-~--~--~-
-~--
Tff~'.
0C,
AN INVESTIGATION OF He 3 -INDUCED REACTIONS ON Ca4 0 AT 12 MeV
by
DOUGLAS MURRAY SHEPPARD
B.Sc., McMaster University
(1960)
SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September, 1964
Signature of Author
Department of Physics, September 1964
Certified By
Thesis Supervisor
Accepted by
"i
Chairmani
partmental Committee on Graduate Students
I
AN INVESTIGATION OF He 3-INDUCED REACTIONS ON Ca4
0
At 12 MeV
by
DOUGLAS M. SHEPPARD
Submitted to the Department of Physics in September, 1964 in Partial
Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
ABSTRACT
Angular distributions and absolute differential cross sections have
been obtained for the reactions Ca 4 0 (He3 ,He3 )Ca 4 0 , Ca4 0 (He3 ,d)Sc 4 1
Ca 4 0 (He3 ,a)Ca 39 induced by 12 MeV He 3 ions.
The energy levels and the
angular distributions for several excited states in Sc4 1 and Ca 3 9 are
given.
The ground state Q-values for the reactions Ca4 0 (He3 ,d)Sc 41 and
Ca 4 0 (He3 ,a)Ca 3 9 are -4.406±0.010 MeV and 4.939±0.010 MeV respectively.
Excited states in Sc 4 1 have been found at 1.718 MeV and 2.415 MeV and at
2.471, 2.787, 3.632, 3.812, 3.871, 3.939, 4.016, 5.124, 6.149 MeV in Ca 39 .
An attempt has been made to interpret single nucleon transfer reactions
induced by He 3 ions in the framework of the distorted wave Born approximation calculation (code JULIE, unpublished) of Bassel, Drisko and
Satchler; the values of the nucleon angular momenta transfer which produce
the "best" agreement with the experimental differential cross section
have been determined.
The 12 MeV He 3 beam was obtained from the acceleration of doubly
ionized He 3+ + in a radio-frequency source in the terminal of the MIT-ONR
electrostatic generator.
It was possible to obtain 10 to 15 nanoamperes
of incident He 3+ + ions with peak currents to 26 nanoamperes.
Thesis Supervisor:
Harald A. Enge
Title:
Professor of Physics
_L~
ACKNOWLEDGMENTS
The author wishes to express his gratitude to Professor Harald A.
Enge for his continual advice and encouragement, to Professors A.M.
Bernstein, R.H. Lemmer, and N.S. Wall for many helpful discussions.
He owes many thanks to the laboratory and staff of the High Voltage
Laboratory for their assistance during the completion of the thesis.
Partic-
ular thanks must go to Dr. W.H. Moore, whose knowledge of the "pulse"
of the generator proved most valuable; to Mr. M.K. Salomaa and Mr. A. Vaudo
for their technical initiative and assistance; to everyone who took a shift
at the control panel during the experiment; to the scanning staff, especially
Mrs. Barbara Saccone and Miss Naomi Elba Rosso for their careful plate
scanning.
Thanks are given to Mrs. Nancy Spencer for her assistance in
the final preparation of the manuscript.
TABLE OF CONTENTS
Page
I.
II.
INTRODUCTION . . . . . . . . . .
IV.
9
. . . . . . . . . . . . . . . . . . . .
12
The MIT-ONR Electrostatic Generator and
Analyzing Magnet . . . . . . . . . . . . . .
12
EXPERIMENTAL PROCEDURE . . . . . . . . . . . . . .
22
1.
Target Preparation . . . . . . . . . . . . .
22
2.
Determination of the Incident Energy . .
23
3.
4.
5.
Determination of the Q-values . . . . . . .
Data Reduction and Error Analysis . . . . .
The Experimental Procedure . . . . . . . . .
24
24
26
. . . . .
29
APPARATUS
1.
III.
. . . . . . . . .
EXPERIMENTAL RESULTS AND DATA REDUCTION
1.
The Ca40(He
3
..
,He 3 )Ca 4 0 Reaction . . . . . . .
29
Optical Model Potential Parameters for
2.
He 3 at 12 MeV . . . . . . . . . . . .
The Analysis of Nucleon Transfer Reactions .
a.
b.
3.
The Ca 4 0 (He3 ,d)Sc 4 1 Reaction . . . . . . . .
. . . . . . . . . . . .
39
44
46
46
a.
Previous Work
b.
c.
The Ca4 0 (He 3 ,d)Sc 4 1 Reaction .....
The Assignment of k and the
p
Calculation of Relative Spectroscopic
..............
Factors . . . ..
Shell Model Implications and
. . . . . . . . . . .
Interpretations
47
The Ca40(He 3 ,a)Ca 3 9 Reaction . . . . . . . .
60
d.
4.
The General Nucleon Transfer
Reaction . . . . . . . . . . . . . . .
The Single Nucleon Transfer
Reaction at a Closed-Shell Nucleus . .
29
39
52
55
a.
Previous Work
. . . . . . . . . . . .
60
b.
The Ca40(He 3 ,a)Ca 3 9 Reaction . . . . .
60
L_
Page
c.
d.
V.
The Assignment of kn and the
Calculation of Relative Spectroscopic
Factors . . . . ..............
.
Shell Model Implications and
Interpretations . . . . . . . . . . .
THE VALIDITY OF THE DWBA METHOD IN He 3-INDUCED
REACTIONS . . . . . . . . . . . . . . . . . . . .
64
72
75
a.
The Assumptions in the DWBA Method . .
75
b.
The Interest in Further He3
Experiments . . . . . . . . . . . . .
83
APPENDIX I . . . . . . . . . . . . . . . . . . . . . . . .
85
The Operation of the MIT-ONR Electrostatic
Generator Using He 3
APPENDIX II
Ions . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
85
89
Suggestions for Improvements in the Experiment . .
89
APPENDIX III . . . . . . . . . . . . . . . . . . . . . . .
91
Over He3+ . . . . . . . . . . .
91
REFERENCES . . . . . . . . . . . . . . . . . . . .
92
. . . . . . . . . . . . . . . . . . . .
95
Advantage of He
VI.
BIOGRAPHICAL NOTE
+
r
LIST OF FIGURES
Page
Figure 1.
MIT-ONR Generator . . . . . . . . . . . . . .
13
Figure 2.
Generator, Analyzing Magnet, and MultipleGap Spectrograph . . . . . .........
14
Figure 3.
Multiple-Gap Spectrograph: Side View
. . . .
15
Figure 4.
Multiple-Gap Spectrograph: Top View ......
17
Figure 5.
Multiple-Gap Spectrograph Slit System: Top
View . . . . . . . . . . . . . . . . . . . .
19
Multiple-Gap Spectrograph Solid Angle
Variation . . . . . . . . . . . . . . . . . .
20
Multiple-Gap Spectrograph Slit System: Side
View . . . . . . . . . . . . . . . . . . . .
21
Ca4 O(He3,He3)Ca 4 O Elastic Angular
Distribution . . . . . . . . . . . . . . . .
34
Sensitivity of Optical Model Calculation to
Variation in Optical Model Parameters . . . .
35
Figure 10.
Typical Deuteron Spectrum . . . . . . . . . .
48
Figure 11.
Deuteron Angular Distribution, Ground State .
49
Figure 12.
Deuteron Angular Distribution, Ex
Figure 13.
Deuteron Angular Distribution, Ex = 2.415 MeV
51
Figure 14.
A = 41 Mirror Nuclear States
57
Figure 15.
Summed Alpha Particle Spectrum
Figure 16.
Energy Levels of A = 39 Mirror Nuclei . . . .
63
Figure 17.
Alpha Particle Angular Distribution,
Ground State . . . . . . .
. ... . . . . .
65
Figure 18.
Alpha Particle Angular Distribution,
.................
......
.
Ex = 2.471 .
66
Figure 6.
Figure 7.
Figure 8.
Figure 9.
=
1.718 MeV
........
. . . . . . .
50
61
_~ __
Page
Figure 19.
Alpha Particle Angular Distribution,
E = 2.787 MeV . . . . . . . . . . .. ..
..
S
67
Alpha Particle Angular Distribution,
E = 5.124 MeV . . . . . . . . . . . . . . .
x
S
68
x
Figure 20.
Figure 21.
Alpha Particle Angular Distribution,
E = 6.149 MeV . . . . .
. ...............
69
x
Figure 22.
Figure 23.
Nuclear Cutoff Sensitivity, Ca4 0 (He3 ,d)Sc 4 1
Ground State . . . . . . . . . . . . . . . .
Figure 25.
77
Nuclear Cutoff Sensitivity, Ca 4 0 (He3 ,d)Sc 4 1
E
Figure 24.
S
= 1.718 MeV .
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
Nuclear Cutoff Sensitivity, Ca 4 0 ( He 3 ,, t)Ca 3 9 Ground State . . . . . . . . . . . . . . . .
79
kn-Value
Sensitivity, Ca4 0 (He3 ,a)Ca 39 ,
n
Q = 2.468 MeV . . . . . . . . . . ..
.....
Figure 26.
Q-Value Sensitivity, Ca4 0 (He3 ,a)Ca 39
Figure 27.
Schematic Diagram for Elimination of Gas
Contamination in the Terminal of the MITONR Generator . . . . . . . . . . . . . . . .
kn = 2
.
LIST OF TABLES
Page
I.
II.
III.
IV.
V.
Variation of Optical Model Parameters . . . . . . .
He3-Optical Model Parameters
. . . . . . . . . .
DWBA Parameters for Ca 4 0 (He 3 ,d)Sc
S 38
41
Angles Used in Summation of Alpha Particle Yield
DWBA Parameters for Ca4 0 (He 3 ,a)Ca
36
39
.
62
ii;-
L ----
I. INTRODUCTION
The objectives which form a basis for the work presented here may
be classified in four categories: the installation and testing of a
modified radio-frequency ion source in the terminal of the MIT-ONR
generator, the experimental study of a nuclear reaction using a multiplyionized He
beam in a problem in low-energy nuclear physics, the invest-
igation of single nucleon transfer reactions in the vicinity of a doublymagic nucleus, and a test of the applicability of the distorted wave
Born approximation method of data reduction in a He -induced reaction.
The experimental modification consists of the replacement of a
low power radio-frequency ion source with one into which a large amount
of rf power can be introduced, sufficient to produce a doubly-ionized
3
He
beam of sufficient intensity to be useful in a nuclear reaction exper-
iment.
The large data-handling capability of the multiple-gap broadrange magnetic spectrograph, which can collect the data required for
the analysis of a complete angular distribution for a number of states
in a single exposure, was expected to partially offset the expected
small current of incident He3+
ions.
The analysis of a nuclear reaction was expected to be more straightforward in a process where the transfer of a single nucleon takes place.
If this nucleon transfer results in the formation of a nucleus which can
be described in terms of closed shells plus or minus one nucleon, the
10
analysis is even more simplified.
This leads to the selection of the
3 9 as especially
reactions Ca4 0 (He3 ,d)Sc 4 1and Ca4 0 (He3,
suitable
3a)Ca
examples for investigating the He3 stripping and pickup mechanism.
The nuclear shell model has been very successful in its predictions
of the ground-state spins and parities of nuclei (Ma 55).
Those nuclei
with a closed-shell ground-state configuration are known to be more
stable than their neighbors in the periodic table.
The properties of
nuclei formed by the addition or removal of a single nucleon from such
a configuration provide an interesting check of the shell model predictions.
A nuclear model consisting of a closed-shell core plus a single
particle (or hole) in an orbit about the core should provide an adequate
description for certain nuclear properties at low excitation energies
or when the energy separation of the next higher (or lower) shell is
fairly large.
The excited states of the nucleus may be interpreted
in terms of the single nucleon (or hole) in orbits with other angular
momenta.
-shell ( He4) ,
The neutron and proton shell closures at the ls
1/2
lp
3/2
-shell ( C1 2 ),
6 6
lp
1/2
-shell ( 016),
ld
8 8
5/2
-shell (
2
2
Si2 8) have been
14
14
investigated rather extensively since they can be excited by nuclear
projectiles of relatively low incident energies.
closed shell occurs when the 2s
The nuclear systems
Sc41l(and
- and ld
1/2
Ca41 )
and
3/2
The next higher doubly-
-shells (
Ca3 9 (and
K3 9 )
Ca4 0) are filled.
20
20
are particularly
interesting subjects for investigation in terms of a simplified shell
model as having a single particle or a single hole together with the
"doubly-magic" Ca4 0 core of closed neutron ls-, lp-, ld-, 2s-shells.
The usefulness of a stripping or a pickup reaction in obtaining
information about the spins and parities of the nuclear levels for
both target and residual nuclei has been pointed out by Butler (Bu 51).
In deuteron stripping reactions the angular distribution of neutrons
or protons are characterized by a high differential cross section in the
forward direction.
The reaction yield tends to have a prominent maximum
in the forward direction and secondary maxima at larger angles.
The
angular momentum, £M, carried into the target nucleus by the captured
nucleon is the major factor in determining the structure of the angular
distribution.
From the position of the primary maximum in the angular
distribution of reaction products, it has been possible in (d,p) and
(d,n) reactions to find the relative parities of the target and residual
nuclei and to determine possible values of the total angular momentum
of one of these two nuclei if the other has a known spin and parity.
When the absolute reaction cross section is measured, it is possible to
extract information on the spectroscopic factor, measuring the amount
of nuclear target state which is present in the nuclear states of the
residual nucleus.
II. APPARATUS
1.
MIT-ONR Electrostatic Generator and Analyzing Magnet
The MIT-ONR Electrostatic Generator, a vertical, pressure-insulated
3+ +
beam in this investVan de Graaff generator (Fig. 1) accelerated the He
igation.
This accelerator has been operated in recent years at terminal
potentials in the range between 4.0 and 8.5 MV as an accelerator of proton,
deuteron, helium-3, and helium-4 beams.
In this investigation a terminal
potential of 6.0 MV was used as determined by the 900 analyzing magnet of
60.5 cm.
Van de Graaff (Va 48) and Herb (He 59) have given a general description
of Van de Graaff generators.
Braams (Bm 56) has discussed in detail the
properties of the analyzing magnet, and Moore (Mo 63) has described the
stabilizing corona load and built the voltage stabilizer which proved
invaluable during this study since it was necessary to stabilize a four
nanoampere beam.
The energy spread of the beam is defined by the analyzing
magnet and the two sets of slits labelled in Figure 2 as xj and x4. The
position of the beam can be determined by the current readings on the
slits, and can be controlled by two sets of electrostatic deflectors
which are located above the slit xj.
The vertical direction is defined
by the slits x 4 which provide the energy stabilization through an error
signal which is applied to the generator terminal.
The detection and analyzing system used in the experiment is the
broad range, multiple-gap magnetic spectrograph (Fig. 3) described by
REMOVABLE TANI
ELECTRONIC CIR(
BUILT-IN 2kw PC
CHARGE COLLEC7
LINER
GENERATING V(
EQUIPOTENTIAL
INSULATING COL
2 - 50hp ,1800
BELT TENSION
DRY ICE
MERCURY
PUMPING
0
2
4
6
8
FEET
MIT- ONR GENERATOR
Figure 1
1
Corona Triode
Tank Liner
S -Grid
Liner
Arno.
100 KV Focusing
Supply
PS.
6BK4
____
11
-I
IonSource
GV
CV
I
C
jI
Ii
II
Resistor
Current
ii
ifi
I I
j
III
Alarm
-
Relay
Ind Plate
PS
II
I I
I
•
P.
• ! •
I
I
"
Belt
-,Current:
.. ..
--
___
_Fluxmeterl_. Freq.
Volt.Stab.
i Control ! Meter!
---
- --.-.
SDefltControl,
Meters J
P.S.
P S
•
..
x, xF C.F
ToVacuum Pump
x2
Freq. I'Fluxmeter
LMeter , Control
Osc
Slit Current'
Meters
+C.F
CF
C.
CF
Targe
'Gu
Xý
X3
X_
X
X4
X5
Os
X6 X7
Currenta
C.n
a
0
FEET
Velm
Selector
nLens
_Control.es
Figure 2
Q.4
CStr
Cure
Integrator:
Control
PLATE HOLDE
POSITION OF B
ELECTR(
a-----~-·
n~-~l·
~OEM
Ill.
Imi'l
i
I-
-
11.- - "I'll--~;--
.
----
·---
-
-
;;:
---
16
Enge and Buechner (En 63).
Briefly, this instrument records simultaneously,
on nuclear track plates, the spectra of charged particles at twentythree different angles divided into 7.50 increments over the range 7.50 to
172.50.
Each of the twenty-three gaps is similar to the single gap
broad range magnetic spectrograph described by Buechner (Br 56).
An
energy ratio of 2.4:1 with a resolving power in excess of 1000 can be
simultaneously recorded.
Without the broad coverage of this instrument,
in angle and energy, an angular distribution experiment, using an incident
beam current of about 20 nanoamperes would have been impractical.
Before striking the target, the beam must pass through an electrostatic quadrupole lens which focuses the beam on the target with a
magnification approximately one-third in the vertical direction and unity
in the horizontal direction.
Two further sets of slits inside a wedge-
shaped collimator are adjusted so that they will not intercept the main
beam but will remove the halo of scattered particles.
Once through the
target the beam is collected in a Faraday cup, as shown in Fig. 4.
The
current in the Faraday cup is measured with a combined microammeter and
integrator.
A 300 volt negative voltage, applied at the entrance to
the Faraday cup, prevents external electrons from entering the Faraday
cup and suppresses the secondary electrons produced inside the cup.
The rear of the Faraday cup can be rotated out of the beam direction,
allowing the beam to be collected on photographic plates at zero degrees,
and permitting visual alignment of the target with the beam direction.
The target holder will accomodate four separate solid target films
mounted on one inch diameter frames.
The targets are rotated at
approximately 260 r.p.m. in order to prevent heat accumulation and to
minimize the effect of carbon and oxygen build-up during the exposure.
Generally the target is placed at 450 to the incident beam.
There is
I~;
~-~
~~-----~-----
---
· '~i~
~--F"'
~
~~--~--5t?~·~I~T-t-;
;-·iL-
-·-·"
IY·i
" -~
---
--
18
also provision for gaseous targets.
In order to monitor the condition and
position of the target, a solid state detector is located at 900 to the
beam direction and can be controlled from the generator panel.
A closed-
circuit television system and a periscope provide a visual check on the
target condition.
Particles emitted from the target pass through a remotely
controlled shutter and four sets of slits in each gap before they strike
the emulsion.
The a-slits which provide vertical collimation are 28.8 cm from the
target; the other three slits
(Fig. 5).
01i
2
,
3
provide horizontal collimation
The combination of the a-slits and the ý 3 -slits determine the
solid angle of acceptance.
The absolute solid angle is approximately
.35 millisteradians at a plate distance of 55 cm (Fig. 6).
For gaseous
targets the combination of slits 0 1 /2 ,a,>3 determine the solid angle
(Fig. 7) and effective target thickness.
The reaction products are detected on nuclear track plates, Kodak
NTA with 50 micron emulsion thickness, which are placed on the focal surface,
a hyperbola of slight curvature.
The radius of curvature p, as a function
of distance along the plate, is determined by the position on the plate
of Po 2 1 0 alpha particles (Bp=331.75) for various magnetic fields.
The
spectrograph magnetic field is monitored by proton resonance fluxmeter
probes at several points in the spectrograph.
Each of the twenty-four
plate holders contain three 2 in. by 10 in. plates.
Provision is made
to rotate the carousel where the plate holders are located so that three
exposures may be made on each plate.
The exposed photographic plates are scanned under a microscope
equipped with a 20x objective and 12x eyepiece.
The nuclear tracks in
0.5 mm strips across the exposed zone are counted in dark field illumination
and the number recorded.
L
M.G. SLIT SYSTEM
TOP VIEW
Ný
>132.9
< 201.8
T
40
53.4
-
-
w
> 122.0
< 135.6
--- 13.0 ----
Pole Piece
al
!fl,
'fl2
Pole Piece
00
I0
Scale 1'5
"
Dimensions in crm.
- -- · ,~-
·- i ·
=~-F-------I~---P--L-=T--LLILI~-~··CI__
hA
42
40
-4
38
3.8 x 10
36
3.6
S34
3.4
32
3.2
30
3.0
28
2.8
26
2.6
24
2.4
22
2.2
20
2.0
C"
,-iF12 •
0"O
0
C-
O,,
'P
0
5
10
15
20
25
Distance Along
I
--
~--~~~-
--
r---I~..Y,,
------ -----
-I.I- -L·~1~_~__
30
35
40
The Nuclear-Track
i.
-FI
-~LI~
I
45
50
Plate
55
60
65
75
80'C'
D (cm)
- -~---~-
~~ L·-·.
-~---e
-
--
-
;~.-
~
--
------
---
- ---- --------~
·-------------------
·~
__
V~L~IC_
_
M.G. SLIT SYSTEM
SIDE VIEW
Z
2.14
13.0
28.8
Dimensions in cm.
Scale 1:2
--
u·~l~
·;--I----·
i~------------~--~-t----~----·--~-·-~·rc
MM"
III. EXPERIMENTAL PROCEDURE
1.
Target Preparation
The two calcium targets used in these experiments were prepared
by the electron bombardment of Ca 4 0 CO 3 * using an electron gun evaporator.
The Ca4 0 CO 3 (or possibly Ca4 0 0) was deposited onto a thin film of carbon
which had been prepared in advance by vacuum evaporation of carbon onto
a glass slide to which a solution of Teepol** had been applied.
When the
glass slide is immersed carefully into a distilled water bath, the
carbon film plus the calcium layer will float off the slide.
This is
then mounted in a one-inch diameter circular frame and supported by a
thin formvar (polyvinyl formal) backing.
Formvar is by weight 33 per
cent oxygen, 59.1 per cent carbon, and 7.8 per cent hydrogen, plus
traces of sulphur and nitrogen.
A thicker target was prepared by the
evaporation of Ca4 0 CO 3 onto a self-supporting carbon film which had
first been mounted in a circular frame.
The thin target on the formvar
backing was used in the elastic scattering experiment.
For the longer
exposure it was felt a target using the self-supporting carbon foil
backing would be more satisfactory since it was physically stronger and
could withstand an extensive exposure to the incident beam.
The target
thickness was measured by assuming that the elastic cross section of 12MeV
He 3 ions at 22.50 follows the usual Rutherford scattering formula.
The
SThe separated isotope was obtained from Union Carbide Nuclear Company,
Oak Ridge, Tennessee
**Sodium secondary alkyl sulphate, Shell Chemical Company, Teepol 610
target thickness for the thin target was measured to be 4.1 pg/cm 2 .
The thickness of the second target (10.6 pg/cm 2 ) was measured by comparing
relative yields at 82.50.
2.
Determination of the Incident Energy
There are two ways that the incident energy may be determined:
from the energy of reaction products from a reaction with a known
Q-value or from the determination of the magnetic rigidity of the
incident particle in the analyzing magnet.
Hysteresis effects in the
analyzing magnet can introduce an uncertainty in the expected incident
energy.
Since the magnetic field is measured by a proton resonance flux-
meter probe at a point which is different from any along the particle
trajectory, the value of the magnetic rigidity will be uncertain.
The
analysis of nuclear reaction products of a known Q-value provides a
more accurate method of determining the incident energy through the
investigation of ground state reactions from the carbon and oxygen
present in the target.
The incident energy, as calculated from the
latter case, was 12.001 ~0.0l0 MeV.
The magnetic field in the spectro-
graph was measured by observing the position of the yields from the
contaminant reactions
(giving the radius of curvature (p) in the magnetic
field) and the calculation of the energy (Bp) of the reaction products
from their known Q-values.
This confirmed the magnetic field setting
to be 10338 gauss (proton resonance frequency is 44.015 mcs).
Recent difficulties with the multiple-gap calibration have been
attributed to the improper cycling of the magnet (Ra 63).
This has been
avoided to a large degree by cycling the spectrograph before and after
each exposure.
To insure that there would be no accumulated energy
IIM
uncertainty during the exposure the magnets were kept continually
operating.
This would also tend to minimize any drift during warm-up
of the equipment.
3.
The Determination of the Q-Value
The difference between the rest masses of the incident particle
plus target nucleus and the masses of the residual nucleus plus the
observed particle is defined to be the Q-value of the nuclear reaction.
The Q-value is related to the incident energy E., the outgoing energy
E
and the recoiling energy E
o
by the relation (Ba 53)
r
Q=
M + M
r
oE
M
o
-
E2 + E2 - E2
M - M
r
iE. + i
o
r
1
M
r
1/2
1/2
2(M oM.) 1/2 (E.E
O1
0
M
where
2M c 2
r
r
10
)
)1/2
1/2 cosa(
E.
1 +
)1/2
(1 +
0
___
r
1/2
E
2M.c 2
2M c2
1
0
M. = mass of the incident particle
1
M00 = mass of the outgoing particle
M r = mass of the recoiling nucleus
6
= laboratory reaction angle
The relativistic effects can be treated as small corrections, because
in the present experiment E./2M.c 2 is of the order 1/500.
1
1
They give
rise to corrections in the Q-value of magnitude about 4 key at
4.
00.
Data Reduction and Error Analysis
After a precise value for the incident energy and spectrograph
magnetic field is determined from the reactions with C 1 2 and 016 it is
possible to calibrate the distance along the photographic plates as a
function of the Q-value for the particular reaction under study.
This
relativistically correct calculation was carried out for each of the
twenty-three angles at which the reaction was observed.
All unknown reac-
tion peaks were labelled according to this Q-value calculation.
It
was now possible to pick out those peaks which have the same Q-value
at a majority of the angles of observation and identify them with the
residual nucleus under study.
The position of all known or expected
contaminants was also calculated at all twenty-three angles and this made
possible the rapid elimination from consideration of many of those peaks
which result from contaminant reactions.
The differential cross section do/dQ (millibarns per steradian) is
determined by the relation
do
_
N
N (dQ)t
0
N
N
0
t
=
number of detected particles in the reaction peak
=
number of incident particles (lpC
=
solid angle at the spectrograph distance for the reaction peak
=
the effective target thickness in nuclei per millibarn
=
3.12 x 10
12
helium ions)
The uncertainty in the scanning and the assumption of Rutherford
scattering are estimated to introduce errors in the absolute cross-sections
of the order of 20%.
A 10% relative uncertainty is shown on all reaction
angular distributions except on those where the statistical error is
greater than 10%.
The correction for solid angle variation along the
plate was made in the forward angles only.
The statistical uncertainty in
the backward angles was much larger than the correction for solid angle.
Within the total uncertainty in the yield, the yield in the laboratory coordinate system is the same as the center-of-mass system except at the
_W"M
I
1
very forward angles where there is a correction of about 10%.
The
maximum deviation of the laboratory angle from center-of-mass angles is four
degrees at a laboratory angle of ninety degrees, and one degree at a
laboratory angle of thirty degrees.
5.
The Experimental Procedure
The elastic scattering data and the reaction data were collected on the
same set of plates in successive exposures.
used in the collection of the data.
All twenty-three angles were
The plate holders at every third
angle from 7.50 were loaded with Kodak NTA emulsions of 100p thickness
so that better track length discrimination could be made at those angles.
No aluminum foils were used to slow down the particles.
holders were loaded with Kodak NTA emulsions
All other plate
of 50p thickness.
Two elastic exposures of 10 and 100 pCoulombs were made on two of
the available zones, the magnetic field being set so that the elastic
yield at 172.50 would be collected.
For the He 3 reaction experiment,
39
3
0
the magnetic field was set so that the yield for the Ca4 (He ,ot)Ca
and Ca4 0 (He3 ,d)Sc 4 1 ground-state reactions and reactions to as many
excited states as possible could be collected in a single exposure.
total exposure was 3134 pCoulombs.
The
After the elastic scattering
experiment, the thin target on formvar appeared to be ruptured so the
self-supporting carbon-backed target was substituted for the longer
exposure.
The targets were rotated continually during the He 3 -bombardment.
The average He 3+ + beam current was 10-15 nanoamperes and it was
found only 300 to 500 pCoulombs of charge could be collected in a day's
operation of the generator before the temperature of the ion bottle rose
enough to make the discharge unstable.
The generator was then given an
TC
8-hour rest; however, all magnets were kept in operation.
The experiment lasted nine days during which time the photographic
emulsions were left in the evacuated spectrograph.
The experiment was
terminated partially because of concern that the nuclear tracks in the
emulsion would begin to fade and because of some instability in the
beam analyzing magnet.
In order to offset any effect due to fading, the emulsions were
developed in a concentrated solution, two parts D19 developer, one
part water for 15 minutes (twice the normal time).
It was felt that some
fading did take place in the tracks (particularly in proton tracks which
were also on the plates).
The 100p emulsions proved very troublesome,
the emulsion tended to lift off the glass backing when normal drying
procedure was followed.
However, the application of a solution of
benzene and Canada balsam to the glass-emulsion boundary was successful in
keeping the emulsion on the photographic plates.
The grain density of the
tracks in the 100p emulsions was less than that in the 501
emulsions.
This
may possibly be attributed to a difference in the emulsion batch and/or
age.
In general, all the tracks were fainter than the tracks from other
experiments carried out with the generator because of the increased
incident energy.
The scanning for deuterons, alphas, and He 3 posed a problem only
at lower energies where the deuterons and elastic He 3++ tracks had
approximately the same length and grain density in the emulsion.
Some
variation in grain density took place from plate to plate so that care
was necessary when counting was begun on a different plate.
When
possible the photographic plates at a given angle were developed together,
thereby minimizing grain density variation at a given angle.
The use
28
of oil on the emulsion during the scanning procedure helped to enhance
the nuclear track contrast.
Since the He3 ++ elastic scattering yield occurred on both experiments it was possible to relate the two target thicknesses.
No 11e3 +
tracks were expected on the plates but very short tracks in the emulsion
were attributed to the background for the Ca'4O(He3~,He3+)Cal4Oreaction.
IV. EXPERIMENTAL RESULTS AND DATA REDUCTION
1.
The Ca4 0 (He3 ,He3 )Ca4 0 Reaction-optical Model Potential Parameters
for He 3 at 12 MeV
The calculation of theoretical angular distributions for the
stripping and pickup of a single nucleon depends critically on the
potential between the projectile and target, between the residual
fragment and residual nucleus, and on the potential in which the stripped
nucleon is bound to the target.
At the present time, the optical
model potential seems to provide the most practical description in
direct reactions of this sort where only a few internal degrees of
freedom, expressible in terms of a collective or single particle
model, are involved.
Experimentally it is found that elastic scattering is the most
important process which can occur and all other processes, both reaction
and inelastic events, can be treated as perturbations.
This approach
(the so-called weak coupling) leads, for example, to the calculation
of the transition amplitude in the distorted-wave Born approximation
(To 54, To 59).
It is important that the effect of the nuclear potential on
the elastically scattered projectiles be considered since the shape of
the differential reaction cross-section can be modified substantially
by the effect of absorption in the nuclear interior.
Any absorption
present is taken care of by using the appropriate distorted waves.
The absolute differential elastic scattering cross-sections for
He 3 ions on Ca4 0 were then measured to provide information on the
3
nuclear optical model potential parameters for He at 12 MeV.
wwý
The ratio of elastic-to-Coulomb cross sections deviates from unity,
as expected, and there are indications of diffraction-like oscillations.
The experimental data was treated with the ABACUS-2 code of Auerbach
(Au 62) in order to extract the optical model parameters.
The ABACUS-2
code combines an optical model calculation of the differential scattering
cross section and the associated elastic-to-Coulomb ratios with an
automatic search of the optical model parameters.
An indication of the
quality of the calculated cross-section and the corresponding experimental data is the calculation of a chi-square (X2 ) test.
The ABACUS-2
prcgram provides for the use of a variety of interaction potentials
but the analysis was carried out using an optical potential of the
Woods-Saxon form plus the Coulomb potential.
Since the optical model
parameters were to be used in a distorted-wave Born approximation (DWBA)
calculation of He 3 -induced direct reactions, using a Woods-Saxon potential
with no spin-orbit potential (the code JULIE (Sa 62)), it was decided
not to include a spin-orbit interaction term in the optical potential
in ABACUS-2.
The optical model potential used in the search for the "best" fit
to the experimental elastic differential angular distribution, then,
has the form
V=
V + iW
1 + e+(rR)/a
where
+
V
c
V
= the real part of the central nuclear potential
W
= the imaginary part of the central nuclear potential
R
A
/ 3
= R A 1/3
fermi, the nuclear radius
A =atomic weight
= atomic weight
a
V
=
the nuclear surface diffusivity parameter
=
the Coulomb potential
c
The target nucleus was assumed to be a sphere of uniform charge
density
Z1 Z2 e2
2R'
___
c
V
3
}
-
Z1 Z2 e2
r
=
c
{
___
1/3
r
<
r
>
the charge radius
=
R A
c
=
projectile nuclear charge
=
target nuclear charge
,
For calculation purposes, the charge radius was set equal to the
nuclear radius.
That this is a satisfactory assumption would seem to
be borne out by the recent series of experiments of Klingensmith et al.
(Kl 64) at 20 MeV.
Their calculations indicate the insensitivity of
the calculated differential cross-section to changes in the charge
radius.
Their experiments also indicate that the elastic scattering
cross-sections are relatively insensitive to changes in the nuclear
spin-orbit potential.
Until recently, the optical model parameters for He3 elastic
scattering indicated that the real and imaginary parts of the potential
had similar depths (approximately 220/A 11 2 14eV) and of Woods-Saxon form
(a
=
0.65f, R
0
=
1.6 f) (Ho 62).
However, these results were based on
less than a complete energy range (only 5.5 and 29 14eV).
Lacking any
further data, it was decided to use these ~besttt parameters of Hodgson
I
m
as initial values in the search for a more accurate optical model
potential at 12 MeV.
A maximum of fifteen partial waves was sufficient in the calculation.
At this point (Z = 15) there was no more interference between Coulomb
The Woods-Saxon potential was
and nuclear scattering amplitudes.
cut off after 8.5 f.
central depth.
At this point the depth is less than 1% of the
The use of a larger cut off radius did not affect the
calculation to a significant degree.
The value of X2 in the calculation
was not absolutely minimized but was relatively small in the search
For 12 MeV He 3 ions on Ca4 0 these are
for the "best" parameters.
V = 43.1 MeV
W=
7.1 MeV
a = 0.658 f
=
R = 5.28 f
1.54 A1 /
3
f
The quantity X2 is defined by
N
2
X X
1i
N
1 cm
i
1
2
(
do.lcalc
1
-
d exp
do.l
1
i
where di
is the ABACUS
calculation at each angle in mb/sr
dexp is the experimental cross-section at each angle in mb/sr
1
w.
1
is the weighting factor
N
is the number of angles considered
-1/2
is taken to be the statistical error in the cross section.
w.
1
Since the data at the backward angles have greater statistical un-
2
certainty, it was felt that the standard X2 weighting factor (
-1/2
/2
experimental cross-section) would put too much weight on the crosssection in the backward angles.
=
,"Qq
It is difficult to assign a physical interpretation to the value
of X2 except to say that it indicates a rough
fit to the experimental data.
idea of the quality of
In a test calculation of X 2 , it was found
that one or two angles would give the major contribution to X2 .
Figure 8 shows the measured cross-sections as a function of laboratory scattering angle.
The error flags associated with the experiAny errors
mental points are determined from counting statistics.
arising from charge integration and solid angle uncertainties have
not been shown.
However, the measured differential cross-sections
will have an error arising from the assumption that the elastic scattering cross-section will be Rutherford in nature at 22.50.
The "best"
calculated elastic differential cross-section is illustrated together
with the coulomb differential cross-section.
The sensitivity of the calculation to variations in the "best"
parameters is illustrated in Figure 9 and a comparison of relative
values of X2 is given in Table 1.
A change in the real part of the potential, V, by ±10 MeV results
in a deviation from the experimental data; the tendency is for the
calculated cross-section to be reduced in the backward angles.
The fit is relatively insensitive to increases in the imaginary
part of the potential W, even though the calculatedX
2
is increased.
A decrease in W produces very distinctive diffraction-like oscillations
while an increase in W tends to wash out these oscillations.
Absorption
of a nuclear particle can be described in terms of an imaginary part
in the nuclear potential; these results, then, indicate that Ca 4 0 is
somewhat opaque to He 3 at 12 MeV bombarding energy.
10o
8
6
4
2
3
10
8
6
4
2
E -o102
8
6
4
2
I0
8
6
4
2
I
20
40
120
100
80
60
Lab. Angle (Degrees)
Figure 8
140
160
LII
Ca 4 0 ( He3 , He3 ) Ca 4 0
EH,
OPTICAL
= 12.00 MeV
MODEL CALCULATION
104
I0
102
I0
E
I
-o
SIO"
lo,
10
3
2
10
10
--
20
--
60
--
100
140
20
Lab. Angle (Degrees)
Figure 9
60
100
140
I
V
(MeV)
"Best"W
43.1
W
R
A
(MeV)
=(R0 A1/3f)
7.1
5.28
0.658
7.1
5.28
0.658
5.28
0.658
(f)
4.05
Fit
'V'
Variation
53.0
33.0
12.0
'W'
Variation
'R'
43.1
43.1
43.1
2.0
7.1
7.2
5.64
'A'
Variation
0.658
23.4
23.9
0.75
11.3
0.55
29.5
4.96
Variation
43.1
7.1
18.5
5.28
Variation of Optical Model Parameters
Table I
Changes in R o (= RA
-1/3
/ 3
) by 0.1 f reduce the cross-section in
the backward direction and in general the fit is not satisfactory.
Changes in the diffuseness by 0.1 f tend to result in an increase
in the differential cross-section toward Coulomb (for a decrease in the
diffuseness) and in a decrease in the differential cross-section
(for an increase in the diffuseness).
This can be interpreted in terms
The oscillations
of the scattering in the fringe of the nuclear field.
in the elastic cross-section arise from diffraction effects in the
bulk of the nucleus.
For He 3 these are confined to the region of the
surface because of the low mean free path of He 3 in nuclear matter.
If the He 3 energy were decreased (or the nuclear charge increased)
the diffractions would become less and less pronounced until only
Coulomb scattering would result.
It is in this intermediate region
between bulk nuclear and Coulomb scattering where the cross-section
is especially sensitive to the fringe of the nuclear field (Ho 60).
These variations of the optical model parameters indicate that the
"best" angular distributions are more sensitive to changes in the
nuclear surface diffusivity and the imaginary part of the optical
potential than in
changes in
the other two parameters.
The tendency for W to be considerably less than V is
not consistent
with the calculations of Greenlees et al. (Gr 60, Gr 61) but is consistent
with the recent He 3 experiments of Klingensmith et al. (Kl 64) at 20
MeV.
Klingensmith found two distinct trends for V, R, a and no correlation
for the imaginary part of the nuclear potential except that it appears
to be at least one-half the value of the real part.
parameters for Ca 4 0 are listed in Table II.
For 20 MeV, their
Present
Trend A
Trend B
Hodgson (Ho 62)
(20 MeV)
(20 MeV)
(5,
Work
(12 MeV)
V (MeV)
W (MeV)
RA-1/3(f)
A (f)
Table II
3
He -
Optical Model Parameters
29 MeV)
2.
of Nucleon Transfer Reactions
The Analysis
l
the calculation
an interaction
rn approximation
these techniques
the reduction
imation considers
g and outgoing
.
The trans-
inal fragment
he target with
as a free parthat the Butlerlly considerably
ell model of
represent the
plane waves
en much more
tributions and
wave theory
following expression for the differential cross section in the reaction A(a,b)B (Ba 62)
da
mmb
-
do
k2
mamb
(27h
#
where M(6) = <bH(-) (kb)
2
)
2
kb
ka
(+)
a
V
IM ( 6) 2
( 2 JA+1)(
2
s +l)
(ka)>
The summation is over all magnetic substates; ma, mb are the reduced masses
of a,b; 6 is the angle between ka
initial
and
final momenta.
a and kb
kb' the
h
nta
ndfnlmmna
In the stripping reaction 'a' is considered to be composed of 'b' + 'x';
then 'B = A + x', where 'x' is the transferred nucleon.
The potential V contains all the information on nuclear structure,
angular momentum selection rules and the type of reaction.
V is the difference
between the interaction potential in the exit channel, Vbx + VbA
optical model potential in the exit channel VbB (To 61).
,
and the
The contribution
VbA - VbB is difficult to handle but in the limiting case of an infinite
mass target the term will vanish.
The neglect of this term is still prob-
ably a good assumption when m A >> mb, mx.
In a stripping reaction the
potential, then, is generally assumed to be Vbx, the interaction between
the transferred particle 'x' and the final fragment 'b'.
state wave function
a
The initial
(+ ) is the solution of a SchrUdinger equation for
a
the total system with the condition that at very large distances the wave
function reduces to an incident wave in the direction ka on the target.
Likewise
b) (kb) is the Schridinger solution which corresponds at large
distances to a final wave leaving the target in direction
kb.
The Born approximation consists of replacing
a
a
(ka) by
a
a
(k),'
a
the elastic scattering wave function for particle 'a'; this approximation
is equivalent to the neglect of all the possible outgoing channels except
that corresponding to elastic scattering.
Since the wave function ý(()
cannot yet be calculated from first
principles, it is further assumed that an optical model potential can generate wave functions, in the Schridinger equation, which are good approximations to the actual wave functions.
The optical model wave functions
are found from the solution of the Schridinger equation with a potential
of Woods-Saxon form.
U(r)
U3(r)
=(V
=
+ iW)
+
1 e(r-R)/a
The reaction cross section for A(a,b)B reaction will depend on the four
parameters V, W, R, a.
There are two sets of these parameters, one for
the particle 'a' and one for the particle 'b'.
The ideal procedure for
determining these eight parameters would be to carry out the elastic scattering experiments A(a,a)A and B(b,b)B.
In practice, however, it is not always possible to carry out both of
these experiments at the appropriate energies.
(Note that the second
experiment must be carried out at the laboratory energy in the frame where
the residual nucleus, if it exists naturally is at rest, and in an excited
state.)
Often, these four parameters can be approximated by using data
from stable nuclei of about the same A. The calculation proceeds under
the assumption that 'x' is bound in a Saxon well to 'A' with total angular
momentum j. The bound state wave function is composed of an orbital and
radial part specified by 'nV'and a spin part specified by 's'.
If no
spin-orbit potential is considered only 'nt' need be specified for the
bound state wave function.
The value of 'n' determines the number of nodes
in the radial part of the wave function.
The transferred particle 'x'
is assumed to be bound to the target in the Saxon well with the correct
nucleon binding energy.
The bound state wave function for 'x' to 'A' must
match smoothly at some distance outside the nuclear radius to the bound
42
state wave function for 'x' in 'B' (this external function is a Hankel
function for an uncharged 'x' or a negative-energy Coulomb function for
a charged 'x').
The differential cross section for stripping is given in the form (Ba 62).
do
() = constant
dQ
B +
S(sj)do
d.
+ 1
2J~s(Zsj)
A
d £sj
isj
DWBA
term contains the distorted wave integrals; S(£sj)
The do
dG
2J
2 B + 1
Ssj DWBA
is the spectroscopic
factor and contains the nuclear structure information.
In particular, it is a measure of the probability that, in the final
nuclear state, all but 'x' of the nucleons find themselves in an arrangement corresponding to the initial state.
orbit (2sj) enters the calculation.
Usually only a single nucleon
The constant term contains information
relating to the break-up probability of the incident particle 'a' into 'b'
and 'x' and factors relating to the relative momentum of 'b' and 'x',
their spins, and the number of ways 'b' and 'x' can be formed from 'a'.
In the stripping reaction Vbx is considered to be non-zero only when
rb = rx (this is the usual zero range approximation). This assumption,
bx
however, is not reasonable when the target is stripped instead of the
projectile (heavy particle stripping) (Ba 62).
If a small separation (for example, a Hulthen form at large distances)
exists between 'b' and 'x', the zero-range transition amplitude must be
corrected for finite momentum components.
For deuteron-induced reactions,
the result is still proportional to the zero range approximation.
Pick-up reactions are just the inverse of stripping reactions.
The
differential cross section for pick-up B(b,a)A is simply related to the
stripping differential cross section A(a,b)B (Ba 62).
do
d-- ()
(2s
=
constant
2s
2s b +d1
+1
+ 1s_
b
Ssj
d
d-
dsj
isj
DWBA
The advantage of the DWBA theory in deuteron-induced reactions is that
all terms may be calculated and an absolute value for the spectroscopic
This is not possible yet with more complex incident
factor can be extracted.
particles, such as He3 , since the breakup probability and the nucleonfragment interaction are unknown.
However, it should be possible to extract
relative spectroscopic factors providing the probability for breakup is
not too dependent on the internal relative momentum at the instant of
nucleon transfer.
Therefore, for stripping reactions, the experimental cross-section for
two excited states in the same nucleus are related:
(da
V
+ I1sji
2
2
j2 + 1
(do/dS)s
2j(d_/d_)£
ii
S
2
(da/dQ)
2
and for pick-up reactions,
d
1
S
sjl (da/dQ) lsji
DWBA
ACE
S
(da/d)
DWBA
32
2sj2
2sj
2
1 sj1
1 DWBA
2sj2 DWBA
ell Nucleus
rom the point of
is a single nucleon
protons, or in
nce the closed
lar momentum of
xtra particle or
seful tools in the
f the nuclear shell
found in two
the formation
he angular distributions of the final particle corresponding to these states.
The angular
distributions give a determination of the orbital angular momentum of the
stripped or picked-up nucleon, enabling one to assign both the spin and
parity of the nuclear level except for the ambiguity j = k•
1/2.
A
measure of the amount of shell model state in each nuclear level of the proper
spin and parity is given by the total cross-section for the formation of
the level.
The entire single-particle state does not necessarily manifest itself
in a single nuclear level but the single-particle states of nuclei in the
same region apparently tend to share their single particle strengths with
other levels in the nucleus and cover, perhaps, an excitation of several
MeV in width.
This phenomenon has been observed by Bockelman, Kashy, and
Rapaport in this mass region (Bo 57, Ka 64, Ra 63).
(Be 64) has reported the splitting of the
2
Recently, Belote
p3/ 2 single particle state in
45
41
the mirror nucleus, Ca
In order to find the center-of-gravity of the levels belonging to a
given shell model state one weights those levels, not according to their
cross-section, but rather by their spectroscopic factor which is the ratio
of the observed (do/dQ)exp to the DWBA (do/dQ) calc for the shell model
state assumed to be at that energy.
This eliminates the Q-value dependence
of the cross-section (a problem of the nuclear reaction) from the shell
model calculation (a problem of nuclear structure).
_
m
3.
II_~~~_____~
_
The Ca4 0 (He3 ,d)Sc 4 1 Reaction
a) Previous Work
The Sc4 1-Ca4 1 energy difference has been the object of some disagreeThe first measurements were carried out on the positron
ment in the past.
decay of Sc4 1 after its formation in the deuteron bombardment of Ca4 0.
The
+
energy was 4.94 MeV.
This result seemed to be confirmed by the
results of a later Ca4 0 (d,n)Sc 4 1 experiment (Pl 55, 59) where the Q-value
of the reaction was measured to be -0.57±0.05 MeV.
These two experiments
indicated the Sc4 1-Ca4 1 mass difference to be about 5.94 MeV.
The measure-
ment of the Q-value for the reaction Ca4 0 (He3 ,d)Sc 4 1 by Wegner and Hall
(We 60) of -4.47±0.10 MeV was in considerable disagreement with the earlier
work; this indicated the Sc4 1-Ca4 1 mass difference to be 6.56 MeV.
A
remeasurement of the Q-value of Ca4 0 (d,n)Sc 4 1 using a time-of-flight
method (Ma 60) resulted in a change in the previously accepted Q-value.
The new value was -1.14±0.20 MeV.
Further Q-value measurements by Hinds
and Middleton for the Ca4 0 (He3 ,d)Sc 4 1 reaction (quoted in Bu 61) and Butler
for the Ca4 0 (p,y)Sc 4 1 reaction (Bu 61) confirm the mass difference to be
about 6.49 MeV.
The Q-value reported by Hinds and Middleton is -4.414±0.010
MeV.
Wegner and Hall also found four excited states in Sc4 1 at 1.69, 3.35,
5.10 and 6.01 MeV.
Only the lowest of these is contained in the energy
range of the present experiment.
Macefield et al. (Ma 61) using the
Ca4 0 (d,n)Sc 4 1 reaction found excited states at 1.709±0.030 MeV and
2.476±0.030 MeV.
41
Butler (Bu 61) has examined the excited states of Sc
in the reaction Ca4 0 (p,y)Sc 4 1 and reports four low-lying levels at excitation
energies of 1.72, 2.59, 2.67, and 2.88 MeV. Youngblood (Yo 64) confirms
.·.L.· ...·^I
----------- -·--·~-
-I
these excitation energies and reports further levels to 6 MeV in the
Ca4 0 (p,y)Sc 4 1 reaction.
Recently, Brown (Br 63) has carried out an exten-
4 1 by studying the Ca4 0 (p,p)Ca4 0
sive study of the higher excited states of Sc
reaction.
b) The Ca4 0 (He3 ,d)Sc 4 1 reaction
The typical energy spectrum of the emitted deuterons at a laboratory
angle of 52.50 with respect to the incident beam is shown in Figure 10.
In this spectrum there are three resolved deuteron groups corresponding to
levels in Sc4 1 at 1.718 and 2.415 MeV excitation.
The deuteron energy
4 1 to 2.6 MeV.
range corresponds to excitation energies in Sc
4 1 are shown in
3
40
The deuteron angular distributions in Ca (He ,d)Sc
Figures 11 to 13.
In these figures all the experimental data are shown
with laboratory differential cross sections plotted versus laboratory angle.
A 10% uncertainty is assumed in the relative cross section.
It was not
possible to obtain forward angle data on the 1.718 MeV level because of the
contaminant reactions.
Each angular distribution is identified by the
41
excitation energy of the corresponding level in Sc .
Also shown in these figures are the predictions of the distorted
wave Born approximation calculations.
The curves shown are those calc-
ulated with the stated proton angular momentum capture determined principally by the shape of the calculated angular distribution and guided by
41
40
the predictions of the shell model and the results of the Ca (d,p)Ca
experiment.
Some comments on the validity of the assignment in view of the
approximations in the calculations will be made later.
The Q-values measured in this investigation are given as the average
values obtained from measurements at several different angles in the
multiple-gapspectrograph.
The ground state Q-value is -4.406± 0.010MeV.
This is in close agreement with the value reported by Hinds and Middleton.
EXCITATION
2
2
2_.5
---
T-~-~-~--
~---.`----~-.--
-
2..0
,=,
-- r
·---
----
ENERGY (MeV)
Ca4 (He3, d) Sc
TARGET: Ca
LAB. ANGLE
- 400
EXPOSURE =
EHeS
12.00
SPECTROGR
- 300
Sc
41
(I)
- 200
F17
(I)
- 100
1
!
41
Sc
(2)
1
00ao
- -oo
15
35
DISTANCE ALONG PLATE (cm)
-
I
1
-L
-
-
-?lu
._
40
RUN 156-Z
-'.------------------~
-
~
-
-
-
I0
E
.0
0
-i
,_.J
_-
OFQ5
-I°
Sb
CD
0
30
60
C--g-~
90
Lab. Angle (Degrees)
i
120
__
180
150
; i. ~,
i..--...."'".~·.~-Y.
-YY
II~-
I
Ca40 (He3 d) Sc4'
Level No. I
Q = -6.124
Ex
-
MeV
t =1
-p
5
= 1. 7 18
MeV
E
4i
b
2.5
0
0
00
30
30
30
60
60
90
90
1
Lab. Angle (Degrees)
120
120
00
150
150
180
180
~
r~~
~
-..---
,
~-~
·
-V
-~V-w--C.r"-R~~F~·
0.5
(0
E
.0
0
-J
0.4
0.3
*0
0.2
0.1
0
I
30
I
Y ..
60
90
Lab. Angle (Degrees)
.....
....~~~~~~~~~~~~
-. ............
·.--.--
· ·- ----
120
-- · ·- ·-·-; · ·
180
150
--
--
·
c) The Assignment of 'Lp' and the Estimation of Relative Spectroscopic
Factors
4 1 are listed
3
The DWBA parameters used in the analysis of Ca4 0 (He ,d)Sc
in Table III.
The deuteron parameters are estimates from the trends given
by Melkanoff et al. (Me 62); the cutoff radius was determined from the
"best" fit to the angular distributions.
The cutoff radius is defined to be that distance from the residual
nucleus at which the integration of the overlap integral was begun (and
carried out to very large distances).
This integral is the degree of over-
lap of the wave functions of the bound state nucleon, the final fragment,
and the initial projectile (in the zero range approximation).
The use of a
cutoff radius means that the contribution of the bound state nucleon wave
function inside that radius is ignored.
The DWBA calculation has not included the effect of a spin-orbit
potential on the calculated angular distribution.
However, it has been pointed
out that changes in the spin-orbit potential do not appear to affect the
3
calculations of cross-sections for elastically scattered He to a signif-
icant degree (Kl 64).
This could mean that the exclusion of a spin-orbit
potential from the calculation of He 3-induced reactions may not bring about
important differences in the cross-sections.
However, recently, Lee and
Schiffer (Le 64) have pointed out a difference in the Zn= 1 transitions in
(d,p) reactions for the two different values of j=n~l1/2 for medium weight
nuclei at angles greater than 90* .
This observation may mean that (He3 ,d)
reactions may also exhibit such a j-dependence in the angular distributions.
Negative Q-values can be very undesirable in DWBA calculations.
Tobocman (To 62) has indicated a procedure which was used here in the calc4
ulation of the angular distributions for the excited states of Sc l(all of
which are unbound).
3
The binding energy of the proton in He is adjusted
Bound State Proton
Ca 4 0 (He
3
,He 3) Ca40
Sc41(d,d) Sc
4 1
Q=-6.821 MeV
S=2
Q=-6.124 MeV
1
S=
p
Q=-4.406 MeV
S=3
p
P
40.6
60.7
58.8
V (MeV)
43.1
W (MeV)
7.1
RA-1/3(f)
1.54
1.5
1.25
1.25
1.25
a (f)
0.658
0.65
0.65
0.65
0.65
1.06
1.00
1.00
60.
15.
Binding
Energy (MeV)
Cutoff Radius
6.0 f
Table III
41
3
40
DWBA Parameters for Ca (He ,d)Sc
~
_
~-,...~,,.;.~~
.,,~,.,.,UYIU~I~-*~Y~L~i·
-
- I -- ~ d~ -~- - - ---
·· YYI~I~LIYYVYY*Y·IIWylY·*I~U·Y~Y~LJ~YI~U
. -.--YII·YI·IYYYYIYI-I II·----I···(II1.I
1
I
I
- e-~I
F-- L
- --C`-~-`3-L
___
from its physically correct value rather than adjusting the Q-value of the
reaction, since the outgoing deuteron energy depends on the Q-value of the
state under investigation.
The assumption of a binding energy of 1.00
MeV for these unbound states guarantees that the bound state proton wave
function is fairly well localized around the target nucleus at least until
the outgoing deuteron is well away from the residual nucleus.
The DWBA calculation, in the zero-range approximation, can result
in qualitative fits for the angular distribution for the ground
state reaction using no cutoff radius.
However, for the excited
states of Sc4 1 a cutoff radius was necessary for a good fit to be
obtained.
For each angular distribution only the value of 'nZ' and the cut-
off radius were taken to be free parameters.
The cutoff radius was not
allowed to vary from level to level so that once the cutoff radius for
"best" fits was determined only the value 'nk' of the bound state proton
was varied.
The DWBA code finds the depth of a Saxon well which will
bind the proton to Ca 4 0 at the correct energy.
These have been listed
in Table III for completeness.
The calculated cross section could vary up to 20% depending on the
The cutoff radius
exact value of the cutoff radius used in the calculation.
could be taken to be somewhere from 4.5 f to 6 f and qualitative agreement with the distributions would still occur.
when no cutoff radius was assumed.
Poorer agreement resulted
The effect of a cutoff radius is
illustrated later when the validity of the DWBA method is examined.
A comparison of the angular distributions for the ground and first
excited states can give the relative spectroscopic factors assuming
that the DWBA approach is valid, and that only 1f7 /2 and
2
p3/ 2 nucleon
transfer takes place
2J3/2 + 1
S(2p3/
2J7/2 + 1
S(1f 7 / 2 )
S(2p3/
)
=
7/2
3/2
or
2
2
)
=
0.59
,
1.2 S(1f 7 /2)
Likewise, if the level at E = 2.415 MeV is a proton capture in an
kP = 2 state
p
2J + 1
S(£=2)
2J7/2 + 1
S(1f 7/2)
(2J + 1) S(£=2)
=
0.096
= 0.77 S(1f 7 /2)
S(d3 /2 ) = 0.2 S(1f 7 /2)
S(d5 / 2 )
=
0.13 S(1f 7 /2)
d) Shell Model Implications and Interpretations
On the basis of the shell model, Sc4 1 consists of filled is-, lp-,
ld-, and 2s-shells and a twenty-first proton in the If7/2 shell in the
ground state; this is consistent with the known spin and parity (7/2)
of the ground state (Cr 62).
As the higher excited states of Sc4 1 are
reached, they would be expected to include the
2 p3/
2
, 2pl/ 2 , if5 /2 shell
model configurations among the states with low excitation, but the exact
location of these single particle states cannot be predicted.
Because
of the relative stability of the Ca4 0 core (the first excited state lies
41
at 3.35 MeV), one might expect that the low-lying excited states of Sc
would be formed by the promotion of the extra proton to these orbits.
56
The corresponding energy levels would then directly give, among other things,
41
a measure of the magnitude of the spin-orbit coupling in Sc .
Sc4 1 is somewhat unique among nuclei because of its small proton
separation energy; there are no bound states other than the ground state.
The measured Q-value of the (He3 ,d) reaction is -4.406±0.010 MeV which
(The proton
corresponds to a proton separation energy of 1.087 MeV.
separation energy of He 3 is taken to be 5.493 MeV).
41
It is possible to attempt some comparisons of the levels in Sc
with those seen in Ca4 1 by Bockelman (Bo 57) and Belote (Be 64).
ground states of both nuclei are known to be k = 3, J
= 7/2.
The
The level
at 1.718 MeV has been observed in the Ca4 0 (d,n)Sc 4 1 reaction (Ma 61)
and in the Ca4 0 (He3 ,d)Sc 4 1 reaction (We 60) and was assigned k = 1
by Wegner and Hall.
The orbital angular momentum necessary for the
"best" fit by DWBA calculation is in agreement with their value.
This
state is identified with the two low-lying kn = 1 nucleon transfer states
in Ca4 0 at 1.949 and 2.471 MeV (Be 64).
(See Figure 14).
4 1 compared with
The spectroscopic factors for these two states in Ca
the spectroscopic factor for the 1f7 /2 ground state are 0.94 and 0.28;
the total spectroscopic factor for the
ground state.
2 p3/
2
is 1.22 times that for the
The shell model state is at 2.08 MeV (Be 64).
It might
then be expected that the mirror state in Sc4 1 would have a similar
relative spectroscopic factor.
The relative spectroscopic factor for
the first excited state of Sc4 1 is 1.2, assuming a
2 p3/
2
proton transfer
occurs for the formation of that state.
Brown (Br 63) has indicated the gamma-ray energy width of the 2p3/2
state in Sc4 1 to be less than I eV.
This indicates that the lifetime
of this state is long, likely because of the high barrier for proton
0
A=41 MIRROR NUCLEI
ENERGY SPECTRUM UP TO 3 MeV
4.12
L2p/
2
2
2.967
2.890
2.972
2.882
2.677
2.677
2.588
2.582 2.612
2.471
=2
2.415
*
ln=l
2.08
2.017
n =22
n
1.949 ln=l
2p3/ 2
ln= 3
41
.72
If7 /2
Sc 4
Ca
*
Figure 14
1.718
1
41
Not Seen In Ca4 0 (p,() Sc
IM
This observation supports the DWBA calculation assumption that
emission.
It is not possible to evaluate
this level can be considered a bound state.
the assumption of a bound state for the level at 2.415 MeV.
States at
about 6 MeV excitation in Sc'l however, have an energy width of 50 to 250
keV as seen in the Ca4 0 (p,y)Sc 4 1 experiment, an indication that proton
emission is still somewhat inhibited by the Coulomb barrier.
In the Ca4 0 (p,y)Sc 4 1 reactions (Yo 64), no gamma rays corresponding
to a level at 2.415 MeV were observed.
The most intense gamma radiation
should come from E-1 transitions to the ground state of Sc4 1 (7/2-)
= 2, 5/2
i.e. from k
+
and from Z
p
p
= 4, 9/2
+
and 7/2
+
states.
a possible way of telling the difference between k = 2, 5/2
This provides
and 3/2
states, the latter of which must de-excite by M-2 plus E-3 radiation.
However, since all the excited states of Sc'41 are unbound, the state may
decay by particle emission before radiation can occur.
This means that
the absence of an observed radiative transition does not necessarily
imply that the k = 2 state is 3/2
for a 5/2
but presence is a strong indication
assignment.
It is expected that in Sc'41 there should be a state corresponding
to the 2
n
= 2 state at 2.014 MeV in Ca4 1 ; this state has a relative
spectroscopic factor of 0.08 with respect to the ground state (Be 64).
The "best" calculated agreement with the experimental data for the level
at E =2.415 MeV required the assumption of an 9 P = 2 transfer leading to a
relative spectroscopic factor of 0.2 (if the spin corresponds to a d3/2
state).
Macefield et al. (Ma 61) report that the state they observed
at 2.476 MeV could be kP = 1 or 2 proton transfer.
Armstrong and Blair (Ar 64) observed a weak level at 2.28 MeV excitation
in the Ca48(He3,d)Sc49reaction at 12 MeV (peak cross-section of 0.27 mb/sr)
which resembled the prediction of a distorted wave calculation for an
S=
2 transfer indicating a very small admixture of (id
could be in the Ca4 8 ground-state wave function.
3
/2)
2
configuration
The existence of a
level at an excitation energy of 2.415 MeV in Sc Il with a peak crosssection of 0.45 mb/sr may be the analogous state in Sc
1
.
On theoretical grounds, the levels in Sc4 1 formed by £P = 3, 1, 2
would correspond to the formation of the single particle states in if-,
However, any weakly excited ZP = 2 level lying at low
2p-, 2d- orbits.
excitation could be an instance of the stripping of a proton to a Id-state,
if the ground state of Ca4 0 contains some 2 particle-2 hole excitation,
say (1d
a22
3
/2 2 ) (1f
7
/
2
2
).
It is interesting to note that the level in Sc4 1 which appears to
have the major part of the 2p3/ 2 strength is depressed in energy below
the probable mirror state in Ca 4 1 .
This gives the probable
2
p3/
2
-
If7/2 proton single-particle separation energy to be 1.72 MeV in Sc'41 as compared with the neutron single-particle separation energy of 2.08 MeV in
Ca C41
Thomas (Th 52) has shown that this depression can be explained,
while still retaining charge independence of nucleon forces by a difference
in the single nucleon wave function outside the nucleus; this becomes
particularly marked when lightly bound particles occur in nuclei.
I I
4.
The Ca40(He 3 ,a)Ca 39 Reaction
a) Previous Work
Recently, the energy levels of Ca 3 9 and angular distributions of
the reaction products leading to Ca 3 9 from single nucleon capture from Ca 4 0
have been
given.
The Ca4 0 (He3 ,a)Ca 3 9 reaction up to 5.1 MeV excitation
has been carried out by Hinds and Middleton (Hi 60) using He 3 ions at 10.1
MeV bombarding energy.
No angular distributions have been reported but
the measured Q-value was 4.960±0.040 MeV.
The Ca 4 0 (p,d)Ca
39
reaction
has been carried out at 30 MeV (Ka 63, Ca 64), and 155 MeV (Ba 63).
Kavalovski et al. (Ka 63) report levels at 2.6, 3.9, 5.0, 6.3, 8.3, 9.0,
10.0, 10.9 MeV.
The ground state reaction occurs through an
neutron transfer.
The levels at 2.6, 3.9 MeV appear to be k = 0 and
n
the level at 6.3 MeV, £
level at 5.0 MeV.
nn = 2
n
= 2.
No k
n
assignment could be made for the
Cavanagh et al. (Ca 64) reported levels at 0, 2.6, 6.0,
8.3 MeV; levels in Ca 39 at 2.5 and 3.0 MeV (unresolved), 4.5 and 7.0 MeV
(unresolved), and 8.0 MeV were reported by Bachelier et al. using 155 MeV
protons (Ba 63).
In general, the (p,d) reaction is rather difficult to perform.
The
Q-values are usually less than -10 MeV, requiring the use of an accelerator
with a moderately high energy proton beam.
The problem of high proton
background also exists in these reactions.
The super-allowed ý -decay Ca 9(o )K 39 identifies the ground state
of Ca 3 9 as 3/2
(Ki 58).
b) The Ca4 0 (He3 ,a)Ca 3 9 Reaction
A summed energy spectrum of alpha particles from the reaction
Ca 4 0 (He 3 ,a)Ca 3 9 to an excitation energy of 6.2 MeV is given in Fig. 15.
Only the reactions leading to states at 0.0, 2.471, 2.787, 5.124, 6.149
MeV in Ca 39 were intense enough to give an angular distribution.
The
_______________
y _____
*
C -
-1YIELD SUMMED OVER SEVERAL
ANGLES
39
Ca40 (He3 , oc) Ca
TARGET = Ca 40 C 03
EXPOSURE = 3134 pC
EHe3
12.00 MeV
SPECTROGRAPH FIELD = 10338 GAUSS
150
E
=
Ca5 9
(I)
100
E
50
0o
F
00
o 9
0)
0.V:
°
ooo
o
I
I
6500
6000
0
0
0
o
0
1
ob
0
00
,13
o0.1
J,
o,,
0
5000
EXCITATION
5500
o
0r
•
g°oo,,--o
0
"
0
"
° >
6'•C•
°
o
----
-- na~----,
----
i
~a~ClslC-~LIII
----
..- ~I
-·I
L-
-
I
r
---
-L---~i--.
~-~iYaii·.l
I
I
I
4000
4500
ENERGY (keV)
·
----
(3)
0D
a
o0,
o
(7) (6)(5)(4)
(8)
3000
3500
·
-.-..Y'.
-.
-·-·-----··-·- · ·-- ·-·---- ------ -- ----- -
·e
·--
-·-~I
2500
·;--l:i;--~..·--·
·-- -
energy spectrum was summed over about eight separate angles in each group
as indicated on Fig. 15.
The existence of large reaction yields from
contaminant reactions together with the long tails behind the main peak
made it necessary to use different groups of angles in the summation.
Since the tails never completely disappear, a particular angle would not
be included again in the summation until the background count was low and
constant (about 2 tracks per half-mm strip).
The angles summed over in
a given group are listed below: (half angle is omitted, e.g. 37.5- 37)
Table IV
Group
Angles
A
30, 37, 45, 52, 60, 67, 75, 82
B
22, 45, 52, 60, 67, 75, 82, 97
C
22,
D
60, 67, 75, 82, 89, 105,
E
45,
67, 75, 82, 89, 105, 120
F
22,
52, 75, 82, 89, 105, 120
30, 52, 60, 67,
75, 82, 97
120
Angles Used in Summation of Alpha Particle Yield
Nine excited states in Ca 3 9 were definitely identified at 2.471, 2.787,
3.632, 3.812, 3.871, 3.939, 4.016, 5.124, 6.149 MeV, several other
possible states have been indicated in Fig. 16.
The uncertainty about these
latter states comes from their low yield even when summed over several
angles and from the fact that when contaminant levels are averaged in this
way, they can produce a level which is too broad when compared to the other
yields from states in Ca39 .
· I
WE
_·
=
(In 1,3)
6.149
(5.74)
(5.54)
n =-2
5.124
5.071
5.13
5.07
(4.90)
(4.82)
4.92
(double ?)
(4.43)
(4.33)
4.71
4.61
4.49
4.43
4.32
---- - --
In= 2
4n=
5.168
5.010
4.928
4.737
4.678
4.511
4.472
4.122
4.092
4.078
3.935
4.016
3.939
3.876
3.812
4.02
3.95
3.88
3.84
3.632
3.66
(3.0)
3.032
3.021
2.787
2.799
2.817
2.471
2.473
3.879
3.603
0
39
CQ
PRESENT WORK
2.526
Ca39
K39
HINDS AND
MIDDLETON
SPERDUTO AND
BUECHNER
Figure 16
64
The alpha particle angular distributions for the five most intense
states are shown in Fig. 17 to 21.
In these figures, the experimental data
are shown with laboratory differential cross sections plotted versus laboratory angle; a 10% uncertainty is assumed in the relative cross sections.
Each angular distribution is identified by the excitation energy of the
corresponding level in Ca39
.
The smooth curves are the predictions of
the distorted wave Born approximation.
The angular momentum of the stripped
neutron from Ca4 0 which would give the "best" agreement of the DWBA
calculation with the experimental data is also shown.
The validity of
the DWBA assignment will be discussed later.
The Q-value of the Ca40(He3 ,a)Ca3 9 ground state reaction, obtained
from an average over several different angles in the multiple-gap spectrograph, is 4.939±0.010 MeV.
This is in agreement with the value of 4.960
±0.040 quoted by Hinds and Middleton (Hi 60).
The excitation energies
are known with an uncertainty of 0.010 MeV.
Figure 16 summarizes the results of the experiment and makes a comparison with the results of Hinds and Middleton for the Ca40(He3 ,a)Ca3 9 reaction at 10.1 MeV and the energy levels of K3 9 as observed in the K 3 9 (p,p')K 3 9
reaction (Sp 58).
The values of k. are those which produced the best
n
calculated agreement with the experimental data.
c) The Assignment of k. and the Estimation of Relative Spectroscopic Factors
n
The Ca4 0 (He3 ,a)Ca 39 case was treated as the stripping of a neutron
from an alpha particle to combine with Ca3 9 to form Ca4 0 in the ground
state.
The incident energy of the alpha particle was adjusted for each
nuclear state so that it
alpha particle in
would have the same center-of-mass energy as the
the actual (He
3
,a)
reaction.
The binding energy of the
neutron to excited states of Ca 3 9 to form Ca4 0 in the ground
state is
I
-
t
-
ic\.-
-_-
5.0
E
ja
b~
2.5
ON
2.5
Lab. Anqle (Deqrees)
w
--
39
5.0
Ca40 (He3, cc) Ca
Level No. I
Q = 2.468 MeV
Ex = 2. 471 MeV
In:= O
0
-j
2.5
-
V
b
V
n
L
uc)
I
I
30
I
I
60
w
90
120
18(
150
Lab. Angle (Degrees)
............
.
----
I
I.
...
..-
"
~I~
1.0
39
Ca40 (He 3,c) Ca
Level No. 2
Q = 2.152 MeV
Ex =2.787 MeV
n:= 2
..0
iJ
V
0.5
o(
30
60
90
120
I
30
Lab. Angle (Degrees)
-" ' ~Y^
- '~~~----- ~~~~~~-~11-~--1'~'
;--I
-u^~'~~--·~· -·
- --_I
·. ~L.·-L-l-_l~ii~LL.
I iiTi ·_i ~
I
---- --I~LL-
"'
_.--.-
· i _I- · .
·-. -i--II · _-ii~l_~~l-.i
L
4
I~-
1.0
0.5
30
60
120
90
150
180
Lab. Angle (Degrees)
___
~___,
,,
~·*-;aanrrrrr
L-- ---~
-cllC·II~--~.LY
.r.~hsLdacl;Ll~a~i'
.... ._i~.L,.
._~
._..__~_....._._~ .__....~-l.--i -~'-
--
-r-
"' r- '
'
-~.Y..--.Y--,
1III····---··-
--~
lit--
--
1.0
h.
U,
05
b
-o
d'
0
_
60
30
·
;r
~ ~'~
~i~iii~iCS~~-l~~-.~.1.
-
-·
90
Lab. Angle (Degrees)
~--
---
·
-·---r
-
~-
·-
-·-····---
120
150
180
I
I
I
different from the neutron separation energy of Ca4 0 in the ground state;
it is that separation energy plus the excitation energy of Ca3 9 which is
the appropriate binding energy for these inverse reactions.
The DWBA parameters used in calculation of theoretical angular dis39
39
tributions for Ca4 0 (He3 ,a)Ca 39 are listed in Table V. The Ca (a,a)Ca
parameters are the suggested parameters of Hodgson (Ho 62).
The cutoff
radius was taken to be that value which produced the "best" fit to the
ground state angular distribution.
The depth of the Saxon well for binding
a neutron to Ca39 with the correct binding energy is given for completeness.
A calculation for the ground state angular distribution with no cutoff
in the integration did not produce an acceptable fit.
This indicates that
in the DWBA code, using the zero range approximation, the nuclear interior
contributed strongly to the reaction angular distribution.
However,
it is generally believed that the effect of the nuclear interior should be
considerably weakened in a more exact calculation using a finite range.
It was thus felt that it was more physically reasonable to presume that
the dominant contribution to the interaction should occur in the region
close to the nuclear surface; in fact, the calculations indicated that
a radial cutoff in the vicinity of the nuclear surface was necessary
for even qualitative agreement and that the "best" agreement was obtained
for a cutoff about one-half fermi outside the nuclear surface although
good qualitative agreement was achieved for a cutoff about one fermi
inside the nuclear surface.
The shape of the distribution at the forward angles
was not too sensitive on the precise value of the cutoff but the magnitude
of cross section could vary up to a factor of two depending on the exact
cutoff value.
The effect of a spin-orbit potential was not considered in the
Bound State Proton
Ca 40(He 3,He 3)Ca
4 0
Ca 3 9( c, o)Ca3 9 Q=4.939MeV
Q=2.468MeV
Q=2.152MeV
Q=-0.185MeV
Z =2
n
, =0
n
z =2
n
k =2
n
51.4
56.1
53.5
57.8
Q=-1.210MeV
[
n = 1,3]
135
135
83
43.1
40.
W (MeV)
7.1
12.
RA-1/3(f)
1.54
1.7
1.25
1.25
1.25
1.25
1.25
a (f)
0.658
0.65
0.65
0.65
0.65
0.65
0.65
(MeV)
Binding
21.8
17.1
17.1
15.7
Energy
21.8
(MeV)
Cutoff Radius
6.0 f
Table V
3
3
40
DWBA Parameters for Ca (He ,a)Ca
9
1
- -;-- I~---,
~-;j~j--·-~--~-cl-~-~--cE~-~n~--~----~- -------
I
--
--
'-=
-*-*l·r*lrr~a.rr.s*--I
-
~3PH
II
-
-·
-I
I
~-----~--~~---
--
I
reaction angular distribution.
The exclusion from the calculation of a
spin-orbit term by Alford et al. (Al 64) in 0 1 6 (He 3 ,a)0
small effect on the angular distribution.
t
I.
15
had only a
However, the remarks by Lee
and Schiffer (Le 64) on the j-dependence of the backward angle cross
sections may still be applicable to (He3 ,a) reactions (Le 64).
The comparison of the angular distributions of alpha particles from
the various excited states with the angular distribution from the ground
state can give the spectroscopic factors of these states relative to the
ground state spectroscopic factor.
However, it must be remembered that
these relative spectroscopic factors can be meaningfulonly so far as the
DWBA calculation and the approximations associated with the calculation
are valid.
Ex
S(nij)/S(ld3/2)
(MeV)
2.471
0.24
(A = 0)
2.787
0.11
(2 = 2)
5.124
0.068
(2 = 2)
6.149
0.018
(2 = 1)
0.071
(2 = 3)
d) Shell Model Implications and Interpretation
In the (He3 ,a) reaction on Ca4 0 , a neutron can be lifted from the
2s- and ld- shells which ordinarily are completely filled.
residual nucleus with a hole in these shells.
This leaves the
On the basis of this simp-
lified picture one would expect three alpha particle energy groups arising
from these hole states and having spin values 3/2 , 1/2
,
5/2
corresponding
-1
-1
-1
to the configurations 1d3 / 2 , 2s1 / 2 , Id5/
.
According to the shell model
-1
the energy state corresponding to Id3/2 should have the lowest energy.
3/2
In fact, evidence from the
+ -decay of Ca 39 to K 39 indicates this state
(the ground state) to have the spin 3/2
(Ki 58).
The spin-orbit splitting is usually assumed to be about 5 to 6 MeV
in this region.
-1
Thus, one would expect the d5/2 state to have this
5/2-
separation from the ground state.
The 2sl/2
state is expected to lie
-I
somewhere in the energy region between the two Id1 states.
Kavalovski et al. (Ka 63) have measured the reduced widths for
levels for the ground state of the Ca40(p,d)Ca
39
reaction.
The reduced
width 62 is related to the spectroscopic factor, specifically 62 = Se 2
0
where S is the spectroscopic factor.
62 is a measure of the probability
0
that, if the final state is composed of the initial state plus one nucleon,
the two components will actually unite.
Macfarlane and French (Mac 60)
list the values for these probabilities.
A comparison between the first excited state and the ground state
can only be made because of differences in the experimental resolution.
0 2 (ld3/2)
3/2
= 0.085
62 (2s/2) = 0.044
1/2
S(1d3/2
) =
4
8 2 (ld) = 0.021
0
62 (2s) = 0.04
0
S(2sl/2)
= 1.1
The relative spectroscopic factor, 0.28, would tend to confirm the
value 0.24 from the DWBA calculation presented here.
The results of
Cavanagh et al. (Ca 64) indicate this level to have a relative spectroscopic factor of 0.48 (much closer to the theoretical value of 0.50).
It must be noted that the
Z= 0 state measurement by Kavalovski
contains only about 50% of the expected single-hole strength indicating
that there could be other weak Zn = 0 states which have not been observed
39
is 2.47 MeV;
The probable Id3/2 - 2s1/2 energy separation in Ca
this compares favorably with the value 2.65±0.2 MeV reported by Kavalovski
(Ka 63) and with the finite nuclear matter calculation of 2.6 MeV by Brueckner
(Br 61).
The existence of any low-lying
n
n = 2 state is probably not evidence
-1
for Id5/2
5/2
since a comparison of single-hole states in neighboring nuclei
-1
would indicate that the ld5/2
5/2
ld
-1
3/2
state (Co 63).
strength should be about 7.0 MeV above the
Such a state could possibly arise from the coupling
-1
+
4
hole with a 0 core excitation of Ca4 0 .
3/2
of a d
At excitation energies
in Ca 39 about 3.0 MeV, one might expect to be able to interpret levels
-1
40
core.
in terms of the coupling of a d3/2 hole to an excited Ca
If the
core is excited to a state with J > 2, one would expect four states in
this region.
V.
3
The Validity of the DWBA Method in He -Induced Reactions
a) The Assumptions in the DWBA Method
The use of the distorted wave Born approximation method of data
reduction as presented by Bassel, Drisko and Satchler (Ba 62) can result
in the extraction of absolute spectroscopic factors from the experimental
The
cross-section for (d,p), (d,n) and the inverse pickup reactions.
calculations have to be adjusted if other than very low components of
internal momentum in the deuteron contribute to the total momentum
transfer to the target nucleus.
If low values only of the nucleon internal
momentum in the deuteron contribute to the nucleon transfer, then the socalled zero-range approximation becomes exact.
The zero-range approximation
in the standard (d,p) case consists of the assumption that the stripping
reaction is local; that is, the separation of the neutron and the proton
in the deuteron is zero when the stripping takes place.
The introduction of
a small separation (represented by a Hulthen form) in the deuteron results
in an increase in the overall cross-section.
It is necessary to view the implication of these assumptions in
He 3 -induced reactions.
They may be summed up as follows.
The He 3 nucleus
is assumed to be a point particle composed of a deuteron plus a proton.
3
The effect of a finite size for He is expected to result in a constant
factor in the calculation.
3
The break-up probability of He into a deuteron
and a proton is assumed to be constant, independent of momentum transferred to the target nucleus.
For He 3 the probability of finding a deuteron
and a proton has been found to be about 73% (Cu 62) compared to the
probability of finding H 3 as a deuteron and a neutron of 11% (We 56);
3
3
the maximum theoretical probability for the breakup of He or H into
a deuteron plus a nucleon is 75%.
The probability of finding He 4 as a
neutron plus He 3 has recently been found to be about 14% (El 64).
There
is assumed to be no rapid variation in the probability of finding nucleon
internal momentum in He 3 or H 3 , at least in the region of total momentum
transfer of interest here.
The interaction between He 3 and neutron in
He 4 and between deuteron and proton in He 3 is assumed to be effective only
when the separation is zero (zero-range approximation).
The effect of a cutoff radius, the sensitivity of the distribution
to the value of 'nk' of the transferred nucleon, and the sensitivity
of the calculated angular distribution to the Q-value are examined in
Figures 22 to 26.
The calculated DWBA distributions for the (He3 ,d)
stripping reaction give qualitatively the same shape for the angular
distribution although variations of up to 20% can be seen depending on
the exact value of the cutoff radius used in the integration (Fig. 22, 23).
For the (He3 ,a) reaction the use of a cutoff radius produces a substantially different result from the no cutoff case.
to produce the acceptable fit.
The cutoff was necessary
The shape of the distribution is qualita-
tively the same at the forward angles when a cutoff is used but a factor
of two depending on the exact radius used can occur in the calculation.
In Figure 24 a variation in the peak position by about ±7.50 will mean
This is illus-
an uncertainty in the transferred angular momentum of ±1.
trated in Figure 25.
The Q-value dependence on the shape of the calculated
distribution is small but the calculated peak position may shift by 100
(Fig. 26).
No spin-orbit potential has been considered in these calculations.
The spin-orbit effect on the angular distribution has been seen in the
backward angles (Le 64).
The only case where a comparison could be made
-1
in the states excited in this experiment would be the ld3/2
hole states in Ca4 0 .
-1
and ld5/2
Unfortunately, the statistical uncertainty in the
__
-
~-~---~~--
15,
c-
:D
>,0
L..
I,,
CD
0
n5
b
O0
30
0
i
~ -- l.~=I-~.li;-i~;TiiS~;;l~~~i~~.
i~99-"I·..
~
60
90
Reaction Angle (Degrees)
._L_.__ I---
lili----~-C~~~~~~fll_
_i g
_
lii_
120
fl~.
Y~nM
150
i.. .
. J-ýLIULLMA !I&l
180
i _--
i__·
)Lpp~----~-~-·-
15
-
Nuclear Cutoff Sensitivity
Relative DWBA Calculation
Ca4 0 (He3, d ) Sc41
Ex = 1.718 MeV
No Cutoff
Cutoff At 4.0f
Cutoff At 6.0 f
U,
4-
10
D
-
'
0
4-
n
p= I
' \
\
0
\,
,c~
5
~0
V
-
JI
-
--
aJI
I .'
.
,
.
.-
i.
"
..
180
150
120
90
60
Reaction Angle (Degrees)
30
tt i
-
-
...
.
. .. ."-'.
...
.....
.
L
15
Nuclear Cutoff Sensitivity
Relative DWBA Calculation
I
c/
7\
S.
\\
/
\
/
/
10 0
Ca 4 0 (He 3 ,oc) Ca39
\
\
"
A
..
No Cutoff
. Cutoff At 4.0 f
Cutoff At
.
"
\
I.\ H
Ground State
6.0 f
4
=2
\.
5N
bo
00
O0
I
S30
I
I
I
I
" -".. .I
90
60
Reaction Angle (Degrees)
'--•.
--
120
r -
--
150
---,,,-- -
180
15
n
In- VALUE SENSITIVITY
NORMALIZED DWBA CALCULATION
Ca40 (He3,oc) Ca39
Q =2.468 MeV
I
i-
SI
0
2s
2p
lb
I'
I
I
l
lb
Id
If
lb
lb
t
I
5
H-
-\
E'
0)
-
-
Constant Neutron Binding Energy
At 17. IMeV
\
-
1
Cutoff at 6.0 f
-
0
-- '
Qn= 0
In = I
An=2
In=3
·---
30
-·.
60
90
Reaction Angle (Degrees)
-- -I
I-IC;-~i
120
II
--
L-
---
-~-~~---L1I~--~-,-
7C-Y---C
V
--
C'
Q-VALUE
y
-~-
~--
SENSITIVITY
RELATIVE DWBA CALCULATION
Ca40 ( He3 , <c)Ca3 9
10
In= 2
-
4-
Q = 4.939 MeV
00
--
-
e•V
Q= 2c..2 Mr-
\
-o
%m
<
5
[Ca 3 9 (OC, He3 )Ca 40 C olculation]
Cutoff at 6.0 f
I
I
30
~L- ~
L
I
I
I
,
90
60
Reaction Angle (Degrees)
.^-~-.I-L~~--
~
·.
_~__._·a~~dPillY
I
-
120
·- · ~--1 ·
-
Illll··UII=
_
_
'isons from being made.
eraction in relation to compound
nucleus formation is difficult to evaluate.
One could be more certain
of a direct interaction for an incident energy substantially above the
coulomb barrier or in the range above 30 or 40 MeV.
The assumption of a bound state for the "almost bound" states in
Sc 4 1 is probably valid in view of the relatively small gamma widths for
de-excitation of Sc41 .
40
3
The Woods-Saxon parameters for the incident 12 MeV He ions on Ca
are probably not unique.
It was found that if
the ABACUS-2 program was
allowed to search over the values of nuclear radius and nuclear diffusiv2
2
ity parameter simultaneously for the minimum 'X ', the same X could be
obtained for a set of parameters covering about 0.5 fermi.
It
was decided
that the best approach would be to not allow these two to vary at the
same time but attempt a minimum search with one fixed and then with the
other fixed and "ease" up to the "best" optical model parameters.
It
was assumed that in these calculations that the charge radius was equal
to the nuclear radius;
this assumption is not inconsistent with the
results of Klingensmith (Kl 64).
The program of data analysis used here has been very inflexible because
it was felt that excessive adjustment of the parameters to achieve the
"best" fit to the experimental angular distributions would result in a
very ambiguous result.
After the cutoff radius for the integration had
been determined it was kept constant in all calculations.
The only
variable parameters for a given state of a given binding energy were the
values 'nk' of the bound state nucleon and the depth of the Saxon potential
necessary for binding the nucleon to the target with the correct binding
energy.
P
b) The Interest in
The measured aný
reactions in this wo:
lations toward the bý
direct interaction.
The high Q-valu(
reactions make possil
However, the reactior
icantly below the col
accelerators, reactic
periodic table.
Thei
to carry out any det,
or in the nuclei neabe particularly simp.
taken before He 3 -indi
induced reactions in
Once the mechan'
some information on 1
three body problem ot
The nature of tl
of a single nucleon c
More data in the
direction would be us
particularly to exami
in the potential.
It is hoped thai
finite range effects
84
experiments, will result in a better understanding of the spectroscopic
factor, particularly its variation as a function of the incident energy
of the projectile.
4
__~__
__
I
_
APPENDIX I
3
The Operation of the MIT-ONR Electrostatic Generator with He
-
Ions
Before the modifications in the terminal of the MIT-ONR generator
were undertaken, a series of preliminary tests were carried out in order
to obtain an estimate of the beam current that could be expected and
an indication of the source gas pressure that would produce the maximum
He3++ beam.
3
A search for the doubly-ionized component of the He beam was
undertaken using the maximum radio-frequency power (about 8 watts) that
could be coupled into the ion source.
The minimum He 3 gas pressure that
would keep the source stable was picked.
It had been reported (Br 59)
that this pressure would give the best results for both singly and doubly
ionized beams.
Evidence for contamination of the He 3+ + beam with components
of hydrogen and deuterium has been obtained in previous experiments (He 61)
+
in this laboratory. This HD component of the beam was expected to be
troublesome but was not expected to completely prevent the observation
of He 3
.
The difficulties of low power available and gas contamination meant
that any beam detected ought to be the lower limit on the expected beam
after terminal modifications.
A small beam was detected
on the control
slits but the peak current was less than one nanoampere and the beam
was quite unstable.
It was not clear that this beam was doubly-ionized
He 3 .
The hydrogen and deuterium contamination probably originated from residual
gases in the ion source and from cold leaks through the palladium gas bottles.
The hydrogen gas leakage was eliminated by inserting solenoid valves in the
_
_
~~____~_
__
hydrogen gas lines as indicated in Fig. 27.
__
When the hydrogen or deuterium
palladium leak is not heated the solenoid valves are not activated; this
guarantees no hydrogen leakage.
The solenoid valves are enclosed in
stainless steel cans filled with helium gas at one atmosphere pressure
so that if any small leaks should develop in the stems the pressure
differential across the stem would be only one atmosphere (the tank gas
pressure is about 13 atmospheres).
Also, if helium gas should leak into
the source it would be as equally difficult to ionize as He 3 and this
type of leak should not cause the same source problems as hydrogen
(which has the effect of lowering the plasma temperature).
A new radio-frequency power supply, capable of delivering 100 watts
at 100 mcs, was installed so that considerably more power was available for
raising the plasma temperature.
The circuit is similar to one described
elsewhere (Sa 62'). The major difference in design was in the use of
lower power ceramic tubes (RCA type 7203/4CX250B).
A new "long source" ion source bottle (available from HVEC*,
type SO-77)
was installed together with a solenoidal magnet providing an axial
magnetic field.
This field increases the electron-He 3 atom collision
probability and ought to result in a higher percentage of doublyionized output.
A similar source geometry has been described previously
(Be 62) and a more complete description is to be published (Sa 62').
In order to obtain maximum doubly-ionized He 3 , it was necessary to
operate the source at close to maximum output; this also resulted in
a considerable current of singly-ionized helium together with a large
secondary electron loading on the terminal.
This limited the potential
of the terminal to around 6MV in order for stable operation.
* High Voltage Engineering Corporation,Burlington, Massachusetts
I
--
-L31141Lllt
-·- -
_ I
,UGE
)N SOURCE
fnul\E
VL.Vr
VIE
CYLINDER
SOLENOID VALVE OPENS ONLY WHEN PALLADIUM LEAK IN CORRESPONDING
GAS CYLINDER IS BEING HEATED
-
_
~
~
~~
~__
~
I
_
________
_I__) ___I__ ______
During the experiment, a maximum beam current of 26 nanoamperes was
attained, and typical operating currents were 10 to 15 nanoamperes of
He 3+ +
.
A "ghost" beam about 300 key lower in energy was observed when
it activated the stabilizing equipment but it did not record on the
current integrator.
The reason could be defocussing of this beam in
the electrostatic quadrupole lens.
The exact content of this "ghost" beam
is uncertain at the present time.
After the experiment (about 100 hours operation) it was noticed
that there was more damage to the source walls and canal than after
100 hours of normal operation with hydrogen and deuterium beams.
_____~__
_;_
_~___
_
q
I
·i-
APPENDIX II
Suggestions for Improvements in the Experiment
There are several suggested improvements for these experiments.
Enriched calcium carbonate targets, while certainly preferred over natural
calcium targets, are not the best targets for helium-3 induced reactions.
Self-supporting Ca4 0 targets, without carbon or oxygen compounds, in the
target or as backing, are more desirable since the reaction yields from
C 12 and 016 obscure wide areas of the energy spectrum.
In an effort to distinguish between the various nuclear reaction
products, it was felt that some Kodak NTA 100p nuclear emulsions would
provide better track length discrimination, at least at the forward angles,
and at those energies where the nuclear projectiles would completely
traverse a 50p emulsion.
Unfortunately the 100p emulsions did not prove
too satisfactory for several reasons.
They did not respond as well as
the 50p emulsions to the incident ionizing particles, with the result
that the grain density in the emulsion was somewhat lower than expected.
This may have been further complicated by the time required to perform
the experiment (about 250 hours),
further fading.
a factor which possibly resulted in
Proton tracks, were almost invisible in those 100p plates,
and the alpha and deuteron tracks were considerably fainter than in the 50P
emulsions.
There is also a problem involved in the adhesion of emulsions
to their glass backings.
A test development of the 100p emulsions was
carried out after one-half the total exposure.
The emulsions showed a
tendency to lift from the glass if the normal drying procedure was
1
90
followed.
A few glues were tried to keep the emulsion down but the best
procedure to save the emulsions was to use a mixture of Canada Balsam
diluted with benzene and to paint the emulsion-to-glass boundaries.
An experiment using aluminum foils in front of the emulsions would
have permitted a better track length discrimination and would have made
possible a distinction between He 3 and deuteron tracks, a problem which
occurred only in a small portion of the energy spectrum.
The experiments used the thickest Ca4 0 C0 3 target that was available
at the time but a further increase in target thickness would be desirable.
Recently we have been successful in producing thicker, self-supporting
natural calcium targets and the problem of thickness has been overcome.
The elimination of carbon and oxygen would make possible longer
exposures if care were taken to restrict their buildup on the targets
during the exposure.
Robertshaw (Ro 61) has discussed this build up in
some detail.
Naturally, an experiment at even higher energies is to be preferred
since this would result in higher excitation in the final nuclei, and the
determination of more spectroscopic data near the closed shell nuclei.
An increase in the total beam current would keep the time required to
carry out an experiment to a minimum.
I
r
I
APPENDIX III
3+ + over He3+
The Advantage of He
For a given incident energy the helium ion beams have a lower velocity
and higher charge than the hydrogen ion beams.
These factors result
in larger energy loss due to ionization in the target films.
The energy
loss can be reduced by decreasing the target thickness but this procedure
results in a more structurely fragile target and lower yield in a reaction
from a given exposure.
The energy dissipated in the target results in an increase in the
thermal strain.
This strain can be reduced by rotating the target or by
increasing the thermal conductivity of the target.
Usually the target
material is deposited on a thin backing film such as formar and previous
experience (He 61) indicates that these films do not stand up well under
the bombardment of a helium-3 beam.
Thin self-supporting carbon films
are especially suited for helium-3 reactions if self-supporting targets
are unavailable or impossible to manufacture.
3
The He3+ + beam does not cause as much a problem as the He + beams
with the same accelerator since almost a factor of two in energy can be
achieved.
The doubly-ionized He 3 has a magnetic rigidity (Bp) which
makes it readily separable from all hydrogen ions which are accelerated
through the same potential.
Then, beam contamination is less of a problem
when a He 3+ + beam is used.
The relative probability of inducing a nuclear reaction is higher
if
a higher energy is available and the Coulomb barrier penetrability
will increase with increasing energy.
This opens up wider areas of the
periodic table for possible experiments.
on
1
VI.
REFERENCES
Ar 64
D.D. Armstrong, A.G. Blair, Phys. Lett. 10, 204 (1964).
Au 62
E.H. Auerbach, ABACUS-2 (unpublished).
Ba 53
K.T. Bainbridge, Experimental Nuclear Physics, ed. E. Segre,
John Wiley and Sons, New York, 1953.
Ba 62
R.H. Bassel, R.M. Drisko, G.R. Satchler, ONRL-3240 (unpublished).
Ba 63
D. Bachelier, M. Bernas, I. Brissaud, C. Detras, P. Radvanyi,
J. Phys. (Paris) 24, 1055 (1963).
Be 62
L.E. Beghian, M.K. Salomaa, Nucl. Inst. and Meth. 17, 181 (1962).
Be 64
T.A. Belote, E. Kashy, A. Sperduto, H.A. Enge, W.W. Buechner,
ANL-6848 p. 1 7 2 (1964) Phys. Rev. to be published.
Bo 57
C.K. Bockelman, W.W. Buechner, Phys. Rev. 107, 1366 (1957).
Bm 56
C.M. Braams, Ph.D. Thesis, University of Utrecht, 1956 (unpublished).
Br 59
D.A. Bromley, E. Almqvist, AECL No. 950 (1959).
Br 61
K.A. Brueckner, A.M. Lockett, M. Rotenberg, Phys. Rev. 121,
255 (1961).
Br 63
N.A. Brown, Ph.D. Thesis, Rice Institute, 1963 (unpublished).
Bu 51
S.T. Butler, Proc. Roy. Soc. (London) 208, 559 (1951).
Bu 61
J.W. Butler, Phys. Rev. 123, 873 (1961).
Ca 64
P.E. Cavanagh, C.F. Coleman, G.A. Gard, B.W. Ridley, J.F. Turner,
Nucl. Phys. 50, 49 (1964).
Co 63
B.L. Cohen, Phys. Rev. 130, 227 (1963).
Cr 62
J.G. Cramer, C.M. Class, Nucl. Phys. 34, 580 (1962).
Cu 62
B. Cujec, Phys. Rev. 128, 2303 (1962).
E1 41
D.R. Elliott, L.D.P. King, Phys. Rev. 60, 489 (1941).
El 64
M. El Nadi, F. Riad, Nucl. Phys. 50, 33 (1964).
En 62
P.M. Endt, C. Van der Leun, Nucl. Phys. 34, 1, (1962).
.. I;.
~~l.l...~-i-T-------.
-~--.~~~-
---
-·----
---------.. ,
I
m
I
I
En 63
H.A. Enge, W.W. Buechner, Rev. Sci. Inst. 34, 155 (1963).
He 59
R.G. Herb, Handbuch der Physik XLIV, Springer Verlag Berlin, (1959).
He 61
H.J. Hennecke, Ph.D. Thesis, Massachusetts Institute of
Technology, 1961 (unpublished).
Hi 60
S. Hinds, R. Middleton, Proceedings of the Kingston Conf. on
Nuclear Structure ed. D.A. Bromley, University of Toronto
Press, Toronto, (1960) reported in En 62.
Ho 53
J. Horowitz, A.M.L. Messiah, J. Phys. Radium 14, 695 (1953).
Ho 60
P.E. Hodgson, Nucl. Phys. 21, 28 (1960).
Ho 62
P.E. Hodgson, Proceedings of the International Conference on
Direct Interactions, ed. E. Clementel, C. Villi, Gordon and
Breach Publishers, New York (1962).
Ka 63
C.D. Kavalovski, G. Bassini, N.M. Hintz, Proceedings of the
Padua Conference on Direct Interactions and Nuclear Reaction
Mechanisms ed. E. Clementel, C. Villi, Gordon and Breach
Publishers, New York, (1962).
Phys. Rev. 132, 813 (1963).
Ka 64
E. Kashy, A. Sperduto, H.A. Enge, W.W. Buechner, Phys. Rev.
(to be published) (1964).
Ki 58
0.C. Kistner, BM. Rustad, Phys. Rev. 112, 1972 (1958).
K1 64
R.W. Klingensmith, H.J. Hausman, W.D. Ploughe, Phys. Rev.
134 B 1220 (1964).
Le 64
L.L. Lee, J.P. Schiffer, private communication (1964).
Ma 55
M.G. Mayer, J.H.D. Jensen, Elementary Theory of Nuclear Shell
Structure, John Wiley and Sons, Inc., New York, (1955).
Ma 60
B.E.F. Macefield, J.H. Towle, Proc. Phys. Soc. (London)
76, 56 (1960).
Ma 61
B.E.F. Macefield, J.H. Towle, W.B. Gilboy, Proc. Phys. Soc.
(London) 77, 1050 (1961).
Mac 60
M.H. Macfarlane, J.B. French, Rev. Mod. Phys. 32, 567 (1960).
Me 62
M.A. Melkanoff, T. Sawada, N. Cindro, Phys. Lett. 2, 98 (1962).
Mo 63
W.H. Moore, Ph.D. Thesis, Massachusetts Institute of Technology,
(1963) (unpublished).
I
-
.
~1-
-~
-i`.
I;.~-.
.
. ~..~I
. -"-
1.1~.11--
P1 55
H.S. Plendl, F.E. Stergert, Phys. Rev. 98, 1538A (1955).
P1 59
H.S. Plendl, F.E. Steigert, Phys. Rev. 116, 1535 (1959).
Ra 63
J. Rapaport, Ph.D. Thesis, Massachusetts Institute of
Technology, 1963 (unpublished) and Phys. Rev., to be published
(1964).
Ro 61
J.E. Robertshaw, Ph.D. Thesis, Massachusetts Institute of
Technology, 1961 (unpublished).
Sa 62'
M.K. Salomaa, Nucl. Inst. and Meth. 15, 113 (1962)
M.K. Salomaa, to be published.
Sa 62
G.R. Satchler, the Code JULIE, unpublished.
Sp 58
A. Sperduto, W.W. Buechner, Phys. Rev. 109, 462 (1958).
Th 52
R.G. Thomas, Phys. Rev. 88, 1109 (1952).
To 54
W. Tobocman, Phys. Rev. 94, 1655 (1954).
To 55
W. Tobocman, M.H. Kalos, Phys. Rev. 97, 132 (1955).
To 59
W. Tobocman, Phys. Rev. 115, 99 (1959).
To 61
W. Tobocman, Theory of Direct Nuclear Reactions, Oxford
University Press, (1961).
To 62
W. Tobocman, private communication (1962).
Va 48
R.J. Van de Graaff, J.G. Trump, W.W. Buechner, Reports on
Progress in Physics XI, 1 (1948).
We 56
A. Werner, Nucl. Phys. 1, 9 (1956).
We 60
H.E. Wegner, W.S. Hall, Phys. Rev. 119, 1654 (1960).
Yo 64
D.A. Youngblood, private communication (1964).
1
_
·
__
_~_I_______~ ___ ___~;_II_1C____
__~__~ __
BIOGRAPHICAL NOTE
The author, Douglas Murray Sheppard, was born September 19, 1939 in
Hamilton Canada.
He received his elementary school and high school education
in West Flamboro Township public schools.
He obtained the degree Bachelor
of Science from McMaster University in May 1960, and was enrolled in the
graduate school of the Massachusetts Institute of Technology from September
1960 to September 1964.
Mr. Sheppard has been a half-time graduate assistant while continuing
his graduate studies.
He was an instructor in the freshman physics laboratory
and a research assistant in the High Voltage Laboratory.
Mr. Sheppard was recently elected to full membership in the Society
of the Sigma Xi and is a member of the American Physical Society and an
associate member of the Canadian Association of Physicists.
m