Mt
b . w
'“ 1.79
1’ 1.51 2
v
A 1m ‘9‘“

0* m m m m m

8813-905 INTEWTIM miss-mo INTENTION! WT
1388 1388 1388
R R H

Figure 14. Comparison of results of shell model calculations
discugged in Sec. IVA.with experimentally known energy levels
in Ba.

“mitt. ._‘_ . .-.‘_'

96

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1‘09
‘09 .
4L— 13.7 5° 3.3.
3' av’u”
IL 3.13 3.20
3.“
3.“ 3.12
b w
-JL————-——-su1&“. i,.
E! 'fll __
i, 22.37: w a
'19 1'
w w ;
—‘?—— 2.“ no t.“ I
_h———_. 1.“ ii
it w w L—‘_ “ ~
.L ..
-n n L— 373 w w m—_ m
J-
-“———————-un . LN
h um -1L——————-Lu
A—___ m
.h—_—_ m It m
kit!” WT!“ WT

HHS" HHS"

Figure 15. Comparison of.results of shell model calculations
disguised in Sec. IVA with experimentally known energy levels
in Sm. .

97

 
  
 

 
 
 
     
 

 

 

 

 

 

 

 

L . L .
10° __ L436. 2 * 1 10° __ 1
§ 3 E |.898. 4* i
D .. I h ‘
is 10-1: 1 10-1: 1
q . 4 L 4
Q ¢+ j A, J
.8 101 : d7 10_1_ a... 2.090. 6+ -7
E 2.88I. 3- E i
. J >- .
10° _. 1 10'3L .
s s E a
10-1 WM 10-3 WW
0 20 "lo 60 80 100 130 0 20 ‘00 60 80 100 130
9cm(deg) 9c.m.(deg)

Figure 16. Results oflggllective model calculations for in-
elastic scattering on Ba The optical model parameters
given in Table 2, Set SII were used, and both the real and im-
aginary parts of the optical mode1+wer§ degormed. The calcu-
lations shown are for the lowest 2 , 3 , 4 and 6 states.

 
  
 

 
 
  

 

 

 

 
 
   

 

 

 

’ ' ' T
F |.66|. 2 * ‘ ’ ‘
. . Z . .
t : t - :
L .. .
210".- . 10":- -.
i; E 0'5 E E
v I 1 C I
<3 ‘7 :7
v P d I-
\ A, 4, '
- - 2. 24. *
'3 101 .- 1 10 1.- +. 3 6 1
* - 1 E :
: L84L13 . _ I. .
u- d u- q
r- d L- . -<
100 b 1 10-2 r 1
.. .. .- q
C 1 Z 3.
I I .4 P ..
10-1lllllLLlllLllllllLdlll 10-3 llLlLlllllelll'LlLL
0 20 90 80 80 100 120 0 80 90 60 80 100 120
9mm”) 9mm)
Figure 17. Results oflggllective model calculations for in-
elastic scattering on Sm. The optical model parameters

given in Table 2, Set SI were used, and both the real and im-
aginary parts of the optical model were degormed.+ The calcu-
lations shown are for the lowest 2 , 3 , 4 and 6 states.

W“"‘-"—"-r_'
.4.

99

—Dm10m

--- DIECT
1° . I:\
.7 \ . 10936. 2+
1 1
b ‘ \ .
’ 0
16' - ‘x (
Jr v .
9 \
,5 . 9 \ I
. I 1.898. Q"
32 H? E:-' \\ '0
3 5 ‘
. \ __
.2 . .

10“

t1 ‘fiq— r
l

 

10.; .— . 2.090. 6+

: ‘00

g 09

x ’ \
F " \° 0
\

10‘ __ \ \

E \

: \
10"

0 20 110 60 00100120
9cm(d”’

Figure 18. Microscopic model DEBA calculations foi3§he in-

10"

10“

10"

10"

10"

elastic scattering to the 2 , 41 and 6
calculations are identical except for
left hand column employed the A=136-l40 set while the A=l36-

145 set was used for the right hand column.
wave functions predict very similar angular distributions.

 
 

l
I r ‘1'"

I 1 WTVUII'

V I 'I'IU'

b

 

I'IIUIU'] T11

—— OIKCT o m
-- - DIRECT

‘ 290909 6+
0..
...
C.
x I A \
0
\ 0
\ ~
\

\

0 20 ~10 so so 100 120

in

9mm)

states in

Ba.

Both sets of

The

e wave functions; the

dc/da (rub/5r)

Figure 19.

10'

10"

10‘

10"

10"

10'

10‘

10"

10"

10“

 

100

-—-qnmmro£xmnaz
---0mcr

2.218. 24-

2.308. 9+

2.201. [6+]

20 40 60 80 100120
Hammeg)

10'

10"

10'

10"

10"

10"

10*

10"

xcept for

—- DIRECT 0 am
- - '- OIKCT

I I. 20218. 2*
I

20308. 9*

/

7"

V I U""'

T V IIIUV"
I
\
/\

 

20 90 60 80 100 120

Microscopic mod 1 D cal ulations fo he in-
elastic scattering to the 2?, 4gngnd BS f3§
calculations are identical 3
left hand column employed the A=l36~l40 set while the A=136-
145 set was used for the right hand column.
indicate that the A=136-l40 set of wave functi

states in Ba.

ghe wave functions; the

better description of the low lying states in Ba.

 

These calculations
gag provides the

._

‘ 71‘ fifr—W‘T‘w
" :.=
..|

.<

101

M0,. —A'l36-|40 m _A'l36'|40
-—-A cuss-145 —--A=136-145

1'2 F LSJ'TM

 

 

 

Figure 20.133ransition densities for the 2+ 2, 4+ and 6+ 2
states in+ Ba alculated with both sets of wave’gunctiohs.

The 2 , 4 and 6 densities are very similar, and+predict vegy
similér c}oss se tions. The differences in the 2 , 4 and 6

cross sections are directly related to the differenceg in thg
transition densities for these states.

‘L:_o-gu-u.n an...- .‘J-_ 9 “AW
.

102

114 .
I;

. n
:3... {9.1.1 1041--...- . J i.“ In L”

 

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semen 0:» means new oeauemane 0:» sues emumasoaeo we 0950mm came some one .meofiuoesm m>ez

 

 

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E E
b b
qu cad .839 99m . 0Q; cad
HI, .q a 1 .4 fil1 . q .. 46fi91

 

 

 

 

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103

 

 

 

 

 

 

 

CENTRRL+TENSOR+LS
— - — CENTRFIL ONLY
10"" _ 1
5 2.415, [5+]:
: + ***+++ 1
10'3 '7 1
Q * “
13 b e
.5. ‘t \
10.2 _ 2.4%, 3+ .2
c: : It :‘
Q : + + +++++
t) .
U 1-
- / \
I.
b \
lifq 11111R11l111.1111111111111
0 80 ‘+0 60 80 100180
Gunfideg)
Figure 22. Microscopic model DWBA calculations for inelastic
scattering to the 3 and 5 states in Ba calculated with the

A=136-140 set of wa$e functions. The two-body force included
central,tensor and L15 terms, but no collective enhancement
was included. Only the Direct + Exchange calculations are shown.

.~ a: v” “3‘" ‘- 2’

'JmnnLub‘

104

-— DIRECT 0 5m

-- - DIECT
10 ' —— once? 0 cm 10 ° .
- - -' DIECT . .
I 23*23. 2+
I .3
. 9

a
l
,

r M...__._—__._'...e_m—_ sou-aw“ ‘
r _ _ . ..- . g .
n

da/dn (mb/sr)

.. 2.32». (6+)
0

 

10“
02090608010030 020‘106080100120
Figure 23. Microscopic model DWBA+calculations £25 inelastic

scattering to the 21 , 41 and 6 states in Sm calculated
with the A=136-145 ség of wgve funétgons. The inadequacy of

the basis space is demo strated in+the calculated angular
distributions for the 22, 42 and 62 states.

APPENDIX C

COMPILATION OF EXPERIMENTAL ANGULAR DISTRIBUTIONS

105

on+m:m.m n.m0a oo+mhho. nosmmoo. oo+mmm.m ¢.m0a
noommm.m o.nofi oo+mo:o. no+umno. oo+mmm.a 3.00”
Honm:m.m o.mm «o-umom. «onwamo. ao-mmm.m ¢.mm.
o+mma.m 0.0m oo+ummo. 004mmdo. oo.mma.m ¢.om
oo+m:s.m o.mm on+momm. oo.m«:o. oo+um¢.w ¢.mm
Hn+mma.m o.nm ao+msmo. do.mono. Ho+umd.a 3.0m
oo+mon.m o.mn ooomumm. oo+mmoo. oo+mno.m ¢.mn
no.woa.m 0.0a oo.mm««. no.m¢mo. oo.m¢n.m ¢.ou
nn+m¢s.m o.mo oo+umma. no+m¢mo. oo+mo¢.o ¢.mo
Ho+mcm.m 0.00 do+uomo. ao+mmao. doamsm.m ¢.oo
Ho+m¢m.m o.mm ao+mmmfl. «osmmmo. doomom.m :.mm
ao+mcm.m 0.0m do+mmn«. do+mmflo. aoomdm.m m.om
«oomom.m o.m: «commOa. Hoommmo. uo+mmm.m m.m:
do+mmh.m m.m¢ so.msdd. so.mdmo. fio+wma.m m.m:
Ho+mom.o 0.0: ao+mnma. ao+mmmo. ao+mma.o m.o:
mo.m0a.~ o.mm mo+mmoo. mo.maso. mosmmo.m m.mm
mo+mmm.e n.om moouosd. moommmo. mo+ums.: m.om
mo+men.m o.mm mo+ummo. moommoo. mooumo.a N.mm
mo+m::.m o.nm moomda”. mo+mm¢o. mo.mmm.m a.om
:nouoa.w n.md :o+mo¢o. :o+m«mo. ¢o+m¢H.a «.ma
.mm\mr. lawn. .«mxmr. Ammxmr. .amxmz. .omo.
.m¢4.<rmemn .m<4.m7< .mam J<p0p .wmm.»«»m .yu.<ym~mo lzu.oz<

.>mr ooo. .xm .>mr o.om.nm

zmnkumm mmoau unkm<4m .a.a.mma<m

mo.mmm.s o.mo« mo.mmnn. mo.mmmo. mo.mos.¢ s.mo~
mo.mmm.a 0.00" mo.mmmm. mo.mamn. mo.mfim.a s.oo«
mo.uon.m o.mm moumsmo.d mo.wmmm. moamdo.m ¢.mm
Ho.m¢fi.fi 0.0m so.mmmH. Ho.mflda. Ho.w:fi.fi ¢.om
mo.mmm.m o.mw mo.mflmm.« mo.mamo.a mo.m«o.m s.mw
no.mmm.a o.ow an.mmofi. Ho.mmmo. Ho.mfim.fl ¢.om
so.mfim.M o.m~ Ho.mmms. Ho.umwo. «o.mom.d ¢.ma
«o.mm:.m 0.05 do.mmaa. «oumamo. do.m:¢.m :.on
Ho.wam.m o.mo o.mnon. «o.mmmo. «o.mom.~ s.mo
“0.“:m.fl o.no do.mm-. Hoummmo. so.mmm.d :.oo
Houmnm.m 0.0m anommos. ”oumHmo. Ho.mdm.m m.om
aoomom.w m.m¢ Hormone. acumen”. «oommm.o wcm:
Ho.mfi~.m 0.0: Ho.mamm. do.mmmo. so.u¢o.m m.o:
do.m:m.m o.mm Ho.umds. so.mmmo. do.maw.m m.mm
on+mma.a 0.0m oosmmno. nosmmoo. oo+mfia.a m.om
oo+mmm.a o.mm oo+mmmd. oo.mooo. oo.mmm.~ m.mm
oo.mmm.a 0.0m oo.mmma. oo.w¢do. oo.mmm.s ~.om
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