
Citation 
 Permanent Link:
 https://ufdc.ufl.edu/UF00099393/00001
Material Information
 Title:
 Proton radiative capture by tritium below 30 MeV
 Creator:
 McBroom, Robert Chism, 1947
 Copyright Date:
 1977
 Language:
 English
 Physical Description:
 vii, 93 leaves : ill. ; 28 cm.
Subjects
 Subjects / Keywords:
 Angular distribution ( jstor )
Asymmetry ( jstor ) Center of mass ( jstor ) Crystals ( jstor ) Electronics ( jstor ) Energy ( jstor ) Experimental data ( jstor ) Proton energy ( jstor ) Protons ( jstor ) Signals ( jstor ) Dissertations, Academic  Physics  UF Physics thesis Ph. D Protons  Capture ( lcsh ) Tritium ( lcsh )
 Genre:
 bibliography ( marcgt )
nonfiction ( marcgt )
Notes
 Thesis:
 ThesisUniversity of Florida.
 Bibliography:
 Bibliography: leaves 8792.
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by Robert Chism McBroom.
Record Information
 Source Institution:
 University of Florida
 Holding Location:
 University of Florida
 Rights Management:
 Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
 Resource Identifier:
 026333657 ( AlephBibNum )
04080406 ( OCLC ) AAX4722 ( NOTIS )

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Full Text 
PROTON RADIATIVE CAPTURE BY TRITIUM BELOW 30 MeV
By
ROBERT CHISM McBROOM
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1977
ACKNOWLEDGEMENT
This project was conducted under the guidance of
Dr. H. R. Weller. His patience and direction are deeply
appreciated. A special thanks to Dr. N. R. Roberson and the
members of the Duke Capture Gamma Group for the privilege of
working with them. The author is especially grateful for
the Group's assistance with the acquisition of the experi
mental data. Thanks to Dr. C. P. Cameron for the use of
some of his computer programs. Also, thanks to Mrs. M.
Bailey for her care in drafting the figures, Mr. D. Turner
for the photography, and Mrs. J. Ogden for the typing. The
author is thankful to Mr. D. C. Pound and The General
Atomic Company for their support during the final stages
of the dissertation.
Financial support for this study was provided by the
University of Florida.
Final and most heartfelt thanks to the author's
wife, Mrs. Karen McBroom, for her love throughout all the
trials.
TABLE OF CONTENTS
ACKNOWLEDGEMENTS . . . . . .
LIST OF TABLES . . . . . .
LIST OF FIGURES . . . . . .
ABSTRACT . . . . . . . .
INTRODUCTION . . . . . . . . .
CHAPTER
I. SUMMARY OF LITERATURE . . . .
II. EXPERIMENTAL METHODS AND MEASUREMENTS
Experimental Details . . .
Experimental Data . . . .
III. DATA ANALYSIS . . . . . .
Kinematics . . . . . .
Excitation Curve Analysis . .
Angular Distributions . . .
Results . . . . . . .
SUMMARY AND CONCLUSION . . . . . .
APPENDICES . . . . . . . . .
A . . . . . .
B . . . . . .
BIBLIOGRAPHY . . . . .
BIOGRAPHY . . . . . .
81
S . . . . 85
. . . . 87
92
Page
ii
iv
v
vi
1
LIST OF TABLES
Table Page
1.1 4He Photodisintegration Reactions . . .. 11
1.2 Sum Rules for 4He and Their Shell Model
Representations . . . . . .. 13
1.3 He Experimental Studies . . . . .. 16
1.4 Relations Between the Legendre Polynomial
and Sin 6, cos 6 Angular Distribution
Expansions . . . . . . .. 26
2.1 Measured Differential Cross Sections . . 47
3.1 Direct and Resonance Strengths of E2
Interactions . . . . . . ... 70
3.2 ai Coefficients of Measured Angular
Distributions for 3H(p,y)4He ..... . 73
3.3 Cross Sections Integrated from 22 to 44
MeV Excitation . . . . . ... .77
LIST OF FIGURES
Figure Page
1.1 Excited States of 4He. . . . . .. 7
1.2 4He Photodisintegration Cross Sections . . 24
2.1 Detector Assembly for 25.4 x 25.4cm.
Cylindrical NaI(Tl) High Resolution
Detector . . . . . . . .. .32
2.2 Detector Assembly for 10.2 x 17.8cm.
Cylindrical NaI(Tl) Detector for
Small Angle Measurements . . . ... .35
2.3 Electronics Block Diagram for High
Resolution Detector . . . . . 38
2.4 Electronics Block Diagram for Small
Angle Experiment . . . . . .. 41.
2.5 3H(p,y) He Center of Mass Cross Section
for l1ab = 900 . . . . . ... .46
2.6 H(p,y) He Center of Mass Cross Sections
of 8 = 550 and 1250 . . . ... .46
cm
2.7 3H(p,y)4He Angular Distributions . . .. .53
3.1 Laboratory and Center of Momentum
Coordinates . . . . . . ... .56
3 4
3.2 Asymmetry of the H(p,y) He Angular
Distribution . . . . . . .. .72
3.3 Coefficients of the Legendre Expansion
of o(6) . . . . . . . ... 75
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy
PROTON RADIATIVE CAPTURE BY TRITIUM BELOW 30 MeV
By
Robert Chism McBroom
June 1977
Chairman: Henry R. Weller
Major Department: Physics
The 3(p,y) 4He reaction is studied for incident
proton energies from 830 MeV. The experimental data is
composed of excitation functions measured at the angles of
550, 90 and 125 in the center of mass system in 1 MeV
steps from 815 MeV proton energy and in 0.5 MeV steps from
1730 MeV proton energy. These data measured the energy
dependence of the cross section and its deviation from the
sin 6 angular dependence of the electric dipole (El)
radiation.
Additional angular distributions were taken at
selected energies in the range from 1330 MeV with up to
nine angles measured between 200 and 1420 in the laboratory.
This number of angles was taken to insure statistical signi
ficance for determining the coefficients of the Legendre
expansion of the differential cross section through a4.
The experimental data is shown to be consistent with the
assumption that the observed radiation is primarily El with
an additional electric quadrupole (E2) component and
negligible contributions from higher order multipoles. The
E2 radiation is shown to be consistent with a model of the
E2 cross section composed of the sum of a direct interaction
component, and two BreitWigner resonances. A resonance of
width 12 MeV at a 4He energy of 35 MeV is observed in agree
ment with previous experimenters and a second previously
unobserved resonance is observed with a width of 3.8 MeV
at a 4He energy of 39.5 MeV.
INTRODUCTION
Investigations of the properties of the 4He nucleus
are an important method of improving the understanding of
the nuclear force. Consisting of four nucleons, 4He falls
in a transition region between describing the nuclear
interaction in terms of nucleonnucleon interactions and
statistical descriptions of many nucleon interactions. The
properties of the 4He nucleus show a number of differences
from predictions based on the twonucleon interaction and
the assumption that the twonucleon forces are additive.
Similar difficulties are observed with models that are
based on systems with a large number of nucleons.
The application of models of the nuclear inter
action to 4He seeks to predict the observed properties of
this nucleus. One of these properties is the fact that 4He
is the most tightly bound of all the nuclei.1 Another
property to be considered is the rootmeansquare (rms)
charge radius measured using the elastic scattering of
electrons.24 Indications of the presence of many body
forces are obtained in measurements of the charge distri
bution of the nucleus by electron scattering. The charge
distribution deviates significantly from a Gaussian shape.4
Also to be considered in a description of 4He is evidence
that the ground state is not a pure spin zero (S=0) shell
model state.5
Many models have been presented to describe the
observed properties of 4He. Their success has been varied.
In general the models proceed by incorporating short range
correlations and noncentral forces into the interaction.67
Further information on the importance of ground state
correlations is sought from the angular distributions of
4He photonuclear reactions.9
Several measurements of the 4He photonuclear
reactions have been made. A number of inconsistencies were
observed in the early works, some of which have been
resolved by more recent measurements. The behavior of the
angular distributions has been interpreted to give suppor
ting evidence for the presence of ground state correlations.1012
In addition, evidence exists for the presence of resonant
excited states in 4He whose influence should be considered
13,14
in a complete description of the angular distributions.1314
Evidence for excited states in He is presented in
the analysis of many experiments. This evidence is derived
primarily from reactions in which 4He is formed as the
compound nucleus.15 Evidence for several states has been
obtained from phase shift analyses of nucleon scattering
data. Other states are seen in inelastic electron
scattering.16,17 All of these states have positive energy.
There exists no evidence for bound states of 4He. Isospin
selection rules indicate that the photodisintegration of
4He can be useful in the investigation of negative parity,
J = 1 isospin T = 1 states and positive parity J7 = 2,
T = 0, 1 states. Some studies of the He (y,p) H reaction
have exhibited resonancelike structures, but 3H (p,y)4He
measurements have failed to confirm the existence of the
proposed levels. 1821 Measurements of the He (y,n)3He
reaction present evidence for a J = 2 T = 0 resonance at
a gamma energy of E, 35 MeV.22 It will be shown that the
Y
present work provides additional evidence for the existence
of this level at 35 MeV in 4He.
Most of the 4He photodisintegration experimental
efforts for excitation energy in the range from 2040 Mey
have been devoted to the He (y,n) He reaction. The early
studies of the 4He (y,p) 3H reaction suffered from diffi
culties due to poor statistics. For energies below 35 MeV
where the triton tract becomes too short to measure, the
diffusion chamber and cloud chamber measurements have
difficulty distinguishing protons which arise from multi
11,12
particle final states.112 Studies of the inverse
reaction 3H (p,Y)He have been conducted for limited
excitation energy ranges from 20 MeV to 34 MeV. In the
present experiment measurements of the energy and angular
dependence of the 3H (p,y) He reaction were taken for
excitation energies from 26 to 42 MeV.
The proton beam from the Triangle Universities
Nuclear Laboratory CycloGraaff was used to provide
incident protons in the energy range from 8 to 30.5 MeV.
The beam was incident on a 5p selfsupporting tritiated
titanium foil target containing about 0.14 mg/cm2 of
tritium. The emitted photons were detected with a thallium
doped sodium iodide NaI(TI) detector for up to nine labora
tory angles between 200 and 1420. It will be shown that
these measurements indicate the existence of a previously
unobserved level in 4He at about 40 MeV excitation having
a J7 of 2+ and a width of about 3 MeV in the center of mass.
An appendix is included which describes a peak
fitting program. This program was designed to extract the
low energy tail of the Nal detector response curve from
beneath the peaks due to gamma rays from transitions to
states other than the ground state. The shape of the Nal
response curve was based on the measured response to the
3 4
H (p,y) He reaction.
CHAPTER I
SUMMARY OF LITERATURE
The 4He nucleus has been the subject of intense
interest. The resulting body of literature is extensive
in both experimental and theoretical studies. This body
of literature is reviewed in a number of recent works,
particularly the survey by Fiarman and Meyerhof (1973).15
Many different theoretical models have met with qualitative
success in describing the observed ground state properties
of 4He, but have been unable to satisfactorily describe all
the observed properties. Further input for model deter
mination is provided by experimental studies of the excited
states of He. While there are no bound states of He,
experimental evidence exists for virtual states as shown in
Figure 1.1. Particular consideration is given here to the
photon excitation states having spin and parity (JI) and
isopin (T) quantum numbers as follows:
J= = = 1 1 and
Jr = 2+, T = 0,1
These states have been studied by photodisintegration and
the inverse photonuclear reaction. The experimental
results to date have not definitively established the
Figure 1.1 Excited States of 4He. The energy levels of
He and the general form of the cross sections for the four
reactions, which form He as the compound nucleus, are shown
(Fiarman and Meyerhof5).
b: 7
wr
C I
u~QA
C
c 4
0
Cu
CI d
existence of any of these states; although studies of the
4He(Y,n)3He reaction indicate the existence of a Jn = 2+,
T = 0 state for an excitation energy of 35 MeV.22 In
addition to the question of the existence of excited states,
there are fundamental questions regarding the observed
shapes of the photoproton and photoneutron angular distri
bution as a function of the gamma ray energy. The (y,n)
distribution is predicted to be backward peaked while
experimentally it is observed to be forward peaked for
excitations less than 27 MeV and then to become backward
peaked between 27 and 35 MeV excitation.22 For energies
greater than 35 MeV excitation the angular distribution of
the (y,n) reaction is forward peaked and the asymmetry
approaches the asymmetry of the (y,p) reaction channel.012
The (y,n) forward peaking may be interpreted to be due to
correlations in the ground state wave functions.23
The experimental observations of the photodisinte
gration reaction have limited resolution. These
measurements are averaged quantities in a given energy
interval. One method of relating these averaged quantities
to the properties of the Hamiltonian and the ground state
wave function is the sum rule approach.24 The theory of
sum rules has undergone extensive development starting from
the study of atomic photoabsorption processes. For the
electric dipole (El) radiation in the long wave length
approximation the integrated cross section is given by the
sum rule
El
El = fo(E))dE
00
int (1.1)
= k ek <01 [Zk' [H, Zk]]10>
k
1 60 NZ ( = 60 MeV mb for 4He).24
A
The wave vector 0 > is the ground state wave function, H is
the nuclear Hamiltonian, Zk the position operator, and ek
is the effective charge for the kth nucleon. The equality
in the last term is satisfied if the position operator com
mutes with the potential energy operator. This condition
is violated by Majorana or Heisenberg exchange forces or
velocity dependent forces. Each of these three interactions
would tend to increase o El Electron scattering experi
int
ments indicate that the contribution due to exchange
currents may be included as follows:
El 2527
S. = 60 NZ(1 + a), a = 0.40. (1.2)
int.
A
(= 84 MeVmb for 4He)
Under the assumption that the ground state is spatially
symmetric and including the charge radius of the proton,
the Levinger, Bethe (28) bremsstrahlungweighted cross
section is related to the nuclear radius by the following
dipole sum rule:
El "I1 El
b = foE (E)dE
b oE
S0.096 (R2 R2)
Ai z p
( = 2.48 mb for he)
where R and R are the nuclear and proton rms radii
z p
respectively.29
The quantity Ea (E) referred to in the sum rules
is the total cross section for all electric dipole photo
disintegrations. For the 4He reactions there are five
competing reactions. The yields for these five reactions
were measured by Gorbunov and coworkers.10, A 170 MeV
Bremsstrahlung beam incident on a cloud chamber was used
in the study. Their results for the relative yield of the
various reactions are shown in Table 1.1.
The measured angular distributions indicated that
the El interaction comprises r94% of the total integrated
cross section up to 170 MeV. Approximating oE by the
total cross section, integration of Gorbunov's data for
the reactions of Table 1.1 gives:
El El
int = 95 7 MeV mb and = 2.4 .15 mb
int.
This value is larger than that predicted by the simple
sum rules (Eq. 1.11.3) indicating contributions due to
exchange and velocity dependent forces. When these forces
are considered as in the sum rule analysis of Quarati34
o
C3
OL
0
1) I
O)
0
U 0
4 1O
i
r >
HO
S1
cia
3
0)
E1 0
0
4J
N
a,
iN
'd +
> 0)
 3
C I
00
o
ti
0)
>
an
03
0
'd >
U U rN 0)
o o C N o C )
LO
C!)
C0
o
rl
ro
;,
0
00 >
g 0
0
o
(N 'd
rl H
H H x
Cd 'd
 0
SdO
H 'd H C H O
Cd 0 d C d
S H rl ) H
 '4 
> 0 r 0 d r.
l Ca a3 n O
O Q 0 .0 4 0
Cd .0 Cd a) 'd U
for the shell model (Table 1.1), the experimental cross
section up to 170 MeV excitation exhausts 95% of the sum
rule.
These results indicate the usefulness of sum rules.
This usefulness may be increased by the removal of the
approximation that the total cross reaction is due to the
El interaction. In Table 1.2 are listed the sum rules
commonly appearing in the literature along with their
representations in the shell model formulism. The results
of these equations for the shell model parameters are
11 1 '
=1.75 fm2, kEl 1 fm = 36 MeV and kE2 fm
These values give the rms charge radius of 4He to be
= 1.62 fm. It has been suggested by Leonardi and
38
Lipparini38 that the final entry in Table 1.2, a be
3
E2
measured, as the E2 sum rule for the electric quadrupole
3
strength is dependent only on the ground state wave function.
These features of the four nucleon system have
received extensive theoretical treatment. Many different
models have been successful to varying degrees in descri
bing these features. The common factor in most of these
analyses is an attempt to develop a method to properly
account for the contributions of many body and velocity
dependent forces in an orderly way. Many of the models
are based on "realistic" potentials that are derived from
13
>>
S":
S E Ln
0 r 0 +
SICo co
ul l ro 4i
c* 00 +1
S 0H 1 L (D 0
C Qir in CM o
0 X n o o
+ r oo o< o
O H I o o
4 II II II II
aJ C
0)
r.,
Q)
0)
4 I
a N
0 j H 0
H I c
0 c0 I 0
SrQll 0 N T
) ( : 1
.0 '0 43 4 +
H : 0) r: W ar
Url E
N , m
a) I I 0
rl ) C cD (0 I Q) 5
ID 13 *D
 + I + + I H
3Q I ( 441
C OC H o aI c 
ra d l (N N N cv
1) rHl (1) I 40D *Il
H() r( rM fM (U
00 I N N. rI
H c N oJ c 3 0' X
0 a) N dc NM H (
I II i i 3
H
0o
M0 3 +
3 3 d 3 3
TV V 3 V 3
) f 33 3 3 3 33 3
II II II II II II
U l 4J
0 N 3
o o o I o
14
d)
0
1 4J
00 0
Ea QO
4> O 4H
( OO
Qa 00
O Q) m 1 a)
04,4 u1 0
0 p
4J 4J
o) a
S0 0
G 0
0 1 4 o 0
0 O0 0 '.
o3 o e0
0 0
0 44
.0 0
U) OH
0O1 0
O0 0
rI H 4 1
* ei 1 2 0
0 d o 0
H I 0
Qo
0 c0 () c (C
co r a)) e
0 4' 0
H 0
1 (0 3
041 (di I H
0rl 0 0
Ul (H) a) 44
14 0 0
Q 0 a U3 )4 U)
014 7 4 H *
a)c1 (1) *H *H (c
H 0 0 0 I g
4 1)4 00 I
4 (a r 0
w o a
c! 0 0
n Y ca + 4
3 ~ ~ ^ a ; .
the nucleonnucleon scattering phase shifts.7,8,39,40 These
potentials fit the onshell matrix elements equally well.
Their differences are embodied in their treatment of off
shell matrix elements. One such potential is the Sussex
potential which was used by Szydlik, et al.36 to calculate
the spectrum of 4He.
Photodisintegration studies of the 4He nucleus span
the excitation energy range from threshold to 500 MeV. The
reaction is characterized by the asymmetries in the proton
and neutron angular distributions. At low excitation
energies the asymmetries exhibit structure which has been
interpreted as being due to interference between the electric
dipole (El) and electric quadrupole (E2) absorption ampli
tudes.22,41 For excitations above 100 MeV the (y,n) and
(y,p) asymmetries approach one another.2 The absorption
of photons of energies of 100 MeV or higher is expected to
occur predominantly by the quasideuteron mechanism discussed
by Levinger.9 Recent calculations by Gari and Hebach account
well for the total cross section and asymmetry in the range
50 < E < 140 MeV.23
Many measurements have been made of the various 4He
photodisintegration processes. An overview of these experi
ments is shown in Table 1.3. The history of the experimental
4144
measurements is one of controversy. The early measure
ments gave the general character of the energy and
Co
co ~
C'
00 '.0
CM (
a)
O m
4 .4J
..
0)
ao
* a
CN
0 H
0 >
4)
4
a
0
Dri
4) O
0) 0
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r 4)
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en en en a }. 4e
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n >1 > r >1 ><1 r 4
0
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4 p4
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rO l H r r m U u en
F m H1 H U l E h 4 E 4 1 4
.0 fft r1 .00 H H f H .0
, 0 ) 4 0 r ., ,40
0 4p 0 4J J n^J a oa R: 4u4,
In LA IV
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o
2n 0
en >1 4
> w
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^ro
00
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S i m 0
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1 P Q
m , C o
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4
n.
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4 o) 0W N N 0 C o C i
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0 4 44 44 LI 44 ir 4 4) 4.) 1
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NI LI r0
a r' n P
angular dependence of the excitation. For the cloud chamber
and diffusion chamber results there are difficulties relat
ing to small numbers of events in a given excitation range
and in analyzing the tracks of the product particles for
excitations below 30 MeV. Particular difficulty is associ
ated with the measurement of the 3He recoil for the (y,n)
measurement. All the photodisintegration experiments suffer
from difficulties associated with the spectral distribution
of the photon beam. These early measurements indicated the
existence of a number of structures in the cross sections,
but could give little definitive information concerning the
(y,n) assymmetry.
The measurements by the different experimenters are
in general agreement concerning the magnitude of the (y,p)
cross section. However, the (y,n) cross section is quite
variable. Charge independence considerations support the
results obtained by Gorbunov that the (y,n), (y,p) cross
sections are the same.11,7780 The measurements are in
agreement for an excitation energy greater than 35 MeV.
5052
The results reported by Berman, et al. and
48,49
Busso, et al.4849, give a lower (y,n) cross section for
energies between 23 and 35 MeV. In the case of Berman's
results, the cross section would agree with the measure
ments of Gorbunov0 if multiplied by a factor of 1.8.41
This disagreement has been clarified by a series of experi
ments by Irish, et al.53'54 which show that the variation was
due to a reduced target thickness from bubble formation in
liquid He.80 Measurements with liquid targets by Malcom,
et al.22give the (y,n) cross section in agreement with
Gorbunov. The 5 atm diffusion chamber measurements by Busso
agree with other experimenters for excitations above 30 MeV
but fall below the (y,p) cross section at lower excitation.
Recent 1.5 atm diffusion chamber measurements by Arkatov,
et al.55 with improved accuracy in the low energy 3He
recoil measurement give the (y,n) cross section in agreement
with the (y,p) cross section. These results for the 4He (y,p)
and (y,n) cross sections give experimental evidence for the
charge independence of the nuclear force.79
The existence of evidence for excited states of 4He
has been reported by many experimenters as seen in Table 1.3.
There is little agreement in the structure seen by various
early experimenters. The recent experiments with better
statistics exhibit much less structure than seen previously.
The experimental cross sections are shown in Figure 1.2. The
significant features are the peak of the cross section for an
excitation energy of 26 MeV and a broad peak for 28 MeV exci
tation. Peaks in this region are consistent with the
existence of two states with angular momentum, parity JT = 1
and isospin T = 1. These states correspond to a mixture of
the spectroscopic states singlet P1 and triplet 3P13
4
Figure 1.2 He Photodisintegration Cross Sections.
This figure is a compilation of the results reported by
the indicated authors.
20 30 40 50 60 70
EYM (MeV)
Evidence for a third excited state at 35 MeV excitation
with angular momentum, parity J" = 2+ and isospin T = 0 is
obtained from (y,n) asymmetry measurements.22
The (y,n) and (y,p) angular distributions are given
in terms of the Legendre polynomial expansion as
= Ao ( + jk k (cos 6)).81 (1.4)
The coefficients contain contributions from photons of
multipolarity L for k < 2L, k even and k < L + L k odd
where L is the multipolarity of an interfering photon.
The equivalent expansion in terms of sin e and cos 6
prevalent in the literature is
do = B(a + sin2 B + B sin2 6 cos 8 + 6 sin2 6 cos2 +
cos 6 + . .) (1.5)
The coefficients Ba, B, B3, B6 and Be correspond to ampli
tudes and cross terms for multipolarities of 0, 1, and 2.
The respective relations between the coefficients of the two
expansions are shown in Table 1.4. Electric operators in
volve only space coordinates and magnetic operators only
spin coordinates in the central force approximation. For a
pure IS ground state the allowed final states of the out
going particles are 3S1, 1P, and 1D2 for angular momentum
changes less than two. The admixture of D states in the
ground state indicated by variational calculations and
Table 1.4
Relations Between the Legendre Polynomial and
sin 6, cos 6 Angular Distribution Expansions
3 5
B = A (3 + a)
a= 1+ a2 + a4 +
3 5
2 8 a4 + .
5
S= 2 a3 7a5 +
23 5
3 5
6 = 3 a4 +
3 5
2 a2 8 4 +
3 5
= a + a a4 + .
3 a 5 a +
2 a2 8 a4
4He (p,p) He measurements complicates the interpretation of
the coefficients of the angular distribution. Possible
additional contributions may be observed as an increase in
the isotropic component from El transitions from the D state
to P and F states and a contribution from the cos 0 term
which arises from interference between El and nonspinflip
Ml transitions.41 The angular distribution measurements by
Wait12 Gorbunov 10 and Meyerhofl3 yield negligible
values for the coefficients a and E. Their results are
dominated by the El term B.
The asymmetry (as) of the angular distributions about
9 = 900 can be defined in terms of the difference between the
cross section at 550 and 1250. This quantity, as will be
shown in Chapter II, can be written as
a = al P (cos 55 a= a .68 a3
for the Legendre polynomial expansion (or equivalently the
term for the sin 6, cos 8 expansion) where P2(cos 550) = 0
eliminating a2. These terms arise from interference between
the electric dipole and quadrupole interactions. At low
energy the observed angular distributions are very close to
a pure sin 6 distribution.56 This distribution indicates a
predominately dipole interaction. The isotropic terms were
observed by Gorbunov to be small for photon energies up to
170 MeV providing a basis for the assumption that only El, E2
28
radiation contribute significantly to the yield. The
interference terms producing the asymmetry are more
sensitive to smaller amounts of E2 interaction than the
E2 amplitude itself under these conditions. This result
is due to the fact that the coefficients al and a3 are
proportional to the product of the dipole and quadrupole
amplitudes.
CHAPTER II
EXPERIMENTAL METHODS AND MEASUREMENTS
This study consists of a series of experimental
measurements designed to obtain the electric quadrupole E2
strength as a function of the excitation of the compound
3 4
nucleus for the reaction H(p,Y) He. The methods used to
determine the E2 strength consisted of measuring the asym
metry of the angular distribution about 90 and the measure
ment of the detailed angular distributions of the outgoing
Yrays. Shell model calculations predict a number of states
of angular momentum and parity J" = 2+ with isospin T = 0,1
in the energy range from 3050 MeV excitation. The proton
energies available from the Triangle Universities Nuclear
Laboratory (TUNL) CycloGraaff allow study of the energy
range from 22 to 42 MeV excitation of the compound nucleus.
Experimental Details
Using a 15 MeV isochronous Cyclotron as a source,
the TUNL FN Tandem provides a pulsed 25 MHz proton beam.
The beam energy is continuously variable from 1731 MeV.
The beam pulse width is 2 ns and the energy spread is 30 KeV.
The rootmeansquare beam current delivered to the target
was kept in the range of 20 to 100 nanoamperes depending on
the total count rate in the sodium iodide detector. The
count rate in the detector was kept below 350k counts per
second because pileup problems and a deterioration of the
resolution occurred for higher count rates.
Two separate detector assemblies were used in the
experiment. The first detector assembly was a high resolu
tion system and was used for the yield curve measurements
and the angles of the angular distributions between 420 and
1420. The second detector assembly allowed measurements to
be obtained for angles between 100 and 900 although with
reduced resolution. The experimental arrangement was
designed in both cases to minimize the background events
that were not associated with the target.
The detector assembly is based on a cylindrical
25.4 x 25.4 cm thallium doped sodium iodide crystal. This
crystal is surrounded by a well type Ne 110 plastic scintil
lator. The center crystal is viewed by six RCA8575 photo
multiplier tubes and the plastic scintillator by eight XP1031
photomultiplier tubes.82 A schematic of the detector arrange
ment is shown in Figure 2.1. Active shielding is provided
by detecting coincidences between the plastic detector and
the centercrystal detector which were processed by the
electronics to reject cosmic ray events. Passive shielding
for the detector is provided by four inches of lead and eight
inches of paraffin loaded with lithium carbonate (%50% by
weight). The detector is mounted on a carriage that allows
Figure 2.1 Detector Assembly for 25.4 x 25.4 cm Cylindrical
NaI(T1) High Resolution Detector. This illustration shows
the arrangement of the active and passive shielding from the
cosmic rays and beam associated background. The collimator
arrangement that was used to restrict the cone of gamma rays
to that just illuminating the back face of the detector
crystal is shown (Weller, et al.82).
CP
L
0
radial and angular positioning about the target location.
Two radial positions of the detector were used in the
experimental series. These distances placed the rear face
of the NaI crystal at the positions of 82 cm and 106 cm
respectively. Collimators were provided which restricted
the observed solid angles to those subtended by the rear
_2
crystal face corresponding to solid angles of 7.42 x 102 Sr
and 4.46 x 102 Sr respectively. Excitation curves were
measured with the detector at the 82 cm distance in the
interest of higher yield rates. The angular distributions
were measured with the detector at the 106 position to allow
the inclusion of measurements of the angles at 420 and 142 .
To determine the angular distribution as the labora
tory angle approaches zero, additional measurements were
taken for the laboratory angle of 200 using a separate
detector system. This system was based on a 10.2 x 17.8 cm
NaI(Tl) crystal surrounded by a 22.9 x 22.9 cm NaI(Tl)
annulus for active cosmic ray rejection. The experimental
arrangement is shown in Figure 2.2. Passive shielding is
provided by a 20.3 cm thick copper annulus and 4.4 cm of
lead. A cadmium sheet was located between the lead and
copper for thermal neutron absorption and the collimator was
filled with paraffin loaded with lithium carbonate. The
distance from the target to the crystal front face was
93.3 cm. The collimator restricts the photon flux to the
Figure 2.2 Detector Assembly for 10.2 x 17.8 cm Cylindrical
NaI(Tl) Detector for Small Angle Measurement. The active
and passive shielding for cosmic rays and beam associated
background is shown. The beam stop shielding for measurements
at 61ab = 200 is shown.
lab
14
z z
 I
:) z
z i
z L
6.57 x 103 Sr solid angle subtended by the back face of
the crystal. The entrance aperture was formed from tungsten
for attenuation of fast neutrons. The remainder of the
collimator was lined with lead.
Further measures to minimize the background events
not associated with the target consisted of the use of a
single insulated collimator close to the target and adequate
shielding of the beam dump. The insulated collimator allows
monitoring of the beam current striking it. This current
was kept to a minimum by adjustments of the focusing of the
beam transport system. The beam stop for the large crystal
arrangement is located three meters beyond the target.
Originally this beam stop was shielded by about 10 cm of
lead and 70 cm of paraffin. The shielding was changed to
about 10 cm of copper and 40 cm of lithium loaded paraffin
to more effectively attenuate neutrons produced in the beam
stop. For the small crystal assembly, the beam stop was
located 33 cm behind the target. The shielding was provided
by the copper shield of the detector itself and a tungsten
shadow bar.
The electronics for the two detector systems were
similar; however, the use of the Nal annular cosmic ray
detector on the smaller crystal required a different elec
tronic arrangement than the plastic shield of the 25.4 x
25.4 cm system. The electronic arrangement for the large
crystal is shown in Figure 2.3 and that for the small crystal
Figure 2.3 Electronics Block Diagram for High Resolution
Detector. This diagram illustrates the signal routing used
to determine the coincidences between the active shield
detector and the center detector. The additional electronics
used to determine the number of coincidences are not shown
(Weller, et al.82)
38
oc
C C 
(/)
co
(I))
o
L .Ln .cn
0 LC ILO
C~ CP
c C

x w 0: c
0O C:)
c\l
EO
< x
Z o 1
clo ZMO (
in Figure 2.4. Typically, the count rate due to low energy
events is several orders of magnitude greater than the count
rate due to events of interest. Common to both systems is
the arrangement of the electronics to minimize the events
due to the pileup of lower energy events. This minimization
is accomplished by timeclipping the detector signals. The
signal is then passed through a fast linear gate (250 ns)
before processing. During the 10 ps processing time, the
gate is held closed.
Further reduction of the background due to unwanted
events was possible in this experimental series due to the
pulsed nature of the CycloGraaff beam. A time of flight
requirement (TOF) was set up between a signal derived from
a radiofrequency inductive pickup loop at the Cyclotron
and the gamma ray events in the center crystal. The RF
signal was used for the stop signal for a time to amplitude
converter (TAC). The TAC start signal is derived from the
linear signal of the Nal center crystal. The TAC was gated
by the pileup rejection circuitry to be active only for
the time during which the linear gate was open. Both
detectors are pulse stabilized by a calibrated light emit
ting diode flash into the crystal. This pulse was used to
correct for slow changes in the system gain due to thermal
drifts. The pulse stabilization was found to be unnecessary
for the large detector system. The cosmic ray rejection was
accomplished differently for the two detector systems. The
Figure 2.4
Experiment.
to determine
detector and
Electronics Block Diagram for Small Angle
Illustrated here are the electronics used
the coincidences between the active shield
the center detector for the yield measurement
at 68ab = 200. The electronics associated with the pulse
stabilization, accidental monitoring and beam time of
flight circuits are also shown.
41
L) w
W 14 LL 14 Z
0Z J Wlj
1414 14 4 I 
z, z
ui m
L)L
1 4 Z1 1401
UJ >.
0 W W Fia
I in Lnn 
M 0 0
0 z4 0
L) o )
FC1 
r I
In W
o ILZ ~ c IL
C3 UJ L
00
jj
uw~1 4Z 4
7 >
~z z
0 Lj
a 0
U)U
3W cn z ~
Do
own
ocr z
M W; (
W 0  cr 03
OnO
0 Ej
W ulW I I
co
JW
r P
WOO
th a
:3 UZ
(L cr cr
in ,o W
response of the plastic scintillator is faster than that of
Nal. For the large detector with the plastic shield, the
shield signals were delay line clipped and timed for an
overlap coincidence with the center crystal events. When
a coincidence occurs, a routing signal is sent to the
computer for storage of the event in a coincidence spectrum.
The small crystal system utilizes the linear signal from the
center crystal as the start signal for a time to amplitude
converter. The shield pulses are then the stop signal for
the TAC. The TAC output goes to two single channel analyzers,
one of which is set to provide a signal for events occurring
in the peak of the coincidence TAC spectrum. The other
analyzer is set off the peak to determine the counts due to
accidental coincidences. In each case the resulting signals
are processed by two analog to digital converter's (ADC's)
in coincidence and passed to the online computer. For
maximum resolution, the threshold of the coincidence circuit
could be set to give a coincidence for events due to annihi
lation yrays from events in the center crystal.
The information stored in the computer consisted of
eight 512 channel spectra. A window lightpenned on the
yflash in the TAC spectrum separated the Yray events into
three pairs of spectra. These pairs were stored as short
time events, true data events and long time events. Each
pair corresponds to center crystal events and coincidences
respectively. The eighth spectrum stored was used with the
large crystal system to store the elastic proton spectrum
from a 500p silicon surface barrier detector located at a
laboratory angle of 1600
The targets used in this experiment were tritiated
titanium foils. The foils were 5p thick and the activity
2
was 6 Curies/in This activity corresponds to an atomic
ratio of approximately one between the tritium and the
titanium. The tritium contained in the foils was deter
mined by comparing the yield obtained by the solid state
detector with the known yield of 3H(p,p) 3H. The measured
tritium values for the two targets used were 0.14 mg/cm2
2
and 0.11 mg/cm The error in these concentrations is
determined by the uncertainty in the experimental geometry
and is estimated to be 10%. The errors associated with the
statistics, the H(p,p) H cross section and the beam charge
measurement combined contribute about 1% to this error.
Experimental Data
The experimental data were taken in four series. The
first series consisted of a set of excitation curves which
were utilized to determine the experimental asymmetry of the
angular distribution. The second series consisted of angular
distributions measured at eight angles between 42 and 1420
These measurements were extended in the third series to in
clude a measurement at an angle of 20 The fourth series
was an extension of the excitation curve data to lower
energies than covered in the first series. This data was
then analyzed to determine the effects due to contributions
from the electric quadrupole interaction.
In the first series, the excitation curves were
measured for laboratory angles that transform approximately
to the center of mass angles of 550, 930, and 1250 (Figures
2.5, 2.6). The measurement was taken in 500 keV steps over
the energy range from 1730 MeV. The resulting cross sections
are shown in Table 2.1. The center of mass angle in the for
ward direction (bf = 550) and in the backward direction
(eB = 125) were chosen to be approximately the zeros of the
second order Legendre polynomial P2 (cos 0). This condition
on Of and OB requires that OB = 1806f which yields the
following results for the Legendre polynomials:
P1 (cos 6B) = P1(cos )
P2 (cos (cos = P2( O ) = 0
(2.1)
P3 (cos 6B) = P3(cos )
P4 (cos B ) = P4(cos f )
The observed yield as a function of angle can be written as
an expansion of Legendre polynomials,
Y(8,) = Ao(l + ,akPk(cos 9)).
k=l
3 4
Figure 2.5 H(p,y) He Center of Mass Cross Section for
6 ab = 900. The data from this series of experiments is
shown as the solid dots. The error bars represent the
statistical error associated with the data points. The
data is plotted both as a function of the laboratory energy
of the incident proton and the center of mass energy of the
emitted gamma ray. The experimental cross section reported
by Meyerhof, et al.13 is included as the open squares. The
solid curve is a least squares fit of the data in the proton
energy range from 5 to 30.5 MeV to a cubic polynomial. The
cross section was obtained by normalizing to the data of
Meyerhof.13 The measured target thickness, solid angle
and collected charge was combined with the normalization to
obtain the detector efficiency. The value of 27 + 3% obtained
is consistent with the efficiency 26 6% reported by Weller,
et al.82
3 4
Figure 2.6 H(p,y) He Center of Mass Cross Sections for
o = 550 and 125. The experimental data is shown as the
cm
solid dots with the error bars indicating the associated
statistical error. The data is plotted both as a function
of the laboratory energy of the incident proton and the
center of mass energy of the emitted gamma ray. The solid
lines are fits to the data in terms of the model discussed
in Chapter III.
o Meyerhof
* TUNL
 Polynomiol Fil
5 10 15 20 25 30
E (MeV)
E,(MeV)
20 25 30
E (MeV)
35 40

o a
in
*0
+1
\D
C(
rIT
H
tol
OD
+1
D
In
In
"0
+1
+I4
CN
CN
CN
+41
0.
0
Q)
r.
14
U
a)
4i
c4
(d
a4
CM 0
a)
,
(o
4
3 r1
H c
r N
H (N
H H
o 0 0 0 0 0 0
iI r4 ii il ri
0 1 0 (N m
H H H H Hmp
N4 N
+1 +1
o 0
+1 +1
Hl r
+1 +
m
+1 +1
co m
m CN
CM C
. (N
N N
+1 +1
l m
m m
r ,
c; o
+1 +1
H O
N
+1 +1
co Ln
0 1n
N~ \D
r( r NO m ( 
CN CN CM CM (n
cO Ln
N4 (N
+1 +1
lN r
NO o
o o
+I +l
o 10
+1 +1
* *
l e
CM r
H n
H o
+1 +1
N N
m m
+1 +1
o 10
a N
0 0
+1 +1
10 r
mn CM
(1 (N
O
In 0
Ln
II
;>
00
F: ) C
n O 11 CO C 048
C. ko in In %0 In 'T 48
+1 +1 +1 +1 +1 +1 +1 +1
in Co uI in fm CO I rN
co Ln o m c) r Ln C o
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N
Il
m c t rm N m No N
+1 +1 +1 +1 +1 +1 +1 +1
rI 0 n r H O CD D
+1 +1 +1 +1 +1 +1 +1 +1
0) C m Co in N r z
rN co N o In Ln '
+ + + +1 +1 +1 +1 +1 +1
0 i l l l ^
rl 0o < 0 0
C D CM r H H 9 In 9 O N
+ 1] +i +i +i +i +i +i +i
So ril ii ro in co r
4N CN CO o T rm H (O
8 In In In m m r ^ u,
CN co In In In In NM C LI
Q) 'I r IV m c m
S+I +i +1 +1 +1 +1 +1 +1
Q I 'm CO N C C O
n r rN H In co D
co rN r r rN N In In
rn
in
H V m m n Cl N Co
+1 +1 +1 +1 +1 +1 +1 +1
CN rN CN CN N 'T m co
In In In T Ln IV V '
CO H! I n cO CN CO O
C mn rn fn r "r in
mo 00o 0 co mo o o c
l rl l rl l Cl I CD
O O cO CO CO CO O C
SH rH H rH H 'IN N
S m ,' r 49
+1 +1 +1 +1 +1 +1
00 r r A I LA
o 0 0 0 w in
N
e(
mr N N N m
0 I r i 1 rI I I
+1 +1 +1 +1 +1 +1
c N C CO in
(N ( 0 C rH
LA N LA C0 LA L1
NM r N N N N
+1 +1 +1 + +1 +1
S I! O N LA
i IT T 0 r .D
'Z 'IT ) CM4 N N N
+1 +1 +1 +1 +1 +1
oa o N D LA
D0 0 0 0 0
+I +I +I +I +I +I
J 00 IT Cl CO C CN
r! ml 0 r ko LO n
0 H H r H O N
Cl
(N
a 0 0 0
+1 +1 +1 +I +1 +1
I re m l ClI C
0l l Cl Cl C al Cl
in inl il in a0 0
( N N C N
co 0 o cD O N
(N (N NM C
0
Cn
1r
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II
o
,
0O CO O 0co 0o co
0\ ^ 0) '^ 01 ^
ll 1 lo lo
m 'T LO LA I'D
(N (N (N (N CN (N
0 LA 0 LA
LA LA D l
(N (N (N 4
+1 +1 +1 +1 4 +1
 C r H H H
l rl rH H H 
+1 +1 +1
rN Ln 10
Sm m
o o 0
+i +1 +I
m r a
+1 +1 +1
0 0 0
,' C )
H H O
CM
N N
+1
(N
0
rO1
n0 0
O H
+1 +
0 O0
co rl
(N CN m
+l +i +i
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+I +I +I
(N H1 M
N cN (N
iA 0 Ln
n 0 o
NM en m
Under the assumption that multipolarities of L > 2 contri
bute negligibly to the cross section, the sum of the yields
for the angles 6f and B reduces to
Y(Of) + Y(OB) = 2Ao(l+a4P4(cos 6f)). (2.2)
where a and a4 are defined in equation (1.4). The dif
ference between the yields is then
Y(f )Y(gB) = Ao(alP(cos )+aos f)+ s f)alP (cos gB)
a3P3(cos GB)) (2.3)
= 2Ao(alPl(cos f )+a3P3(cos a )).
Therefore, assuming the condition a4P4(cos f)<
should be expected if the E2 intensity is small compared to
the El intensity, the total cross section is given by
at = 4nAo = 2n(Y(Of)+Y(6B)) (2.4)
and
Ao = (Y(f)+Y( B)). (2.5)
Under these conditions, the experimental asymmetry in terms
of the yields Y( f) and Y(B ) is equivalent to that defined
in terms of the coefficients of the Legendre expansion of
the angular distribution (Figure 2.7). This equivalence is
shown by the following relations:
S Y(e ) Y(B) = 2Ao(alPl(cos f)+a3P3(cos 9f))
s P1(cos 6 )(Y( )+Y(OB)) P1(cos f)(2Ao(l+a4 4(cos Of))
(2.6)
= a1 + P3(cos )a3 a4P4(cos ef)<<1.
P1(cos f)
Figure 2.7 3H(p,y) 4He Angular Distributions. The
experimental data is shown as the solid dots with the
error bars indicating the associated statistical error.
The solid lines are least squares fits to the data in
terms of the Legendre Polynomials as discussed in
Chapter III. The measured values for the cross sections
are included in Appendix B.
0 V ( .
)
LO
>
0
o
L) 0)
0
Io
it
D CN
0 .0
As will be shown in Chapter III, the measured asymmetry and
the measured cross section can be used to obtain an experi
mental E2 cross section.
The El cross section is obtained by observing that
the terms of the multiple expansion (equation 1.5) for
do
multipolarities up to L = 4 reduce to = B (a +1) at
6 =900
cm
Therefore, taking a<< 1 (no spinflip or magnetic
dipole radiation) the yield at this angle is the El cross
section.13'19'33 The angle measured was 8lab = 900 so that
the results could be readily compared with those of previous
experimenters.13,56
A second experimental series was conducted to con
firm the energy behavior of the E2 cross section and to
provide a direct measurement of the E2 amplitude. This
series consisted of a set of angular distribution measure
ments taken every 2 MeV between 17 and 27 MeV incident
proton energy, and included angular distributions measured
for proton energies of 13 and 30 MeV. For each angular
distribution, yields were measured for a minimum of seven
angles in the range 420142 covered by the detector.
Additional angles were included as needed to define the
shape of the peak and the slope of the sides of the angular
distribution. The minimum number of measurement points was
chosen so that the number of degrees of freedom associated
witha Legendre polynomial least squares fit to four
coefficients is greater than zero. The amount of charge
collected was varied so that a minimum of 200 counts was
observed for the 1420 measurement. The yield was larger
for all other angles observed (Figure 2.7).
The third series of measurements was conducted
using the 10.2 x 17.8 cm Nal detector. The purpose of
this experimental series was to determine the shape of the
angular distribution as the angle approaches zero. Measure
ments were made every 2 MeV in the proton energy range
17 < E < 27 MeV. Yields were measured for three laboratory
angles including 200, 900 and an intermediate angle corres
ponding to an angle at each energy for which the yield was
measured in the second series. These measurements were
normalized to the data of the second series, and an effective
yield for the 25.4 x 25.4 cm detector at the laboratory angle
of 200 was obtained. This yield was incorporated into the
angular distributions used to determine the Legendre
expansion coefficients.
The fourth series of measurements extended the
asymmetry data to lower energy. The TUNL FN tandem was used
to measure the foreaft asymmetry every 1 MeV in the proton
energy range from 8 < E < 15 MeV. This information was
p 
obtained for comparison with the published results of
earlier experimenters.13
CHAPTER III
DATA ANALYSIS
Kinematics
The kinematics of the reaction 3H(p,y) He for
proton energies above 10 MeV are subject to relativistic
corrections. Therefore, the kinematics were explicitly
treated according to special relativity following the treat
ment by J. J. Jackson.84 The reaction coordinates in the
laboratory and center of momentum frames are shown in
Figure 3.1.
Laboratory Center of Momentum
P qY
m m )6z m m
yp pt y P 7t
+ q
hk
Figure 3.1 Laboratory and Center of Momentum Coordinates
The conservation equation for the square of the four momen
tum of the alpha particle in its rest frame and in the
laboratory system is
(0,j cmac2)2 = ( 2p 1 m ckT, j/c (ypmc2+ 2 2E ))2 (3.1)
where y. is the Lorentz transformation parameter. For the
proton in the laboratory, this parameter is y = //lv2/c2.
p p
Solving the conservation equation for the laboratory gamma
ray energy as a function of laboratory angle gives
E (E')2 m2c4
S (3.2)
2(ypmc 2+m tc2 yl m C2COSO)
pp t p p Y
where the energy
E' = (m2c4+m2c4+2ypm2mc2 ) (3.3)
p t p p t
is the total energy in the center of mass system.
The gamma ray energy in the center of mass is
S (E') 2 _m2 c
E = a (3.4)
2E'
The excitation energy of the compound nucleus is found by
subtracting the alpha particle rest energy from the total
center of mass energy
E = E' m c2 (3.5)
The relationship between the angle in the center of
momentum and in the laboratory is given by
I 2 1
sin G p
G' = tan ( Y ) where = p (3.6)
y (cos 8) Y2
cm ( P
where the Lorentz transformation parameter for the center of
momentum ycm is related to the total laboratory energy and
the total energy of the compound nucleus by
y = y m C2 + mtc2
Ycm yp mpc +t (3.7)
E'
These relations were incorporated in the computer programs
used for data analysis.
The relationship between the measured cross section
in the laboratory reference frame and that of the center of
momentum reference frame is also necessary for comparison of
the experiment to theory. This relation can be obtained
from the equality of the yield in the two frames following
Marion.85 This equality is
lab (6y)dQ = Qcm (0')df'. (3.8)
Using the kinematic relations between 6 and 6' with the
definitions of the solid angle increments, which are
dQ = 2r sin9 d6 and dQ' = 2n sine'dO' the relation be
Y Y
tween the two differential cross sections is found to be
as follows:
sin36' cos6'cos6y+ c) (3.9)
S, (9 ) = o (9') ( y+ y ). (3.9)
lab y cm sin9 sin'sin cm
This form of the relation between the two frames was con
venient to use as the analysis programs calculated 0' and
ym for measured gamma ray angle 6 .
Excitation Curve Analysis
Assuming only El and E2 radiation with the
additional assumption that the El and E2 spinflip terms
are negligible, the cross section may be written in terms
of the spherical harmonics Y11 and Y21. The interaction is
modeled by assuming that it consists of direct and resonance
terms. The center of mass cross section expression is
do (8) i61 i62 i(Di+ni)
'P =eDie Y11 + ( e2D2e + Rie
(3.10)
41
i(02+rI2)
+R2e )Y211 2
This expression for the cross section implicitly includes
the relations a3 = a and 1 + a2 + a4 = 0 between the coef
ficients in the Legendre expansion. The first term of the
cross section is due to a direct El interaction. The
second term is due to a direct E2 interaction, The factors
86
e1 and E2 are the kinematic effective charges. The third
and fourth terms are resolved Breit Wigner resonances cen
tered around energies E1 and E2 respectively. Each term of
this expression will be examined further in the following
paragraphs.
The expressions for the resonance terms in the
center of mass are
= g() p (3.11)
((E' ) 2+F/4)
and
R2 = g2(EI)/2 p/2
p2 (3.12)
((E' E )2+r /4)
Y 2
The energy dependence of the resonance partial widths is
explicitly included in these expressions where P is the
proton penetrability calculated by the method of B. Buck,
87
et al. and
2L+l F
(E ( (B') p
(E k2 ) E
describes the energy dependence of the transition amplitude
(F ) of a L = 2 multiple. The resonance strength for the
gamma ray resonant absorption is then
2L+1
C2 C2(2L+1) F F 4g2(E) P (3.13)
ty p (3.13)
(E') 2 (E') 2
The energy dependence of the total width of the resonance
is included by using the proton and neutron penetrabilities
and their respective reduced widths, i.e.
r = F + F = 2yn + 2yp (3.14)
n p n n pp
The deuteron and gamma partial widths are assumed to be
negligible with respect to the neutron and proton partial
widths. The reduced widths are solutions to the equations
(E) = 2y2Pn(E) + 2y Pp(E) (3.15)
n n pp
and
Yn Yp
for the resonance centered about the energy E. The neutron
and proton contributions are here assumed to differ only by
coulomb effects. The phase 4 in each case is the resonance
phase, given by
E E
n = tan E i) (3.16)
1i/2
The phases n1 and n2 account for the relative phases be
tween the resonances and the direct terms.
The functional form for the direct terms was taken
from the analysis of Flowers and Mandl.88 They assume
gaussian type wave functions and obtain relations of the
form
i bE i6
DE e = a(EQ E 6 e P e (3.17)
Da e = (y) p
and
i62 3E bE i62
D e = Ca (E )2E / e e (3.18)
2 y p
The gaussian wave functions parameters are chosen to fit
the 4He rms. radius of 1.61 fm. given by electron scatter
ing giving = .101 (mb/MeV/2), a2 = 5.13 x 105
(mb/MeV ) and b = .043 MeV .41 For comparison to the
experimental data, the direct El term was replaced with an
empirical function and the strength of the E2 term, a2, was
allowed to vary. The phases 61 and 62 were varied to ob
tain the relative phase between the direct El and E2 terms
that best fit the data.
The above expression for the differential cross
section is in the form of products of spherical harmonics.
These products are expanded in terms of single spherical
harmonics. The product expansion is
+' L
Y ()Y () = (1)m i (2t+1) (2i'+1)
L= I 'I M=L 4 (2L+1) (3.19)
Cooo m' M M
CZ'ZL CP'L LM (,)
where the ClebshGordon coefficient CmmM is zero for
M / m'm. The resulting expansion is in terms of the YLO
functions alone. The functions YL0 are simply related to
the Legendre polynomials by the function
YL0(6,) =  PL (cose) (3.20)
Collecting the terms of the polynomials and comparing with
the Legendre expansion of the angular distribution allows
identification of the expansion coefficients in terms of the
direct and resonance contributions of the model. In order
to make this identification the following three definitions
are made:
B = (EcDI)2
B1 = [2EDIE2D2Cos (662)2c1DI (Ricos (i.+ni61)+R2COS(42+ 26) ]
B2 = [(C2D2) 22e2D2(Rlcos(4l+nl5c2)+R2cos(2+r 262)) +12
+2R1R2cos(01+qir2n2) + R2].
(3.21)
In these definitions B appears in terms of only El ampli
tudes, B1 appears in terms of products of El and E2
amplitudes, and B2 appears in terms of E2 amplitudes only.
The coefficients of the cross section expansion may then be
written as follows:
1 3 1 5
A = (B +B2), A a = B, A a2 (Bo+ B)
4x/5 o 4 o
12
Aoa3 = A .a, Aa4 = 1 B2
0 6a 7(4[) (3.22)
In this form the El and E2 contributions are explicitly
included.
The Legendre expansion gives the differential cross
section as a function of angle 6 in the limit of infinitesmal
solid angle; however, the actual detector solid angle is
finite. In order to compare the model to experiment, it is
therefore necessary to transform to finite geometry. This
transformation introduces geometric correction factors Q.
(Ferguson 89 ) into the expansion. The resulting expression
for the experimental cross section is
yI = A (Qo+aiQ1Pi(cos6)+a2Q2P2(cos9)
0A (3.23)
+a3Q3P3 (cose)+a4Q4P4(cos6))
The yield from the 3H(p,y)4He reaction is converted
to the photonuclear reaction 4He(y,p) 3H by the principle of
detailed balance. The relationship between the cross
sections is given by
1 ( ( H)+1)(21I(3H)+1) 2m cpE (p90
y,p 2 (2I(4He)+l) E2 pY
(3.24)
where I(A) is the nuclear spin of nucleus A.
Using these relations, a computer program was de
veloped which plotted the model cross sections against the
measured cross sections as a function of energy and angle.
The model parameters for the curves that best represented
the plotted experimental data were used to obtain the
resonance parameters and the integrated strength of the
various components of the model. These results are
presented at the end of this Chapter.
Angular Distributions
The angular distribution data were measured in an
attempt to obtain a direct measurement of the E2 strength
through a determination of the coefficient a4. The magni
tude of the cross section for e = 0 180 also provides a
determination of the contribution of spinflip interactions
to the cross section. The experimental data are fit by
linear least squares to the first four terms of the Legendre
cross section expansion for finite geometry as discussed in
the previous section. For some measurements, the resulting
fits were negative for 6 = 0, 1800. A nonlinear least
squares search with the additional requirement that the cross
section be positive was then used to obtain the values for
the coefficients Ao, al, a2, a3, and a4. The error associa
ted with each coefficient is then obtained from the
curvature of the Chisquare space.
The assumption that the El and E2 spin flip is
negligible is tested by the angular distributions. The
truncated Legendre expansion reduces to
do
d (6) = Ao(l+a +a +a3+a,) (3.25)
at 8 = 00 and
do
dg (0) = Ao(la1+a2a3+a4)
at 6 = 1800. The cross section is equal at these two angles for
a3 = al and zero if in addition l+a +a4 = 0. Assuming only El
and E2 radiation, the coefficients Aoa, and Aoa3 can be
written in terms of the complex vectors Ijj JJ> for total
angular momentum (p3) coupling where + S is the
total angular momentum of the incident particle, = S
is the total angular momentum of the target and J = 3 + 8
is the total angular momentum of the system. The nuclear
spins are S and S of the projectile and target respec
tivelywhile a is the orbital angular momentum. The R
matrix elements () for radiation of multi
polarity e? and compound nucleus of ground state spin I = 0
are the following for El and E2 radiation:
<011 ITI 1> = Pe 2, <011 T i3/21/21> = P3/ e
(3.27)
i2Q2/ 1
<022 [TI1 //2/2> = d/2e /2, <022 TI 5/'/2 = d5e
The resulting expression for the coefficients are as follows:
i(23 11/2 ))
Aoa1 = {/f0 Re(Pd /2 e2
3 /2 1 3/2) (3.28)
5 Re(Pd/2
+33/3 i(25/21/2)
3 Re (P d 5/2d ,/ e
11
A a3 = { 15 Re(Pd3/2 e )
+o i3i2 3/Y)
5 Re(P2d 3/2 e13/) (3.29)
5 3 /2/2
5 Re(P 32dse /)
The relationship a3 = al can be obtained from these equa
tions by imposing the following conditions on the phases
13/2 = '1/2 + nm
/2 /2
n = 0,2,4, ... (3.30)
2 2 + nT
With these conditions there are two possible classes of
solutions to equations 3.28 and 3.29 that lead to the
result a3 = al. One class is obtained when the ratio be
tween the magnitudes d3 and ds/ equals /6/3 in which
case the ratio between the magnitudes Pi/ and P3 is
indeterminate. The other possible class of solutions is
obtained when the ratio between the magnitudes P1/ and P/2
equals /2/2 in which case the ratio of the magnitudes d/2
and d/2 is indeterminate.
Alternatively the system may be described in terms
of the (AS) coupling vectors ( i S X S LSJ>). The total
angular momentum is + = A + and the total spin is
A = A + B. The total angular momentum is then + = A + A.
These vectors can be related to the 33 coupling vectors
of equations 3.27 by the use of the 9j symbols.9 The
relationship is as follows:
IL S LSJ> = I(2S+1) (2L+1) (2j +1)(2j +1)]
SL I ja(3.31)
ja J J
The results for total angular momentum and parity J' = 1
are
10I 1 101 > = 7 1 1 > + /I 3/,1/ >
3 3
(3.32)
0 1 111 > = /j 1 > + 3/21/2
3 3
The results for total angular momentum and parity J" = 2+
are
10 2 202 > = /S /21/22 > + /T15 21/22 >
5 5
(3.33)
10 2 212 > = /1 3/2'/22 > + /I [ 5/21/22
5 5
The two cases leading to the relation a3 =al are obtained
by either I01111> or I02212> being equal to zero.
Studies with polarized protons indicate that I01111> is
92 15,19,33
nonzero. As seen by previous experimenters 15,19,33
the contribution of the spin flip terms 101111> and
1.02212> to the unpolarized cross section is zero to
within the experimental error. The data in this experiment
are consistent with neglecting spinflip as can be seen in
Figure 2.7 and 3.3. Therefore, these terms were not
included in the analysis.
Results
The experimental data is consistent with the
assumption that the reaction is dominated by the electric
dipole radiation. Considering only electric dipole and
quadrupole radiation, the magnitude of the cross section
for 6 = 900 is a measure of the electric dipole strength
cm
if the angular distribution of the quadrupole radiation is
of the form of the spherical harmonic Y21 (which vanishes
at 900). Using these assumptions, an empirical equation
for the electric dipole strength was derived from the ex
perimental data, using a linear least squares fit to the
experimental 3H(p,y) 4He cross section between 23 and 42 MeV
excitation. The cross section in this energy range was
found to be best represented by a cubic polynomial with a
reduced chisquare of 0.02. The resulting expression for
the El radiation in the model is
3 5 2 7 3
2 4m c2E {1.518x10 +3.953x10 E 3.921x10 E +5.419x10 E }
D= p p x x x
E2
Y (3.34)
This expression gives the 90 cross section shown as the
solid curve in Figure 2.5. Using this result for D1 the
model parameters were obtained to produce the solid curves
in Figure 2.6 for the angles cm = 530, 1230. The asymmetry
(as) resulting from this calculation is displayed as the
solid curve in Figure 3.2. The parameters of the calculation
are shown in Table 3.1 for the resonances observed in the
data.
Table 3.1
Direct and Resonance Parameters of the E2 Cross Section
E(Mev)* r (keV) F (MeV) n 6 a(mb )
p 2 MeV
35. 2.6 12.8 500 650 1.24x10 46
39.5 0.60 3.2 350
*all energies are cm
The least squares fits to the angular distributions are shown
in Figure 2.7 as the solid lines. The coefficients for the
Legendre polynomials as a function of the incident proton
energy at the center of the target are displayed in Table 3.2.
The Legendre coefficients obtained from the excitation curve
analysis are shown as the solid curves on Figure 3.3 overlaid
with the experimental results. The experimental results are
shown to support the hypothesis that a3 =al and l+a2+a4=0 to
within experimental error at all measured energies with the
exception of E = 23.0 MeV. The agreement between
P
3 4
Figure 3.2 Asymmetry of the H(p,y) He Angular Distribution.
The asymmetry data from the measurements at cm = 550 and
1250 are the solid dots with the error bars showing the
statistical error. The asymmetry from the angular Distri
bution of Figure 2.7 are shown as x's with arrowheads
indicating the error propagated from al and a3. The results
reported by previous experimenters is also plotted.136058
I
'10
0
OD
N
Cd
C\J,
Jlo
0
>, >
w
M (U
Figure 3.3 Coefficients of the Legendre Expansion of
o(9). The a. coefficients from the angular distributions
of Figure 2.7 are shown as x's in the upper plot. The
error bars are the errors of the coefficients a. obtained
1
from the least squares fit of the data to the Legendre
expansion. The lower plot shows the experimental A
obtained from the cross sections of Figure 2.6. The
solid curves in both plots are the results obtained from
the model (Equation 3.10). The energy E is in the center
of mass.
0
It
0r)
0
LO
r()
If)
c(J
0
CJ
the experimental data and the model of direct E2 and two E2
resonances superimposed on a dominant El background inter
action is seen to be quite satisfactory.
For comparison to theoretical calculations in the
literature, the integrated cross sections due to the indi
vidual components of the model were evaluated. The total
photonuclear cross section is obtained by integrating the
Legendre polynomial expansion of the angular distribution
(Equation 1.4) over all solid angleS. The result is
Fl E2
IdQ do = 47A = B + B = o + o
dn
where Bo and B2 are defined by Equation 3.21. As seen in
Equation 3.21, the individual contributions of the direct
and resonance terms used in the model may be evaluated
separately. The results of this evaluation are shown in
Table 3.3.
Table 3.3
Cross Sections Integrated From 22 to 44 MeV Excitation
fa dE = 30.6 mb MeV / E1 dE = 29.6 mb MeV
YP
foE2 dE = 1.01 mb MeV
I Rk2 dE = 0.226 mb MeV
/ R22 dE = 0.047 mb Mev
f 2 dE = 0.969 mb MeV
I/ E2 dE = 8.12x104 mb/Me
E2
S^R1 dE = 1.78x10 inb/
MeV
E2
R2 4
/ R dE = 3.07x10 mb/
MeV
odE2
E2
S= 6.69x104
dE = 6.69x10 mb/Me
SUMMARY AND CONCLUSION
The TUNL CycloGraaff was used to obtain data from
the H(p,y) He reaction for protons of energies from 830
MeV in the laboratory. The data were taken as a series of
excitation curves and angular distributions. The data
obtained can be adequately accounted for if the radiation
is viewed as being predominantly electric dipole with an
admixture of direct and resonant electric quadrupole radia
tion. Two electric quadrupole resonances are observed and
the resonance width obtained for each. The resonance at a
compound nucleus excitation of 35 MeV with a width of 12
MeV is in good agreement with the result reported by Malcom,
et al.22 in the 4He photoneutron experiments at the
Saskatchewan Linear Accelerator Laboratory. A second,
previously unobserved, resonance is seen with a width of
"3.2 MeV at an excitation energy of 39.5 MeV. A survey of
the experimental literature shows a possible indication of
4 2
structure at this energy in the He(y,d) H reaction measure
ments reported by Skopik and Dodge69 and in 4He(y,p) H
assymetry measurements by Arkatov, et al.,19 thus lending
credibility to the result reported here.
The observed resonances are in good agreement with
the (JC,T) = (2+,0) and (2+,1) levels reported by Szydlik,
et al.36 for shell model calculations with modified Sussex93
potentials, where T is the isospin quantum number.94 This
reference quotes levels of (2+,0) located at about 34 MeV
and (2+,1) located at about 44 MeV excitation. The inte
grated strengths of these two levels are reported as 0.86
and 0.96 mb MeV respectively giving a total E2 strength of
1.86 mb MeV from threshold to 50 MeV. This strength com
pares with 0.23 and 0.047 mb MeV for the two levels observed
and 0.97 mb MeV for the E2 direct term. The total E2
strength from this experiment is found to be 1.01 mb MeV
(Table 3.3).
The above reference would imply that the newly
observed level at 39.5 MeV would have an isospin assignment
of T = 1. A definite assignment of the isospin quantum
number can not be made from this experiment as both T = 0
and T = 1 quadrupole states can be formed in the H(p,y)4He
reaction. The narrow width, however, suggests the T = 1
assignment since fewer decay channels are available.
A recent microscopic continuum calculation by
Bevelacqua and Philpott9 indicates a (2 ,0) resonance
of 3.8 MeV width at about 39 MeV in 4He. Bevelacqua and
Philpott compare this level to the experimental data of
Lin, et al.96 Using inelastic scattering of deuterons from
4He and 6Li, Lin and Associates suggest the existence of a
level of width from 2.8 to 5.6 MeV at an excitation energy
of 31.7 MeV to fit the trend of their data at the upper end
of their observed energy range. Since the data of Lin, et al.
includes only the low energy tail of this resonance, their
width assignment of this level must remain questionable.
While it is tempting to associate this narrow level with the
level observed at 39.5 MeV in this experiment, the level at
22
35 MeV observed both here and by Malcom, et al. cannot be
ignored. If the level in the reported calculation of
Bevelacqua and Philpott corresponds to the 35 MeV resonance
observed here, then their level corresponding to the 39.5
MeV resonance would be at an energy beyond the range of their
calculation.
It is apparent that further theoretical work is
necessary in order to explain and understand the existence
of the 2+ level at 39.5 MeV observed in this experiment.
This series of experiments extends the energy range covered
by the 3H(p,y)4He reaction measurements from 18 to 30 MeV
incident proton energy. The (p,y) reaction lends itself to
better resolution and simpler analysis than the 4He(y,p)3H
or 4He(e,e'p)3H experiments that comprise the data in the
literature in this energy range. The effects of an E2
component in the radiation are seen in the measured angular
distributions.
APPENDIX A
The spectra for each energy and angle measurement
were written on magnetic tape for off line computer analysis.
The spectra were fit to the response curve of the detector
for monochromatic yray. This response curve was obtained
3 4
by accumulating a H(p,y) He spectrum with 2741 counts in
the full width at half maximum of the peak. A least squares
fit to the spectrum was obtained for the composite function
F(x) = (l0(xCg+e)exp (C1+C2X+C3X2+C4x3)
+ 8(xC9C)exp (Cs+C6x+C7X2+C8X3)
where
o x < x
6(xx ) =
1{ x > x
and xo = C9 is the channel where the two functions have equal
magnitude and slope.97 The first term is the low energy tail
of the peak and the second is the peak and high energy edge
of the spectrum. The coefficients Ci obtained from this fit
determined the standard curve shape. The experimental
measurements were then fit to the equation
F(X) = B (l9(XC9+c)exp (CI+C2X+C3X2+C4X3)
+ 6(XCgE)exp (C5+C6X+C7X2+CsX3)
XP
where X = + X The fitting parameters are the
R max
amplitude B, the gain R and the peak position P. The
variable x corresponds to the channel numbers, and the term
X is the peak of the standard spectrum. The fitting is
max
done by a parabolic extrapolation of the chisquared hyper
surface routine that is a modification of the routine Chifit
described by Bevington. This routine was found to be
reasonably fast while yielding reproducable fits to the
experimental spectra. The coefficients of the fitting
equation are as follows:
3
Cl = 3.6393 C5 = 1.4269x10
2 1
C2 = 8.9808x10 C6 = 1.4674x10
2. 2
4 2
C3 = 4.8943x10 C7 = 5.0242x10
7 5
C4 = 9.1427x10 C = 5.7104x10
4 8
Xax = 312.5 C = 294.
2
R = (x2xl) 2.5x10 B = N/95.
The peak channel counts N, and half width x2 xl are
initially selected by the operator using a light pen on
the displayed spectrum.
A sample fit is shown in Figure A.1 for the gamma
ray emitted at the laboratory angle of 650 for an incident
proton energy of 30 MeV.
3 4
Figure A.1 H(p,y) He Spectrum for 30 MeV Incident Proton
Energy. The full energy peak, first escape peak, and
second escape peak are indicated. The solid curve indi
cates the fit obtained using the peak fitting program
Ln
CD Ln
Ln
SILNnOD
APPENDIX B
Table B.1
Experimental Angular Distributions*
E 29.5 32.4 33.9 35.4
8 26. 21. 21.
o(6) 2.34 + .18 1.57 .15 1.41 .14
6 44. 44. 44. 44.
0(6) 4.95 .20 4.48 .15 4.42 .15 4.14 .14
8 52. 52. 52.
a(6) 5.63 .17 5.93 .17 4.80 .16
8 57.
0(8) 7.59 .25
0 72. 73. 73. 73.
o(6) 8.50 .27 7.17 .19 6.28 .18 5.47 .14
8 82.
o(6) 7.99 .27
8 92. 93. 93. 93.
0(e) 6.70 .24 6.02 .18 5.19 .17 4.78 .27
6 112. 113. 113. 113.
a(6) 6.19 .24 4.86 .16 4.36 .16 3.85 .15
6 127. 132. 132. 127.
0(6) 4.28 .20 2.80 .13 2.69 .12 2.50 .12
6 143. 144. 144. 144.
a(6) 2.17 .14 1.74 .07 1.74 .10 1.33 .07
86
Table B.1 (continued)
E 36.8 38.3 39.8 42.0
y
8 21. 21. 21.
o(e) 1.31 .13 1.42 .14 .86 .11
o 44. 44. 44.
0(o) 3.43 .11 3.68 .11 3.18 .11
6 50. 50. 51. 51.
0(8) 3.86 .12 3.96 .12 3.45 .11 4.09 .18
9 58. 58. 58. 61.
o(6) 4.80 .13 4.70 .13 3.92 .12 4.66 .14
e 73. 73. 73. 68.
o(8) 6.68 .17 6.15 .17 5.00 .15 4.71 .14
6 84.
a(6) 5.33 .22
8 93. 93. 93. 97.
0(8) 4.34 .13 3.42 .11 3.29 .11 3.38 .19
o 113. 113. 113. 113.
S0() 3.24 .11 2.63 .10 2.86 .11 3.01 .12
o 128. 128. 128. 128.
Co() 2.28 .10 1.72 .08 1.68 .08 2.01 .10
o 144. 144. 144. 144.
o0() 1.32 .06 .99 .04 1.04 .05 .96 .07
* all entries are center of mass, cross sections are pb
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BIOGRAPHY
The place of my birth is Magdalena, New Mexico, and
the date was May 24, 1947. I am the eldest of four children.
I attended the Quemado, New Mexico, grade school until the
seventh grade when my family moved to Robert Lee, Texas,
where I completed my primary and secondary education in the
Robert Lee Schools. I played the flute in the high school
band and was elected to the Beta Club National Honor Society.
In May, 1964, I graduated second in a class of twentytwo.
After graduating from high school, I entered San Angelo
College, later to become Angelo State University. I was a
member of the college band, was elected into Alpha Chi
National Honor Society, and was awarded a Certificate of
Scholastic Excellence. In June, 1968 I was awarded the
degree of Bachelor of Science with a major in physics. Upon
being awarded a NASA Traineeship by Illinois Institute of
Technology, I entered Graduate School. While attending IIT,
I met my wife to be. When I had completed the requirements
for the degree of Master of Science at IIT, I transferred
with my new bride to the University of Florida to complete
my education.

Full Text 
PAGE 1
PROTON RADIATIVE CAPTURE BY TRITIUM BELOW 30 MeV By ROBERT CHISM Mc BROOM A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1977
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ACKNOWLEDGEMENT This project was conducted under the guidance of Dr. H. R. Weller. His patience and direction are deeply appreciated. A special thanks to Dr. N. R. Roberson and the members of the Duke Capture Gamma Group for the privilege of working with them. The author is especially grateful for the Group's assistance with the acquisition of the experimental data. Thanks to Dr. C. P. Cameron for the use of some of his computer programs. Also, thanks to Mrs. M. Bailey for her care in drafting the figures, Mr. D. Turner for the photography, and Mrs. J. Ogden for the typing. The author is thankful to Mr . D. C. Pound and The General Atomic Company for their support during the final stages of the dissertation. Financial support for this study was provided by the University of Florida. Final and most heartfelt thanks to the author's wife, Mrs. Karen McBroom, for her love throughout all the trials .
PAGE 3
TABLE OF CONTENTS Page ACKNOWLEDGEMENTS ii LIST OF TABLES iv LIST OF FIGURES v ABSTRACT vi INTRODUCTION 1 CHAPTER I. SUMMARY OF LITERATURE 5 II. EXPERIMENTAL METHODS AND MEASUREMENTS .... 29 Experimental Details 29 Experimental Data 43 III. DATA ANALYSIS 56 Kinematics 56 Excitation Curve Analysis 59 Angular Distributions 65 Results 69 SUMMARY AND CONCLUSION 78 APPENDICES 81 A 81 B 85 BIBLIOGRAPHY 87 BIOGRAPHY 9 2
PAGE 4
LIST OF TABLES Table Page 4 1.1 He Photodis integration Reactions 11 4 1.2 Sum Rules for He and Their Shell Model Representations 13 4 1.3 He Experimental Studies 16 1.4 Relations Between the Legendre Polynomial and Sin 6, cos 6 Angular Distribution Expansions 26 2.1 Measured Differential Cross Sections .... 47 3.1 Direct and Resonance Strengths of E2 Interactions 70 3.2 a. Coefficients of Measured Angular ... 34 Distributions for H(p,y) He 73 3.3 Cross Sections Integrated from 22 to 44 MeV Excitation 77
PAGE 5
LIST OF FIGURES Figure Page 4 1.1 Excited States of He 7 4 1.2 He Photodisintegration Cross Sections .... 24 2.1 Detector Assembly for 25.4 x 25.4cm. Cylindrical Nal (Tl) High Resolution Detector 32 2.2 Detector Assembly for 10.2 x 17.8cm. Cylindrical Nal(Tl) Detector for Small Angle Measurements 35 2.3 Electronics Block Diagram for High Resolution Detector 38 2.4 Electronics Block Diagram for Small Angle Experiment 41. 3 4 2.5 H(p,y) He Center of Mass Cross Section for 9. . = 90Â° 46 lab 3 4 2.6 H(p,y) He Center of Mass Cross Sections of 6 = 55Â° and 125Â° 46 cm 3 4 2.7 H(p,Y) He Angular Distributions 53 3.1 Laboratory and Center of Momentum Coordinates 56 3 4 3.2 Asymmetry of the H(p,y) He Angular Distribution 72 3.3 Coefficients of the Legendre Expansion of 0(B) 75
PAGE 6
Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy PROTON RADIATIVE CAPTURE BY TRITIUM BELOW 30 MeV By Robert Chism McBroom June 1977 Chairman: Henry R. Weller Major Department: Physics 3 4 The H(p,y) He reaction is studied for incident proton energies from 830 MeV. The experimental data is' composed of excitation functions measured at the angles of 55Â°, 90Â° and 125Â° in the center of mass system in 1 MeV steps from 815 MeV proton energy and in 0.5 MeV steps from 1730 MeV proton energy. These data measured the energy dependence of the cross section and its deviation from the sin 8 angular dependence of the electric dipole (El) radiation. Additional angular distributions were taken at selected energies in the range from 1330 MeV with up to nine angles measured between 20 and 14 2 in the laboratory.
PAGE 7
This number of angles was taken to insure statistical significance for determining the coefficients of the Legendre expansion of the differential cross section through a.. The experimental data is shown to be consistent with the assumption that the observed radiation is primarily El with an additional electric quadrupole (E2) component and negligible contributions from higher order multipoles. The E2 radiation is shown to be consistent with a model of the E2 cross section composed of the sum of a direct interaction component, and two BreitWigner resonances. A resonance of width 12 MeV at a He energy of 35 MeV is observed in agreement with previous experimenters and a second previously unobserved resonance is observed with a width of 3.8 MeV 4 at a He energy of 39.5 MeV.
PAGE 8
INTRODUCTION Investigations of the properties of the 4 He nucleus are an important method of improving the understanding of the nuclear force. Consisting of four nucleons, He falls in a transition region between describing the nuclear interaction in terms of nucleonnucleon interactions and statistical descriptions of many nucleon interactions. The properties of the He nucleus show a number of differences from predictions based on the twonucleon interaction and the assumption that the twonucleon forces are additive. Similar difficulties are observed with models that are based on systems with a large number of nucleons. The application of models of the nuclear interaction to 4 He seeks to predict the observed properties of 4 this nucleus. One of these properties is the fact that He is the most tightly bound of all the nuclei. Another property to be considered is the rootmeansquare (rms) charge radius measured using the elastic scattering of electrons. 24 Indications of the presence of many body forces are obtained in measurements of the charge distribution of the nucleus by electron scattering. The charge distribution deviates significantly from a Gaussian shape. 4
PAGE 9
Also to be considered in a description of He is evidence that the ground state is not a pure spin zero (S=0) shell model state. Many models have been presented to describe the observed properties of He. Their success has been varied. In general the models proceed by incorporating short range fi Â— 7 correlations and noncentral forces into the interaction. Further information on the importance of ground state correlations is sought from the angular distributions of a Â• 9 H He photonuclear reactions. Several measurements of the He photonuclear reactions have been made. A number of inconsistencies were observed in the early works, some of which have been resolved by more recent measurements. The behavior of the angular distributions has been interpreted to give supporting evidence for the presence of ground state correlations. In addition, evidence exists for the presence of resonant excited states in He whose influence should be considered 13 14 in a complete description of the angular distributions. ' Evidence for excited states in He is presented in the analysis of many experiments. This evidence is derived primarily from reactions in which 4 He is formed as the compound nucleus. Evidence for several states has been obtained from phase shift analyses of nucleon scattering data. Other states are seen in inelastic electron 1 c 17 scattering. ' All of these states have positive energy.
PAGE 10
4 There exists no evidence for bound states of He. Isospin selection rules indicate that the photodisintegration of He can be useful in the investigation of negative parity, j 71 = l", isospin T = 1 states and positive parity J = 2 , 4 3 T = 0, 1 states. Some studies of the He (y,p) H reaction 3 4 have exhibited resonancelike structures, but H (p,y) He measurements have failed to confirm the existence of the 18Â—21 4 3 proposed levels. Measurements of the He (Y,n) He TT + reaction present evidence for a J = 2 , T = resonance at 22 a gamma energy of E^ 35 MeV. It will be shown that the present work provides additional evidence for the existence 4 of this level at 35 MeV in He. Most of the He photodisintegration experimental efforts for excitation energy in the range from 2040 MeV have been devoted to the He (Y,n) He reaction. The early studies of the He (y,p) H reaction suffered from difficulties due to poor statistics. For energies below 35 MeV where the triton tract becomes too short to measure, the diffusion chamber and cloud chamber measurements have difficulty distinguishing protons which arise from multi11 12 particle final states. ' Studies of the inverse reaction 3 H (p,y) He have been conducted for limited excitation energy ranges from 20 MeV to 34 MeV. In the present experiment measurements of the energy and angular dependence of the H (p,Y) He reaction were taken for excitation energies from 26 to 42 MeV.
PAGE 11
The proton beam from the Triangle Universities Nuclear Laboratory CycloGraaff was used to provide incident protons in the energy range from 8 to 30.5 MeV. The beam was incident on a 5y selfsupporting tritiated titanium foil target containing about 0.14 mg/cm 2 of tritium. The emitted photons were detected with a thallium doped sodium iodide Nal(TI) detector for up to nine laboratory angles between 20Â° and 142Â°. It will be shown that these measurements indicate the existence of a previously unobserved level in He at about 40 MeV excitation having a J 11 of 2 + and a width of about 3 MeV in the center of mass, An appendix is included which describes a peak fitting program. This program was designed to extract the low energy tail of the Nal detector response curve from , beneath the peaks due to gamma rays from transitions to states other than the ground state. The shape of the Nal response curve was based on the measured response to the 3 4 H (p,y) He reaction.
PAGE 12
CHAPTER I SUMMARY OF LITERATURE 4 The He nucleus has been the subject of intense interest. The resulting body of literature is extensive in both experimental and theoretical studies. This body of literature is reviewed in a number of recent works, particularly the survey by Fiarman and Meyerhof (1973) . 15 Many different theoretical models have met with qualitative success in describing the observed ground state properties 4 of He, but have been unable to satisfactorily describe all the observed properties. Further input for model determination is provided by experimental studies of the excited 4 4 states of He. While there are no bound states of He, experimental evidence exists for virtual states as shown in Figure 1.1. Particular consideration is given here to the photon excitation states having spin and parity (J 77 ) and isopin (T) quantum numbers as follows: jtt = 1"., t = 1 and jtt = 2 + , T = 0,1 These states have been studied by photodisintegration and the inverse photonuclear reaction. The experimental results to date have not definitively established the
PAGE 13
Figure 1.1 Excited States of He. The energy levels of 4 He and the general form of the cross sections for the four reactions, which form He as the compound nucleus, are shown (Fiarman and Meyerhof ) .
PAGE 14
^: f=Z^ rO' O' rotro* OJ Â— 1
PAGE 15
existence of any of these states; although studies of the 4 He(Y,n) He reaction indicate the existence of a J 77 = 2 , 22 T = state for an excitation energy of 35 MeV. * In addition to the question of the existence of excited states, there are fundamental questions regarding the observed shapes of the photoproton and photoneutron angular distribution as a function of the gamma ray energy. The (y,n) distribution is predicted to be backward peaked while experimentally it is observed to be forward peaked for excitations less than 27 MeV and then to become backward 22 peaked between 27 and 35 MeV excitation. For energies greater than 35 MeV excitation the angular distribution of the (Y/n) reaction is forward peaked and the asymmetry approaches the asymmetry of the (y/P) reaction channel. The (Y,n) forward peaking may be interpreted to be due to 23 correlations in the ground state v/ave functions. J The experimental observations of the photodisintegration reaction have limited resolution. These measurements are averaged quantities in a given energy interval. One method of relating these averaged quantities to the properties of the Hamiltonian and the ground state wave function is the sum rule approach. 24 The theory of sum rules has undergone extensive development starting from the study of atomic photoabsorption processes. For the electric dipole (El) radiation in the long wave length
PAGE 16
9 approximation the integrated cross section is given by the sum rule El El = /Â°Â°a(e)dE a int (1.1) = I e* <0 [Z k , [H, Z k ]]0> 1 60 NZ ( = 60 MeV mb for 4 He). 24 The wave vector  > is the ground state wave function, H is the nuclear Hamiltonian, Z k the position operator, and e k is the effective charge for the kth nucleon. The equality in the last term is satisfied if the position operator commutes with the potential energy operator. This condition is violated by Majorana or Heisenberg exchange forces or velocity dependent forces. Each of these three interactions would tend to increase o . .. Electron scattering experimt ments indicate that the contribution due to exchange currents may be included as follows: El 2527 a . j. =60 NZ(1 + a), a = 0.40. ~ (1.2) "A (= 84 MeVmb for ^He) Under the assumption that the ground state is spatially symmetric and including the charge radius of the proton, the Levinger, Bethe (28) bremsstrahlungweighted cross section is related to the nuclear radius by the following dipole sum rule:
PAGE 17
10 El r Â°Â°l El ,,..,_ Â°b =/ oEÂ° {L)dE = 0.096 Â£Â£=(R 2 R 2 ) Al z p ( = 2.48 mb for He) where R and R are the nuclear and proton rms radii z p respectively. F] The quantity a (E) referred to in the sum rules is the total cross section for all electric dipole photo4 disintegrations. For the He reactions there are five competing reactions. The yields for these five reactions were measured by Gorbunov and coworkers. ' A 170 MeV Bremsstrahlung beam incident on a cloud chamber was used in the study. Their results for the relative yield of the various reactions are shown in Table 1.1. The measured angular distributions indicated that the El interaction comprises %94% of the total integrated El cross section up to 170 MeV. Approximating a by the total cross section, integration of Gorbunov ' s data for the reactions of Table 1.1 gives: a El 1 = 9 5 Â± 7 MeV mb and o = 2.4 i .15 mb int. This value is larger than that predicted by the simple sum rules (Eq. 1.11.3) indicating contributions due to exchange and velocity dependent forces. When these forces . 34 are considered as in the sum rule analysis of Quarati
PAGE 18
11 CO c O H P O (tl d) a o Â•H +J cd U tr cu p c Â•H CO "H 13 O P O 45 ft CU a H
PAGE 19
12 for the shell model (Table 1.1), the experimental cross section up to 170 MeV excitation exhausts 95% of the sum rule. These results indicate the usefulness of sum rules. This usefulness may be increased by the removal of the approximation that the total cross reaction is due to the El interaction. In Table 1.2 are listed the sum rules commonly appearing in the literature along with their representations in the shell model formulism. The results of these equations for the shell model parameters are i _ i I i S = 1.75 fm 2 , k,. = ^jfm = 36 MeV and k Â„ = ^fm El 5 . 4 E2 2 . 5 4 These values give the rms charge radius of He to be % =1.62 fm. It has been suggested by Leonardi and 38 Lipparini that the final entry in Table 1.2, a , be 3 E2 measured, as the a sum rule for the electric quadrupole ~3 strength is dependent only on the ground state wave function. These features of the four nucleon system have received extensive theoretical treatment. Many different models have been successful to varying degrees in describing these features. The common factor in most of these analyses is an attempt to develop a method to properly account for the contributions of many body and velocity dependent forces in an orderly way. Many of the models are based on "realistic" potentials that are derived from
PAGE 20
13 o H P rfl P c 0) 03 (U M cu QJ Pi rH QJ 13 o a Ifl
PAGE 21
14 id
PAGE 22
15 7 8 39 40 the nucleonnucleon scattering phase shifts. ' ' ' These potentials fit the onshell matrix elements equally well. Their differences are embodied in their treatment of offshell matrix elements. One such potential is the Sussex potential which was used by Szydlik, et al. to calculate 4 the spectrum of He. 4 Photodisintegration studies of the He nucleus span the excitation energy range from threshold to 500 MeV. The reaction is characterized by the asymmetries in the proton and neutron angular distributions. At low excitation energies the asymmetries exhibit structure which has been interpreted as being due to interference between the electric dipole (El) and electric quadrupole (E2) absorption amplitudes. 22. For excitations above 100 MeV the (y,n) and 42 (y,p) asymmetries approach one another. The absorption of photons of energies of 100 MeV or higher is expected to occur predominantly by the quasideuteron mechanism discussed by Levinger. Recent calculations by Gari and Hebach account well for the total cross section and asymmetry in the range 50 < E < 140 MeV. 23 4 Many measurements have been made of the various He photodisintegration processes. An overview of these experiments is shown in Table 1.3. The history of the experimental 4144 measurements is one of controversy. The early measurements gave the general character of the energy and
PAGE 24
Tf
PAGE 26
19
PAGE 28
21 angular dependence of the excitation. For the cloud chamber and diffusion chamber results there are difficulties relating to small numbers of events in a given excitation range and in analyzing the tracks of the product particles for excitations below 30 MeV. Particular difficulty is associated with the measurement of the He recoil for the (y ,n) measurement. All the photodisintegration experiments suffer from difficulties associated with the spectral distribution of the photon beam. These early measurements indicated the existence of a number of structures in the cross sections, but could give little definitive information concerning the (y,n) assymmetry. The measurements by the different experimenters are in general agreement concerning the magnitude of the (Y/P) cross section. However, the (y,n) cross section is quite variable. Charge independence considerations support the results obtained by Gorbunov that the (y,n) , (YfP) cross sections are the same. 11,7 ~ The measurements are in agreement for an excitation energy greater than 35 MeV. , 5052 , The results reported by Berman, et al. and Busso, et al. 48,49 , give a lower (y,n) cross section for energies between 23 and 35 MeV. In the case of Berman' s results, the cross section would agree with the measurements of Gorbunov if multiplied by a factor of 1.8. This disagreement has been clarified by a series of experiments by Irish, et al. 53,54 which show that the variation was
PAGE 29
22 due to a reduced target thickness from bubble formation in liquid He. Measurements with liquid targets by Malcom, 22 et al. give the (y,n) cross section in agreement with Gorbunov. The 5 atm diffusion chamber measurements by Busso agree with other experimenters for excitations above 30 MeV but fall below the (y,p) cross section at lower excitation. Recent 1.5 atm diffusion chamber measurements by Arkatov, C.C. 3 et al. with improved accuracy in the low energy He recoil measurement give the (y,n) cross section in agreement 4 with the (y,p) cross section. These results for the He (y,p) ind (y,n) cross sections give experiment; charge independence of the nuclear force. and (y,n) cross sections give experimental evidence for the 79 4 The existence of evidence for excited states of He has been reported by many experimenters as seen in Table 1.3, There is little agreement in the structure seen by various early experimenters. The recent experiments with better statistics exhibit much less structure than seen previously. The experimental cross sections are shown in Figure 1.2. The significant features are the peak of the cross section for an excitation energy of 26 MeV and a broad peak for 28 MeV excitation. Peaks in this region are consistent with the existence of two states with angular momentum, parity Jtt = 1 and isospin T = 1. These states correspond to a mixture of 1 3 13 the spectroscopic states singlet P, and triplet P 1 
PAGE 30
4 ... Figure 1.2 He Photodisintegration Cross Sections. This figure is a compilation of the results reported by the indicated authors.
PAGE 31
24 to \ JD E 2.5 2.0 1.5 1.0 0.5 4 He(/,n) ' x Gorbunov(l968) Â• Irish (1975) 53 . I o Malcom(1972) 22 10 J\_Gorbunov(l968) Meyerhof (1971) 13 i^o 20 30 40 50 60 70 E r (MeV) 'CM
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25 Evidence for a third excited state at 35 MeV excitation with angular momentum, parity J = 2 and isospin T = is 22 obtained from (y,n) asymmetry measurements. The (Y/n) and (y,p) angular distributions are given in terms of the Legendre polynomial expansion as da = A (1 + Ja k P k {cos 6 )). 81 (1.4) dft o k 1 The coefficients contain contributions from photons of multipolarity L for k < 2L, k even and k < L + L , k odd where L is the multipolarity of an interfering photon. The equivalent expansion in terms of sin 6 and cos 9 prevalent in the literature is 4^ = B(ct + sin 2 6 + 6 sin 2 cos 6+6 sin 2 6 cos 2 6 + dfi e cos 8 + Â• Â• Â• ) (1.5) The coefficients Ba, B, BÂ£, B6 and Be correspond to amplitudes and cross terms for multipolarities of 0, 1, and 2. The respective relations between the coefficients of the two expansions are shown in Table 1.4. Electric operators involve only space coordinates and magnetic operators only spin coordinates in the central force approximation. For a pure 1 S ground state the allowed final states of the outgoing particles are S,, P.^ and D 2 for angular momentum changes less than two. The admixture of D states in the ground state indicated by variational calculations and
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26 Table 1.4 Relations Between the Legendre Polynomial and sin 6, cos 6 Angular Distribution Expansions B = A o (Z f a 2 " I a 4 + a = 1 + a 2 + a 4 + 3 5 2 a 2 8 a 4 + *Â•Â• = = a 3 7a 5 + ... ~2 a 2 " 8 H 4 + ' ' jc "35 6 = ~8 a 4 + Â•Â• 3 5 a , ~2 a 2 " 8 a 4 + e = a, + a, + a c + . . Â± 3 o 3 5 "2 a 2 " 8 a 4 +
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27 4 4 n He (p,p) He measurements complicates the interpretation of the coefficients of the angular distribution. Possible additional contributions may be observed as an increase in the isotropic component from El transitions from the D state to P and F states and a contribution from the cos term which arises from interference between El and nonspinflip Ml transitions. The angular distribution measurements by Wait 12 , Gorbunov 10 and Meyerhof 13 yield negligible values for the coefficients a and e. Their results are dominated by the El term B. The asymmetry (a ) of the angular distributions about G = 90Â° can be defined in terms of the difference between the cross section at 55Â° and 125Â°. This quantity, as will be shown in Chapter II, can be written as P> I nnÂ°\ 3 (cos 55 ) co Â„ a = a, + =: t C[ ' x a Q = a .68 a., s IP, (cos 55Â°) 3 1 3 for the Legendre polynomial expansion (or equivalently the term 3 for the sin 6, cos 6 expansion) where P 2 (cos 55 ) = eliminating aÂ„. These terms arise from interference between the electric dipole and quadrupole interactions. At low energy the observed angular distributions are very close to a pure sin 6 distribution. D This distribution indicates a predominately dipole interaction. The isotropic terms were observed by Gorbunov to be small for photon energies up to 170 MeV providing a basis for the assumption that only El, E2
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28 radiation contribute significantly to the yield. The interference terms producing the asymmetry are more sensitive to smaller amounts of E2 interaction than the E2 amplitude itself under these conditions. This result is due to the fact that the coefficients a, and a_ are proportional to the product of the dipole and quadrupole amplitudes .
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CHAPTER II EXPERIMENTAL METHODS AND MEASUREMENTS This study consists of a series of experimental measurements designed to obtain the electric quadrupole E2 strength as a function of the excitation of the compound nucleus for the reaction 3 H(p,Y) 4 He. The methods used to determine the E2 strength consisted of measuring the asymmetry of the angular distribution about 90Â° and the measurement of the detailed angular distributions of the outgoing Trays. Shell model calculations predict a number of states of angular momentum and parity J 11 = 2 with isospin T = 0,1 in the energy range from 3050 MeV excitation. The proton energies available from the Triangle Universities Nuclear Laboratory (TUNL) CycloGraaff allow study of the energy range from 22 to 42 MeV excitation of the compound nucleus. Experimental Details Using a 15 MeV isochronous Cyclotron as a source, the TUNL FN Tandem provides a pulsed 25 MHz proton beam. The beam energy is continuously variable from 1731 MeV. The beam pulse width is 2 ns and the energy spread is 30 KeV. The rootmeansquare beam current delivered to the target was kept in the range of 20 to 100 nanoamperes depending on 29
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30 the total count rate in the sodium iodide detector. The count rate in the detector was kept below 350k counts per second because pileup problems and a deterioration of the resolution occurred for higher count rates. Two separate detector assemblies were used in the experiment. The first detector assembly was a high resolution system and was used for the yield curve measurements and the angles of the angular distributions between 42 and 142Â°. The second detector assembly allowed measurements to be obtained for angles between 10Â° and 90 although with reduced resolution. The experimental arrangement was designed in both cases to minimize the background events that were not associated with the target. The detector assembly is based on a cylindrical . 25.4 x 25.4 cm thallium doped sodium iodide crystal. This crystal is surrounded by a well type Ne 110 plastic scintillator. The center crystal is viewed by six RCA857 5 photomultiplier tubes and the plastic scintillator by eight XP1031 photomultiplier tubes. *" A schematic of the detector arrangement is shown in Figure 2.1. Active shielding is provided by detecting coincidences between the plastic detector and the centercrystal detector which were processed by the electronics to reject cosmic ray events. Passive shielding for the detector is provided by four inches of lead and eight inches of paraffin loaded with lithium carbonate (^50% by weight) . The detector is mounted on a carriage that allows
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Figure 2.1 Detector Assembly for 25.4 x 25.4 cm Cylindrical Nal(Tl) High Resolution Detector. This illustration shows the arrangement of the active and passive shielding from the cosmic rays and beam associated background. The collimator arrangement that was used to restrict the cone of gamma rays to that just illuminating the back face of the detector crystal is shown (Weller, et al. 82 ).
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00 32 o i
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33 radial and angular positioning about the target location. Two radial positions of the detector were used in the experimental series. These distances placed the rear face of the Nal crystal at the positions of 82 cm and 106 cm respectively. Collimators were provided which restricted the observed solid angles to those subtended by the rear 2 crystal face corresponding to solid angles of 7.42 x 10 Sr 2 and 4.46 x 10 Sr respectively. Excitation curves were measured with the detector at the 82 cm distance in the interest of higher yield rates. The angular distributions were measured with the detector at the 106 position to allow the inclusion of measurements of the angles at 42 and 142 . To determine the angular distribution as the laboratory angle approaches zero, additional measurements were, taken for the laboratory angle of 20 using a separate detector system. This system was based on a 10.2 x 17.8 cm Nal(Tl) crystal surrounded by a 22.9 x 22.9 cm Nal(Tl) annulus for active cosmic ray rejection. The experimental arrangement is shown in Figure 2.2. Passive shielding is provided by a 20.3 cm thick copper annulus and 4.4 cm of lead. A cadmium sheet was located between the lead and copper for thermal neutron absorption and the collimator was filled with paraffin loaded with lithium carbonate. The distance from the target to the crystal front face was 93.3 cm. The collimator restricts the photon flux to the
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Figure 2.2 Detector Assembly for 10.2 x 17.8 cm Cylindrical Nal(Tl) Detector for Small Angle Measurement. The active and passive shielding for cosmic rays and beam associated background is shown. The beam stop shielding for measurements at 8, , = 20Â° is shown, lab
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35 _l
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36 6.57 x 10 Sr solid angle subtended by the back face of the crystal. The entrance aperture was formed from tungsten for attenuation of fast neutrons. The remainder of the collimator was lined with lead. Further measures to minimize the background events not associated with the target consisted of the use of a single insulated collimator close to the target and adequate shielding of the beam dump. The insulated collimator allows monitoring of the beam current striking it. This current was kept to a minimum by adjustments of the focusing of the beam transport system. The beam stop for the large crystal arrangement is located three meters beyond the target. Originally this beam stop was shielded by about 10 cm of lead and 70 cm of paraffin. The shielding was changed to about 10 cm of copper and 4 cm of lithium loaded paraffin to more effectively attenuate neutrons produced in the beam stop. For the small crystal assembly, the beam stop was located 33 cm behind the target. The shielding was provided by the copper shield of the detector itself and a tungsten shadow bar. The electronics for the two detector systems were similar; however, the use of the Nal annular cosmic ray detector on the smaller crystal required a different electronic arrangement than the plastic shield of the 25.4 x 25.4 cm system. The electronic arrangement for the large crystal is shown in Figure 2.3 and that for the small crystal
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Figure 2.3 Electronics Block Diagram for High Resolution Detector. This diagram illustrates the signal routing used to determine the coincidences between the active shield detector and the center detector. The additional electronics used to determine the number of coincidences are not shown (Weller, et al. 82 ) .
PAGE 45
38 O C Â° Q Â— WO w> u. aÂ» c.9c 'CCT 01 Â»Â« pi AAA AAAAA O a> Q a a> ^_ Â£ o 3 oo < x AAAAA A LU I a) if) c oTn o Â— jss (D QQ O ' S a> O Qc 0.9Jj w 5
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39 in Figure 2.4. Typically, the count rate due to low energy events is several orders of magnitude greater than the count rate due to events of interest. Common to both systems is the arrangement of the electronics to minimize the events due to the pileup of lower energy events. This minimization is accomplished by timeclipping the detector signals. The signal is then passed through a fast linear gate (250 ns) before processing. During the 10 ys processing time, the gate is held closed. Further reduction of the background due to unwanted events was possible in this experimental series due to the pulsed nature of the CycloGraaff beam. A time of flight requirement (TOF) was set up between a signal derived from a radiofrequency inductive pickup loop at the Cyclotron and the gamma ray events in the center crystal. The RF signal was used for the stop signal for a time to amplitude converter (TAC) . The TAC start signal is derived from the linear signal of the Nal center crystal. The TAC was gated by the pileup rejection circuitry to be active only for the time during which the linear gate was open. Both detectors are pulse stabilized by a calibrated light emitting diode flash into the crystal. This pulse was used to correct for slow changes in the system gain due to thermal drifts. The pulse stabilization was found to be unnecessary for the large detector system. The cosmic ray rejection was accomplished differently for the two detector systems. The
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Figure 2.4 Electronics Block Diagram for Small Angle Experiment. Illustrated here are the electronics used to determine the coincidences between the active shield detector and the center detector for the yield measurement at 6. , = 20 . The electronics associated with the pulse stabilization, accidental monitoring and beam time of flight circuits are also shown.
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41 z iu_ O T i
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42 response of the plastic scintillator is faster than that of Nal. For the large detector with the plastic shield, the shield signals were delay line clipped and timed for an overlap coincidence with the center crystal events. When a coincidence occurs, a routing signal is sent to the computer for storage of the event in a coincidence spectrum. The small crystal system utilizes the linear signal from the center crystal as the start signal for a time to amplitude converter. The shield pulses are then the stop signal for the TAC. The TAC output goes to two single channel analyzers, one of which is set to provide a signal for events occurring in the peak of the coincidence TAC spectrum. The other analyzer is set off the peak to determine the counts due to accidental coincidences. In each case the resulting signals are processed by two analog to digital converter's (ADC's) in coincidence and passed to the online computer. For maximum resolution, the threshold of the coincidence circuit could be set to give a coincidence for events due to annihilation yrays from events in the center crystal. The information stored in the computer consisted of eight 512 channel spectra. A window lightpenned on the yflash in the TAC spectrum separated the Yray events into three pairs of spectra. These pairs were stored as short time events, true data events and long time events. Each pair corresponds to center crystal events and coincidences respectively. The eighth spectrum stored was used with the
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43 large crystal system to store the elastic proton spectrum from a 500y silicon surface barrier detector located at a laboratory angle of 160 . The targets used in this experiment were tritiated titanium foils. The foils were 5y thick and the activity 2 was 6 Curies/in . This activity corresponds to an atomic ratio of approximately one between the tritium and the titanium. The tritium contained in the foils was determined by comparing the yield obtained by the solid state 3 3 detector with the known yield of H(p,p) H. The measured 2 tritium values for the two targets used were 0.14 mg/cm and 0.11 mg/cm . The error in these concentrations is determined by the uncertainty in the experimental geometry and is estimated to be 10%. The errors associated with the 3 3 statistics, the H(p,p) H cross section and the beam charge measurement combined contribute about 1% to this error. Experimental Data The experimental data were taken in four series. The first series consisted of a set of excitation curves which were utilized to determine the experimental asymmetry of the angular distribution. The second series consisted of angular distributions measured at eight angles between 42 and 142 . These measurements were extended in the third series to include a measurement at an angle of 20 . The fourth series was an extension of the excitation curve data to lower
PAGE 51
44 energies than covered in the first series. This data was then analyzed to determine the effects due to contributions from the electric quadrupole interaction. In the first series, the excitation curves were measured for laboratory angles that transform approximately to the center of mass angles of 55Â°, 93Â°, and 125Â° (Figures 2.5, 2.6). The measurement was taken in 500 keV steps over the energy range from 1730 MeV. The resulting cross sections are shown in Table 2.1. The center of mass angle in the forward direction (9 f 55Â°) and in the backward direction (0 B ^ 125Â°) were chosen to be approximately the zeros of the second order Legendre polynomial P~ (cos 6). This condition on 6f and 6 B requires that 6 B = 1806 f which yields the following results for the Legendre polynomials: P 1 (cos 6 B ) = P 1 (cos G f ) pÂ„ (cos e R ) = p,(cos e f ) = o Z Z (2.1) P (cos 9 B ) = P 3 (cos 6 f ) P 4 (cos 9 B ) = P 4 (cos 6 f ) The observed yield as a function of angle can be written as an expansion of Legendre polynomials, Y(8.) = A Q (1 + Â£,a k P k (cos 6)). k=l
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3 4 Figure 2.5 H(p,y) He Center of Mass Cross Section for 9. , = 90Â°. The data from this series of experiments is lab shown as the solid dots. The error bars represent the statistical error associated with the data points. The data is plotted both as a function of the laboratory energy of the incident proton and the center of mass energy of the emitted gamma ray. The experimental cross section reported 13 by Meyerhof, et al. is included as the open squares. The solid curve is a least squares fit of the data in the proton energy range from 5 to 30.5 MeV to a cubic polynomial. The cross section was obtained by normalizing to the data of 13 Meyerhof. The measured target thickness, solid angle and collected charge was combined with the normalization to obtain the detector efficiency. The value of 27 Â± 3% obtained is consistent with the efficiency 26 Â± 6% reported by Weller, 4t 8 2 et al. 3 4 Figure 2.6 H(p,y) He Center of Mass Cross Sections for 6 =55 and 125Â°. The experimental data is shown as the cm c solid dots with the error bars indicating the associated statistical error. The data is plotted both as a function of the laboratory energy of the incident proton and the center of mass energy of the emitted gamma ray. The solid lines are fits to the data in terms of the model discussed in Chapter III.
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E y (MeV) 46 15 20 E p (MeV) E y (MeV) ,^20 25 ion Â— ' Â— ' Â— ' Â— ' Â— r 30 35 40 E p (MeV)'
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Â§ 3 a a "3 CD XI a 39 H XI
PAGE 55
a a 48
PAGE 56
3 a a al
PAGE 57
3 8 9 50
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51 Under the assumption that multipolarities of L > 2 contribute negligibly to the cross section, the sum of the yields for the angles 8^ and 9 reduces to Y(6 f ) + Y(6 B ) = 2A o (l+a 4 P 4 (cos 6 f ) ) . (2.2) where a and a. are defined in equation (1.4). The difference between the yields is then Y(8 )y(0 B ) = A Q (a 1 P 1 (cos e f )+a 3 P 3 (cos 6 f )a 1 P 1 (cos 6g) a 3 P 3 (cos 9 B )) (2.3) = 2A (a,P 1 (cos 6 f )+a 3 P 3 (cos f )). Therefore, assuming the condition a,P.(cos 8^)<<1, which should be expected if the E2 intensity is small compared to the El intensity, the total cross section is given by a. = 4ttA c: 27t(Y(0.)+Y(6 t ,)) (2.4) u O LB and A Q h (Y(6 f )+Y(6 B )) . (2.5) Under these conditions, the experimental asymmetry in terms of the yields Y(6 f ) and Y(6 ) is equivalent to that defined in terms of the coefficients of the Legendre expansion of the angular distribution (Figure 2.7). This equivalence is shown by the following relations: Y(0 f ) Y(0 B ) = 2A o (a 1 P 1 (cos 6 f )+a 3 P 3 (cos f )) P 1 (cos f ) (Y(6 f )+Y(0 B )) P 1 (cos f ) (2A o (l+a 4 P 4 (cos 6 f ) ) ( 2 fi ) b a + P 3 (cos f )a 3 t b (cos e )Â«1. p 1 (cos e f )
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T 4 Â„, Figure 2.7 H(p,y) He Angular Distributions. The experimental data is shown as the solid dots with the error bars indicating the associated statistical error. The solid lines are least squares fits to the data in terms of the Legendre Polynomials as discussed in Chapter III. The measured values for the cross sections are included in Appendix B.
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53 v/(e)o v/(e)o
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54 As will be shown in Chapter III, the measured asymmetry and the measured cross section can be used to obtain an experimental E2 cross section. The El cross section is obtained by observing that the terms of the multipole expansion (equation 1.5) for da dil multipolarities up to L = 4 reduce to ^r= B (a + 1) at = 90Â°. cm Therefore, taking a<< 1 (no spinflip or magnetic dipole radiation) the yield at this angle is the El cross section. ' ' The angle measured was 8, . = 9 so that the results could be readily compared with those of previous 13,56 experimenters . A second experimental series was conducted to confirm the energy behavior of the E2 cross section and to provide a direct measurement of the E2 amplitude. This series consisted of a set of angular distribution measurements taken every 2 MeV between 17 and 27 MeV incident proton energy, and included angular distributions measured for proton energies of 13 and 30 MeV. For each angular distribution, yields were measured for a minimum of seven angles in the range 42 142 covered by the detector. Additional angles were included as needed to define the shape of the peak and the slope of the sides of the angular distribution. The minimum number of measurement points was chosen so that the number of degrees of freedom associated witha Legendre polynomial least squares fit to four
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55 coefficients is greater than zero. The amount of charge collected was varied so that a minimum of 200 counts was observed for the 142Â° measurement. The yield was larger for all other angles observed (Figure 2.7). The third series of measurements was conducted using the 10.2 x 17.8 cm Nal detector. The purpose of this experimental series was to determine the shape of the angular distribution as the angle approaches zero. Measurements were made every 2 MeV in the proton energy range 17 < E < 27 MeV. Yields were measured for three laboratory p angles including 20Â° , 9 0Â° and an intermediate angle corresponding to an angle at each energy for which the yield was measured in the second series. These measurements were normalized to the data of the second series, and an effective yield for the 25.4 x 25.4 cm detector at the laboratory angle of 20Â° was obtained. This yield was incorporated into the angular distributions used to determine the Legendre expansion coefficients. The fourth series of measurements extended the asymmetry data to lower energy. The TUNL FN tandem was used to measure the foreaft asymmetry every 1 MeV in the proton energy range from 8 < E < 15 MeV. This information was obtained for comparison with the published results of earlier experimenters.
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CHAPTER III DATA ANALYSIS Kinematics The kinematics of the reaction H(p,y) He for proton energies above 10 MeV are subject to relativistic corrections. Therefore, the kinematics were explicitly treated according to special relativity following the treat84 ment by J. J. Jackson. The reaction coordinates in the laboratory and center of momentum frames are shown in Figure 3.1. Laboratory Center of Momentum Figure 3.1 Laboratory and Center of Momentum Coordinates The conservation equation for the square of the four momentum of the alpha particle in its rest frame and in the laboratory system is /Z* (0,j m c ) = ( 'y 2 _! m ckhk, j (y m c 2 +m.c 2 E )) 2 (3.1) ' c ^ P /Â„PPty 56
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57 where y is the Lorentz transformation parameter. For the c proton in the laboratory, this parameter is Y = lv 2 /c 2 * c p p Solving the conservation equation for the laboratory gamma ray energy as a function of laboratory angle gives E Y 5 ^ (3.2) 2 (y m c 2 +m,c 2 ^y 2! m c 2 cos0 ) P P t 'p p y where the energy E" = (m^c' t +m 2 c' t + 2Y p m p c 2 m t c 2 ) J2 (3.3) is the total energy in the center of mass system. The gamma ray energy in the center of mass is (KM'^C _ (34) ' 2E' The excitation energy of the compound nucleus is found by subtracting the alpha particle rest energy from the total center of mass energy E x E' m a c 2 (3.5) The relationship between the angle in the center of momentum and in the laboratory is given by i _ 1 Y 1 ,sin 6, . , ,. 'p ,_ _, tan (_ _j_ ) where 3 = ^ Â— , (3.6) Y (cos6 3) Y 2 cm Y p where the Lorentz transformation parameter for the center of momentum y is related to the total laboratory enerqy and cm J r,J
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58 the total energy of the compound nucleus by v = y m c 2 + m.c 2 , . .,> ' cm 'p p t_ (3.7) E' These relations were incorporated in the computer programs used for data analysis. The relationship between the measured cross section in the laboratory reference frame and that of the center of momentum reference frame is also necessary for comparison of the experiment to theory. This relation can be obtained from the equality of the yield in the two frames following Marion. This equality is ^lab < 6 Y) d " = Q cm (e')dft 1 . (3.8) Using the kinematic relations between 6 and 6 ' with the definitions of the solid angle increments, which are 6.Q = 2tt sine d9 and dQ, ' = 2tt sinG'dG' , the relation beY Y tween the two differential cross sections is found to be as follows: /nx rr /ni\ sin 3 8' ,cos8'cos9 . , , Q . Â°lah { \ ) = a cm (9 > sTHeÂ— ( ... . T + W" (3 ' 9) 1 Y smB sm8 1 Y This form of the relation between the two frames was convenient to use as the analysis programs calculated 6 ' and Y for measured gamma ray angle 9 .
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59 Excitation Curve Analysis Assuming only El and E2 radiation with the additional assumption that the El and E2 spinflip terms are negligible, the cross section may be written in terms of the spherical harmonics Y, , and YÂ„, . The interaction is modeled by assuming that it consists of direct and resonance terms. The center of mass cross section expression is da (6) i6i iS 2 i($i+ni) Â— *jg =ejDie Y X i + ( e 2 D 2 e + R x e i(^2+Tl2) +R 2 e )Y 21 (3.10) 41 This expression for the cross section implicitly includes the relations a. = a, and 1 + aÂ„ + a. = between the coefficients in the Legendre expansion. The first term of the cross section is due to a direct El interaction. The second term is due to a direct E2 interaction, The factors e, and eÂ„ are the kinematic effective charges. The third and fourth terms are resolved Breit Wigner resonances centered around energies E, and E* respectively. Each term of this expression will be examined further in the following paragraphs .
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60 The expressions for the resonance terms in the center of mass are R x g^;) 372 pV p (3.ii) ((Â£' E ) 2 + r 1 2 /4) T and 3/ 1/ R 2 = g 2 (E') ' 2 P (3>12) ((Â£' E ) 2 +r 2 /4) 2 Y 2 The energy dependence of the resonance partial widths is explicitly included in these expressions where P is the proton penetrability calculated by the method of B. Buck, et al . and 2L+1 h 3/ * y' k v Y k 2 E y describes the energy dependence of the transition amplitude (T ) of a L = 2 multipole. The resonance strength for the gamma ray resonant absorption is then .Vc'UL+l) r /p . 4 ^' (E ; >2L+1P P ,3.13, (E') ! (E') ! The energy dependence of the total width of the resonance is included by using the proton and neutron penetrabilities and their respective reduced widths, i.e. r = r + r = 2y 2 p + 2y 2 p . (3.14) n p 'n n 'p p
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61 The deuteron and gamma partial widths are assumed to be negligible with respect to the neutron and proton partial widths. The reduced widths are solutions to the equations r(E) = 2v 2 P (E) + 2v 2 P (E) (3.15) ' n n PP and 2 Â— 2 for the resonance centered about the energy E. The neutron and proton contributions are here assumed to differ only by coulomb effects. The phase $ in each case is the resonance phase, given by $. = tan" ( E " E i ) (3.16) 1 r Â± /2 The phases n, and ru account for the relative phases between the resonances and the direct terms. The functional form for the direct terms was taken Q Q from the analysis of Flowers and Mandl. They assume gaussian type wave functions and obtain relations of the form Â±& ,3/ bE i 6 X D, e = ai (E')V e P e (3.17) and 16 3/o s, bE i6 2 D 2 e = a 2 (E) /2 E A e P e . (3.18)
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62 The gaussian wave functions parameters are chosen to fit the 4 He rras. radius of 1.61 fm. given by electron scatter5 / k 5 ing giving a Â± = .101 (mb/MeV /2 )^, a 2 = 5.13 x 10 (mb/MeV 1 ^ 2 ) ii and b = .043 MeV 1 . 41 For comparison to the experimental data, the direct El term was replaced with an empirical function and the strength of the E2 term, a 2 , was allowed to vary. The phases & Â± and 6 2 were varied to obtain the relative phase between the direct El and E2 terms that best fit the data. The above expression for the differential cross section is in the form of products of spherical harmonics. These products are expanded in terms of single spherical harmonics. The product expansion is *" *"" *.Â» r ML ' 4 "< 2L+1) ] (319) ooo m'mM ffl ,, where the ClebshGordon coefficient CÂ™ lÂ£L is zero for M f m'm. The resulting expansion is in terms of the Y LQ functions alone. The functions Y LQ are simply related to the Legendre polynomials by the function Y L0 (e,*) = / ^ P L (cose) (3 . 20) Collecting the terms of the polynomials and comparing with the Legendre expansion of the angular distribution allows identification of the expansion coefficients in terms of the
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63 direct and resonance contributions of the model. In order to make this identification the following three definitions are made: B o = UjDj) 2 B = [2e 1 D 1 e 2 D 2 cos (6i6 2 )2 Â£i D i (Ricos ($ r +n 1 6 i ) +R 2 cos ($ 2 + n 2 6 i ) ) B = [ (e 2 D 2 ) 2 2e 2 D 2 (Ricos ($! + ni i+n i  2~n 2 ) + R2 2 ]Â« (3.21) In these definitions B appears in terms of only El amplitudes, B, appears in terms of products of El and E2 amplitudes, and B 2 appears in terms of E2 amplitudes only. The coefficients of the cross section expansion may then be written as follows: A o = va
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6 4 transformation introduces geometric correction factors Q i (Ferguson 89 ) into the expansion. The resulting expression for the experimental cross section is \p a(e) = A (Q +aiQiPi (cos6)+a 2 Q 2 P 2 (cos0) "M Â° Â° (3.23) +a 3 Q 3 P 3 (cos6)+a 4 Q 4 Pi, (cos8) ) . The yield from the 3 H(p,y) 4 He reaction is converted to the photonuclear reaction He(y,p) H by the principle of detailed balance. The relationship between the cross sections is given by a (e) 1 ( (2I(^H) + 1)(2I(3H) + 1) ) ( !V!!P } (e) .90 Y,P 2 (2I( 4 He)+l) E* P ' Y (3.24) where 1(A) is the nuclear spin of nucleus A. Using these relations, a computer program was developed which plotted the model cross sections against the measured cross sections as a function of energy and angle. The model parameters for the curves that best represented the plotted experimental data were used to obtain the resonance parameters and the integrated strength of the various components of the model. These results are presented at the end of this Chapter.
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65 Angular Distributions The angular distribution data were measured in an attempt to obtain a direct measurement of the E2 strength through a determination of the coefficient a^ . The magnitude of the cross section for 6=0Â°, 180 also provides a determination of the contribution of spinflip interactions to the cross section. The experimental data are fit by linear least squares to the first four terms of the Legendre cross section expansion for finite geometry as discussed in the previous section. For some measurements, the resulting fits were negative for 6 = 0Â°, 180Â°. A nonlinear least squares search with the additional requirement that the cross section be positive was then used to obtain the values for the coefficients A , a 1# a 2 , a 3# and a 4 . The error associated with each coefficient is then obtained from the curvature of the Chisquare space. The assumption that the El and E2 spin flip is negligible is tested by the angular distributions. The truncated Legendre expansion reduces to g (e) = A (l+a 1 +a,+a,+a,) (3.25) at 6 = 0Â° and 2. ( 9 ) = A o (la 1 +a 2 a 3 +a lt ) at 6 = 180Â°. The cross section is equal at these two angles for a 3 = a : and zero if in addition l+a^a^ = 0. Assuming only El
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66 and E2 Radiation, the coefficients A a, and A a can be o 1 o J written in terms of the complex vectors j j R J> for total angular momentum (jj) coupling where j = i + S is the + > total angular momentum of the incident particle, j = SÂ„ is the total angular momentum of the target and J = ] + ]Â„ is the total angular momentum of the system. The nuclear spins are S and S R of the projectile and target respectively/ while I is the orbital angular momentum. The R matrix elements ( ) for radiation of multipolarity Â£Â£ and compound nucleus of ground state spin 1=0 are the following for El and E2 radiation: i 1 ^ i$3/ <011TW> = P^e H , <011 T I 3 / 2 1 / 2 l> = P 3/2 e /2 (3.27) l'$3, 1 9 5/ <022 T  3 / 2 V 2 2> = d 3/2 e h , <022 t  5 / 2 V 2 2> = d^e /2 . The resulting expression for the coefficients are as follows: i( 2 $ 3/ x *i, )) A o a l = I {/I Â° Re( V 3 / 2 e /5 , , 1{H %~ H %\ (3.28) Â— Re(P 3/2 d3 /2 e ) ( 2 $ 5/ "^Sy ) + 3/30 _ ,_ , Â„ / 2 /; tRe(p v 2 d5 / 2 e ] >
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67 2 * . _ 1 ,7, A o a 3 =  {/I5 Re(P^d 3/2 e /2 /2 ) + i^Re(P 3/ d 3/ .*' '* "% ) > /2 /2 (3.29) 9^n i( 1$ 3 / 2 *s / ) Â— = Â— Re(P 3/ d 5/ e ) } > n /2 The relationship a, = a, can be obtained from these equations by imposing the following conditions on the phases J $3/ = l $1/ + nTT h n n = 0,2,4, (3.30) 5 / 2 3 / /2 + nit With these conditions there are two possible classes of solutions to equations 3.28 and 3.29 that lead to the result a., = a, . One class is obtained when the ratio between the magnitudes d 3/ and d 5 . equals /6/3 in which /2 h case the ratio between the magnitudes Pi. and P 3 is /2 /2 indeterminate. The other possible class of solutions is obtained when the ratio between the magnitudes P,. and P 3 /2 /2 equals /2/2 in which case the ratio of the magnitudes d 3 . and d 5/ is indeterminate. /2 Alternatively the system may be described in terms of the (LS) coupling vectors ( I Â£ S Â£ S LSJ>). The total Q d p p anqular momentum is t, = 1 + ~t a and the total spin is r a p , " S = Â£ + Â§ . The total angular momentum is then J = L + S,
PAGE 75
68 These vectors can be related to the ] Â• 3 coupling vectors of equations 3.27 by the us. relationship is as follows: 91 of equations 3.27 by the use of the 9j symbols. ' The I hl D hLSJ> = J [(2S+1) (2L+1) (2j +1) (2j +l)] h a p Â• a p J a J 3 k k S f (3.31) ' e [ I V 6 J> ' The results for total angular momentum and parity J = 1 are I Oh \h 101 > = /3~ I â€¢sl > + S5 I V2V2I > ~T 3 (3.32) 1 0^ 13s ill > = ft I hhl > + /I I V2V2I > ~3 3 IT + The results for total angular momentum and parity J = 2 I 03s 2k 202 > = /TO" s A/2 2 > + ^ I A/2 2 > "IT 5 (3.33) I Ois 23s 212 > /15 I V2V2 2 > + /H I 5 / 2 7 2 2 Â• The two cases leading to the relation a 3 =~a 1 are obtained by either  0^1^111> or  0^2J5212> being equal to zero. Studies with polarized protons indicate that  0Jsl J slll> is 92 Â• 15,19,33 nonzero. As seen by previous experimenters
PAGE 76
69 the contribution of the spin flip terms 0J5l^lll> and .0^2Js212> to the unpolarized cross section is zero to within the experimental error. The data in this experiment are consistent with neglecting spinflip as can be seen in Figure 2.7 and 3.3. Therefore, these terms were not included in the analysis. Results The experimental data is consistent with the assumption that the reaction is dominated by the electric dipole radiation. Considering only electric dipole and quadrupole radiation, the magnitude of the cross section for = 90Â° is a measure of the electric dipole strength cm if the angular distribution of the quadrupole radiation is of the form of the spherical harmonic YÂ„, (which vanishes at 90Â°). Using these assumptions, an empirical equation for the electric dipole strength was derived from the experimental data, using a linear least squares fit to the experimental H(p,y) He cross section between 23 and 42 MeV excitation. The cross section in this energy range was found to be best represented by a cubic polynomial with a reduced chisquare of 0.02. The resulting expression for the El radiation in the model is _3 _4 _5 2 7 3 ? 4m c 2 E {1.518x10 +3.953x10 E 3.921x10 E +5.419x10 E } D = p p x x x E 2 Y (3.34)
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70 This expression gives the 90 cross section shown as the solid curve in Figure 2.5. Using this result for D, , the model parameters were obtained to produce the solid curves in Figure 2.6 for the angles = 53Â°, 123Â°. The asymmetry (a ) resulting from this calculation is displayed as the solid curve in Figure 3.2. The parameters of the calculation are shown in Table 3.1 for the resonances observed in the data. Table 3.1 Direct and Resonance Parameters of the E2 Cross Section E (Mev) *
PAGE 78
Figure 3.2 Asymmetry of the 3 H(p,Y) 4 He Angular Distribution. The asymmetry data from the measurements at 6 cm = 55 and 125Â° are the solid dots with the error bars showing the statistical error. The asymmetry from the angular Distribution of Figure 2.7 are shown as x's with arrowheads indicating the error propagated from a., and a 3 > The results i 4+a 13,60,58 reported by previous experimenters is also plotted.
PAGE 79
72 >Â« C7> CD """ ~ C7> Â£ a) to E "5 J" < < CO 0) <1> o c Z. Z. c 3 3Â°lIÂ— CO o o 00 d CD
PAGE 80
ft U O m 10 c o Â•H p XI H M p CO H Q M n3 rH Cn 0) sl CO id a o CO p C d) Â•H U 73
PAGE 81
Figure 3.3 Coefficients of the Legendre Expansion of a(8). The a. coefficients from the angular distributions of Figure 2.7 are shown as x's in the upper plot. The error bars are the errors of the coefficients a.^ obtained from the least squares fit of the data to the Legendre expansion. The lower plot shows the experimental A Q obtained from the cross sections of Figure 2.6. The solid curves in both plots are the results obtained from the model (Equation 3.10). The energy E is in the center of mass.
PAGE 82
75
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76 the experimental data and the model of direct E2 and two E2 resonances superimposed on a dominant El background interaction is seen to be quite satisfactory. For comparison to theoretical calculations in the literature, the integrated cross sections due to the individual components of the model were evaluated. The total photonuclear cross section is obtained by integrating the Legendre polynomial expansion of the angular distribution (Equation 1.4) over all solid angled, The result is /dfl da = 4ttA = B q + B 2 = a + a where B and B~ are defined by Equation 3.21. As seen in Equation 3.21, the individual contributions of the direct and resonance terms used in the model may be evaluated separately. The results of this evaluation are shown in Table 3.3.
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77 Table 3.3 Cross Sections Integrated From 22 to 44 MeV Excitation fa dE = 3 0.6 mb MeV / a E1 dE = 2 9.6 mb MeV Y / P fo E2 dE = 1.01 mb MeV / a E2 dE = 8.12x10 " mb/ MeV 2 E / Ri 2 dE = 0.226 mb MeV / Rl dE = 1.78xl0~ mb/ Â— MeV E 2 2 / R 2 dE = 0.04 7 mb Mev / 2 dE = 3.07x10 mb/ Â— MeV E 2 / a d dE = 0.969 mb MeV /a, dE = 6.69xl0~ mb/ d MeV
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SUMMARY AND CONCLUSION The TUNL CycloGraaff was used to obtain data from the H(p,y) He reaction for protons of energies from 830 MeV in the laboratory. The data were taken as a series of excitation curves and angular distributions. The data obtained can be adequately accounted for if the radiation is viewed as being predominantly electric dipole with an admixture of direct and resonant electric quadrupole radiation. Two electric quadrupole resonances are observed and the resonance width obtained for each. The resonance at a compound nucleus excitation of 35 MeV with a width of 12 MeV is in good agreement with the result reported by Malcom, et al. in the He photoneutron experiments at the Saskatchewan Linear Accelerator Laboratory. A second, previously unobserved, resonance is seen with a width of ^3.2 MeV at an excitation energy of 3 9.5 MeV. A survey of the experimental literature shows a possible indication of 4 2 structure at this energy in the He(y,d) H reaction measure69 4 3 ments reported by Skopik and Dodge and in He(y/P) H 19 assymetry measurements by Arkatov, et al., thus lending credibility to the result reported here. 78
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79 The observed resonances are in good agreement with the (J 77 ,! 1 ) = (2 + ,0) and (2 + ,l) levels reported by Szydlik, o fi 93 et al. for shell model calculations with modified Sussex 94 potentials, where T is the isospin quantum number. This reference quotes levels of (2 + ,0) located at about 34 MeV and (2 + ,l) located at about 44 MeV excitation. The integrated strengths of these two levels are reported as 0.86 and 0.96 mb MeV respectively giving a total E2 strength of 1.86 mb MeV from threshold to 50 MeV. This strength compares with 0.23 and 0.047 mb MeV for the two levels observed and 0.97 mb MeV for the E2 direct term. The total E2 strength from this experiment is found to be 1.01 mb MeV (Table 3.3) . The above reference would imply that the newly observed level at 39.5 MeV would have an isospin assignment of T = 1. A definite assignment of the isospin quantum number can not be made from this experiment as both T = 3 4 and T = 1 quadrupole states can be formed in the H(p,y) He reaction. The narrow width, however, suggests the T = 1 assignment since fewer decay channels are available. A recent microscopic continuum calculation by 95 + Bevelacqua and Philpott indicates a (2 ,0) resonance 4 of 3.8 MeV width at about 39 MeV in He. Bevelacqua and Philpott compare this level to the experimental data of Lin, et al. 96 Using inelastic scattering of deuterons from
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80 He and ^Li, Lin and Associates suggest the existence of a level of width from 2.8 to 5.6 MeV at an excitation energy of 31.7 MeV to fit the trend of their data at the upper end of their observed energy range. Since the data of Lin, et al. includes only the low energy tail of this resonance, their width assignment of this level must remain questionable. While it is tempting to associate this narrow level with the level observed at 39.5 MeV in this experiment, the level at 22 35 MeV observed both here and by Malcom, et al. cannot be ignored. If the level in the reported calculation of Bevelacqua and Philpott corresponds to the 35 MeV resonance observed here, then their level corresponding to the 39.5 MeV resonance would be at an energy beyond the range of their calculation. It is apparent that further theoretical work is necessary in order to explain and understand the existence of the 2+ level at 39.5 MeV observed in this experiment. This series of experiments extends the energy range covered by the ^H(p,y)^He reaction measurements from 18 to 30 MeV incident proton energy. The (p,Y) reaction lends itself to better resolution and simpler analysis than the 4He(Y/P)^H or ^He(e,e'p)^H experiments that comprise the data in the literature in this energy range. The effects of an E2 component in the radiation are seen in the measured angular distributions .
PAGE 88
APPENDIX A The spectra for each energy and angle measurement were written on magnetic tape for off line computer analysis. The spectra were fit to the response curve of the detector for monochromatic yray. This response curve was obtained 3 4 by accumulating a H(p,y) He spectrum with 2741 counts in the full width at half maximum of the peak. A least squares fit to the spectrum was obtained for the composite function F(x) = (18 (xC 9 + e)exp (C !+C 2 x+C 3 x 2 +C\x 3 ) + 9 (xC 9 e)exp (C 5 +C 6 x+C7X 2 +C 8 x 3 ) where r, I \ o x < x^ 6 (XX ) = r O l l X > X o and x = Cg is the channel where the two functions have equal magnitude and slope. 97 The first term is the low energy tail of the peak and the second is the peak and high energy edge of the spectrum. The coefficients C. obtained from this fit determined the standard curve shape. The experimental measurements were then fit to the equation F(X) =B (16 (XC 9 +e)exp (Ci+C 2 X+C 3 X 2 +C^X 3 ) + 6(XC 9 e)exp (C 5 +C 6 X+C 7 X 2 +C 8 X 3 ) 81
PAGE 89
82 xP where X = + X . The fitting parameters are the R max amplitude B, the gain R and the peak position P. The variable x corresponds to the channel numbers, and the term X is the peak of the standard spectrum. The fitting is max c done by a parabolic extrapolation of the chisquared hypersurface routine that is a modification of the routine Chifit 9 8 described by Bevington. This routine was found to be reasonably fast while yielding reproducable fits to the experimental spectra. The coefficients of the fitting equation are as follows: 3 C, = 3.6393 C c = 1.4269x10 _2 1 C, = 8. 9808x10 C, 1.4674x10 2 6 1* _2 C = 4.8943x10 C ? = 5.0242x10 _7 _ 5 C 4 = 9.1427x10 C 8 = 5.7104x10 X = 312.5 C Q = 294. max 9 _ 2 R = (x 2 Xi) 2.5x10 B = N/95. The peak channel counts N, and half width x 2 x, are initially selected by the operator using a light pen on the displayed spectrum. A sample fit is shown in Figure A.l for the gamma ray emitted at the laboratory angle of 65 for an incident proton energy of 30 MeV.
PAGE 90
Figure A.l H(p,y) He Spectrum for 30 MeV Incident Proton Energy. The full energy peak, first escape peak, and second escape peak are indicated. The solid curve indicates the fit obtained using the peak fitting program
PAGE 91
84 SiNDOD
PAGE 92
APPENDIX B Table B.l Experimental Angular Distributions* E 29.5 32.4 33.9 35.4 Y 6 26. 21. 21. o(6) 2.34 Â± .18 1.57 Â± .15 1.41 Â± .14 9 44. 44. 44. 44. 0(6) 4.95 Â± .20 4.48 Â± .15 4.42 Â± .15 4.14 Â± .14 6 52. 52. 52. o(9) 5.63 Â± .17 5.93 Â± .17 4.80 Â± .16 6 57. o(6) 7.59 Â± .25 6 72. 73. 73. 73. 0(6) 8.50 Â± .27 7.17 Â± .19 6.28 Â± .18 5.47 Â± .14 6 82. o(6) 7.99 Â± .27 6 92. 93. 93. 93. 0(6) 6.70 Â± .24 6.02 + .18 5.19 + .17 4.78 Â± .27 6 112. 113. 113. 113. o(6) 6.19 Â± .24 4.86 Â± .16 4.36 + .16 3.85 Â± .15 6 127. 132. 132. 127. o(8) 4.28 Â± .20 2.80 Â± .13 2.69 Â± .12 2.50 Â± .12 6 143. 144. 144. 144. 0(6) 2.17 + .14 1.74 Â± .07 1.74 Â± .10 1.33 Â± .07 85
PAGE 93
86 Table B.l (continued) E Y
PAGE 94
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BIOGRAPHY The place of my birth is Magdalena, New Mexico, and the date was May 24, 1947. I am the eldest of four children. I attended the Quemado , New Mexico, grade school until the seventh grade when my family moved to Robert Lee, Texas, where I completed my primary and secondary education in the Robert Lee Schools. I played the flute in the high school band and was elected to the Beta Club National Honor Society. In May, 1964, I graduated second in a class of twentytwo. After graduating from high school, I entered San Angelo College, later to become Angelo State University. I was a member of the college band, was elected into Alpha Chi National Honor Society, and was awarded a Certificate of Scholastic Excellence. In June, 1968 I was awarded the degree of Bachelor of Science with a major in physics. Upon being awarded a NASA Traineeship by Illinois Institute of Technology, I entered Graduate School. While attending IIT, I met my wife to be. When I had completed the requirements for the degree of Master of Science at IIT, I transferred with my new bride to the University of Florida to complete my education. 93
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. lijjlj.^ h H. R. Werller, Chairman Associate Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. // / 7 / 'i1 R. A. Blue Associate Professor of Physics I certify that 1 have read this study and that in my opinion it conforms to acceptable standards of scholarlypresentation and is fully adequate, in cope and quality, as a dissertation for the degree of Doctor of Philosophy. T. L. Bailey Professor 6'f Physics I. certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. W^lxl.}'i.*~/< J. W. Dufty Associate Professor of Physics
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a disseration for the degree of Doctor of Philosophy. j.P.Qlfc^ G. R. Dalton Professor of Nuclear Engineering Sciences This disseration was submitted to the Graduate Faculty of the Department of Physics in the College of Arts and Sciences and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August 1977 Dean, Graduate School
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