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ABSTRACT

A STUDY OF THE (p,d) REACTION
ON THE GERMANIUM ISOTOPES

By

David Lawrence Show

 

t“. b ."
1'

States in 69Ge, 71Ge, 72Ge, 73Ge and 75Ge have been
studied with the (p,d) reaction at a proton energy of 35 MeV.
Deuterons were analyzed with an Enge split—pole magnetic
spectrograph and detected in nuclear emulsions with an energy
resolution of about 8 keV FWHM. Excitation energies were
measured relative to a calibration calculated from the known
characteristics of the spectrograph and previously well
established excited state energies in the various nuclei.

The resulting uncertainties were about 1.5 keV per MeV excita—
tion. Angular distributions were measured between the angles

of 6° and 60°. 2-transfer and spectrosc0pic factors were
extracted by normalizing DWBA calculations to the angular
distributions. These data allow the assignment of some spins
and parities for the final states and the determination of the
amount of filling of the active shell model orbits in the target

nuclei.

A STUDY OF THE (p,d) REACTION
ON THE GERMANIUM ISOTOPES

By

David Lawrence Show

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics

1971+

ACKNOWLEDGMENTS

I would like to thank the entire Cyclotron staff for their
assistance, without which these experiments could not have been
possible. Specifically, the following peOple have been extra
helpful:

My thesis adviser, Professor Hobson Wildenthal, for his
indispensible counsel and guidance, and his wisdom to see
when to let me wrestle on my own and when to come to my
assistance;

Professor Jerry Nolen for his aid in setting up the
Cyclotron for high resolution and for helpful discussions;

The plate scanners for their accurate and efficient
scanning of the plates;

Richard Au and the computer staff for their assistance
in maintaining good relations with the computer;

The National Science Foundation for its financial support
during my work at the laboratory;

Finally, my wife for her moral and financial support and
her tolerance of many lonely hours while I was working late or

early.

ii

TABLE OF CONTENTS

Page

LIST OF TABLES......................................... v
LIST OF FIGURES.......................... ..... .....,... vi
I. INTRODUCTION....................................
II. EXPERIMENTAL PROCEDURE..........................

II.1 Target Preparation........................
II.2 Proton Beam...............................

<D~Q x: «a O4

II.3 Deuteron Detection........................

III. ANALYSIS OF DATA................................ 19
111.1 Spectra Reduction........................ 19
III.2 Excitation Energies...................... 19
III.3 Normalization............................ 21
III.h DWBA Calculations and Fits............... 21
111.5 Spin—parity Assignments.................. 25

IV. THE 7OGe(p,d) REACTION.......................... 29
IV.l General Comments.......................... 29
IV.2 Comments on Individual States............. 38

v. THE 726e(p,d) REACTION.......................... 43
V.l General Comments........................... #3
v.2 Comments on Individual States.............. 53

VI. THE 73Ge(p,d) REACTION.......................... 58
VIC]- Genera-l commentSOCOOOCOOCCCOCCOOCOCCOCOOOO 58
VI.2 Comments on Individual States............. 69

VII. THE 7“Ge(p,d) REACTION.......................... 74
VII01 General CommentSOOOOCC.....C.0...’....... 7A
VII.2 Comments on Individual States............ 74

VIII. THE 76Ge(p,d) REACTIONOOOOOCCCOOOOOOOOOOOOOOOOOO 91
VIII.l General Comments........................ 91
VIII.2 Comments on Individual States........... 98

iii

Page

IX. DISCUSSION OF RESULTS......................... 104
11.1 General Features........................ 10h
IX.2 Summed Spectroscopic Factors............ 10h
IX.3 Wave Functions.......................... 122
IX.L Order of Filling of Orbits.............. 126

X. SUGGESTIONS FOR FURTHER STUDY................. 128
LIST OF RWERENCES.0......00000000000.0.0....00...... 130

iv

Table

LIST OF TABLES

Page
Percentage isotOpic contents of the
target from the Oak Ridge National
Laboratory separated isotopes division

specification sheets........................ 8

Summary cu? results for states in 69Ge...... 30

71

Summary of results for states in Ge....... 44

72

Summary of results for states in Ge....... 59

Summary of results for states in 73Ge....... 75

75Ge....... 92

Summary of results for states in
Summed spectroscopic factors................ 119

Fractional fillings of orbits in the even

iSOtOpeSoooo000000000.000000000000000ooooooo 121

Figure

LIST OF FIGURES

Page
Optical model predictions for the 72Ge(p,p)
reaction in the regions used for the
absolute normalizations (upper curve) and
for the monitor counter (lower curve).
Boxes on the upper curve are experimental
points from data taken for the absolute
normalization of cross~sections for the
72Ge(p,d) reaction........................... 11
Sample spectra from the 7OGe(p,d) and
72Ge(p,d) reactions at 90 and 60 respectively.
Energies shown are those obtained in the
present work. Peaks due to germanium im-
purities are labelled with the mass number
of the final nucleus......................... 14
Sample spectra from the 73Ge(p,d) and
7[*Ge(p,d) reactions at 180 and 120 respectively.
Energies shown are those obtained in the
present work. Peaks due to germanium im—
purities are labelled with the mass number
of the final nucleus......................... 16

76

Sample spectrum from the Ge(p,d) reaction
at 15°. Energies shown are those obtained in
the present work. Peaks due to germanium im-
purities are labelled with the mass number of

the final nuCleuSOOO0.00000000000000000...... 18

vi

Figure

10

Page

DWBA predications for g_transfers of

1,3, and h at three excitation energies.

The magnitudes have been normalized to
correspond to an experimental angular
distribution with a spectroscopic factor

of one...................................... 24
Experimental angular distributions obtained

from the 7OGe(p,d) reaction at 35 MeV. The
solid curves are fits to the data of DWBA
predictions.................................. 35
Experimental angular distributions obtained

from the 7OGe(p,d) reaction at 35 MeV. The
solid curves are fits to the data of DWBA
predictions.................................. 37
Experimental angular distributions for

three states observed in the 7OGe(p,d) re-
action at 25 MeV showing the 2:1 j—dependence.
The solid curves are drawn through the points

to guide the eye............................. 41
Experimental angular distributions obtained

from the 72Ge(p,d) reaction at 35 MeV. The
solid curves are fits to the data of DWBA
predictions.................................. 50
Experimental angular distributions obtained

from the 72Ge(p,d) reaction at 35 MeV. The
solid curves are fits to the data of DWBA
predictions.................................. 52

vii

Figure
11

12

13

1h

Page
Experimental angular distributions
obtained from the 73Ge(p,d) reaction
at 35 MeV. The solid curves are fits
to the data of DWBA predictions. For
mixed z-transfers, the dashed lines show
the contribution of each component.......... 66
Experimental angular distributions
obtained from the 73Ge(p,d) reaction at
35 MeV. The solid curves are fits to the
data of DWBA predictions. For mixed £-
transfers, the dashed lines show the con-
tribution of each component................. 68
Experimental angular distributions
obtained from the 7hGe(p,d) reaction
at 35 MeV. The solid curves are fits
to the data of DWBA predictions. For
mixed fi—transfers, the dashed lines show
the contribution of each component.......... 82
Experimental angular distributions ob-
tained from the 7[*Ge(p,d) reaction at
35 MeV. The solid curves are fits to
the data of DWBA predictions. For
mixed z—transfers, the dashed lines
show the contribution of each com—

ponent...0.0.00.0...OOOOOOOOOOOOOOOOO00...... 81+

viii

Figure Page
15 Experimental angular distributions
76

obtained from the Ge(p,d) reaction

at 35 MeV. The solid curves are fits

to the data of DWBA predictions. For

mixed z-transfers, the dashed lines show

the contribution of each component.......... 97
16 Spectroscopic strength plotted as a

function of excitation energy foryg—trans-

fers of 1,3, and h for the (p,d) reaction

on the even targets.......................... 106
17 Energy level diagram for states below

2 MeV in 69Ge showing energy, z-transfer,

spectroscopic factor, and sample spectrum

for states populated in the (p,d) reaction... 108
18 Energy level diagram for states below 2

MeV in 7lGe showing energy, l-transfer,

spectroscopic factor, and sample spectrum

for states pOpulated in the (p,d) re-

action....................................... 110
19 Energy level diagram for states below

3.5 MeV in 72Ge showing energy, g-transfer,

spectrosc0pic factor, and sample spectrum

for states pOpulated in the (p,d) re-

action.0..OOOCOOOOOOOOOOOO0.0.0.0000...I0....-112

ix

Figure Page

20

21

Energy level diagram for states below

2 MeV in 73Ge showing energy, 2-transfer,
spectroscopic factor, and sample spectrum

for states populated in the (p,d) re-
action....................................... 11h
Energy level diagram for states below

2 MeV in 75Ge showing energy, H—transfer,

spectroscopic factor, and sample spectrum

'for states populated in the (p,d) re-

action.OOOOOOOOOOOOOOOOOO0.0.0.0000...0...... 116

I. INTRODUCTION

The nuclear shell model treats the nucleus as a small
number of "active" nucleons moving in a potential produced
by the remaining "core" nucleons. In its simplest form
the nucleons fill one "orbit" after another in a prescribed
order and these orbits are grouped into shells. Each orbit
is described by its principle quantum number, orbital angular
momentum, (2) and total spin (3': 9.11/2) so that the representa-
tion for the first orbit with three units of orbital angular
momentum and total spin of 5/2 would be lf5/2' The total
number of protons and neutrons allowed in a given orbit is
restricted to 2j+1 each. This model has been used with
considerable success, especially near filled shells and in
the p and s—d shells. With these successes there is a grow-
ing interest in applying it to the upper f-p shell, which
is comprised of the 2p3/2, lfS/Z’ 2pl/2’ lg9/2 orbits and
includes the germanium isotopes. In the past, attempts have
been made at applying the rotational model to this region,
but without much success.l’2)
In order to facilitate a shell model calculation it
is desirable to have available considerable experimental
information of the type that can be obtained from single
particle transfer experiments. Included are:

l. Excitation energies. Once the energies of the

emitted reaction particles are measured, it is a
simple matter to calculate excitation energies from the
incident beam energy, the reaction ground-state Q-

value, and the reaction kinematics.

2. Orbital angular momenta of transferred particles.

It has long been known that in particle-transfer ex-
periments a plot of the probability of populating a
given state as a function of the scattering angle has

a shape characteristic of the H-value of the transferred
particle. By making such plots (called angular distribu~
tions) for all the states observed, it is possible to
determine which orbits contain nucleons (neutrons for
(p,d) and (d,p) reactions) in the target nucleus.

This in turn yields information about the wave function
of the target nucleus. Information is also obtained
about the wave functions of the final states. Parities
are determined, since they are the same as that of the
target nucleus for even-2 transfer and opposite for
odd—x transfer. In addition, restrictions are placed
on the spins of the final states since they must be
vector sums of the target total angular momentum and
the total angular momentum of the transferred particles.
In most cases, the angular distributions are not sig-
nificantly dependent upon the total angular momentum

of the transferred particle, so that in general

"j—transferred" cannot be experimentally determined

directly. The exception is the case of an i=0 transfer,
in which case the j—transferred must be 1/2. If the
target spin is not zero, the spin of the final state

is not significantly restricted by the determination

of 2, especially if the target spin is large (j 22)

and the H-transfer is greater than 1. On the other
hand, if the target spin is zero, as is often the case,
the spin of the final state is restricted to two pos-
sibilities regardless of the H—transfer, and there are
sometimes reasons for preferring one over the other

so that tentative spin assignments can be made.

3. Spectrosc0pic factors. Further information is
available from the overall probability of populating
a final state. By taking the ratio of the experimental
intensity of a transition to the Distorted Wave Born
Approximation (DWBA) prediction for the intensity,

one extracts a number called the spectroscopic factor.
This again gives information about both the target
nucleus and the final state. The sum of the spectro—
scopic factors for a given H-j transfer is a measure
of the total population of that orbit in the target
nucleus (assuming that all the states that can be
populated by that H—j transfer are observed). The
individual spectroscopic factors are a measure of the
overlap of the target state wave function with the

final state wave function plus (or minus) a particle

of the appropriate er. For example, in the (p,d)
reaction a large spectroscopic factor indicates a wave
function for the final state that is just the target
wave function with one neutron removed, while a small
spectroscopic factor indicates further rearrangement
or recoupling between the particles in the target and
in the final state or that no particles of that th are
present in the target.
Information of types 2) and 3) are useful in determining which
orbits must be considered in the active space of a shell model
and perhaps whether the calculation is even feasible, since
an active Space that is too large becomes impossible to use
because of finite computer size and time.
Experimentalists have recently shown an increasing
interest in the upper f-p shell mass region, in particular
the germanium isotopes. The y—decays of these nuclei have
been studied by Hasselgren and by Weishaupt e; 31., with
the (n,y) reaction,3’h) by Malan e£_al. with the (p,ny)

7)

reaction,5’6) by Salzman 22.El- with the (a,a'y) reaction,

8)

by Jones £3 31. by stopping 73Ge ions in germanium, and
by Murray et 21" Van Hise 33 $1., MDszynski et 31., Chen,
gt 31., Camp et_al., and Rester £3.31. in B—decay experi-

9’13 3 2)

ments. The results of these experiments provide ac-
curate excitation energies and, for some states, spin-parity
assignment or limitations. Particle transfer experiments

that have been performed include those of Goldman, Heyman

1h,l5,3,16)

§§.a1., Hasselgren, and Kato using the (d,p) reaction
and of Hsu, gt a1. and Fournier, §£_al. using the (p,d)

17.18)

reaction. It is expected that the structure of the
low—lying states in the odd isotopes is predominately de—
termined by the neutrons due to the presence of an unpaired
neutron. For this reason, the neutron transfer experiments
should be essential for studying these states.

The present work was undertaken to refine and extend
the previous neutron transfer experiments on the germanium
isotopes. The MSU Cyclotron Laboratory is especially well
equipped for (p,d) experiments because of its high quality
proton beam and the Enge split-pole magnetic spectrograph.
Using dispersion—matching techniques it has become a routine
procedure to get deuteron spectra with a resolution of better
than 10 keV FWHM at a bombarding energy of 35 MeV. This type
of good resolution is highly desirable if accurate results
are to be obtained in the present experiment, for two reasons
First, the level structure of the odd germanium isotopes is
fairly dense, with several cases where the level separation
is less than 15 or 20 keV.) Second, in the region of excita-
tion above 2.5 or 3 MeV the levels are usually weakly popu—
lated and the high resolution helps considerably in causing
these levels to stand out from any background that may be
present. HasselgrenB) has obtained an energy resolution of
about 8 keV in the (d,p) reaction while the best resolution
previously obtained in the (p,d) reaction was about 35 keV

18)

by Fournier §£_al. Our high resolution (p,d) data can

be expected to clear up many ambiguities attendent in the

previous poorer resolution.

II. EXPERIMENTAL PROCEDURE

II.l Target Preparation

The germanium used for making the targets was obtained
from Oak Ridge National Laboratory in the form of GeOZ.
Table 1 shows the quoted relative abundances of each of the
five stable germanium isotopes in the material used for each
target. The standard techniques for forming the targets are
outlined as follows:

1. Carbon foils (m25 ug/cmz) were floated on water

and then picked up so as to cover a 3/4" hole in an

aluminum frame.

2. About 8 mg of GeO2 was placed in a 1/4" I.D. tantalum
tube with both ends crimped and a hole (.012" diameter)

drilled midway between the ends.

3. The tube and target frame were mounted in a vacuum
evaporator and by passing a current of about 200 amps
through the‘bube the GeO2 was evaporated onto the carbon

at an evaporating distance of about 2 cm.

This procedure resulted in targets that were very uniform and

about 175 ug/cm2 thick.

II.2 Proton Beam
The targets were bombarded with the 35 MeV proton beam
from the MSU cyclotron. Currents were in the range from AOO

to 700 nanoamps. The energy of the beam was measured

Table 1. Percentage isotopic contents of the targets from
the Oak Ridge National Laboratory separated isotopes
division specification sheets.

 

 

 

Isotope
Target 7OGe 72Ge 73Ge 7hGe 76Ge
"7069" 81.62 5.54 1.u7 6.36 2-01
"72Ge" 2.70 90.88 1.27 4.23 0-93
n73Gen 2.11 5.06 85.58 6.61 0.62
"7“Ge" 1.71 2.21 0.90 94.48 0-70

"7608" 7.69 6.65 1.69 10.08 73.89

 

 

using two calibrated A50 bending magnets in the beam trans-
port system, with slits before, between, and after the mag-
nets. A typical measured value for the energy was 35.050
MeV, accurate to N25 keV.

After passing through the target, the protons were
collected in a Faraday cup connected to a charge integrator
which monitored beam current and total charge. These data
were used for relative normalization of the cross sections
from one angle to another. Elastically scattered protons
were detected in a NaI detector (monitor counter) mounted
at 900 with respect to the beam. These data were used as a
check on the normalization determined from the Faraday cup
and in the cross section normalization procedure.

The lower part of Figure 1 shows the DWBA prediction
for elastic scattering from 720e in the region of 900. Since
there is a peak in the cross section near 900 the monitor
data can be expected to be stable against small changes in

the beam position on the target.

II.3 Deuteron Detection

The deuterons were analyzed in an Enge split-pole mag—
netic spectrograph and detected in nuclear emulsions placed
in the focal plane of the magnet. Using dispersion matching
techniques a resolution of about 8 keV FWHM was obtained
for most of the spectra from the plate data. The plates
were scanned with the aid of a computer-linked microscope

system that facilitated scanning in .1 mm steps and in the

0. Mayo ! a!.{.‘nfla

Figure l.

10

Optical model predictions for the 72Ge(p,p)
reaction in the regions used for the absolute
normalizations (upper curve) and for the monitor
counter (lower curve). Boxes on the upper curve
are eXperimental points from data taken for the
absolute normalization of cross—sections for the

72Ge(p,d) reaction.

11

100-

 

 

 

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straightening out of the tilted images on some plates. The
small step size is necessary because the high quality of the
data produces peaks that are only about .3 mm wide, FWHM,

and a larger step size would prevent observation of peak
broadening in the case of closely spaced doublets. Typical
spectra for each of the targets are shown in Figures 2—h.

In all of the experiments with plates, two plates were

placed end to end in the focal plane of the spectrograph.

The spectra shown in the figures are from the first plate
only. The second plate was used primarily to determine if
there were any strong peaks at high excitation energies.
Supplementary data was also taken using a position-sensitive
wire proportional counter instead of the plates, but the
counter-limited resolution of about 60 keV severely restricted
the use of these data, except for cross section normalization
purposes. In connection with the normalization process we
took angular distributions of the elastically scattered
protons in the region of the maximum at about 35°. The use

of these data will be described in the next chapter.

Figure 2.

13

Sample spectra from the 7OGe(p,d) and 72Ge(p,d)
reactions at 90 and 60 respectively. Energies
shown are those obtained in the present work.

peaks due to germanium impurities are labelled

with the mass number of the final nucleus.

excnation energg [keV]

3000

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EDD 2000 2500

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15

Sample spectra from the 73Ge(p,d) and 7"’Ge(p,d)
reactions at 180 and 120 respectively. Energies
shown are those obtained in the present work.
Peaks due to germanium impurities are labelled

with the mass number of the final nucleus.

16

  

 

 

 

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l7

76Ge(p,d) reaction at 150.

Figure A. Sample spectrum from the
Energies shown are those obtained in the present
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nucleus.

at 157

S ent

excitation energg [keV]

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III. ANALYSIS

III.l Spectra Reduction
19)

The computer code Sampo was used to extract peak
positions and peak areas from the spectra. This code fits
a gaussian shape whose parameters are determined from a
chosen peak to all the peaks and gives centroids and inte-
grated counts. This type of fitting procedure is indispen-
sible for cases in which the energy separation between two

states is close to the limit of our resolution and the peaks

are not completely separated.

III.2 Excitation Energies

The excitation energies reported for this work were cal-

20) The

culated with the aid of the computer code Monster.
parameters of the calculation include beam energy, spectro-
graph field strength, observation angle, focal plane position,
peak centroids and areas, centroids and energies of calibra-
tion peaks, and reaction Q-value. Calibration peaks are
those corresponding to states whose energies are well known
(<1 keV uncertainties) regardless of the reaction that pro-
duced them. Typical calibration peaks in the present experi-
ment were the ground state and strong excited states of any
germanium isotope observed, and the ground state of 313

from the 32S(p,d) reaction. The excitation energies used

for the germanium calibration peaks are those from ybray

l9

20

experiments. These calibration peaks were spaced along most
of the plate.

Mbnster uses the known p vs. D characteristics of the
spectrograph in conjunction with these calibration peaks to
determine the excitation energy of each peak given. Since
the program allows input of the Q—values for the various
reactions, these can be adjusted to give the best fit to the
calibration peaks. This allows us to compensate for small
errors in the Q—values calculated from the mass tables of

21) which would otherwise ruin the calibra-

Wapstra and Gove
tion. In the 7OGe and 76Ge targets there was sufficient con-
tamination with the other germanium isotopes to allow the
determination of relative Q-values for all the reactions
except the 73Ge(p,d) reaction. However, since 7“Ge, 73Ge,
and 720e are all stable, their masses are well known and
hence the 71‘*Ge(p,d) and 73Ge(p,d) reaction Q-values were
assumed to be correct and the others were adjusted as needed.
In all cases the adjustments needed were within the limits

of the quoted uncertainties in the masses of the other odd
isotopes. The uncertainty in the excitation energies cal-
culated in this work is about 1.5 keV/MeV excitation. For
the purposes of plotting the angular distributions, Monster

also calculates the center of mass scattering angle and

cross section.

21

III.3 Normalization

As was previously mentioned, the relative angle to angle
normalization for the data on a given target was done using
the integrated charge from the Faraday cup. In comparing
the monitor counter normalization with this method, they
were found to agree within 5%.

For the purpose of establishing a cross section scale
we used proton elastic scattering angular distributions.
These data were normalized to Optical model calculations

made with the proton parameters of Becchetti-Greenlees.22)

The data and calculation for 72

Ge are shown in the tap part
of Figure l and indicate a good fit to the data by the
theoretical curve. Assuming the Optical model predictions

to be correct, we can now use the normalization factor thus
obtained to convert counts in the wire-counter per count

in the monitor counter to mb/sr. Since deuteron spectra

were taken under identical conditions we thus have a cross
section normalization factor for the deuteron angular distri-
butions. This procedure yields an estimated experimental

uncertainty of about 20% in the absolute spectroscopic

factors.

111.4 DWBA Calculations and Fits

All DWBA calculations were made with the computer code
DWUCKZB) and made use of its approximate finite range and

nonlocality correction options. The proton parameters used

22

were those of Becchetti-Greenleeszz) and the deuteron
parameters were calculated from the Becchetti—Greenlees
proton and neutron parameters according to the adiabatic
model of Johnson and Soperzh) with Satchler's prescription.25)
Figure 5 shows the calculated shapes for thransfers of

1,3, and A at three excitation energies. In fitting the
theory to the data, a minimum X? procedure was used which
considered only the angular range from 00 to 320. The result-
ing fits show good agreement between data and theory. At
higher excitation energies there is a discrepancy in the

m:3 fits, as the calculations show a dip at forward angles
while the data maintains the shape of the low lying 2:3
transitions.

There is occasionally some ambiguity in making Qrassignments.
For example, a mixture of 2;l+h may fit a given transition
in the 73Ge(p,d) reaction about as well as an 2sl+3 mixture,
particularly if the 2:1 component dominates the transition.
In such cases we have preferred the 2Fl+3 assignment since
an RFl+h mixture would require the final state to actually
be a doublet because of the different parity associated
with the two components., In the case of the even targets,
any mixture of 2's implies a doublet (because of different
spins or different parities) and we have not attempted
fitting with a mixture unless there is strong evidence for

it.

23

Figure 5. DWBA predictions for A—transfers of 1,3, and A
at three excitation energies. The magnitudes
have been normalized to correspond to an ex—
perimental angular distribution with a spectro-

scopic factor of one.

21»

AGO“; .I.U¢

 

 

 

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Figure 5

25

III.5 Spin-Parity Assignments

The parity of any state is uniquely determined by the
l-transfer of the transition to that state. Since all the
target states in the present experiment have positive parity,
the parity of a final state is positive for an even £—transfer
and negative for odd A-transfer.

In most cases it is not possible for us to make definite
spin assignments since we cannot measure the spins of the
transferred neutrons except for 2:0 transitions. The situation
is complicated even more for the 73Ge(p,d) reaction since the
spin of the ground state is not zero and even if we knew the
spin of the transferred neutron we would not know the spin
of the final state. We can make spin assignments for states
in 72Ge in the occasional cases where v-decay experiments
have limited the spin to two or three possibilities and our
results are consistent with only one of these. For the odd

isotopes we can make tentative spin suggestions for most of

the states based on the following considerations.

1. All states populated via 2:2 transfer are assumed
to have a spin of 5/2. In order to have a state with
J=3/2 populated with an i=2 transfer we would have to
pick up a neutron from the ld3/2 orbit or the 2d3/2
orbit. The former should only occur (if at all) for
states at high excitation energy since the ld3/2 orbit
is two shells below the f-p shell. Pickup from the

2d3/2 orbit could occur but it is doubtful that this

26

orbit is significantly populated in the target nuclei.
The 2d5/2 orbit should fill appreciably before any
filling of the 2d3/2 orbit began. We see very little

i=2 strength which indicates that even the 2d5/2 orbit

is not populated to any significant extent.

2. All states populated via 2:3 transfer are assumed

to have a spin of 5/2. We might expect to see some pick-
up from the lf7/2 orbit which would yield states with
j=7/2. These states would be expected to occur at a
fairly high excitation energy since the lf7/2 shell is
well below the higher orbits populated in energy, and

we would expect most of the J=5/2- states to be lower

in excitation energy than these J=7/2- states. The

fact that we do not see enough 2:3 strength to exceed
the sum rule limit for lfS/Z pickup tends to indicate
that we see little if any lf7/2 pickup. However, we
cannot rule out the possibility that some of the higher
excited states that we observe to be populated by i=3
transfer may actually be 7/2— states, or that some of
the weakly populated states may be 7/2" states whose
wave functions are dominated by (lf5/2)3 etc. configura-

tions.

3. All states populated via i=4 transfer are assumed
to have spin 9/2. There is one exception to this and

that is the case of the 139 keV state in 75Ge which is

6)

known to have J=7/2? This exception indicates that

 

27

76

at least in Ge there is some filling of the lg7/2

orbit. There are also possible 7/2+ states in both
71Ge7) and 73G829)

71

which we do not Observe. The state
in Ge is definitely not populated to an observable
degree while the state in 73Ge would be obscured in

our data by a strong i=1 transition to a state less than
2 keV away. Any pickup from the lg7/2 orbit would be
expected to populate states generally lower in excitation
energy than pickup from the lg9/2 orbit since the lg7/2
orbit is one shell lower than the lg7/2 orbit. All the
other states in these nuclei below 1 MeV excitation that
-are populated by an £=4 transfer in the (p,d) reaction
are known to be 9/2+ states, therefore the few £=A

transitions to higher excited states that we observe

are quite likely due to lg9/2 pickup.

A. The states populated via i=1 transfer present a much
more difficult problem. Both the 2p2/3 and 2pl/2

neutron orbits are partially filled in the germanium
isotopes and they are close together in energy. There-
fore we can at best make a slight preference for one

spin over the other. The 2pl/2 orbit is slightly

higher than the 2p3/2 orbit so in general the simplest
shell model picture predicts the 1/2- states should occur
at lower excitation energies than the 3/2- states.

On this basis we have preferred a spin of 3/2 for

i=1 transitions to states above 1 MeV excitation.

28

The application of a general argument to this specific
case suggests that since the 2p3/2 orbit should be
mostly filled and the 2pl/2 orbit only partially filled,
a strong (d,p) transition coupled with a weak (p,d)
transition indicates a 1/2- state while the reverse
situation indicates a 3/2- state. The data for the
low-lying states with well known spin supports this
argument and we have used it where it is applicable.
However, the summed spectroscopic factors indicate that
both orbits are about half full rather than the 2p3/2
orbit being almost full and the 2pl/2 orbit being mostly
empty. It therefore appears that the 3/2- and 1/2- spin
suggestions based on the (p,d) and (d,p) data must be
regarded as very tenuous. It should be noted that the
spectroscopic factors for 8:1 transitions extracted in
this work are independent of the spin transfer since

the spin-orbit force is not strong enough to produce

an appreciable difference in the calculated cross

sections.

 

1
a v\ A. 5". _VI \-s_nav.

IV. THE 7OGe(p,d)égGe REACTION

IV.l General Comments

The absolute normalization of the cross sections for
this reaction was obtained with a procedure different than
that employed for the other reactions. For this case the
normalization was calculated from the isotopic abundances

2Ge, and 71"Ge in the respective targets and the

of 70Ge, 7
absolute normalizations for the 72Ge(p,d) and 7hGe(p,d)
reactions. This less direct procedure probably accounts for
the fact that the sum of all the spectroscopic factors for
this reaction exceeds the expected limit by about 25%, while
for all the other isotopes the sum is slightly less than the
limit. For the purpose of comparing the fractional fillings
of the orbits we have normalized the sum for this isotope to
the limit.

The results for this reaction are summarized in Table
2 and the angular distributions are shown in Figures 6 and 7.
In general, where it is possible to make comparisons we find
good agreement between the excitation energies, Advalues,
and spectroscopic factors obtained in the present study and
those of previous work. In addition, many new states and
their characteristics are also observed in the present work
about which nothing was previously known. The second plate
for this reaction showed no peaks stronger than 20 ub/sr at

60 between 3.8 MeV and 7 MeV excitation.

29

 

30

 

 

 

 

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31

 

 

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32

 

 

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33

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.emsaHpaoo i N oHnme

34

Figure 6. Experimental angular distributions obtained from
the 7OGe(p,d) reaction at 35 MeV. The solid

curves are fits to the data of DWBA predictions.

dU/dQ [mb/sr]

 

35

E916” hOV
’ I 3

E nzm
" 1

SheV
. '4

 

 

EfZWl lav
I I 1

 

36

Figure 7. Experimental angular distributions Obtained from
the 7OGe(p,d) reaction at 35 MeV. The solid

curves are fits to the data of DWBA predictions.

OCT/0&2 [nib/81‘]

 

237

Ex=3173

lueV

1-3

Ex=3232

 

Ex=3410

 

keV
l

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keV

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Ex=3980
’ =

Ex=3615
I =

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I:

 

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L}

 

HIV
1

 

keV
1

keV

1:3

38

The distribution of the strong transitions in the (p,d)
reaction on the germanium isotopes is shown in Figure 16.
The 7OGe(p,d) reaction is typical in this respect and shows
the following: there is a strongly populated l/2- state near
or at the ground state, a stronger 3/2- state in the 400-600
keV range, and a moderately strong l/2_ state at about 1 MeV.
The 2:3 strength is mostly concentrated in a single state
below LOO keV with two or three weaker states at about 1.4
MeV. A single state near or at the ground state carries
almost all the 2:4 strength with the rest scattered along
the Spectrum in two or three weak states.

For the purposes of comparing our results with those

17)

reported for the (p,d) reaction by Hsu e§_al. and Fournier

22 1118)

it is useful to point out that the isotopic
purities of the targets used in the present work are comp—
arable to those used in the previous work. Thus, impurity
peaks from germanium isotopes other than the one with which
the target is enriched should be populated with approximately

the same strength in both experiments.

IV.2 Comments on Individual States

813 keV state: The angular distribution that we obtain
17)

for this state shows a pure £=2 shape. Hsu £3.31. were
unable to fit the angular distribution for this state well
and show attempted fits with both £=1+3 and £=l+4 mixtures.

The data of reference.17 fOr this state are of low statistical

39

accuracy and do not permit discrimination between these mix-
tures. Our data, with better statistics, removes the previous
ambiguity and also removes any need for calling this state a
‘doublet, which would arise with a mixed l-transfer. The
resulting 5/2+ assignment for this state is consistent with
the systematics seen in the other odd nuclei in which at least
one 5/2+ state is always seen below 1 MeV excitation.

225 keV state: We have made an attempt to determine the
feasibility of making spin assignments for the £=l transitions

27)

by the J=dependence as described by Lee and Schiffen For

this purpose we measured differential cross sections of

the 7OGe(p,d) and 60

Ni(p,d) reactions at a bombarding energy
of 25 MeV in the angular range of 60° to 90°. These data
were taken with the position-sensitive wire proportional
counter, the resolution of which (W60 keV FWHM at this energy)
only allowed us to resolve a few states. Among those states
resolved were the 87 keV, 233 keV, and 995 keV states in 69Ge,
which are populated by 2:1 transitions and have assigned Spin-
parities of l/2', 3/2— and "unknown" respectively. The angu-
lar distributions for these states are shown in Figure 8.
The characteristic shapes of the known 2p1/2 and 2p3/2
transitions are consistent with the results of the 6ONi(p,d)
reaction and indicate an assignment of 1/2- for the 995
keV state.

1152 keV state: This state, as all states of higher

excitation energy populated by an QFl transition, is assumed

#0

Figure 8. Experimental angular distributions for three states
observed in the 7OGe(p,d) reaction at 25 MeV shows
ing the i=1 j-dependence. The solid curves are

drawn through the points to guide the eye.

41

 

 

 

 

 

10*- l I T I l .-
i 70 i
’ [35(3![‘:)9I(j:]Es;Ea[:3'EB ‘
_ I |=1 j-dependence j
3 l- ..
E
:3
Z1 , 87 keV
o u
.E
:5 ‘\ '2
L a
C3 _ -. t: g'
“’ . 995 keV.
(:3 b “".'..k_ H
.0 .
\\\ P 4
8 _. -
23% keV
0'160 l 70 L 810 1 so
elab [deg]

Figure 8

42

to have a spin-parity of 3/2(l/2)-. The preference for 3/2-
is based on the fact that we would expect less 2pl/2 strength
with this target than with the others and the 87 keV and 995
keV states carry almost as much 2pl/2 strength as is seen in
the 72Ge(p,d) reaction. Of course, since this state (and
several higher excited states) has a fairly small spectro—
scopic factor, this is not conclusive evidence. For those
states with larger 1:1 strengths (i.e. the 2106 keV state with
S=.l6) the 1/2- possibility is less likely.

13§§_kev and 1472 keV states: Our data show that these
states are populated by i=4 and £=3 transitions respectively.

17)

Hsu gt_§1, were unable to fit the angular distribution for
the 1477 keV state, but undoubtedly did not resolve the 1468
keV state. Both an 2:3 and an 2:4 shape are shown with the
data in reference 17 and it appears that an 2=3+4 mixture
would do a satisfactory job of reproducing the experimental
shape as observed there.

22§5_kgylagg_3469_kgl states: These states are weakly
populated and the resulting poor statistics do not allow us
to distinguish well between H-transfers. Therefore, the

z-value assignments for these transitions are tentative on

the basis of our data.

v. THE 72Ge(p,d)7lGe REACTION

V.l General Comments

The results of this reaction are summarized in Table 3 and
the angular distributions are shown in Figures 9 and 10. The
second plate for this reaction showed no peaks stronger than
20 pb/sr at 60 between 4.5 MeV and 8 MeV excitation. We find
excellent agreement between the excitation energies determined
in the present work and those from y-ray experiments. Up
through the 1743 keV state, which is the highest excited
state with which we can make comparisons, the agreement is
within 1 keV. In general the l-values and spectrosc0pic
factors obtained in the present study are in agreement with

18)

the previous (p,d) results of Fournier gg'al. The main
differences are that our 2:1 spectrosc0pic factors are all
about 20% lower than the previous work and our improved
resolution allows us to correct some A-value assignments.
It is difficult to make comparisons with the (d,p) results
beyond the lowest lying states, where we find agreement in
A-values. For the higher excited states, the (d,p) reaction
sees many states populated by 2:2 transitions while the (p,d)
reaction shows almost none, presumably because the 2d5/2
orbit, as well as the 2d3/2 is almost empty.

The pattern of the 1/2- and 3/2- states may be broken
in this nucleus as the (p,d) reaction populates no state near

6)

1 MeV with a definite 1/2' spin assignment. Malan gt a1,

43

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iHN\HVN\m mumsNN -HN\HVN\H mo. H AsoaNN
H+N\mo “HuoaNN sN. N
mnemNN HsomNN
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«HuoaHN Has:
Hinxmo mHnoNHN NH. Hmo
mnwsoN AsmsoN
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Hlm\anN\H++N\H as.aooH +N\H amo. o -HN\HVN\H oH. H -HN\HVN\H HH. H HoaH
N.mmoH uN\m.N\n.uN\H Has:
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-HN\HVN\m oHHNmon 1HN\HVN\H No. H HnNmam
1HN\HVN\H onooon uHN\HvN\m No. H Hmooom
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Has... a... .. a... 3...... sea... 6 s...
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sassHsseo . m sHsas

49

Figure 9. Experimental angular distributions obtained from
the 72Ge(p,d) reaction at 35 MeV. The solid curves

are fits to the data of DWBA predictions.

50

EXBHOB hOV
I = 3

 

‘ 17% keV Ex=1026 keV
I 8 3 I 8 3

0.1

l l

      
 

Ex' 198 50V

       
    
  
  
 

Ex=1095 keV Ex=l559 “v
P a l I = .3

fro:

10

Ext 500 “V Ex=1207 1.0V
1 . 1 1

£91598 IIOV
= 3 I =3 l

 

d cun'éf

dU/dQ [mb/sr]

Ex‘l7‘13 IOV
I ' l

Ex=1780 80V
I 8 3

Ex’1963 50v
I 8 I

 

 

   
 
     

Ex=2931 50V

1 Ex=2709 uv

1-9

 

Ex=2889 1.0V
9 - 1

/

£33238“ IOV
I 8 3

   
 
   
 

E 29131-01!
x‘ I-2

Ex=2987 IneV Ex-3287 110V
I I 0 I I I

Ex=3389 M
I a 1

51

Figure 10. Experimental angular distributions obtained from
the 7ZGe(p,d) reaction at 35 MeV. The solid

curves are fits to the data of DWBA predictions.

do/dQ [mb/sr‘]

52

Ex=3518 1w
1 = 1

    
    
   
     
 

Ex=3789 keV
D = 3
0.1

10"2

Ex=3790 16v
9 = 3

0.1

. 9

10"

Ex=3900 IIOV
I==l

f!

.—

Ex=3932 keV
1 = 1

 

0[c.m.]

Figure 10

53

have assigned a spin of 1/2’ to the 1289 keV state, which we
observe, but the assignment is based an excitation intensities
which are not conclusive in this case. Since the other
possible candidates for the 1/2- state in this region (i.e.
the 708 keV and 1095 keV states) have been confirmed to be
3/2‘ states, the 2:1 systematics support the 1/2- assignment
for the 1289 keV state. However, we do not consider this spin
to be firmly established.

Another feature of the distribution of i=3 strength,
which shows up most clearly in this nucleus, is the presence
of a weak i=3 transition to a state near 3.7 MeV. This state
follows a gap of almost 1 MeV in which there are no states
populated by an i=3 transfer. It might be inferred that
these states near 3.7 MeV are 7/2_ states but there is no

other evidence to support such an idea.

V.2 Comments an Individual States

599_keV state: This state is strongly populated in the
(p,d) reaction by an i=1 transfer. The ratio of (p,d) strength
to (d,p) strength suggests a 3/2— spin assignment for this
state and this is the only state in this nucleus to match
the strongly populated 3/2' states seen in the other odd
nuclei. These considerations are in agreement with the 3/2-
spin assignment of Murray §£_§l.9) Goldman,lh) in disagree-
ment with the above results, assigns a Spin of 1/2— to this
state based on the Lee-Schiffer effect. This discrepancy

in g;1 spin assignments is also seen in a few of the higher

54

excited States. Fournier §§.§l.l8) have suggested that Gold-
man's data do not convincingly display the Lee-Schiffer effect
at 1400 as claimed, since only one i=1 angular distribution
(for the 2040 keV state) really does not show a dip here.
This observation, coupled with poor statistics shown in the
(d,p) data, makes it doubtful that spin assignments can be
made from this data. It would be interesting to look for
the £=l J-dependence in the germanium isotopes with high
resolution and good statistics with both the (p,d) and (d,p)
reactions to help clarify the situation.

89] kg! £22225 While this state has been observed in
y—ray work, we see only a very weak (3.015 mb/sr at 6°)

18)

peak at this energy. Fournier §t_§1, report an i=2
transition to a state at this energy, but this is in disagree-
ment with the l/2,3/2 spin assignment from the y-ray work.

The i=2 transition shown in Reference 18 is probably the sum
of the 831 keV i=1 transition and the 7OGe(p,d) i=3 transition
69Ge.

to the ground state of These two states would not have
been resolved in the previous (p,d) experiment. This would
then remove the conflict with the y—ray results.

831_k31:§£§£§: We observe a weak but distinct 2:1
transition to this state which implies a spin of 1/2- or
3/2-. The negative parity is in disagreement with the results
of Malan §£.§l.6) which indicate a spin of 3/2+ for this state.

There is no evidence for a doublet at this energy, and our

energy calibration is in close agreement with the y-ray work

55

so that we can put an upper limit of 1 keV for the separation
of the two states if there is a doublet here. The positive
parity assignment is based on the excitation intensity at

one bombarding energy only. Also, the y-ray angular distribu-
tion for the decay to the ground state favors a significant
MZ/El mixing ratio which is unlikely. We therefore favor

the 3/2- spin assignment for this state.

1026 keV state: This state is weakly populated in the

 

(p,d) reaction which makes it difficult to make a definite
z-transfer assignment. However, our angular distribution
for this state is sufficiently distinct to establish that
this state is populated by an £=3 transition, which is in

agreement with the 3/2,5/2_ spin assignment of Malan.gtԤ1.6)

l8)

Fournier §§.§l, report at i=2 transition to this state,
but would not have resolved the 7OGe(p,d) 2:1 transition to the
234 keV state in 69Ge. This fact, coupled with their poor
statistics, probably accounts for their 2:2 assignment.
lléfi}kgfli§t§£§: This state is very weakly populated in
our data and is almost obscured (<15 keV separation) by the
stronger 7OGe(p,d) 2:1 transition to the 37k keV state in
6909. 18)

at 1166 keV but most likely it is the impurity which is being

Fournier gp‘gl. report on i=1 transition to a state
seen and not a state in 71Ge.

1202 keV doublet: Examination of the spectra indicate
that there is a doublet at this energy of about 7 keV which

we do not completely resolve. The states are weakly

56

populated which makes it impossible to separate the peaks
by a fitting procedure and also makes it difficult to deter—
mine the l-transfers for the states. The 2:3 fit to the
summed angular distribution is sufficiently good that we
could not justify adding in another component, although an
£=2+3 mixture would fit the data slightly better. An £=2+3
14)

mixture would be in agreement with Goldman's data as the
(d,p) angular distribution appears to be predominately an £=2
shape but would be better fit by an 2=2+3 mixture. The £=2+3
mixture would also fit in well with the results of malanqggflgl.6)
as the higher member of the doublet has been assigned a spin
of 5/2- and the lower member has 5/2+ as one of the possible
Spins.

$29§.E23:§2Q.l22§.E21 states: Our data show distinct
i=3 transitions to both of these states even though the 1558
keV state is weakly populated. Goldmanlh) has assigned
tentative 2:2 transfers for these states but the (d,p) data
exhibit poor statistics. Also, the (d,p) angular distribu-
tion for the 1558 keV state is quite similar to the (d,p)
2:3 angular distributions. It therefore appears that the
£=2(?) assignments are incorrect and these states are indeed
populated by i=3 transitions.

12§3_k32;§2§32: The (p,d) angular distribution for this
state shows a definite 2:1 shape while the (d,p) angular

distribution shown by Goldmanlh)

shows a definite 2:0 shape.
It therefore appears that the (p,d) and (d,p) reactions are

populating different states. It is not surprising that the

57

(p,d) reaction does not strongly populate a 1/2+ state
here, even if it exists, since we do not expect any sig-
nificant filling of the 381/2 orbit in the ground state of
72Ge.

ggzg‘ggy.§ggget Again this state is probably not the
same as Goldman'slh) 2270 keV state as the (d,p) angular
distribution is definitely not compatible with the 15:1
transition shown in our (p,d) data. Our angular distribution
does Show some filling in Of the first minimum which could be
an indication of an E=2 or an 2:3 component, but there is no
other evidence in our data for a doublet. In particular, if
there were two states separated by more than 7 keV we should
observe a broadening of the peak in the spectrum. It is
therefore likely that either the (d,p) data is incorrect or
the (p,d) reaction does not populate the 2270 keV state.

Beyond this point it is impossible to make associations
between those states seen in our data and those reported by

14)

Goldman.

VI. THE 73Ge(p,d)72Ge REACTION

VI.1 General Comments

In this reaction, the nonzero spin of the target nucleus
allows more than one z-transfer to populate a given state.

In particular, we eXpect to see many states populated by an
£=l+3 mixture and a few by an £=2+4 mixture. This is indeed
the case as is shown in the summary of results for this reac—
tion in Table 4 and in the angular distributions shown in
Figures 11 and 12. The excitation energies, H-values, and
spectrosc0pic factors Obtained in the present study are in
general in good agreement with the results of previous work.
We again observe many states and their Characteristics about
which nothing was previously known. The second plate showed
no peaks stronger than 25 ub/sr at 60 between 5.8 MeV and

10 MeV excitation.

Where mixed A-transfers are allowed, the extracted
spectroscopic factors must have a larger uncertainty assoc-
iated with them. In particular, the strength of an 2:3
component mixed in with a predominately i=1 transition is
determined entirely by the filling in of the first minimum
of the 2:1 shape. The quality and quantity Of our data do
not allow us to determine this filling exactly. This uncer—
tainty undoubtedly accounts for the fact that our summed i=3
strength for this nucleus exceeds the sum-rule limit while it

does not exceed the limit for any of the other isotopes.

58

 

59

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60

 

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62

 

 

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63

 

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64

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umsaHpqoo n H mHnme

Figure 11.

65

Experimental angular distributions obtained from
the 73Ge(p,d) reaction at 35 MeV. The solid curves
are fits to the data of DWBA predictions. For
mixed thransfers, the dashed lines show the

contribution of each component.

66

    
    
     
  
    
 
   

 

Ex=3l3l IIOV

E’32516 50V
I = l O = 1+3

   
 
    
         
      
   

£93920 m
l - 2

Ex'z7s7 DOV
I s a

)L"?

 

‘\'

 

£9277» uv

, ‘ 2*” Ex‘3762 30V

I-1+3

Ex'aqos iOV
, 3 1’3

      
    

£92065 50V Eg-3038 50V
I I 2 I =- 2N}

do/dQ (mb/srl

    

    

Ex82;02 av tax-307s M
C

E 3627 “V
I - 2+9 x‘

E 3880 “V
I-1+3 1'

III 103

        
 
 
 
 

1 1 to
E E‘IZ‘OS‘O uv £93098 uv £93658 IIOV
. D I 2 I I 8 1+3

    

".13

I
10'

 

 

 

'2
I
PrfirJi‘To

 

 

Figure 12.

67

Experimental angular distributions obtained from
the 73Ge(p,d) reaction at 35 MeV. The solid

curves are fits to the data of DWBA predictions.
For mixed 2—transfers, the dashed lines show the

contribution of each component.

dG/dQ [mb/sr]

68

Exs‘133s uv tax-4915 “V
I - 1 I :- 1+3

Ex'qfi73 {CV . Ex-QSOZ “V

’3 1+3
‘u‘\
'/

Ex-‘HSS 50V
1 I “’3

a»

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x' I I 0+2

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a)“ I - 1+3

 

9fc.m.]

Figure 12

Ex‘SIS‘. keV
’ ' l

E 5312 11W
x' 1 . 1+3

E 53511ro
”ti-3

 

69

VI.2 Comments on Individual States

2565 keV and 2516 keV states: Our data show a weak 1:2
and a strong 1:1 transition to these states respectively.
These are both in agreement with the Spins and parities
previously reported for these states. Fournier §t_al.l8)
do not report the 2404 keV state, but probably did not resolve
it from the 2516 keV state. The angular distribution shown in
Reference 18 is therefore probably the sum of these two transi-
tions. Fournier 2£.§l° fit the distribution with an £=1+4
mixture, but it appears that an £=l+2 mixture would fit the
data satisfactorily. We see no evidence in our data for an
£=h component to either transition.

2221 keV state: We observe a distinct, although weak,
lzh transition to this state which is in agreement with the

2) l8)

(1-3)+ spin assignment of Rester §t_§l, Fournier §t_§l,
report a stronger 2:1 transition to this state which is not
only in disagreement with our data, but also implies a nega-
tive parity in contradiction to the y-ray work. However, the
2:1 assignment is based primarily on only two data points at
about 15°. It is possible that in the previous (p,d) work
this state was not adequately resolved at these two angles

120.

from the 13C(p,d) transition to the h.h MeV state in

223; keV and 2952 keV states: These states are not
completely resolved in most of our spectra, although there is
a consistent broadening of the peak of about three channels

in all the spectra. The sum of the angular distributions

70

is plotted in the figure under an excitation energy of 2952
keV. The Spin and parity of the 2943 keV state allows an 2:1
component in the transition to this state, and there could be
an 2=2 component in the transition to the 2952 keV state.
However, we cannot determine the strengths of these transi-
tions, or even if they exist, since the shape of the summed
angular distribution is reproduced well by the 2;3+4 mixture.
ggzé’EEXIEEEEE: The angular distribution that we obtain
for this state does not allow us to distinguish well between
an 2F2+4 or an 2;3+4 mixture, but the existence of the fish
component is firmly established. The 5L=3+4 fit has a x2
of about one half that for the m=2+4 fit but both reproduce
the shape well. Based on our data alone we would choose the
2s2+4 mixture since this does not require the state to be a
doublet. These are the results shown in the figure and table.

Fournier gt 31.18)

report an 2:3 transition to this state but
it appears that the angular distribution shown in Reference
18 could be fitted well by either an 2;2+4 or an 2;3+4 mix—
ture. The yhray results present a contradiction as they
indicate a negative parity for this state in disagreement
with the positive parity indicated by the 2:4 component to
the (p,d) transition. If neither the negative parity of

the y—ray work, nor the 2:4 component of our work is incor—
rect, then the state must be a doublet. In such a case we

might best fit the angular distribution with the $=3+4 mix—

ture. This would yield spectroscopic factors of .12 and .10

71

for the 2:3 and 2:4 components respectively.

3098 keV state: This state is weakly populated in the
(p,d) reaction and our angular distribution is only fairly
distinct. It does, however, strongly favor an 2:2 transi—
tion to this state. Rester gt a1.2) have assigned a spin of
(1,2)+ and Camp has tentatively assigned a spin of 2+. The
2:2 transfer rules out the J=l possibility since the trans-
ferred spin of 5/2 cannot couple to the target spin of 9/2
to yield a final spin of less than 2. This then confirms
the 2+ spin assignment of Camp.13)

2495 keV, 3658 keV, and 5162 keV states: In our data
the angular distributions for these states are fitted best
by an 2=1+3 mixture. An 2=l+4 mixture cannot be completely
ruled out, but it is much less likely as the spectra show no
evidence for doublets at these energies.

2268 keV state: The 2:1 component to transition to this
state in conjunction with the y-ray work allows us to restrict
the spin for this state to 3-, 4'. The y-ray work of Rester
gt al,2) has restricted the spin to (2-4). The 2:1 transi—
tion rules out the J=2 possibility and implies negative
parity.

3681 keV, 3822 keV, 3986 keV, and 4040 keV states: For

 

each of these states we can assign a spin of 3'. In each
case the y-ray work2’13) has restricted the spin to 2 or 3
and in each case we observe an 2:1 component to the transition

to the state. The 2:1 component rules out the J=2 possibility

72

and implies negative parity, thus establishing the spin of 3-.
3803 keV state: In our spectra the peak for this state

would be covered by the peak for the 7hGe(p,d) 2:1 transition

18)

to the 393 keV state in 73Ge. Fournier 33 31. report a
state at 3804 keV, but it is likely that it is actually this
impurity which is being observed.

4230 keV state: This state is very weakly populated in
the (p,d) reaction and the angular distribution is indistinct.
However, there appears to be a definite forward angle rise
which would indicate an 220 transition. If the state is
indeed populated by an 2:0 transition its spin would have to
be 4+ or 5+. ’

4322_keV state: Our angular distribution for this state
is not complete, but we see no evidence for either an 2:3
or an 2:4 component to the transition. Fournier §t_al.l8)
report on 2=l+3 or an 2=l+4 transition to this state, but the
data shown in reference 18 is not conclusive either.

18)

Fournier §t_§l, report an 2:1 transition to a state

at 4458 keV. Our data show that the peak for this state would

72Ge(p,d) 2:1 transition to the

be covered by the peak for the
500 keV state in 710s. It is, therefore, doubtful that there
is a state in 720e at 4458 keV.

Also, Fournier §t_al,18) have suggested that the 2:4
components observed in the transitions to the states at 3398

keV, 3659 keV, 3754 keV, 3804 keV, and 4339 keV might be due

to the 6+ or 8+ states that should be populated by 189/2

73

pickup. Our spectra and angular distributions indicate that
these 2:4 components probably do not exist. We do observe
:4 transitions to the states at 2774 keV, 2950 keV, 3036
keV, and 3075 keV which could be populating these high spin
states. However, our data do not contain sufficient informa-

tion to determine the spins of these states.

VII. THE 7“Ge(p,d)73Ge REACTION

VII.1 General Comments

The results for this reaction are summarized in Table
5 and the angular distributions are shown in Figures 13 and
14. The second plate showed no peaks stronger than 15 pb/sr
at 60 between 4.8 MeV and 8.5 MeV excitation. In general there
is good agreement between the excitation energies, 2—values,
and spectrosc0pic factors obtained in the present study and
those obtained in previous work. There are several states
and their characteristics which we observe about which nothing
was previously known.

There is a slight variation in the 2:1 transfer system-
atics for this nucleus. The 3/2— strength near 500 keV is
split into two close states at 364 keV and 394 keV. There is
a 1/2- state at 895 keV to complete the pattern of a 2pl/2
transfer to a state near 1 MeV. There is an 221 transition
to a state at 1044 keV which has been assigned a spin of 3/2-.
Although a l/2- assignment for this state might fit the
systematics better, Fournier §£_al, and Kato have preferred

a 3/2- assignment.

VII.2 Comments on Individual States

392 keV state: It is difficult to determine the spin of
this state, which is populated by an 2:1 transition, but 3/2-

appears to be the best choice. The ratio of (p,d) strength

74

75

 

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Figure 13.

81

Experimental angular distributions obtained fron1
the 7“Ge(p,d) reaction at 35 MeV. The solid curves
are fits to the data of DWBA predictions. For
mixed 2-transfers, the dashed lines show the

contribution of each component.

82

   
 
   
 
  
   
   
   

Ex=1635 I-OV Ex82335 uV
I I 2 I I 1

  

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/'
5.917% 1.0V 5.32982 av
1 - o 1 - 1

  
 

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I 8 3+9

dc/dQ [mb/sr]

Ex-zms luv
1 - 2

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9(c.n1.]

Figure 13

Figureilh.

83

Experimental angular distributions obtained from
the 74Ge(p,d) reaction at 35 MeV. The solid

curves are fits to the data of DWBA predictions.
For mixed flrtransfers, the dashed lines show the

contribution of each component.

81+

Ex=2796 uov Ex=3929 keV
1:3

181

   

Ex=3017 uv Ex=9000 uov
I = 1 I a 1+4

 

 

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1:1

  
 

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r i L in
9[c.m.]

Figure 14

85

to (d,p) strength is ambiguous and the y-ray work gives no
information, but two other considerations favor the 3/2
assignment. First, a J=l/2 assignment leaves us with too
much 2p1/2 strength relative to the other isotOpes, and
second, J=3/2 fits in with the 2:1 systematics as previously
mentioned.

500 keV state: We observe a pure 2:2 transition to
this state which is in agreement with the results of the (d,p)
reaction. Fournier gt al.18) fit the angular distribution for
this state with an £=l+2 mixture which implies that there are
two close states here. We see no evidence for an £=l component
to the transition and our spectra show no evidence for a
second state near this energy. Actually, the angular distribu-
tion for this state that is shown in Reference 18 is almost
indistinguishable from the pure 2:2 shape which is shown
fitted to the 557 keV angular distribution. It is, therefore,
doubtful that there is an £21 component to this transition.

55h keV state: In our spectra the peak for the 72Ge(p,d)

 

g;l transition to the ground state of 71Ge falls right on top
of the peak for this state. In the figure we show a fit to
the sum of these transitions using an £=O+l mixture which
allows us to extract the spectroscopic factor for this state.
The resulting fit is good, supporting an 2:0 assignment for
the transition to this state in agreement with the (d,p)
results. Fournier §t_al,,l8) however, report an £22 transfer

for this transition. It is possible that in the previous (p,d)

86

work the impurity was not subtracted out correctly. Also,
the forward angle rise may have been missed since no data
was taken at angles less than 150. These two factors could
then cause an error in determining the thransfer.

912 keV state: Our data show a fairly weak but distinct
iFZ transition to this state. This is in disagreement with
the (d,p) 2:1 transition to this state reported by Hasselgren.3)
The energy calibrations indicate that the two reactions are
populating states at the same energy, although it may be
possible that the (d,p) reaction is pOpulating the 929 keV
state. According to the y—ray spin assignment, the 929 keV
state should be populated by an 2:1 transfer. Another pos—
sible explanation is that this state is a doublet and each
reaction is populating only one member.

1122 keV state: This state also appears to be a doublet
despite the lack of evidence for two states at this energy‘
in the spectra. Our angular distribution is only fit well
with an 2Fl+3 mixture, with the 2:3 component dominating the

18) show an 2:3 fit to the transi-

transition. Fournier gt El!
tion but the experimental shape is not well reproduced. Also,
the 150 point shows indications of a possible rise at forward

3)

angles. Hasselgren quotes a pure £;l (d,p) shape for this
transition. Thus, the angular distributions of both (p,d)
experiments combined with the (d,p) experiment give evidence
for this state being a doublet. However, the state is weakly

populated and it is possible that there is no 2:1 component

87'

to the (p,d) transition. Hasselgren” does not show an angular
distribution for this state so we cannot comment on its quality.

1122 keV and 1611 keV doublets: Although there is no

 

evidence from the spectra for either of these states being

a doublet, we deduce that they are based on the fact that the
angular distributions can only be fitted by an £=3+h mixture.
Indeed the shapes of the angular distributions are very
similar to that of the transition to the 1699 keV state in
75Ge which is observed to be a doublet from the spectra and

18)

is fit by an 2=3+4 mixture. Fournier §t_§l, also have
trouble fitting these distributions with £=3 shapes, although
their other 2:3 and 2:4 transitions are fitted well. It
appears that these distributions shown in Reference 18 would
only be fit well by an m=3+4 mixture. Pure z=t assignments
to these transitions yield a summed i=3 spectroscopic factor
that is too small relative to the results of the (p,d) reac-
tion on the other isotopes, while either an 2;3+A mixturecu‘a-
pure z=3 transitdcmlgives consistent results. A pure £=3
transition is doubtful since both (p,d) experiments show the
distortion.

Aside from assuming that these states are doublets one
might try to explain the shapes in other ways. One pos-
sibility is that these are really 7/2' states and the distor-
tion is due to the J-dependence in the shape of the angular
distributions. Sherr gt 31.28) have observed an 2:3 J—dependence

56 29)

in the Fe(p,d) reaction at 28 MeV and more recently Nolan

88

and KongBO) have seen it on the (p,d) reaction on the nickel
isotopes at 35 MeV. This J-dependence is characterized by

three effects:

1. DWBA calculations fit the 7/2- transitions but not

the 5/2— transitions.

2. The 5/2— transitions have the first maximum shifted
by about 30 toward lower angles relative to the 7/2"

transitions.

3. The 7/2' transitions exhibit a less pronounced

diffraction pattern (i.e., the minima are filled in).

The shapes we observe for these states would be consistent
with 2) and 3) but not with 1). Our 1:3 DWBA calculations fit
the distributions for transitions to known 5/2- states very
well, but do not fit these shapes. It is possible of course
that the J-dependence effect is dependent on the relative
filling of the lfS/Z and lf7/2 orbits in which case we would
not expect that the effect seen in the nickel region would be
reproduced exactly in the germanium region. However, our
summed 2:3 spectrosc0pic factors (assuming these transitions
to be pure 2:3, j=5/2) do not indicate more strength than we
would expect to see from the sum rule limit for the 1f5/2
orbit, and finally, we would not expect to see significant
lf7/2 strength at this low an excitation energy. Another
possibility is that these shapes are the result of a multi-

step process. We cannot rule out this possibility but it

89

would seem that we should see more evidence of it if multi-
step processes were important enough to produce the effect
seen in these distributions as these transitions are fairly
strong.

115g keV state: The transition to this state is quite
weak, as would be expected for an 2:0 transfer, but the shape
is definitely that of an 2:0 transition. This is in agree-

l8)

ment with the (d,p) work. Fournier gt 31. report an 2:2
transfer to this state but would not have resolved it from
the weak peak of the 1756 keV state. This, coupled with the
poor statistics and lack of data points below 15°, probably
accounts for the discrepancy.

2101 keV state: Katolé)

reports a possible 2:0 transfer
for this state but shows no angular distribution for the
transition. Our data is not very complete for this state,

as the transition is quite weak, but there is no evidence

for an 2:0 component and the 2:2 transfer we observe is in

3)

agreement with Hasselgren's (d,p) results. It is, therefore,

likely that the 2:2 assignment is correct.

18) report states at 730 keV and 1653

Fournier g£_al.
keV, but in both cases our data indicates that these peaks
would be covered by a peak resulting from the (p,d) reaction
on one or the other germanium isotopes. For the one case
(1653 keV state) where an 2rvalue is reported for the transi—

tion, it is the same as that which would populate the impurity.

As previously mentioned, the purity of the targets used in

90

the previous (p,d) work is comparable to ours. Therefore,
it is doubtful that there are states in 730e at these

energies.

76

VIII. THE Ge(p,d)750e REACTION

VIII.1 General Comments

The results for this reaction are summarized in Table
6 and the angular distributions are shown in Figure 15.
The second plate showed no peaks stronger than 15 Lb/sr at
60 between 4 MeV and 9 MeV excitation. In general, excitation
energies, 2rvalues, and spectroscopic factors obtained in the
present study are in good agreement with those of previous
work. It is, however, difficult to make comparisons with the
(d,p) work except at low excitation energies because at higher
excitation energies the (d,p) reaction mostly populates states
by 2:2 and 2:0 transfers.

In doing the excitation energy analysis it was necessary
to adjust the Q—value for this reaction by +11 keV. This

21) mass of 75Ge

indicates a +11 keV correction to the quoted
since we expect the mass of 760e (a stable nucleus) to be
accurately known. This new mass is consistent (within 2 keV)
with the mass for 75Ge determined from the 7L’Ge(n,v) Q-value
measured by Hasselgren.3)
There again appears to be a slight variation in the 1/2—,
3/2- systematics in this nucleus. There is a fairly weak
(S=.17) state which is assumed to have a spin of 1/2'. The
16)

1/2- assignment is preferred by Kato in the (d,p) reaction
and by the ratio of strengths argument. If the state does

indeed have a spin of l/2", it has no counterpart in the other

91

92

 

 

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95

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.voscHucoo u 0 meme

Figure 15.

96

Experimental angular distributions obtained from

76Ge(p,d) reaction at 35 MeV. The solid curves

the
are fits to the data of DWBA predictions. For
mixed 2—transfers, the dashed lines show the

contribution of each component.

97

      

Fx=lHl7 uov

E=23SSLV
1:1 xi=l.

          
  
  
   
       
 
     

Ex=2382 ROV
8 H

Ex=253H IIOV
I = 3

Ex=2572 IOV
I 8 l

Ex=1292 fiOV

E 2687kV
D83 x3981.

Ex'3182 NOV
’ 8 l

Ex'3826 IIOV

Ex” 576 hOV
’ = 1'43

’81

 
   

0.1

 

98

odd nuclei. However, the spin is not firmly established and
it could well be that the spin is actually 3/2_. This would
then be more consistent with the systematics as 690e has a

weak 3/2_ state at 233 keV.

VIII.2 Comments on Individual States

139 keV state: We observe an 2:4 transition to this

 

state which would initially suggest a 9/2+ spin assignment.

However, life-time and conversion coefficient measurements

+ 31)

suggest a spin assignment of 7/2 . There are other

"anomalous" 7/2+ States in neighboring nuclei26)

26)

which have
been attributed to (lg9/2)3 configurations. In particular
there is the 68.2 keV state in 73Ge with a probable spin of
7/2+. The fact that we observe the 139 keV state in our data
indicates that there is some population of the lg7/2 orbit
on 760e. It also indicates that the wave function of the
139 keV state includes a component with a (lg7/2)l term.
The state is nevertheless quite likely dominated by the
(lg9/2)3 configuration since it is not observed in the (d,p)
reaction. Since we do not observe the 68.2 keV state in 73Ge
(it would be covered by the strong transition to the 66.7
keV state) we cannot make comparisons as to the relative 1g7/2
population in 740a and 760e. Neither can we deduce anything
about the wave function of the 68.2 keV state in 730e.

197 keV doublet: This peak is clearly observed to be

a doublet in the spectra at 6°, 90, and 120 by a (one or two

channel) broadening in the peak. This is reflected in the

I”

99

angular distribution by a rise above the pure 22h shape at

these angles. 'This same deviation occurs in the angular

3)

distribution shown by Hasselgren

l8)

and to a lesser degree

16)

those of Fournier gt 31. and Kato. In our data the
deviation is so slight that the fit is not significantly
improved by adding in another 2-component. For this reason,
we cannot distinguish between an 2=l+4 or an 2=1+3 mixture
for this transition, although the 2=l+h fit is slightly better.
The energy separation between the states of this doublet is
about 7 keV. The lower member may be the same state as the
180 keV state reported in the Nuclear Data Sheets, but if it
is, the 180 keV energy is too low.

457 keV state: Our data show a distinct 2=3 transition
to this state which is in agreement with the (p,d) results of
3.)

Fournier §§_al.18) Hasselgren observes a state in the (d,p)

reaction at 457 keV, but gives no 2-transfer for the transi-

16)

tion. However, Kato reports a state at #53 keV which is
listed in the table as being pOpulated by an 2=2 transfer and
is shown in the figure with an 2:0 angular distribution.
Either an 220 or an 2:2 assignment would be in disagreement
with our data. At this low an excitation energy the energy
calibrations for the different experiments are sufficiently
consistent that either this state must be a close doublet or
there has been an error in assigning 2—transfers. The agree-

ment in the two (p,d) experiments supports the 2:3 assign-

ment for this reaction. However, the confusion between figure

100

and table in Reference 16 and the lack of any 2~transfer
assignment in Reference 3 makes it impossible to give a
meaningful critique of the (d,p) 2—transfer assignment.

585 keV state: We observe an 2:2 transfer to this state
which is in agreement with the (d,p) results of Kato.16)
Fournier gt 31.18) did not resolve this state from the 576
keV state and fit the sum of the angular distributions with
an 2=l+3 mixture. However, the summed distribution is domin-
ated by the 2:1 component and it appears that it could be fit
acceptably by an 2=l+2 mixture. This would then be in agree-
ment with the (d,p) results and with our (p,d) data.

675 keV state: We observe a distinct 2:0 transition to
this state which is in agreement with the (d,p) results of

Hasselgren.3) Katolé)

again shows a discrepancy between fig—
ure and table, but the figure shows an 2:0 transition. In
disagreement with these results is the 2:2 transition re—

ported by Fournier 23 31.18)

This contradiction is probably
due to the fact that the previous (p,d) work did not resolve
the 651 keV state from this state and therefore obtained an
angular distribution which is actually an 2=0+3 mixture.

1258 keV state: The uncertainty in the 2-transfer
assignment for this state is due to the fact that angular
distribution is not fit as well as the other 2=h transitions
we see. A small 2:3 component would improve the fit, but
not significantly enough to warrant calling this state a

doublet.

101

1417 keV state: There is slight evidence in our spectra
that this state is actually a doublet. However, the angular
distribution is fit very well by a pure 2:1 shape which in-
dicates that if there are two states here, they are both
populated by 2:1 transfers. Fournier §t_al.l8) fits the
angular distribution for this state with an 2:1+3 mixture,
but did not resolve the 1396 keV state which is populated by
an 2:2 transition. This resolves the contradiction since the
angular distribution shown in Reference 18 is dominated by
the 2:1 component and it appears that it would be fit accept-
ably by an 2:1+2 mixture.

1699 keV doublet: There is a consistent one channel
broadening of the peak for this state in our spectra which
indicates a doublet of about 5 or 6 keV separation. The
angular distribution also indicates a doublet as it can only
be fitted well by an 2:3+h mixture. Fournier gt 31.18) fit
the angular distribution for this state with a pure 9:3 shape,
but the fit is not good. It appears that the angular distribu-
tion shown in Reference 18 is consistent with our data in
that it would be fit well by an 2:3“. mixture.

2359 keV and 2382 keV states: Both of these states are
weakly populated in the (p,d) reaction, but the angular distri-
butions are distinct and are well fitted by an i=1 and an i=4
shape respectively. The (d,p) work appears to be in contra-

l6)

diction to our results as Kato reports an 2:2 transition to

a state at 2359 keV and Hasselgren3) reports an 2:2 transition

102

to a state at 2382 keV. However, neither Hasselgren nor Kato
report seeing both states. Also, there is a growing dis-
crepancy starting at about 1800 keV as to the energy of the
states seen, and even which states are populated, in the (d,p)
reaction. It is likely that the energy calibrations in the
two (d,p) experiments are not consistent, and that the states
reported by HasselgrenB) at 2321 keV, 2382 keV, 2527 keV,

2574 keV, and 2660 keV are the same as those reported by

Katolé)

at 2310 keV, 2359 keV, 2462 keV, 2553 keV, and 2636
keV. Equating these states would present no contradictions
in 2—transfers between the two (d,p) experiments. This would
help to explain the contradiction between the (p,d) and (d,p)
results for the 2359 keV and 2382 keV states. It is possible
that these states were not resolved in the (d,p) work and the
2=l+h was mistaken for an 2:2 transition. If the 2:2 assign-
ment is indeed correct, then the (d,p) reaction must be
populating a different state from the ones we observe in the
(p,d) reaction.

2323 keV and 2572 keV states: Our data show definite
2:1 transitions to both of these states. HasselgrenB)
reports 222 transitions to states at 2321 keV and 257A keV
which would be in disagreement with our results. However,
as has already been pointed out, the energy calibration in
the (d,p) work is suspect and it is therefore not at all
obvious that the same states are being seen in both the (p,d)
and (d,p) reaction. If these two states are the same in both

reactions, then the 2:2 assignment is probably incorrect,

103

since the angular distributions shown in Reference 3 would
be fitted well by the empirical 2:1 shape of the 252 keV state.
18)

Fournier §£.é_- report states at 2043 keV, 2105 keV,
and 2198 keV. However, these states would all be covered by
peaks from germanium isotopes that are populated by the same
g—transfers, and with approximately the same strength as

reported for these states. It is, therefore, doubtful that

there are states in 75Ge at these energies.

IX. DISCUSSION OF RESULTS

IX.1 General Features

Figures 16—21 show the general features of the results
of the present study. Figure 16, which was referred to in
Chapter IV, shows the distribution of the strong transitions
to states in the odd isotopes. Figures 17—21 show energy
level diagrams for each of the nuclei studied. Here we show
the previously observed states up through 2 MeV excitation and
the states we observe in the (p,d) experiment. We observe
most of the states previously reported up to this excitation

energy.

IX.2 Summed Spectroscopic Factors

It was mentioned in the Introduction that the sum of
the spectroscopic factors for a given 2-j transfer is a measure
of the number of particles in that orbit. The maximum number
of neutrons allowed in an orbit is given by 2j + 1 where j is
the total angular momentum of the neutrons in that orbit.
(The same relation holds for protons.) This then gives a
sum-rule limit for the spectrosc0pic factors for pickup from
that orbit. The situation is complicated a little bit however
by isotopic spin considerations. Isotopic spin (T) is
primarily a measure of the relative number of protons and
neutrons in a nucleus and it obeys quantum mechanical

angular momentum algebra. Indeed for all states of all

104

105

Figure 16. Spectroscopic strength plotted as a function
of excitation energy for 2rtransfers of 1,3, and

4 for the (p,d) reaction on the even targets.

106

 

 

 

 

 

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107

Figure 17. Energy level diagram for states below 2 MeV in
69Ge showing energy, 2-transfer, spectroscopic
factor, and sample spectrum for states populated

in the (p,d) reaction.

108

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

33 ‘2
use” 5

M JAJ; A A 1 k4 AAA‘A All A

Figure 17

109

Figure 18. Energy level diagram for states below 2 MeV in
71Ge showing energy, 2rtransfer, spectroscopic
factor, and sample spectrum for states populated

in the (p,d) reaction.

110

 

 

 

 

LA

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

LA ll 1 I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

JL Li A Al I

71
343939

Figure 18

 

 

111

Figure 19. Energy level diagram for states below 3.5 MeV
in 72Ge showing energy, 2—transfer, spectroscopic
factor, and sample spectrum for states populated

in the (p,d) reaction.

112

 

 

 

326990 5

Figure 19

113

Figure 20. Energy level diagram for states below 2 MeV in
730e showing energy, 2rtransfer, spectroscopic
factor, and sample spectrum for states populated

in the (p,d) reaction.

114

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1m

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

73
32 Ge '21 8

Figure 20

115

Figure 21. Energy level diagram for states below 2 MeV in
750a showing energy, 2rtransfer, spectroscopic
factor, and sample spectrum for states pOpulated

in the (p,d) reaction.

 

L LL_ W 21 WAWA

116

 

 

 

 

 

 

 

 

 

 

 

 

 

1320
u ‘9‘}

 

ll

 

 

1137

 

 

 

 

 

 

 

 

 

 

 

7s
32 69 «a

Figure 21

 

 

117

nuclei T2 = (n-z)/2, and usually T = TZ for the states of
low excitation energy. Thus, 69Ge has TZ = 5/2, while the
ground state of 70Ge has TZ = 3, etc. The above relation for
Tz holds for individual protons and neutrons also, so that

for protons Tz = —l/2 and for neutrons TZ = 1/2.

Consider 7OGe which has 4 active protons and 10 active

neutrons and see what happens to T for proton and neutron

pickup. For proton pickup we have an initial isospin T1 = 3

P = l/2 and TP = -l/2

and T: = 3 coupled to a proton isospin T z

to give a final isospin Tf and Ti.

The following relations must hold:

f i P
TZ — Tz - TZ l)
Tf=TiiTP 2)
Tf> Tr." 3)
— z
r r_ . f_ .
Thus, TZ = 7/2 and T — 7/2 Since T — 5/2 does not satisfy

the last relation above. Now the sum-rule for the pickup of

the 4 active protons is
02(7/2.7/2)2:S(3-7/2) = L.

In this equation 02(7/2,7/2) is the Clebsch—Gordon coefficient

for T = 3, T = 3 coupled to T = 1/2, TZ = -l/2 to give T = 7/2.

Z

 

118

T2 = 7/2, and z 8(3-7/2) is the sum of all spectrOSCOpic
factors for proton pickup transitions from 7OGe.
Neutron pickup is slightly more complicated. Using

Equations 1—3 above we find T:

= 5/2 and both Tf = 5/2 (called
T<) and Tr = 7/2 (called T>) are allowed. It will be noted
that a T> state will have the same isospin as states populated
by proton pickup only with a different z-projection. Indeed
there will be pairs of states pOpulated in proton and neutron
pickup whose wave functions differ only in Tz’ These are said
to be analogue states. The sum-rule for neutron pickup has

two terms, one for T> states and one for T< states. Since

we have 10 active neutrons in 70Ge it takes the form:
02(5/2.7/2)zs(3-7/2) + c2(5/2.5/2)23(3-5/2) = 10.

In this equation £S(3—7/2) is the same as for proton pickup
so that it can be calculated from the proton pickup limit
equation and used to calculate the limit for T< states in
neutron pickup. This is useful for our analysis since all
the T> states in the germanium nuclei are calculated to be at
greater than 9 MeV excitation energy and are therefore un-
observed in our experiment. Using this procedure we obtain
'the limits listed in Table 7.‘ Also listed in Table 7 are

the summed spectrosc0pic factors for each 2—transfer in each
reaction. For each reaction the total sums are very close

to the sum—rule limits.

119

Table 7. Summed spectroscopic factors.

 

 

Reaction 7OGe(p,d) 72Ge(p,d) 73Ge(p,d) 7I"Ge(p,d) 76Ge(p,d)

 

2—transfer

 

0 .03 .06 .01 .03 .02
1 3.32 3.64 3.69 3.96 3.92
2 .15 .20 .49 .60 .38
3 6.22 5.4 6.37 4.10 4.65
4 1.69 2.06 1.32 4.81 5.66
total 11.50 11.50 11.90 13.55 14.80

limit 9.43 11.56 12.60 13.64 15.69

 

 

120

While the (p,d) reaction summed spectroscopic factors
measure the number of particles in a given orbit, the (d,p)
reaction summed spectroscopic factors measure the number of
holes in (or emptiness of) a given orbit. Therefore, if the
emptiness measured by the (d,p) reaction is added to the full—
ness measured by the (p,d) reaction, the result should be 1.

Table 8 presents such a comparison between our data on the

14)

even targets and the (d,p) results of Goldman and of

16)

Kato. In general, the agreement is quite good and indicates

that 75-100% of the allowed strength has been observed.

Following are comments on each orbit:

l6) 3)

‘lfi/Z orbit: Neither Kato observed

76

nor Hasselgren
an 2:3 transition in the Ge(d,p) reaction. This would in-
dicate that the lf5/2 orbit is full in the ground state of
760e. Our results indicate that it is only about 80% full,
but it may be that we do not see all the 2:3 strength as we
would expect some 2:3 transitions to higher excited states.

70

For Ge there is clearly too much strength observed. How-

14)

ever, Goldman's data shows only one definite 2:3 transition

in the (d,p) reaction and that one was mixed with an 2:4.

Due to the poor quality of the (d,p) data there must be a

very large uncertainty attached to extracting spectroscopic

factors from mixed—2 doublets. It, therefore, seems likely

that the (p,d) strength is at least approximately correct.
223/2 and 221/2 orbits: For these strengths there is

the difficulty of making accurate spin assignments which

121

 

 

 

'Table 8. Fractional fullness (from (p,d)) and emptiness
(from (d,p)) of orbits.
7OGea) 720e 740e 76Ge
(p,d) .85 .90 .82 .78
lfS/Z (d,p) .70 .18 .20 0.0
sum 1.55 1.08 1.02 .78
(p,d) .50 .69 .81 .73
2p3/2 (d,p) .03 .14 .13 .06
sum .53 .83 .94 .79
(p,d) .36 .43 .36 .49
2p1/2 (d,p) .50 .50 .38 .26
sum .86 .93 .74 .75
(p,d) .14 .21 .40 .57
1g9/2 (d,p) .70 .51 .43 .32
sum .84 .72 .83 .89
(p,d) .02 .03 .09 .06
2d5/2 (d,p) .50 .66 .70 .58
sum .52 .69 .79 .64
(p,d) .01 .03 .02 .01
351/2 (d,p) .50 1.06 .93 .89
sum .51 1.09 .95 .90

 

 

a)(p,d) data has been normalized to sum rule limit.

122

should be reflected in the summed spectroscopic factors.
For example, in 7OGe(d,p) Goldmanlh) sees too much 2p1/2
strength relative to 2p3/2 strength, but this is due at least
partly to incorrect 1/2- spin assignments as discussed pre-
viously. It also appears that there should be more 2p1/2
strength relative 2p3/2 strength in the 7l“Ge(p,d) reaction
indicating that possibly some spin assignments are wrong
in 73Ge. In general it appears that 10-25% of the expected
2:1 strength has not been observed in either the (p,d) or
(d,p) reactions.

lgg/Z orbit: The missing strength here is probably in
the (d,p) reaction, as 2:4 transitions in the (p,d) reaction
should populate low-lying states.

Some of the missing strength noted above could be due
to normalization problems. The absolute spectroscopic factors
are probably only accurate to 15 or 20% and this could account

for some of the discrepancies.

IX.3 Wave Functions

We will now consider what the data tells us about the
wave functions of these nuclei. Based on the simplest shell
model picture we would expect to see the lf5/2 and 2p3/2
orbits filled for the ground states of all the target nuclei.
720e would also have the 2p1/2 orbit filled and for the higher
A targets the 1g9/2 orbit would begin to fill up to 4 particles

for 760s. However, this picture does not take into acCount

123

the binding energy that comes in when two nucleons in the
same orbit pair their spins to produce a net sum of zero
angular momentum. This pairing energy can change the order
of filling in the following way.. If, for example, the pairing
energy of two nucleons in the lg9/2 orbit is greater than
twice the difference in energy between the 1g9/2 and 2pl/2
orbit, two nucleons will populate the lg9/2 orbit rather than
the 2p1/2 orbit. Also, the simple model is based primarily
on a calculation of the order of the orbits and since these
orbits are very close to each other their order may be dif—
ferent in some nuclei.

The spectrosc0pic factors give us a measure of the actual
population of the different orbits and a better idea of what
the wave functions look like. The wave function will actually
be a linear combination of all the possible arrangements of
the active particles in the active Space. In general, our
data give us no information about the proton structure.
However, the fact that we see most of the previously observed
states for these nuclei indicates that to a good approxima—
tion all the protons can be considered part of the core. The
data also indicates that for these nuclei we can consider
all the neutron orbits up through the 1f7/2 orbit as filled
and constituting the remainder of the core nucleons. This
leaves ten active neutrons in 70Ge. The active space in-
cludes the lf5/2, 2p3/2, 2pl/2, lg9/2, 2d5/2, and 331/2

orbits. The summed strengths measured for these orbits

 

 

124

place restrictions on their populations. In particular, the
snail 2d5/2 and 381/2 strengths indicate that these orbits
are only weakly populated and perhaps, to a good approxima-
tion, terms containing them could be ignored. The fact that
they are pOpulated at all indicates that in these nuclei they
rnust be much closer in energy to the 1g9/2 orbit than pre-
dicted by the simple shell model calculation. 0n the other
hand, the lf5/2 orbit is mostly full. Thus, in considering
terms for the wave function, we could ignore those with a
small lfs/2 populati0n. If we made the above approximations,
and assumed that the 1f5/2 orbit always contains at least
four particles, we could write down the wave function for

70

Ge as follows:

70Ge = Cl(lf5/2)6(2p3/2)h + 02(1f5/2)6(2p3/2)2(2pl/2)2

+ C3(lf5/2)6(2p3/2)2(1g9/2)2 + Ch(lf5/2)6(2p1/2)2(189/2)2
+ 05(1f5/2)6(189/2)‘* + C5(lf5/2)4(2p3/2)4(2p1/2)2
+ c7(113/2)“(21:>3/2)‘*(leg/2)2

+ 08(1f5/2)"(2p3/2)2(2p1/2)2(189/2)2

+ C9(1f5/2)l+(2pB/2)2(1g9/2)l+
+ C10(lf5/2)h(2p1/2)2(lg9/2)h

+ Cll(lf5/2)h(lg9/2)6.

The summed spectroscopic factors place restrictions on the

coefficients, but we do not have enough information to

 

 

 

125

determine them uniquely. For example, since the summed 2pl/2

strength is .72 we can write down the following restriction:

2
4

2 2

_ 2 2

2
Similar relations could be written down for the summed

strength for each orbit. This would give us four equations
with eleven unknowns. We can say that terms involving ,

6 4 . . .
(1g9/2) and (lg9/2) must have small coeffiCients Since the

 

summed strength for the lg9/2 orbit is only about 1.4.
Similar wave functions could be written down for each of the
target nuclei but in order to completely determine the co—
efficients one must use the information contained in the
excitation energies of the excited states. This requires

a complete shell model calculation.

For the excited states in the odd nuclei we only have
information about their wave function relative to the target
wave function. In each odd nucleus there is a low lying state
that carries almost all the 2:4 strength for that reaction.
This indicates that the wave functions for these 9/2+ states
are just the same as the target wave function with a 1g9/2
neutron removed. Similar arguments hold for the strongly
pOpulated 5/2—, 3/2_, and 1/2- states. The other excited
states have wave functions which differ more significantly
from the target wave function.

Fournier gt al.18) have tried to do a very simple

_- ___._ AI—r

 

126

calculation assuming that only the 2pl/2 and lg9/2 orbits

are involved in the active space. Based on the summed spectro-
scopic factors, this appears to be too much of an over simpli-
fication to yield any useful information. While it might be
possible to consider the 11‘5/2 orbit as full, it is clear that

the 2p3/2 orbit must be taken as part of the active space. a

IX.4 Order of Filling of Orbits

The summed spectroscopic factors also give us some in—

 

formation about the order in which the orbits are filling as

we add neutrons in the germanium isotopes. Due to the small

‘p0pulation of the 2d5/2 and the 381/2 orbits we cannot say

much about them except that the 2d5/2 orbit is gaining some
76

population as we go from 7OGe to Ge. This leaves the
2p3/2, 2pl/2, 1f5/2, and 1g9/2 orbits for consideration.

For the even isotopes the 1f5/2 orbit maintains a fairly
constant population. The apparent decrease exhibited in

the spectroscopic factors is possibly due to unobserved

2=3 transitions to higher excited states. The pOpulation of
the 2:1 orbits increases slightly with the 2p3/2 orbit getting
most of that increase. The lg9/2 orbit is getting most of the
overall increase in population which is just what we would
expect from the simple shell model picture. The one odd tar—
get, 73Ge, Shows a deviation from this trend of increasing
population of the lg9/2 orbit. In this nucleus the lg9/2

72

population is lower than that for Ge and the lf5/2 pOpula—

tion is significantly larger. The apparent decrease in

 

127

lg9/2 population could be due to 2:4 transitions to highly
excited 6+ and 8+ states in 720e which we do not see, or to
some unobserved 2:4 components in states populated by 2:2

transfers.

 

X. SUGGESTIONS FOR FURTHER STUDY

As has already been mentioned, it would be interesting
and helpful to study the J-dependence of the 2:1 transitions
with both the (p,d) and (d,p) reactions. In doing this, it
would be highly desirable to do the experiment with a counter
instead of plates because of the time and effort involved in
scanning the plates. Accurate excitation energies are not
needed so partial spectra from a small counter would be
acceptable. The immediate availability of the data would
be very helpful in determining that the experiment is indeed
going to be successful before large amounts of time are
expended. If the experiments were successful, it would yield
accurate spin assignments for more of the states populated
by 2:1 transfers and thereby give more information about
the wave functions of these nuclei.

The 2:3 J—dependence observed in the nickel region is
useful in making spin assignments there, but it cannot be
used with confidence in the germanium region since there are
no known 7/2— states in these nuclei. A systematic study of
the 2:3 angular distributions for the nuclei in the mass region
between nickel and germanium might yield more information
about the trends in, or existence of, the effect in these
nuclei when the lf5/2 orbit is more completely filled.

Further spectroscopic information might be gained for

the even nuclei by the use of the (p,t) and (p,p') reactions.

128

 

 

129

Hsu et al.17) have studied the 7OGe(p,t) reaction but the
usefullness of their results was again limited by their

energy resolution.

 

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REFERENCES

 

 

10.

ll.
12.

13.

14.
15.

l6.
l7.

l8.

 

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