SPECTROSCOPY IN THE TITANIUM ITSOTOEPES *
VIA (p,d) AND (m) REACTIONS

Thesis for the Degree of Ph. D‘.
MICHIGAN STATE UNIVERSITY
1 PHILLIP JAMES PLAUGER

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This is to certify that the
thesis entitled

SPECTROSCOPY IN THE TITANIUM ISOTOPES

VIA <p,d) AND (p,t) REACTIONS

presented by

Phillip James Plauger

has been accepted towards fulfillment
of the requirements for

Ph. D. degree ill—mp i

3 A / 7
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Major professor

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ABSTRACT

SPECTROSCOPY IN THE TITANIUM ISOTOPES

VIA (p,d) AND (p,t) REACTIONS

by Phillip James Plauger

Deuteron angular distributions from u8Ti(p,d)u7Ti were
measured at incident proton energies of 2U.80, 29.82, 35.15,
39.97 and 45.05 MeV to detect the presence of any energy
dependence in various methods of extracting spectroscopic
factors. Using optical parameters obtained from the liter-
ature, distorted wave Born approximation calculations were
performed with the Oak Ridge code JULIE for the principal
Ln: 1, 2 and 3 transitions observed. In all cases, signifi—
cantly better agreement with the data was obtained in the
zero-range approximation by using a lower integration cutoff
at the nuclear surface. The variation of spectroscopic
factors with cutoff radius also exhibited a plateau at the
nuclear surface; however energy dependence of extracted
spectroscopic factors was still present. The best agreement
was obtained by applying finite range and non—locality
corrections to the form factors used by JULIE. No integra-
tion cutoffs were required and energy dependence was removed.

Partial angular distributions from u6Ti(p,d)u5Ti at 3U.78

MeV and 50Ti(p,d)u9Ti at ”5.05 MeV were also measured.

Plauger

Using the prescription obtained from the energy dependence

study, spectroscopic factors were obtained for transitions

to “5’47’M9Ti.

A6,A8,50 AA,A6,A8

The reactions Ti(p,t) Ti were performed

from 25 to M5 MeV in an attempt to excite T= TZ+2 analog

states. Only the T= 2 state at 9.31 MeV excitation in uuTi

was observed. The ground state Q-value for u6Ti(p,t)uuTi

was found to be -lU.2M6(0.0ll) MeV.

SPECTROSCOPY IN THE TITANIUM ISOTOPES

VIA (p,d) AND (p,t) REACTIONS

by

Phillip James Plauger

A THESIS

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics

1969

ge/773
1/ -—2 7- 70

ACKNOWLEDGMENTS

While it is a convenient fiction that a doctoral thesis
is the work of one person, in a cyclotron laboratory this
is far from the truth. In any endeavor, the assistance of
Professor Henry Blosser, Harold Hilbert, N. Mercer and their
technical staffs is required throughout. I gratefully
acknowledge receiving their help in many ways. Others have
contributed their time and talents to this thesis in a more
specific fashion:

I wish to thank Ivan Proctor for the use of the detector
bench he developed and built, and Douglas Bayer for the
extensive use of his program TOOTSIE.

Fred Trentelman and Larry Learn helped me acquire the
bulk of the data used in this study. For both their help
in operating the cyclotron and late—night company I am
grateful.

Particular thanks goes to Bryan Horning for the many
hours he spent reducing this data. And to Mrs. Karen Fox
for preparing the illustrations.

Much of my understanding of optical model and distorted
wave calculations I owe to Dr. Barry Preedom. His advice
and suggestions proved invaluable on more than one occasion.
Many conversations with Professor Hubby Sherr have taught
me much about nuclear spectroscopy and stimulated much deep
thought.

ii

iii
Finally and most important, I wish to thank Professor
Edwin Kashy for his extensive assistance throughout this
thesis, and for his sound guidance throughout my collegiate

career.

TABLE OF CONTENTS

List of Tables

List of Figures

1.

2.

7.

Introduction

Nuclear Theory

2.1 Shell Model with Residual Interactions
2.2 Spectroscopic Factors and Sum Rules
2.3 The Distorted Wave Born Approximation
Experimental Procedure

3.1 Beam Production and Handling

3.2 Particle Detection and Identification
3.3 Error Analysis

Optical Model Parameters

Energy Dependence of u8Ti(p,d)u7Ti

5.1 Experimental Results

5.2 Zero—Range DWBA Calculations

5.3 Finite-Range DWBA Calculations
Spectroscopy of (p,d) Reactions

6.1 “8T1(p,d)”7Ti

6.2 “6’50T1(p,d)u5’u9Ti
6.3 Sum Rules and Q Dependence

Spectroscopy of (p,t) Reactions

List of References

iv

vi

10

13
16

22

27

32
37
“1

AA
52
63
68
73

6.3

6.“

LIST OF TABLES

Predicted energies and spectroscopic factors
for lf7/2 pickup.

Sum rule predictions for neutron pickup from
valence orbitals.

Isotopic analysis of targets by atomic per
cents.

Contributions to energy resolution at A0 MeV.

Contributions to uncertainty in cross-
sections.

Proton optical model parameters.
Deuteron Optical model parameters.

Per cent increase in peak cross—section for
ten per cent increase in parameters.

Zero-range spectroscopic factors for the L= 3

transition to “7Ti(O.I6 MeV).

Zero-range speggroscopic factors for the L= 1

transition to Ti(l.5u MeV).
FRNL spectroscopic factors for some of the
principal transitions to ”7Ti.

Summary of results for u8Ti(p,d)u7Ti at
35.15 MeV.

5

Summary of results for OTi(p,d)u9Ti at

“5.05 MeV.

Summary of results for u6Ti(p,d)u5Ti at

3A.78 MeV.

Comparison of summed spectroscopic factors
to predicthns.

Predictions and results of T= Tz + 2
investigation.

Summary of results for u6Ti(p,t)uuTi at

39.2“ MeV.
v

15
22

2A
28
29

31

38

39

U2

U8

56

62

6M

69

69

3.1
3.2
3.3
3.“
“.1

5.1

5.3

5.“

6.3

6.“

6.5

6.6

LIST OF FIGURES

Neutron single particle energy levels.
Floor plan of cyclotron experimental area.
Schematic of detector electronics.

Block diagram of experiment control logic.
Flow chart of data acquisition routine.
Deuteron optical model parameters.

Deuteron spectrum from “8T1 target and levels
observed.

Energy dependence of odd-L transfers in
“8Ti(p,d)u7Ti.

Energy dependence of even-L transfers in
u8Ti(p,d)u7Ti.

Some DWBA fits to u8Ti(p,d)u7Ti for L= 1

(1.5“ MeV), L= 2 (1.82 MeV) and L= 3 (0.16
MeV.

Measured angular distributions of
u8Ti(p,d)u7Ti at 35.15 Mev, Ex= O - “.25 MeV.

Measured angular distributions of

u8Ti(p,d)u7Ti at 35.15 MeV, Ex= “.36 — 6.36
MeV.

Measured angular distributions of

48Ti(p,d)u7Ti at 35.15 MeV, Ex= 6.57 - 8.7“
MeV.

50
Deuteron spectrum from
observed.

Ti target and levels

Measured angular distributions of
50Ti(p,d)u9Ti at “5.05 MeV, Ex= O - 7.33 MeV.

Measured angular distributions of
50Ti(p,d)u9Ti at “5.05 MeV, Ex= 7.“9 - 11.“9
MeV.

vi

1“
18

19
21

3O

33
35

36

“O

“6

“7

53

55

6.7

6.8

6.9

7.2

vii

“6

Deuteron spectrum from Ti target and levels
observed.

Measured angular distributions of
u6Ti(p,d)u5Ti at 3“.78 MeV, Ex= O - 3.37 MeV

Measured angular distributions of
u6Ti(p,d)u5Ti at 3“.78 MeV, Ex= 3.“9 - 6.98
MeV.

Triton spectrum from u6Ti target and levels
observed.

Measured angular distributions of
u6Ti(p,t)uuTi at 39.2“ MeV.

59

61

7O

71

1. INTRODUCTION

The study of (p,d) reactions in the titanium isotopes
provides useful information about both the reaction mecha-
nism and the nuclei themselves. Stable titanium nuclei
have Z= 22, N= 2“ - 28, and so each can be treated as a

“0

Ca core plus particles mainly in the 1f shell. The

7/2
(p,d) reaction consists principally of picking up a neutron
from one of the outer shells of the target, with a minimum

of configuration rearrangement. Therefore one would expect
to observe at least one strong L= 3 transition, correspond-
ing to the pickup of a lf7/2 neutron, and some L= l strength
due to the admixture of the next higher subshell, 2p3/2.

Even L transfers, pickup from the filled 2s-1d shell,

are also expected.

The strength of these transfer reactions is a very direct
measure of the overlap of the final state and target ground
state wave functions. Since the shape of the angular dis-
tribution strongly depends on the L-transfer, the principal
transitions can be sorted fairly well according to the
orbital from which they originate. Given a theory of the
reaction mechanism, one can then determine the occupation
probabilities of the various shell model orbitals. Such a
theory is the distorted wave Born approximation, embodied
in the Oak Ridge computer code JULIE (Ba 62). The transition
strengths measured by comparing experimental cross—sections

1

2

to DWBA predictions are called spectroscopic factors.

Consistency checks exist for these factors. Since the
sum of all spectroscopic factors for picking up a neutron
from a shell gives its total occupancy, the sum has a
rigorous upper bound. One expects inner shells to be essen-
tially filled, closely defining their sums. Still other
"sum rules" exist for other constraints, as will be seen
later. There exist also the calculations, by McCullen,
Bayman and Zamick (Me 6“), of the wave functions of (lf7/2)n
configuration states. These provide a basis for predicting
spectroscopic factors and other nuclear prOperties of the
titanium isotopes.

Such considerations raised questions about the results

”6’5OTi(p,d)uu-u8Ti studies at 17.5 MGV by

obtained from
Kashy and Conlon (Ka 6“). DWBA calculations using the zero-
range approximation consistently predicted less than one

“8

neutron in the 1d shell and, in Ti, nearly two neutrons

3/2
in the 2p3/2 shell.
Still other difficulties were reported by Sherr, et al.
(Sh 65) in a study performed at 28 MeV which included
“8’50Ti(p,d)u7’ugTi. These experiments also excited high-lying
isobaric analog states - states having a total isotopic spin
one greater than the low—lying states and configurations
analogous to the low-lying states in their scandium isobars.
Applying sum rules to the different isospin states, developed

by French and Macfarlane (Fr 61), Sherr et al. were able to

study the dependence of DWBA predictions on reaction Q-value.

3

Using normal procedures they found too large a Q dependence
for L= 3 transfers and were led to an "effective binding"
prescription to improve agreement with the sum rules. This
procedure has since been shown to be theoretically unsound
(Pi 65).

Some hope for improving this state of affairs was raised
by Snelgrove and Kashy (Sn 69) in a study of l6O(p,d)150 over
an energy range of 21 to “5 MeV. They discovered strong energy
dependence in the extraction of spectroscopic factors up to
a deuteron exit energy of about 20 MeV. Consequently, it
was decided to extend this energy dependence investigation
to u8Ti in the hope of better understanding the mechanics
of extracting spectroscopic factors. At the same time
“6’50Ti(p,d)u5’u9Ti measurements were made, each at one
energy, to explore the systematics of the (p,d) reaction
across the titanium isotopes and to apply the knowledge
gained from the energy investigation.

Since data for these studies were acquired on-line by
the laboratory's SDS Sigma 7 computer, it was an easy matter
to retain spectra of the tritons detected during the (p,d)
eXperiments. Thus it was possible to obtain additional
information on isobaric analog states, which have attracted
considerable attention of late, including analogs to states
in calcium. Supplemented by several runs devoted to obtain—
ing good triton statistics, the (p,t) data also permitted a

determination of many of the excited states in uuTi.

2. NUCLEAR THEORY

2.1 The Shell Model with Residual Interactions

The nuclear shell model (de 63) forms the basis for nearly
all calculations in the titanium mass region. In its sim-
plest form, it describes the nucleus as a collection of
neutrons and protons in different orbitals, each nucleon
interacting with the others only through an average potential
well formed by the remaining particles. Figure 2.1 shows
such a well, of the usual Woods-Saxon form, and the calcu—
lated spectrum of stationary neutron single particle states.
These orbitals are subsequently split by a strong spin-orbit
interaction to give the major shell groupings and "magic"
occupation numbers 2, 8, 20, 28. . . shown. The well para-
meters and spin—orbit strength used are fairly realistic
(Be 6“) for A: “8 nuclei.

To pickup a neutron from the highest occupied shell, 1f7/2,
one must provide the separation energy S= 13 MeV, approxi-
mately. Since 2.22 MeV is liberated on forming a deuteron,
the mass energy released in a (p,d) reaction would be Q=
2.22 - S, or about -11 MeV. Pickup from the 2s-ld shell
would then lead to three excited states between seven and
ten MeV, not including the Coulomb analog transitions also
expected. Unfortunately, while the predicted ground state

“

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RADIAL DISTANCE FERMIS

Figure 2.1 Neutron single particle energy levels.

Q value is essentially correct, many more than three excited
states are observed below 15 MeV in (p,d) reactions.

Clearly there must be residual interactions between the
nucleons which split the single particle levels into many
states. One approach to estimating this effect is that

taken by McCullen, Bayman and Zamick (Mc 6“). Confining

their attention to pure (1f7/2)n configurations, they
obtained empirical two—body interaction energies from the
“2

spectrum of Sc. Since this nucleus consists of two

non-identical 1f nucleons outside a closed core, one

7/2
expects eight levels with total spins J= O - 7, giving
the interaction energy differences. These differences
are sufficient to determine the matrix elements of the
residual interactions between all possible orthogonal wave
functions of a given J. Diagonalizing this matrix gives
the physical wave functions of spin J and their relative
energies.

These wave functions are tabulated (Mc 6“a). As an

example of the versatility of even pure (If )r1

“7

configu-

7/2

rations, the spectrum predicted for Ti consists of 133
levels below 15 MeV. Seventeen of these are J= 7/2, easily
reachable by an L= 3 (p,d) transition. Splitting of the
"hole" states, due to 2s-ld pickup, is of course not compu-
ted in the MBZ scheme. Nor is pickup from shells above
lf7/2 considered, since no valence nucleons are assumed

to be there. But the calculations do include the wave

functions of the targets used, and of the principal levels

7

reached by L: 3 transfers. Thus it is possible to compute

several important spectroscopic factors.
2.2 Spectroscopic Factors and Sum Rules

A direct reaction can be characterized by the fact that
only a few internal degrees of freedom of the interacting
system are excited (Ba 62). Since there is no intermediate
stage of compound nucleus formation, the (p,d) interaction
is dominated by the overlap of the initial and final nuclear
states. Thus, in the absence of polarization the (p,d)
differential cross—section can be written (Mc 6“, Ba 62)

73%.; LA.) = ‘3: 5 r‘”. (2.1)
where.S is the spectroscopic factor. The 3/2 arises from
summing over the spin states of the deuteron and averaging
over those of the proton. Sigma is the mechanism—dependent
cross-section for transferring a given sz, assuming one
particle available for pickup. It is this function that
is computed in the distorted wave Born approximation by
the codehJULIE.

The spectroscopic factor contains the overlap information.
Thus it can be written (Mc 6“)-'

5: N (<¢,¢+mlh7)‘ (2.2)
where N is the number of neutrons in the shell from which
pickup occurs, the ket is the target wave function and the
bra is the final state wave function coupled to a neutron

in the picked-up orbital. A straightforward example is

8

OTi(p,d)u9Ti(G.S.). MBZ naturally predicts (Jp Jn J)=

5

(0 0 0) for the target wave function, since the neutron
shell must be closed; and N is eight. From the tables
(Mc 6“a)
(“9T1 G.S.)= -0.9136(0 7/2 7/2) + 0.“058(2 7/2 7/2)

+ 0.0196(“ 7/2 7/2) - 0.01“6(5 7/2 7/2) (2.3)
There is only one way to add a neutron to this wave function,
and only the first term contributes to the inner product.
Thus S= 6.68 in this case. Table 2.1 shows the spectrosco-
pic factors for the principal predicted transitions from
even-A titanium targets.

Table 2.1 Predicted energies and spectroscopic factors
for 1f7/2 pickup.

u6Ti(p,d)u5Ti u8Ti(p,d)u7Ti 50Ti(p,d)u9Ti
Excitation T S Excitation T S Excitation T S

0.28 MeV 1/2 3.10 0 MeV 3/2 “.77 0 MeV 5/2 6.68
3.87 1/2 0.17 2.50 3/2 0.1“ 2.53 5/2 0.59
“.“2 3/2 0.67 2.87 3/2 0.55 “.86 5/2 0.““
“.75 1/2 0.05 “.13 3/2 0.01 8.“0 7/2 0.29

“.55 3/2 0.05

5.51 3/2 0.0“

6.11 3/2 0.02

6.“6 3/2 0.02

6.“8 5/2 0.“0

Basic sum rules follow directly from Equation 2.2 and
the discussion of the previous section. Adding up the
squared amplitudes of all transitions of a given sz is
merely a way of regrouping the single particle strength
split by residual interactions. Since in the simple model
the initial and final states of a (p,d) reaction would

overlap completely, the single particle spectroscopic factor

.9
is Just the number of particles in the orbital.

The sum rules derived by French and Macfarlane (Fr 61)
apply to transitions to the higher T, or analog, states of
the residual nucleus, which are also isotopically allowed
in (p,d). Thus

S= P/(2T + 1) (2.“)
where S is the spectroscopic factor (or sum if the strength
is split) for a transition of given sz to an analog state,
P is the number of protons in the target sz orbital and T
is the total isospin of the target. Table 2.2 gives the
predicted sums for pickup from the outer orbitals of the
even titanium isotopes.

Table 2.2 Sum rule predictions for neutron pickup from

valence orbitals.

Target 1f ld 2s

7/2 3/2 1/2
T-lower T-upper T-lower T-upper T-lower T-upper

“6

Ti 3.33 0.67 2.67 1.33 1.33 0.67
Ll8T1 5.60 0.40 3.20 0.80 1.60 0.u0
50T1 7.71 0.29 3.43 0.57 1.71 0.29

One can obtain the results of French and Macfarlane in
a straightforward manner in the case of analog hole states.
As an example, consider ”6Ti(p,d)u5Ti(T= 3/2 J= 3/2+).
Using the notation (# of lf7/2 protons, # of lf7/2 neutrons)
(# of 1d?”2 protons, # of ld3/2 neutrons), one can express

the corresponding hole state in ”58c as (2 “)(3 “), obtained

by promoting a proton from the ground state configuration

10
(l “)(“ “). Applying the T- ladder operator should leave
T= 3/2 and change scandium into titanium. Thus
T'<2 A)<3 A)= (2/3)“2 <3 3)<3 A)

+ <1/3)1/2 (2 A)<A 3) (2.5)
where the wave function has been normalized. On coupling
a (13/2 neutron, the first term drops out and the second
becomes Just the target ground state wave function. Thus
S= “(l/3) from Equation 2.2, in agreement with French and

Macfarlane.
2.3 The Distorted Wave Born Approximation

The cross-sections defined in Equation 2.1 were computed
on the SDS Sigma 7 by the distorted wave Born approximation
code JULIE (Ba 62, Sa 6“). In the plane wave Born approx-
imation, one treats all scattering as a first order pertur-
bation; i.e. the initial and final states are free particle
wave functions, and the interaction potential is the pertur-
bation that causes the transition. It is far more realistic,
however, to describe the relative motion of the interacting
bodies before and after the transition by waves distorted
by an optical potential. These potentials are chosen to closely
approximate elastic scattering data for the entrance and
exit channels.

Using the reaction notation A(a,b)B, the differential

cross-section is written (Ba 62)

 

515: ,, ,a. at :5? 4/7"‘ (2.6)
abnu" (imn199" 2:2? JEEZEE‘N*‘”41__

(2 In!) (2 n+0

11

where the mus are reduced masses and the k's are momenta.
The transition amplitude T is of the form (Sa 6“)
‘T' = 1421.142"... 3' (14.3»)“0’V't'0 ¢’*’(I.,Ai.) (2.7)
where the phis are the distorted waves, satisfying incoming
(-) and outgoing (+) boundary conditions. The matrix element
describes the effective interaction taken between the inter-
nal states of the colliding pairs. In the case of (p,d)
it is assumed that the proton-neutron interaction Vpn causes
the transition to occur. Thus the matrix element can be
written

4+alvlaa> = <8IA><JIVP~If> (2.8)
where the nuclear overlap is usually taken to be the wave
function of the transferred neutron.

Since Equation 2.7 involves a six-dimensional integral
which is extremely difficult to evaluate numerically, the
zero-range approximation is usually introduced. This is
accomplished by replacing the p-n interaction matrix element
by a constant, Do’ times a delta function in the separation
of the proton and neutron in the deuteron. Aside from col-
lapsing the integral to three dimensions, this approximation
invokes several physical assumptions. Principally, it means
that the deuteron is created at the same point at which the
proton is absorbed. Also, it ignores any tensor forces in
the p-n interaction and neglects all but the S-wave of the
deuteron internal state function. These assumptions are best

met in the case of low momentum transfers, and become increas-

12

ingly worse as the amount transferred increases (Sa 6“).

The effective interaction is expressed in terms of the
sz transferred and the distorted waves are expanded into
a sum of partial waves of different L. The computer code
JULIE calculates the necessary transition amplitudes on this
basis and outputs the (p,d) cross—section in millibarns per
steradian. Because the code uses a factor DO computed on
the basis of a delta-function bound deuteron instead of the
Hulthen wave function, the cross—section computed by JULIE
must be multiplied by 1.5 to give the sigma used in Equation
2.1 (Ba 62).

Calculations involving finite range (Au 6“) and non-local
(Au 65) interactions have been investigated. Both tend
to damp the contribution of the nuclear interior, which
is known to be necessary to improve agreement with data.
Since finite range calculations are difficult and time comsuming,
the "local energy approximation" has been exploited to approxi-
mate finite range (Be 6“a, Bu 6“) and non-local (Pe 6“) effects.
This results in a radial damping function which modifies
the neutron bound-state wave function used in the zero-range
calculation. Comparison between such calculations and full
six-dimensional integrations have been made (Di 65) and are
found to agree reasonably well. Finite range and non-local
(FRNL) damping factors used in this study were computed on

the Sigma 7 by the code WAVDAM (Sa 69).

3. EXPERIMENTAL PROCEDURE

3.1 Beam Production and Handling

Proton beams for these experiments were accelerated in
the Michigan State University sector focussed cyclotron
(BI 66). Between 70 and 100 per cent of the internal H+
beam was extracted via an electrostatic deflector and mag-
netic channel. Figure 3.1 shows a floor plan of the cyclo-
tron experimental area and the beam line used.

The transport system (Ma 67) focussed the extracted
proton beam from the cyclotron on slits SI, and subsequent
foci were formed at slits S3 and at the center of the
36-inch scattering chamber. The energy analysing system,
consisting of “5-degree bending magnets M3 and M“, quad-
rupoles Q5 and Q6 and sextupoles SXl and SX2, was designed
to give an energy resolution of one part in 1000 for 0.130
inch horizontal slit openings at 81 and S3. In these studies
the energy spread in the beam was kept at approximately
25 keV.

Beam divergence was controlled by slits S2, positioned
51.75 inches from 81. A typical Opening of 0.“0 inch
limited beam divergence to +/— “ mrad. Since the total
magnification between 81 and target is just under one, a
properly focussed beam could be kept within a 0.1 inch wide

13

1“

 

 

 

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CYCLOT RON VAULT

 

 

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15
by 0.15 inch high area on target. Closed circuit television
cameras and scintillators at each of the three foci permitted
visual alignment of the beam before each run.

Proton energies were determined from nuclear magnetic
resonance fluxmeters in the central fields of M3 and M“.
These magnets were always set following a standard cycling
procedure to insure proper field shape. Energy reproducibi—
lity has been estimated at one part in 15000 (Sn 67) and
absolute energy at better than one part in 1000, based on
the consistent agreement of several recent measurements
in the laboratory.

Targets used in these experiments are described in Table
3.1. All targets were cut to approximately one inch squares,
weighed and measured before mounting in frames. It is
believed that the thicknesses determined are accurate to

four per cent.

Table 3.1 Isotopic analysis of targets by atomic per cents.

Target (mg/cm2) ”6T1 ”7T1 ”8T1 “9T1 5OT1
“6T1 1.05“ 86.1 1.6 10.6 0.8 1.0
”8T1 0.923 0.17 0.2 99.36 0.17 0.11
50T1 1.090 3.1 2.39 22.8 2.02 69.7

Beam passing through the scattering chamber was collected
in a 2.9 inch diameter by 11.5 inch deep Faraday cup. An
Elcor A3108 current indicator and integrator connected to
the cup was used to monitor beam intensity and to measure

the total charge passed through the target during a run.

16
Currents were kept low enough to insure negligible pileup
of pulses in the detectors and overall system dead time
under four per cent. First order corrections were made to
the integrated charge for the observed dead time. Periodic
checks of the current integrator during the course of this
study showed the absolute calibration to remain within one

per cent.
3.2 Particle Detection and Identification

Charged particles scattered from the target were detected
in a counter telescope consisting of three commercial silicon
surface barrier detectors purchased from Ortec. All were
totally depleted and transmission mounted. The front counter,
260 microns thick, provided a "delta—E" signal. The remain-
ing two counters were operated in parallel to provide the
“000 microns of silicon needed to stop the most energetic
deuterons of interest and thus produce an "E" signal.

The entire telescope was cooled by methanol circulated around
a dry ice bath at -780 C.

A tantalum collimator, between 0.050 and 0.090 inch
thick depending on beam energy, was mounted in front of the
telescope to define the solid angle and angular acceptance.
Typical solid angles were on the order of 10-” steradian
with an angular acceptance of 0.8 degree. The angle of the
detector telescope to the beam was read from a remote digital

voltmeter readout which, if care was taken to eliminate

17
backlash, could be set reproducibly within 0.2 degrees.
The zero degree line was optically determined before each
run and checked by measuring spectra on both sides of the
chamber. These were always found to agree within the limits
of angular uncertainty.

Figure 3.2 is a schematic diagram of the detector elec-
tronics. The 5 M resistor to ground provided a signal
proportional to the sum of the charges deposited in all
detectors, and thus proportional to the total energy of a
particle stopped in the telescope. This method of "charge
summing" eliminates the necessity of closely matching
amplifier gains before forming a sum pulse.

The control logic for the experiment is shown in Figure
3.3. The E and delta-E signals are fed into Ortec ““0
filter amplifiers where they are double delay line clipped
and passed on to Ortec 220 single channel analyzers. A
coincidence pulse is formed in an Ortec “09 linear gate and
slow coincidence to control input to both analog to digital
converters. To keep electronic noise to a minimum, the sum
pulse is amplified in a Tennelec TC200, where it also is
double delay line clipped. To synchronize with the control
logic, the amplified sum pulse is delayed by an Ortec “27
amplifier before passing through the Ortec “26 linear gate.

Particle identification could be performed by plotting
delta-E against the total energy pulse. Particles with
different Z2A fall on separate hyperbolic bands. It is

easier to distinguish between these bands, however, if E

 

TO DATA R00“

18

 

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20
times delta-E is the ordinate, since the bands thus formed
are essentially straight. Thus a field effect transistor
pulse multiplier (Mi 63) is used to form this product,
which is input to an Ortec ““0 and passed through the “09
linear gate.

The gated mass and energy pulses were fed into a pair of
Northern Scientific 629 13-bit analog to digital converters
under control of the cyclotron laboratory's Scientific Data
System Sigma 7 computer. Most of the data in this study
was processed under the on—line acquisition code TOOTSIE,
written by D. Bayer (Ba 69). Final runs were made using
task GEORGE, written by the author (P1 69) to perform a
subset of the TOOTSIE options, operating under the realtime
timesharing system JANUS (Ko 68). The latter code is
described here.

GEORGE operates in two principal modes. The first,
SETUP mode, acquires two parameters and displays them as a
32xl28 two-dimensional array. Gate lines can be defined
by accepting a series of points on the 2D raster and request-
ing that a polynomial of given order be fit through them.
Each pair of lines defines a band, inside which all events
are treated as being the same particle. Once one or more
bands are defined, RUN mode is entered to acquire data as
singles spectra of different particle types. Figure 3.“
is a flow chart of the routine which processes an event in
RUN mode.

Sample values of the gate lines are computed for every

21

NTERRU’T

'ANALYSIS COMPLETE' OCCURS

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22
16 channels along the abscissa. Thus the Sigma 7 computer
is made to act as a large raster of SCA's, each covering
a narrow strip of energy. This not only greatly relaxes
the requirements on the mass identification parameter, it
also permits the simultaneous acquisition of more than
one particle Spectrum without elaborate routing hardware.
In these (p,d) experiments it was common practice to monitor
(p,p) and acquire (p,t) spectra at the same time, using

three 20“8-channel storage areas.
3.3 Error Analysis

Overall energy resolution was between 50 and 60 keV
for all data acquired. Table 3.2 lists the major contri-
butions to the resolution at “0 MeV. Electronic noise
was measured by injecting pulses into the system at the
preamplifier input (with the detectors cooled and connec-
ted), and measuring the analyzed peak width. Straggling
was estimated for a one mg/cm2 titanium target from tabu-

lated energy loss data (Ni 66).

Table 3.2 Contributions to energy resolution at “0 MeV.

Source Contribution
Electronic noise 35 keV
Beam resolution 25
Straggling in target 16
Finite size of collimator 13
Finite size of beam 10
Divergence of beam 8

Added in quadrature “9 keV

Experimental 55

23

Differential cross section is defined experimentally
as the number of product particles scattered into a given
solid angle divided by (l) the number of incident particles,
(2) the number of scattering centers per unit area and
(3) the size of the solid angle. The integrated charge
counts the number of protons, since each has a unit of
charge. Since a gram-molecular weight contains Avogadro's
number of particles, the measured target thickness counts

scattering centers. Combining these factors gives

- - z.
a-g ‘= 2.651on A, m. '32—'9- ‘3‘3' 3,-3- (3.1)

where n is the number of events, theta the angle of the
target foil to the beam, MW the mean molecular weight

of the foil, d the collimator to target distance, Q the
integrated charge in microCoulombs, t the foil thickness
in mg/cm2, f the fraction of foil atoms to be considered
targets, and A is the area of the collimator in the same
units as d. The resulting cross section is in microbarns

30

per steradian (10— cm2).

Table 3.3 summarizes the sources of error in deter-
mining cross—sections, which combine to give 5.1 per cent.
In addition to this systematic error, each measurement
has its own statistical error due to fluctuations in
counting. Errors in counting are Poisson distributed
and so, for a large enough number of counts, A, can be
approximated by a normal distribution whose standard

1/2

deviation is A . In general a background, B, is estimated

for each deuteron or triton group and assigned an error

2“
equal to its square root. Adding errors in quadrature
gives for a net area A-B the statistical error (A+B)l/2.
Error bars on the data presented in this thesis reflect
only this statistical error.

Table 3.3 Contributions to uncertainty in cross-sections.
Source Contribution
Target thickness
Area of collimator
Target angle (sine)

Integrated charge
Collimator to target distance (squared)

per cent

OOUTUTO

\J'l I—‘l—‘I—‘NJ‘:

Added in quadrature .l per cent

Calibration points for the energy scale in each run
were obtained by (p,d) scattering from 0.00025 inch thick
mylar, a plastic containing only carbon, hydrogen and
oxygen, and the (p,d) transitions to the ground states
of “5,“9Ti and to the strong 0.160 MeV state in “7T1.

All of these Q values are well established (Ma 65, Fr 66,
En 67) with assigned errors of only a few keV, and provide
calibration points bracketing the region of interest in
deuteron energy. Gains varied between 15 and 20 keV per
channel for different runs. It was found that the ADC
zero level changed extensively between spectra, necessi-
tating the use of a calibration peak in each spectrum to
define the zero offset. No explanation has been found for
this effect. Evidence also exists for gain shifts, mostly
small, between runs. These correlate strongly with

variations in the building AC line voltage.

25

Another unfortunate effect was observed arising from the
use of two "E" detectors. The inner faces of the counters
represented a dead layer approximately 160 keV thick for
four MeV deuterons. Thus those particles that stopped
in the first counter appeared more energetic, that is at
a lower excitation, than was consistent with the more
energetic calibration groups. Fortunately the major non-
linearities due to this dead layer occurred in the region
of interest only at “0 MeV beam energy, and there in a
region of excitation containing only weak transitions.

The problems in calibrating the energy scale were reflec-
ted in the root mean square deviations of the calibration
line fits and in the scatter of measured excitations for
the same state from one spectrum to another. RMS deviations
were typically eight to 16 keV for the calibrations, and
varied from seven to about 25 keV for excitations, increas-
ing with decreasing cross-section. However the presence
of small quantities of vacuum pump oil, due to backstreaming
from the scattering chamber roughing pump, provided a
fortuitous carbon calibration amidst the high-lying titanium
levels in several runs. From the jitter in the centroids
measured for these small carbon groups, it is estimated
that the systematic error in excitation energies is 0.6
per cent. Within these limits, the data in this thesis is
in agreement with several earlier studies (Ka 6“, Sh 65,

Ra 66).

Other measurements (R0 67, An 69), however, indicate

26
50 . “9 . . .
that the T1(p,d) T1 calibration is low by 1.5 per cent.
This disagreement is probably due to the erroneous identi-

50Ti

fication of carbon groups in this study, since the
foil was exposed the least to pump oil contamination. One
of the measurements (R0 67) also indicates that the energy
calibration is 0.5 per cent low for the other two (p,d)

reactions studied here. Corrected excitation energies

will be presented with the measured values in this thesis.

“. OPTICAL MODEL PARAMETERS

The distorted waves used in code JULIE describe the
relative motion of protons elastically scattered from the
target and of deuterons elastically scattered from the
residual nucleus at the appropriate energies. These waves
are calculated using an optical model, where experimental
scattering data are used to determine the parameters of the
potential. All parameters used in this thesis are taken
from the literature, or are determined by averaging over
several published sets.

There are several contributions to the optical potentials
used. All but the Coulomb term involve either a "volume"
Woods-Saxon geometry (see Figure 2.1) or a "surface" term
formed by differentiating the volume form. Let

f(r,ro,a)= l/(eXp(x) + l) (“.l)
where

x= (r — rOAl/3)/a (“.2)
Then the potential used is a sum of:
a real volume term

-V f(r,ro,a) (“.3)

an imaginary volume term

-i W f(r,roI,aI) (“.“)
an imaginary surface term
“i Wb d/dxI f(r,roI,aI) (“.5)

a Thomas type spin-orbit term

27

28

(M/mc)2 Vso (l/r) d/dr f(r,rs,as) L.s (“.6)
where m is the mass of the pion,
and a Coulomb repulsion term

ZZ'eZ/r outside RC= rCAl/3 (“.7)
and

ZZ'e2/2r (3 - (r/RC)2) inside. (“.8)
The imaginary terms account for loss of probability current
from the elastic channel due to all other processes.

Proton optical model parameters were obtained from the

30 - “0 MeV study of Fricke (Fr 67). The values obtained
were in reasonable agreement with those obtained by Satchler
at 30 MeV (Sa 67), by Greenlees and Pyle at 30 MeV (Gr 66)
and by Fricke and Satchler at “0 MeV (Fr 65). The (p,d)
study by Kashy (Ka 6“) used parameters for 9 - 22 MeV
scattering obtained by Perey (Pe 63a). One of the princi-
pal differences between the Perey and Fricke sets is that a
much faster V(E) dependence is predicted in the earlier set.

Table “.1 lists the proton parameters used in this thesis.

Table “.l Proton optical model parameters.

V* r a r W W

o C D
“6.8 MeV 1.16 fm 0.75 fm 1.25 fm 3 MeV “ MeV
roI aI Vso rs as
1.37 fm 0.63 fm 6.0“ MeV 1.06“ fm 0.738 fm

* V is given for “8Ti at 35 MeV. In general:

V: “9.9 - 0.22E + 0.“Z/Al/3 + 26.“(N-z)/A. MeV

29

Deuteron parameters used in previous (p,d) investigations
(Ka 6“, Sh 65) were taken from the 11 - 27 MeV study by
Perey and Perey (Pe 63) which included no spin-orbit term.
Most recent works include such a term, having the same geometry
as the real well, and indicate that the real well depth
should be approximately equal to the sum of typical proton
and neutron depths (Pe 66). Thus, deuteron parameters for
this thesis were estimated from (1) the 2.5 - 10 MeV set
of Wilhjelm and Hansen (Wi 68), (2) the 11.8 and 21.“ MeV
sets of Perey and Perey (Pe 66) and (3) the 3“.“ MeV set
of Newman, et a1. (Ne 67).

Figure “.1 shows the sample parameters and the choices
made. The Wilhjelm set is plotted at 2.5 MeV where it
gave the best fit. Extra weight was given here to the
20 - 35 MeV parameters, since most of the transitions
studied produced deuterons of these energies. But since
most energy dependence studies of Optical parameter fits
show no justification for varying anything but V, and since
the more widely varying parameters seldom influenced the
calculations strongly, all but V were taken to be independent
of energy. The parameters used are also given in Table “.2.

Note that an imaginary volume term is not used.

Table “.2 Deuteron optical model parameters.

a
V rO a rC WD

101 MeV 1.065 fm 0.81 fm 1.30 fm 10 MeV

r a V r a
OI I so 5 s

1.“1 fm 0.75 fm 7 MeV 1.065 fm 0.81 fm

* V is given at 25 MeV. In general:
V= 117 - 0.627Ed MeV

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Figure “.1 Deuteron optical model parameters.

31

To investigate the sensitivity of JULIE calculations to
errors in the optical parameters, test cases were computed
each with one of the parameters increased by ten per cent.
Table “.3 shows the results Of this study for an L= 3 transfer
at 25 MeV. The first three entries are for the neutron
bound-state well, which employed the same (real) geometry
used to describe elastic proton scattering. As expected,
the dominant effects are caused by the real well depths
and radii. The deuteron real parameters are surprisingly
sensitive - a one per cent increase in radius causes a “.7
per cent drop in peak cross—section (and a notable change
in the shape of the cross-section). Care must be taken,
therefore in interpreting absolute spectroscopic factors
extracted.

Table “.3 Per cent increase in peak (p,d) cross—section
for ten per cent increase in parameter.

Parameter Effect for p Effect for d
rO BSW 28 per cent -—

a BSW 8 —-

rC BSW 0 —-

V 20 —“7 per cent
rO 20 —“9

a -2 5

r 0 -3

WC 0 -—

WD —2 -12

rCI -15 -21

al -3 “‘17

Vso 0 0

rS 0 --

a 0 --

5. ENERGY DEPENDENCE OF u8Ti(p,d)u7Ti

5.1 Experimental Results

Angular distributions of u8Ti(p,d)u7Ti transitions have
been measured over an angular range of eight to 90 degrees
in the laboratory at 2“.80, 29.82, 35.15, 39.97 and “5.05
MeV incident energies. Figure 5.1 shows a typical spectrum,
aligned with the observed energy levels in “7Ti. The L-
transfer, J and T assignments for the strong transitions
are taken from earlier studies (Ka 6“, Sh 65, Ra 66, R0 67).

Note that the principal transition occurs to the 0.16
MeV first excited state and not to the ground state. The
latter is known to have spin 5/2-, and is expected to consist
mostly of three lf7/2 neutrons coupled to 5/2-. Since the
direct pickup of a lf7/2 neutron from a 0+ target cannot
excite this state, the transition can only proceed in a direct
process via lf5/2 configuration admixtures in the target and
final state wave functions. The 1f spin-orbit splitting is
known to be on the order of 5.5 MeV in this mass region (Be 6“)
so the direct reaction cross—section for this channel is
expected to be small (Be 65).

Thus the 0.16 MeV L= 3 transition is of prime importance
in the energy dependence investigation. Also included in
the study are the strong L: 1 transition at 1.5“ MeV measured
excitation and the L= 2 transfer at 1.81 MeV. The latter

32

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Figure 5.1 Deuteron speCtrum from'uaTi target and levels
observed. '

3“
is known to be an unresolved doublet (Ra 66), which includes
an L= 1 transition. This component is relatively weak, as
will be shown later, and does not strongly influence the
shape Of the cross-section.

The three strong transitions between seven and nine MeV
are to T= 5/2 analog states. One can locate such states
by subtracting the n-p mass difference from the mass of
the analogous state in “78c, then adding the Coulomb energy
required to bring an additional unit of charge into the
nucleus. Systematics of Coulomb energies in this region
have been investigated (Sh 67), and usually agree with
observed analog excitations to within 100 keV. Analog state
transitions stand out as concentrated deuteron groups against
a nearly continuous background of states of lower T.

Above these analog states, in fact up to at least 20 MeV
excitation, no significant transitions are observed. Thus
one can characterize u8Ti(p,d)u7Ti as leading principally
to a few low-lying states in each isospin spectrum. Figures
5.2 and 5.3 show the odd-L and even-L transitions described
above at the five incident energies. In addition, the strong
L= 0 transfers measured at 2.35 and 8.7“ MeV are displayed.
Since these distributions are expected to have their princi-
pal maxima at zero degrees, they are not easily compared to
calculated shapes and so play no role in the DWBA investi-
gation.

All L= 3 cross-sections were found to increase quite

 

35

 

 

 

 

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37
linearly with energy up to about 35 MeV, then become fairly
constant (see Figures 5.2 and 5.3). This is consistent
with the data of Kashy and Conlon (Ka 6“) and of Sherr,
et al., at 28 MeV (Sh 65). The shape of these cross-sections,
however, undergoes a marked change, the shape apparently
being a function of the energy of the outgoing deuteron
(compare 7.33 and 0.16 MeV L= 3 distributions). Other
L-transfer cross-sections also increase with energy, but
more uniformly. Such differences in energy dependence
must, of course, be reproduced by the DWBA calculations

to produce consistent spectroscopic factors.
5.2 Zero-Range DWBA Calculations

DWBA calculations were performed for the principal L-
transfers described above at 25, 30, 35, “0 and “5 MeV.
In all cases the neutron bound-state wave function was
computed for a well having the same geometry as the proton
elastic channel optical model. Neutron well depth was
adjusted to bind the specified single-particle orbital with
the correct spearation energy. The real well depths for the
proton and deuteron elastic channels were selected for each
case according to Tables “.1 and “.2.

In the zero—range approximation, the calculations included
a series of lower integration cutoffs from zero to seven fm.
to bracket the nuclear surface, which is at about 1.25A1/3

or “.5“ fm. Table 5.1 lists the spectroscopic factors

38
extracted for the principal L= 3 transition as a function
of cutoff radius and energy. All cutoffs except at three
fm exhibit a noticeable change with energy, usually a 30
per cent decline from 35 to “5 MeV. At each energy, S rises
abruptly between two to four fm, passes through a local
maximum near the surface and climbs steeply. The same
study performed for the high-lying T= 5/2 L: 3 transition

gave essentially identical results.

Table 5.1 Zero-range spectroscopic factors for the L= 3
transition to u7Ti(0.16 MeV).

Cutoff 25 MeV 30 MeV 35 MeV “0 MeV “5 MeV
0 fm 3.0 2.7 2.7 2.3 1.9
2 3.0 2.7 2.7 2.“ 1.9
3 3.1 3.2 3.“ 3.“ 3.0
“ 5.6 5.3 5.“ “.8 3.6
5 5.0 “.8 “.8 “.0 3.3
6 6.6 5.“ 6.“ 5.6 “.9
7 11.8 12.9 l“.l 13.7 11.7

Table 5.2 shows very similar results for L= 1, except
that no energy dependence is Observed out to three fm.
Again a plateau occurs near the nuclear surface exhibiting
energy dependence very similar to L= 3. Finally, a study
of L= 2 spectroscopic factors showed (1) no energy depen-

dence at zero cutoff, (2) a marked increase of S with energy

 

at three fm, (3) a local minimum of S near the surface and
(“) energy dependence for surface cutoffs nearly identical
to that for L= 1 and L= 2.
The ability of DWBA to reproduce the shapes of experimental

angular distributions is displayed in Figure 5.“. The

39
dashed curves are zero-range calculations with no integration
cutoffs. In all cases, such calculations predict far too
much scattering at back angles. Even fits to the principal
maximum become so poor at higher energies that comparison
with the data is moot. The solid lines represent zero-range
calculations where the integration is cutoff near “.5 fm.
These curves track the data much more closely, particularly
at back angles. While the agreement with data also deterior-

ates with increasing energy, it is better than for no cutoff.

Table 5.2 Zero-range spectroscopic factors for the L= 1
transition to u7Ti(l.5“ MeV).

Cutoff 25 MeV 30 MeV 35 MeV “0 MeV “5 MeV

0 fm 0.18 0.15 0.15 0.19 0.17
2 0.23 0.22 0.26 0.30 0.25
3 0.26 0.26 0.29 0.33 0.27
A 0.33 0.31 0.32 No.35 0.22
5 0.23 0.23 0.23 0.22 0.19
6 0.26 0.27 0.29 0.32 0.29
7 0.“9 0.51 0.60 0.72 0.69

Thus, in the zero—range approximation, DWBA calculations
with lower integration cutoffs at the nuclear surface appear
to be the most reliable. They produce shapes which best
approximate the data out of all cutoffs examined. They lead
to spectroscopic factors which are quite constant for small
changes in cutoff radius. Finally, they yield factors
whose variation with energy, while not constant, is known
for 25 - “5 MeV and is apparently independent of L-transfer
or excitation energy. This conclusion agrees with the findings

of Snelgrove (Sn 68) in the oxygen mass region.

“0

  
 
   
  
  

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Figure 5.“ Some DWBA firs to u8Ti(p,d)q7Ti for L= l (1.5“
MeV), L= 2 (1.81 MeV) and L= '3 (0.16 MeV).

“l

5.3 Finite Range DWBA Calculations

DWBA calculations were performed for the principal L-
transfers described in Section 5.1 at 25, 30, 35, “0 and
“5 MeV, using bound-state wave functions corrected for finite
range and non-locality effects. Optical model parameters
for the bound-state well and elastic channels were chosen
as in the preceding section. The range of the interaction
was taken as 1.5 fm, and the non-locality parameters were
0.85 fm for the proton and neutron and 0.5“ fm for the
deuteron.

The effect of integration cutoffs on FRNL spectroscopic
factors has been investigated for the principal L= 3 transfer.
There is no energy dependence for zero cutoff, and surface
cutoffs behave much as those described above. The predicted
shapes, however, are best for no cutoff and become much
worse with increasing cutoff radius. Similar results were
observed for the other L-transfers studied. Figure 5.“
shows the FRNL zero cutoff predictions as dot-dash lines.

At 25 — 30 MeV they are generally better than zero—range
with cutoff, but tend to be not as good at back angles for
“0 — “5 MeV data.

It is concluded, then, that FRNL corrections produce
results superior to the zero-range procedure described in
the preceding section. The physical basis for FRNL is much
better established than the computational artifice of discar-

ding part of an integration. Predicted angular distributions

“2
are of comparable quality. Finally, there is no apparent
energy dependence from 25 to “5 MeV, as Table 5.3 shows for
a variety of transitions. Note that the 0.16 MeV L= 3
measurements differ slightly from those in Table 5.3. This
illustrates the limits of repeatabliity in the curve—matching
process used to measure S. Fluctuations on the order of
five per cent are, then, clearly not significant.

Predicted angular distributions, using FRNL corrections,
deviate most from the data at back angles and at higher
energies. Both are cases of large momentum transfer, or
relatively deep penetration into the nucleus. From the
study of integration cutoffs, it is evident that the devia-
tions are caused by the nuclear interior contributing propor-
tionately too much to the cross-section. Finite range and
non—locality corrections damp this contribution (see Section

2.3), but apparently not enough.

Table 5.3 FRNL spectroscopic factors for some of the
principal transitions to ”7T1.

Ex L J 25 30 35 “0 “5
0 MeV 3 5/2 0.13 0.1“ 0.13 0.1“ 0.13
0.16 3 7/2 3.5 3.5 3.7 3.“ 3.“
1.5“ 1 3/2 0.19 0.18 0.18 0.19 0.17
1.81 2 3/2 2.1 2.2 1.9 1.8 1.9
7.33 3 7/2 0.“6 0.5“ 0.52 0.“7 0.“5

Green (Gr 67) has investigated yet another possible
correction, for the dependence of the p—n interaction on
the density of nuclear matter. As in the case of finite

range and non—locality, this correction tends to damp

 

“3
contributions from the nuclear interior. At present
calculations have been performed only for the oxygen mass
region. There the agreement of DWBA with (p,d) data is
markedly improved by using this correction (Sn 69). The
results of this thesis indicate that performing density-
dependent corrections for titanium would result in similar

improvement.

SPECTROSCOPY OF (p,d) REACTIONS

6.1 ”8

Ti(p,d)u7Ti

The general features of I48Ti(p,d)u7Ti were discussed in
Section 5.1, and are displayed in Figure 5.1. Based on the
conclusions of Chapter 5, it was decided to study the spec—
troscopy of this reaction at 35.15 MeV, using FRNL correc-
tions with no integration cutoffs in all DWBA calculations.
Table 6.1 summarizes the observed data and spectroscopic
analysis. The measured cross-sections for all observed
transitions are displayed in Figures 6.1, 6.2 and 6.3.

Measured excitation energies are given in the first column
of Table 6.1, followed by energies recalibrated to agree with

the magnetic spectrograph study, by Rosner and Pullen (Ro 67),
”5’u7’u9Ti.

are compared with the precision u6Ti(d,p)u7Ti study by

of the analog state spectra of Low-lying states
Rapaport, Sperduto and Buechner (Ra 66) and the (d,p) J-
dependence study by Lee and Schiffer (Le 67). Earlier (p,d)
investigations of u7Ti levels have been made by Kashy and
Conlon (Ka 6“) and by Sherr, et a1. (Sh 65). Finally, there

“He) studies by L'Ecuyer and St.-

are the more recent (3He,
Pierre and by Lutz and Bohn (L'E 67, Lu 68). All previously
established levels are included in the table down to the dashed
line. Below this line the level density is too high for

meaningful comparisons.

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139

35'

NVIOVUBlS/NHVG”

48

Table 6.1 Summary of results for u8Ti(p,d)u7Ti at 35.15 MeV.
MEASURED ASSIGNED PREVIOUS
Ex a Ex b a'max Gmax L J S T Ex J
MeV MeV mb/sr deg. MeV
0 0 0.14 22 3 5/2— 0.12 3/2 0 5/2—
0.16 0.16 5.1 20 3 7/2- 3.6 0.157 7/2-
1.24 1.25 0.02 flat 1.247
1.43 1.44 0.05 30-60 (5) (9/2-) 0.01 1.442
1.54 1.55 1.8 8 1 3/2- 0.15 1.545 3/2—
1.81 1.82 0.48 8 1 (3,1/2) 0.04 1.788 1/2-
1.8 16 2 3/2+ 1.9 1.816 3/2+
2.15 2.16 0.40 10 1 (3/2—) 0.03 2.157 (1,3/2)
2.24 2.26 0.16 @8 2.252
2.292
2.35 2.37 1.6 22c 0 1/2+ (0.59) 2.361 1/2+
2.58 2 60 0.22 220 0 1/2+ (0.08) 1/2+
0.25 20 3 7/2- 0.29
2.80 2.82 0.22 22 3 7/2- 0.25 (L= 3)
3.20 3.22 0.48 22 3 7/2— 0.46 7/2-
3.53 3. 5 0.24 6 1 (3/2-) 0.03 3.545 3/2-
3.68 3.70 0.05 12
3.82 3.84 0.12 18
3.90 3.92 0.10 18
4.13 4.15 0.14 8 (1) 0.02
0.14 16 (2) 0.37
4.25 4.28 0.24 @20
4.36 4.38 0.18 @16
4.46 4.48 0.2 @8
4.58 4.62 0.06 @30
4.63 4.67
4.83 4.86 0.10 @8
4.96 4.99 0.10 @8
5.12 5.16 0.09 @20
5.26 5.30 0.09 @8

Table 6.1 continued

MEASURED

Ex a Ex b d’max Omax L

MeV MeV mb/sr deg.

5.47 5.51 0.14 @20

5.55 5.59 0.01 @20

5.62 5.66 .

5.75 5.79 0.04 @12

6.05 6.09 0.06 @16

6.18 6.23 0.04 @12

6.26 6 31 0.08 @8

6.36 6.40 0.19 @8

6.57 6.61 0.11 @12

6.74 6.78

7.05 7.10 0.02 10—40

7.2 7.25 0.14 @8

7.33 7.38 0.26 28 3

7.57 7.62 0.09 @8

7.89 7.94 0.06 16

8.13 8.18 0.37 14 2

8.74 8.80 0.29 22c 0
Notes: a) Excitations measured

c) Second maximum

49

ASSIGNED PREVIOUS

J S T EX J
MeV

7/2- 0.46 5/2 7.38 7/2-

3/2+ 1.4 5/2 8.18 3/2+

1/2+ (0.80) 5/2 8.80 1/2+

b) Corrected excitations

50

Since DWBA predictions are usually compared to data at
the principal maximum, the cross—section in millibarns per
steradian and the center-of—mass angle in degrees is listed
at this point. If the distribution has no definite peak,
the largest value measured is given, followed by the angle
at which it was measured flagged with an @ Sign. An excep-
tion is any state assigned L= 0, for which the data on the
second maximum is recorded.

Several angular distributions measured were known or
suspected to arise from unresolved doublets of different L-
values. In such a case, least—squares fits were performed
of the Six possible linear combinations of two pure L= 0 - 3
distributions. From the variances of the fits, the most
likely combination was determined. The computed variances
of the coefficients, obtained from the inverted least-squares
matrix (Mo 60), indicated the validity of the fit and the
sensitivity to individual contributions. Where such a fit
has been made to the data, two lines of cross—section are
given, indicating the maximum for each contribution.

L-transfers are generally determined by comparison with
known transitions, aided in some cases by DWBA predictions.
Strong non-zero L-transfers are almost always assigned to the
ld3/2, lf7/2, or 2p3/2 shells because these lie nearest the
Fermi surface. Following the usual convention, parentheses
indicate tentative assignments. Thus, all L= 0 assignments

are given only tentative spectroscopic factors, due to the

5l

difficulty of comparing data to DWBA calculations. Isospin
assignments are always to the lower allowed T unless speci-
fically stated otherwise.

AS Table 6.1 reveals, the qualitative description of this
reaction given in Section 5.1 is quite precise - there are
two isospin spectra, each with only a few strong transitions.
A comparison with the predicted L= 3 transfers in Table 2.1
shows qualitative similarities between the data and the MBZ
Spectrum, but not enough to warrant close scrutiny. The
levels of interest are the two unusual assignments at 1.44
and 1.81 MeV. The former is a very weak transition having
a definite direct reaction character, but peaking far back in
angle. DWBA calculations for L= 5 reproduce the wide change
in peak angle observed from 25 to 45 MeV. If the L assign-
ment is correct, lh9/2 is the lowest-lying candidate, closely
corresponding to a 9/2- state predicted by MBZ (Mc 64a).
The L= 1 component of the doublet at 1.81 MeV has been assigned
spin 1/2- on the basis of J-dependence in back angle scattering
(Le 67). If this much 2pl/2 admixture is known to occur, then
weak L= l assignments cannot be assigned a definite J in this
study.

The large body of states between four and seven MeV are
too small to permit reliable assignment of L values. Many
show a direct reaction distribution, however, having a strong
dependence on scattering angle. Of these, it can only be
concluded that they represent fairly complex configurations

in terms of single particle states, and that they may account

52
for a significant fraction of the total reaction strength.
It should be emphasized that the excitation energies and
distributions reported for this region are not necessarily
for discrete states; rather they represent peaks in the
(p,d) cross-section.

46,50

6.2 Ti(p,d)u5’u9Ti

The 5

OTi(p,d)u9Ti reaction was measured at 45.05 MeV over
an angular range of eight to 60 degrees in the laboratory.
Figure 6.4 Shows a typical deuteron spectrum obtained from
the 50Ti foil, along with the levels observed. Since the
target was 23 per cent u7Ti (see Table 3.1), its strong trans-
itions are also indicated. L, J and T assignments for the
principal transitions are taken from earlier work (Ka 64,
Sh 65, Ba 67, An 69).

Besides the titanium studies mentioned in the previous
section (Ka 64, Sh 65, L'E 67, R0 67, Lu 68), other inves-

tigations of the levels of “9

Ti include the precision (d,p)
work of Barnes, et al., at 6.2 MeV (Ba 67) and the consistent
level scheme developed by Anderson, et al. (An 69). The
measured angular distributions are presented in Figures 6.5
and 6.6, and results of the spectroscopic study are summarized
in Table 6.2. The same general remarks apply to this table

as were made in the preceding section concerning Table 6.1.

All known levels (An 69) are presented, down to the dashed

line, below which only states corresponding to observed

53

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

"Ti 4971 I21
ao'l'i(|:>,d)“9Ti I 3/ (1»
Ep=45.05 MeV 2——— 3/ ”>1 ‘
o”_“\v a»
9925.2 DEG _— <
- ————— I0‘
n>)o v
‘7’) 2 3’ 3 7’ ‘1'),
I400- - nw)3 w 8‘
m d
u:
an
IE
I) 5 .
Z A
d A— 6
g "——-—-_A 5.
< A. “.14 .
I: 3 ‘w ___—A
" 3H wuv_____
o v __
I650- - o /—\ I/ '
2 3/
' y 3/__‘~w 2.
(|-——ll)
I 3/
3————-w ‘
3”—‘\s/
A__ A1 71 ' .
r L, J L. J,
.. , i 5‘
0 ' 500 . I000

COUNTS PER CHANNEL

Figure 6.4 Deuteron Spectrum from 50Ti'target and levels
observed.

54

 

muumouo so m

0' O

1 1 {ll

.
. A
.
A
A A
3.»
A

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n50.

mummomo Sam
on ow o oo 1 ow 1 Jo low ow m
1 . . 1 .
AAA. . . AA
A >331. A A
OAAA* 0‘0 AA* A . A
¢¢ * A pp
A UT
AAA A 095 AA A I"
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80. AAA 2..» AAA 1 . 1... A .

Table 6.2 Summary of results for

EX a

MeV

O

1.

1

FJH
ocamnouaauwcoann~r<—qunfixnunz1::n: QMNLU
O O O O O O

36

.55

.74
.23

.46

MEASURED
Ex b crmax emax
MeV mb/sr deg.

0 9.0 8
1.38 1.3 8
1.58 0.48 8
0.04
1.77 0.12 8
2.27 1.2 8
2.50 2.5 20c
1.3 8
2 66 2.6 10
3.17 0.44 8
3.42 0.20 8
3.84 0.17 8
0.05 20c
4.08 0.16 18
4.56 0.29 18
4.78 0.08 @15
4.97 0.40 12
5.28 0.10 @12
5.67 0.18 8
5.82 0.18 @12
6.03 0.27 8
7.45 0.14 @12
7.61 0.08 @12
7.81 0.11 20
8.24 0.14 @20
8.33 0.10 @20
8.62 0 O7 20
8.75 0 26 22
8.89 0 26 12
9.66 0 15 12
9.95 0 21 8
10.1 0 08 @26
10.4 0.07 @26
10.9 0 47 22c
11.1 0.51 16
11.2 0 12 @8
11.7 0.34 @8

FJHFJ
14440

Notes:

a) Excitations measured
c) Second maximum

56

50Ti(p,d)u9Ti at 45.05 MeV.
ASSIGNED PREVIOUS
L J S T EX J
MeV
3 7/2- 3.6 5/2 0 7/2-
1 3/2- 0.19 1.382 3/2-
1.542
1 3/2_ 0.07 1.586 (3/2—)
1.622
1.724 1/2—
(3) (0.05) 1.762
3 7/2- 0.58 2.261 (7/2—)
2.472
0 1/2+ (1.5) 2.503 1/2+
2.505
(3,2) (0.6,1.2) 2.516 7.5/2—
2.557

2 3/2+ 2.4 2.665 3/2+

1 (1,3/2) 0.08 3.176 1/2—

1 (1,3/2) 0.04 3.430 (1,3/2)
(3) (7/2-) 0.10 3.847 (7.5/2)
(0) <1/2+)(0.03)

3 7/2- 0.23 7/2 8.75 7/2-

2 (5/2+) 0.48

O
2

(1)

1/2+ (0.77) 7/2 10.99 3/2+

3/2+

1.7 7/2 11.12 1/2+

1.0 (7/2)

b) Corrected excitations

57
(p,d) transitions are reported (An 69, R0 67).

Comparing this data to “8Ti(p,d)u7Ti, one is again struck
by the selectivity of the (p,d) reaction. There is the
strong L= 3 transition, this time to the ground state, a
few strong low-lying transitions, a wide gap and then the
analog states. Again the similarity to the MBZ predictions
of table 2.1 is only qualitative.

Of the states previously assigned spin 1/2-, the one at
1.724 MeV is not seen and that at 3.176 apparently only weakly.
For the doublet at 2.46 MeV measured excitation, it was not
possible to distinguish between an L= 0 + 2 and an L= O + 3
assignment. Due to the similarity of L= 2 and L= 3 angular
distributions at this energy, several assignments are uncer-
tain.

A strong L: 2 transition was observed at 8.74 MeV which is
not a candidate for being an analog state. It is believed
that this may be a 1d5/2 hole state, since it lies at about
the excitation predicted by the simple shell model (Figure
2.1). Such a state would be much less likely to be spread
in this nucleus, since promoting a neutron from the core closes
the outer neutron shell and severely restricts the possible
couplings. Finally, a fairly strong transition was measured
at 11.5 MeV which has been tentatively assigned L= 1 and may
be an analog state.

46

The Ti(p,d)u5Ti reaction was measured over an angular

range of eight to 70 degrees in the laboratory at an incident

58
energy of 34.75 MeV. Figure 6.7 shows a typical spectrum,
with the observed levels and principal contaminants. L, J
and T are given for the strongest transitions.

Since uuTi is not stable, (d,p) stripping information is
not available. The principal transitions have been reported
(Ka 64, L'E 67), and the excitations established for the
analog states (R0 67). A low—lying triplet (0 (7/2-), 37
(3/2—), 40 keV) has been reported by Jett, Jones and Ristinen
(Je 68), for which some evidence has been found in this thesis,
but other levels reported at 0.744 and 1.227 MeV are not
observed. A level scheme based on 26 MeV (p,d) data has also
been reported by Jones, Johnson and Jett (Jo 68), which gives
excitation energies as much as seven per cent different than
those reported here. The assigned L-transfers for the first
few states agree, however.

Measured angular distributions are presented in Figures
6.8 and 6.9, and the findings are summarized in Table 6.3.
The reaction strength is more uniformly distributed than for
the other isotopes studied, but bears the same qualitative
features. Only ten states have cross—sections greater than
0.15 mb/sr, four of which are assigned T= 3/2. Thus the (p,d)
reaction is highly selective for all three targets.

The lowest-lying transition has been fit with L= 3 + l
distributions, which is consistent with a low=lying (7/2,
3/2, 5/2) triplet (Je 68). Likewise the lowest analog trans-
ition, clearly a doublet in Figure 6.7, is best fit with L=

3 + 2, as expected (R0 67). An L= 1 transition observed at

59

...?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

47-“ 45'“ 8 A.
45Ti (p,d)45Ti
E. = 34.75Mev ”>1 0 V _ -
9L=25.O DEG (m 2 3/
III: 6~
('0) 3, 7/ o I/(T>)
|=3/(T>) ‘
(T>) 2+3 314-71
5 2: %
CD = 2x -
12 Lu
3 =
Z A.
3 7/ 2-
E: 3 w J 3'
z .—
2 3+: 7l+3/
< 0 l/ A.
I: o l/
0 2+! 3/+3/ 2 3/
ISOO ' MA. 7/ OJ
Ln J
3 w
3 5/
L J
>- 5°Ti (p,d)”Ti "
O . 500 JOOO

COUNTS PER CHANNEL

Figure 6.7 Deuteron spectrum from '
observed.

u6Ti target

and levels

60

.>82 Am.m : o uxm .>mz

mm.:m pm HBmonAQVHB mo mCOHpSQthmHo pmadwcw woLSmmmz m.m opzmflm

 

 

 

 

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61

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mmmmuuoEom
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Table 6.3 Summary of results for

EX a
MeV

O

.32

.54
.77

:24

.48
.53
.80
.86
.90

.04

.16
.28
.37
.49
.56
.64
.78
.94
.01
.25

NFJFHAFJO

.96

.20
.28
.48
.70
.93
.07
.16
.30
.75
.98

O\O\O\C\O\\DU‘IU‘IU‘IU‘I\DJ: :zttwwwwwwww WNNNNNN

Notes:

.72.

MEASURED
Ex b a’max Gmax
MeV mb/sr deg.
O 1.4 22
0.30 12
0.32 1.0 14
1.35 0.01 flat
1.56 0.74 25c
1.79 0.35 12
1.95 0.07 22
2.26 0.07 22
0.15 12
2.50 0.04 22c
2.55 0.09 @8
2.82 0.03 @26
2.88 0.04 10
2.93 0.02 @16
2.98 0.05 18
3.07 0.08 22c
0.09 14
3.19 0.08 18
3.31 0.03 @12
3.40 0.11 20
3.52 0.04 20
3.59 0.03 @8
3.67 0.01 @16
3.82 0.06 22c
3.98 0.03 @8
4.05 0.01 flat
4.29 0.02 @20
4.60 0.02 @20
4.76 0.42 22
0.18 14
5.00 0.03 20
5.16 0.19 12
5.25 0.03 @8
5.33 0.03 @8
5.53 0.02 @16
5.75 0.30 22c
5.98 0.03 @20
6.12 0.06 @20
6.21 0.04 28
6.36 0.04 12
6.81 0.07 8
7.04 0.04 20

a) Excitations measured
c) Second maximum

62

46

ASSIGNED

J T
7/2— 1.2 1/2
3/2- 0.33

3/2+ 1.0

1/2+ (0.33)
(3/2—) 0.05
(7/2-) 0 07
(1/2+)(0.02)
(1/2+)(0.05)
(3/2+) 0.15
(7/2-) 0.14

1/2+ (0.04)
7/2— 0.62 3/2
3/2+ 0.50 3/2
3/2- 0.04 (3/2)
1/2+ (0.40) 3/2

PREVIOUS
Ex J
MeV

0 7/2-
0.037 (3/2-)

3/2+

l/2+

3/2-

7/2-

3/2+

4.74 7/2-

4.81 3/2+

5.75 l/2+

Ti(p,d)uSTi at 34.78 MeV.

b) Corrected excitations

63
5.12 MeV is a good candidate for a 3/2- analog state. No
transitions of any significant strength were observed above

six MeV.

6.3 Sum Rules and Q Dependence

Having made a spectros00pic analysis of the (p,d) data
obtained, one is now in a position to test the sum rules put
forth in Section 2.2 (or, contrariwise, to use the sum rules
to test the spectroscopic factors extracted). Table 6.4
lists the summed spectroscopic factors for the 281/2’ ld3/2,

1f and outer shells for each of the three reactions studied.

7/2
The sums are further divided into lower- and upper-T trans-
itions, and ratios of experimental results to the predictions
of Table 2.2 are obtained. Although it is of questionable
validity, the analysis is carried through for L= 0.

It should be emphasized from the start of this discussion
that the absolute normalization of these sums is open to some
question. The results given in Chapter 4 show that the peak
DWBA cross—section is quite sensitive to small changes in the
optical model parameters used to describe the incident and
exit channels and the bound-state wave function. Systematic
errors on the order of 20 to 30 per cent are not unexpected.
Moreover, an unknown amount of strength can go into many small

levels, and so be lost to the sums.

Even granting these limitations, many of the results in

6M
Table 6.“ Comparison of summed spectroscopic factors to

predictions.

Tgt. Shell TOTAL LOWER-T UPPER—T

Exp. Th. Exp. Th. Ex/Th Exp. Th. Ex/Th

“6T1 251/2 0 8H 2 0 an 1 33 0.33 0 no 0 67 0 60
ld3/2 l 7 4 l 2 2 67 O.U3 O 50 l 33 O 38
lf7/2 2 O u l “l 3 33 0.42 O 62 O 67 O 93
outer O.“ O 0.38 0.04

”8T1 251 1.5 2 0 67 1 60 0.02 0 80 0 00 2 0

/2
ld3/2 3.7 u 2 3 3.20 0.72 l.“ 0.80 1.8
lf7/2 5.1 6 U 6 5 60 0.82 0 U6 0 “O l 2
outer O.“ O 0.90 O
50

Ti 281/2 2.2 2 l 5 l 71 0.88 O 77 O 29 2 7
103/2 0.1 u 2 u 3.03 0.70 1.7 0.57 3.0
lf7/2 5.2 8 5 O 7 71 O 65 0.23 O 29 O 79
outer l.“ O 0.38 l O

65
Table 6.“ are absurd. If the sums are to be believed, fully

half the expected number of particles are missing from the

s—d shell in u6Ti, while essentially all are accounted for

in the heavier two nuclei. The likely explanation here is
that much more of the (p,d) strength is unaccounted for,
proportionately, since the cross—sections are observed to be

more uniform and since no (d,p) spectroscopy is available to

help identify weaker transitions. Doubling all sums for u6Ti

would also bring the lf sum more into line.

7/2

One can argue that the 0.9 lf7/2
48

Ti is within reason, but it is difficult to explain away

neutrons missing from

nearly three from 50Ti, particularly since it is expected
to be a closed—shell nucleus. Again the sums insist this
is not so, indicating that 1.4 of the missing neutrons are
in the 2p shell. Yet the lighter nuclei promote only 0.H
neutrons to this shell. Clearly something is wrong.

The sums of transitions to upper-T states are uniformly
larger than eXpected. An extreme case is the high-lying
L= 1 transfer to ugTi which has S= 1 even though the lower-T
sum is only 0.38. The common feature of these analog states
that could lead to such a discrepancy is their large excitation
energies, i.e. they have Q values significantly more negative
than the lower-T states. Evidently DWBA calculations do not
predict the proper dependence of cross-section on Q.

This is not too surprising. The prescription for obtaining

the single particle bound—state wave function, or "form

66
factor", is to pick the well depth that binds the sz orbital
with the right separation energy. But the data shows that
pickup from the same orbital can lead to levels seven or
eight MeV apart. To bind the particle with seven MeV greater
separation energy requires a well ten MeV deeper. The particle
is bound tighter, there is consequently less overlap in the
transition amplitude integral, the predicted cross—section
is too small and so the resulting spectroscopic factor is
too large. While the simple shell model level scheme of Sec-
tion 2.1 is known to be widely split by Coulomb or other
residual interactions, each level is treated as a pure unper-
turbed single-particle level at the proper energy for the
sake of the calculation.

One approach to this problem is to just ignore the change
in binding energy. Sherr, et al., in their study of isobaric
analog states in the titanium-nickel mass region (Sh 65) were
able to improve agreement with predictions by using an "effec-
tive binding" procedure. In this scheme, the same bound-state
wave function is used for all Q values, effectively ignoring
the energy shifts caused by residual interactions. Unfortu-
nately, this produces an incorrect exponential falloff outside
the nucleus for the form factor, which the "separation energy"
prescription is designed to produce correctly.

The use of a neutron bound—state wave function to represent
the nuclear overlap is strictly correct only for pickup of
a single particle outside a closed core (Fr 68). Thus, while

the form of the overlap in the nuclear interior is open to

67

question, the exponential falloff outside is rather closely
defined by the separation energy. Pinkston and Satchler,
in an investigation of the Q-dependence problem (Pi 65),
conclude that other features of the bound-state well must
be changed, besides the depth, as a function of Q and that
the effective binding procedure is essentially wrong.

Another manifestiation of the Q—dependence problem is the
“8Ti(p,d)u7Ti(G.S.) transition, which proceeds by a small

lf admixture. A well nearly ten MeV deeper than for the

5/2
0.16 MeV state is required to give the proper separation
energy. The predicted spectroscopic factor is almost certainly
too high. Prakash (Fr 68) has ameliorated this problem in
(d,p) stripping by introducing "pseudopotentials", due to
the presence of interacting extra-core nucleons, into the
bound-state wave equation. Likewise Rost (Ro 67a) has devel-
oped a coupled-channels method for computing more realistic
bound-state wave functions, for use in L= 3 transitions to
analog states. The general problem, however, of properly
reproducing Q-dependence in DWBA calculations is still a
topic for discussion.

Thus, the results presented in Table 6.“ are of only
limited reliability. Because in many cases spectroscopic
factors are summed over a wide range of Q values, any syste-
matic agreement with predictions can only be considered
fortuitous. One can conclude, however, from the systematics
of the (p,d) reaction over the titanium isotopes presented
here, that the sum rule predictions presented in Chapter 2

are in good qualitative agreement with the data.

7. SPECTROSCOPY OF (p,t) REACTIONS

An investigation of (p,t) reactions in the titanium
isotopes, paralleling the (p,d) studies, was conducted to
gain additional information on isobaric analog states. For
a target with non-zero T, three different final state T
values can be reached via (p,t), as opposed to two for (p,d).
Garvey and Bayman have suggested (Ga 6“) that many (p,t)
transitions to these highest—T states, having T= TZ + 2,
should be enhanced over lower-T transitions of the same Q.

Using the MBZ wave functions and Coulomb systematics,
they predicted the strengths and approximate Q values for
(p,t) reactions on the even titanium isotopes and other
targets. Shortly thereafter, Garvey, Cerny and Pehl (Ga 6“a)
published preliminary findings supporting the predictions,
including a measurement of the T= 2 state in uuTi. Conse-
quently it was decided to repeat this measurement and extend

“6’u8Ti. Table 7.1 lists

the search for TZ + 2 states to
the predicted excitations and strengths, given in terms of
the ground state strength.

Figure 7.1 Shows a typical spectrum of u6Ti(p,t)uuTi at
39.2“ MeV, indicating the levels observed and contaminants.
Angular distributions for these levels are shown in Figure
'7.2 and summarized in Table 7.2. The few spin assignments
made are based on comparison with angular distributions of
u8Ti(p,t)u6Ti measured at the same

68

known levels (Ho 68) in

69

Table 7.1 Predictions and results of T= TZ

Nucleus

uuTi

“6Ti

“8T1

Table 7.2 Summary of results for

Ex (MeV)

O
1.07
2.87
3.17
3.36
3.79
“.01
“.79
5.05
5.31
6.03
6.56
6.90
7.61
7.88
8.31
8.68
9.31

Note:

Ex predicted

9.8 MeV
l“.02
16.8

6;,(mb/sr)
0.50
.05
.05
.01“
.008
.02
.0“
.015
.01
.016
.01
.03
.03
.03
.01
.0“
.0“
.05a

OOOOOOOOOOOOOOOOO

a) Second maximum

S/S(G.S.)

0.11
0.06

0.0“

fiz£deg.)

22
16
16
21
25
16
15
20
25
16
26
36
l6
16
26
16
16

25

+ 2 investigation.

Ex observed

9.31 MeV

T

0

J
0+
2+

(3-)

(14+)

0+

<r7a'(G.S.)

0.12
40.03

(0.03

“ Ti(p,t)uuTi at 39.2“ MeV.

7O

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

'm m
‘ C(p," C ‘4 . l0q
-9.3l To: 0’
2.. ___. 8‘
:000« ‘ .
'60(p,s)"‘o _-
5
m. -.- 46 . 5..
g i Tu(p.t)44Ti >
z E Ep=39.24MeV — g .
_J E‘ _ at
g , 9,-250056 m
g i=..r — 4-
I .-
o b _ ..
azoo *— _—
: =
L. _
.. __ 2-
k— 2’
———- 2’ ..
. 3'
'. 0°
- ~ 2120 0* ,__
4’ J'
l500 2'
Tu y
o , so ' IOO

COUNTS PER CHANNEL

Figure.7.l Triton spectrum from u6Ti target and levels
observed.

71

.>62 :m.mm pm

 

_

09*

3.0 o ¢

mo mCOHpsoflppmHU awasmcm Uthmmoz maw mhsmfim

Hanzfip.avfie©:
- mmwmowo E... m
o... J 01.. 1 To oo 3 on 0'

 

 

 

Jo o
z . * _ _ .
I _ . _
no.0 * a o
3» . . . .
. * o o u

p

u C
n o
" 2'
o
u.-
*
..-
C.-
‘ - *
a.
%
I
'0
00/
\.
up
—

.0

d
O

WALL“ W
‘ NVIovuaismavefi

.-
'h
—.—

**
u—Q—

72
energy. Clearly, the 0+ level at 9.31 MeV is a prime candidate

for being the analog to the ground state of uuCa. The T: l

analog to ““80 is predicted to lie near 6.5 MeV (Sh 67),
but there is no reason to expect it to be enhanced in (p,t)
nor is there a nearby level definitely having the required
2+ spin. No assignment is made.

It was found that the published mass of uuTi (Mc 65) is
too great by about 120 keV. An unusually good determination
of the (p,t) Q value was possible in this study because the

uuTi ground state is bracketed by states of well-established

Q value in “6Ti (Ho 68). Thus the u6Ti(p,t)uuTi ground
state Q value was found to be -1“.2“6(0.011) MeV.

Studies of the other titanium isotopes failed to locate
the TZ + 2 analogs. The results of this search are listed
in Table 7.1 alongside the predictions. Only the ratio of
experimental cross-sections is given, rather than a ratio of
spectrOSCOpic factors, because of the many questions surroun-
ding DWBA calculations for two—nucleon transfers (Pa 69).

As can be seen from the table, however, the experimental
ratio of cross-sections for uuTi is very similar to the
predicted ratio of strengths. The upper bounds established
for the other two reactions indicate that the levels Should
have been observed if the predictions are at all correct.

A possible explanation is that these levels are split, since
they represent configurations much more complex than uuTi.

Even Splitting into two approximately equal transitions

could render them undetectable in this study.

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An

Au

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Ba

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Be

Be

Be

B1

Bu

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En

Fr

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6“

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62

67

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6“

6“a

65

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63

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Ga

Ga

Gr

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Ho

Je

Jo

Ka

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67

6“

6“a

66

67
68

68

68

6“

68

L'E 67

Le

Lu

Ma

Ma

M0

M0 6“a

67

68

65

67

6“

7“

M.P. Fricke, E.E. Gross, B.J. Morton and A. Zucker,
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G.T. Garvey and B.F. Bayman, Argonne National
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Mi

Mo

Ne

Pa

Pe

Pe

Pe

Pe

Pi

P1

Pr

Ra

R0

R0
Sa
Sa

Sa

Sh

Sh

63

60

67

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63

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1111'

111

175 47

1111111111

3
O
3
A 9
M IIIZ

11111111