,_. “pru-mq‘.muu_—o

6A STUDY OF THE ENERGY LEVELS oF 8;,NE
MG, SI, 3",,334AR AND3 CA BY THE (nUREACflON

Thesis for the Degree of Ph. D.
MICHIGAN STATE UNIVERSITY
ROBERT ALTON PA‘DDOCK
1969

THESIS

 

ABSTRACT

A STUDY OF THE ENERGY LEVELS OF 18Ne,

22Mg, 26s1, 305, 3“Ar, AND 380a
BY THE (p,t) REACTION

By

Robert Alton Paddock

A study of the (p,t) reaction on the even-even
N=Z nuclei in the 2sld shell has been carried out. This
reaction has been used to study the energy levels of

18Ne, 22Mg, 2681, 308, 34

Ar and 38Ca. Until recently
little has been reported about these nuclei. Except for
a few scattered reports of (p,t) experiments, only the
(3He,n) and (3He,ny) reactions have been used to study
these nuclei. The excited states that were observed
are reported along with the spin and parity assignments
when possible.

The two nucleon transfer distorted wave theory of
N. K. Glendenning has been studied with respect to these
(p,t) reactions. It was found that the shapes of the
predicted angular distributions are primarily dependent
on the orbital angular momentum transfer and the Optical
model parameters. This fact was used to make the spin-
parity assignments. It was also found that the magni-

tudes of the predicted cross-sections are strongly

dependent on not only the Optical model parameters, but

Robert Alton Paddock

also the bound state parameters of the transferred neu-
trons and the configuration mixing in the initial and
final nuclear wave functions.

It is concluded that the (p,t) reaction is useful
to study the energy levels of nuclei two nucleons away
from stability. It is also concluded that the two
nucleon transfer distorted wave theory is useful to pre-
dict the general shapes of angular distributions but
that the magnitudes are too dependent on parameters

which are not well known to be predicted successfully.

A STUDY OF THE ENERGY LEVELS OF 18N8,

22 26

Mg, Si, 303, 3uAr, AND 380a

BY THE (p,t) REACTION
By

Robert Alton Paddock

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Physics

1969

ACKNOWLEDGMENTS

I would first of all like to thank Dr. Walter
Benenson for suggesting these experiments as a thesis
tOpic and his help in carrying out the experimental
work. I would also like to thank Dr. B. Freedom for his
helpful discussions on the distorted wave method. My
thanks go to Dr. K. Koltveit for his aid in my under-
standing Of parentage factors. I would also like to
thank Dr. w. Gerace for making a COpy of the code TWOFRM
available to the M.S.U. Cyclotron Laboratory, and

Professor R. M. Drisko for making a copy of the FORTRAN-

.IV version of the JULIE available. Special thanks go to

Dr. P. Locard and Mr. Ivan Proctor for their help in
taking the data as well as to the entire staff of the
Cyclotron Laboratory. I must also thank Mr. P. Plauger,
whose relativistic kinematics computer routine was used
throughout the analysis of the experimental data.

I would also like to acknowledge the financial sup-
port of the National Science Foundation and Michigan
State University throughout my graduate work.

Very special thanks to my wife, Connie, for her
understanding through the past two years, and for the

typing of the rough copies of this thesis.

ii

TABLE OF

ACKNOWLEDGMENTS

LIST OF TABLES.

LIST OF FIGURES

Chapter
10

2.

wwww

INTRODUCTION.

CONTENTS

SELECTION RULES FOR (p,t)

TWO NEUTRON PICKUP AND THE DISTORTED

WAVE METHOD

.1:me

THE

1:" Etttttfi'
oo NONUWEUUNH

The Distorted Wave Method. .
The Two Neutron Pickup Matrix Element.

The (p, t) Form Factor .

The Two Neutron Parentage Factor

EXPERIMENT

The Proton Beam

The Faraday Cup and Charge Collection.

The Scattering Chamber.

Targets.

The Detector Telescope.

Dead Time Corrections
Electronics and Particle

Identification

Triton Energy Spectra and Energy

Resolution.

DATA REDUCTION

EXPERIMENTAL RESULTS

6.1 The Ground State Transitions. .
6. 2 The Transition to the First Excited

6. 3 Other Transitions

States . .

iii

Page

ii

vi

10
2O
22
A2

A2
“3

AA
A6
A8
A8
53
59
63
63

6A
6A

Chapter

7.

DISTORTED WAVE CALCULATIONS

7.1 The Optical Model
7.2 The Ground State Transitions
7.3 Dependence on the Bound State Well
7.“ Dependence on Configuration Mixing
7.5 The Transitions to the First Excited
2+ States. . .

7.6 Transition to States in l8Ne .
7.7 Transitions to States in 22Mg . . .
7.8 Transitions to States in 2681 . . .
7.9 Transitions to States in 308 . . .
7.10 Transitions to States in 3gAr . . . .
7.11 Transitions to States in 3 Ca . . . .

8. SUMMARY AND CONCLUSIONS

APPENDICES

2o 18 ,

A. Ne(p.t) Ne EXPERIMENTAL DATA

B. 2uMg(p,t)22Mg EXPERIMENTAL DATA
'3

C. ”8Si(p,t)26Si EXPERIMENTAL DATA

D. 32S(p,t)3OS EXPERIMENTAL DATA

E. 36Ar(p,t)3uAr EXPERIMENTAL DATA

F. “OCa(p,t)3SCa EXPERIMENTAL DATA . . . .

LIST OF REFERENCES. . . . . . . . . . .

iv

119
121
122
126
131
137
1A2
1A9
157

Table

Energy
Energy
Energy
Energy
Energy

Energy

levels

levels

levels

levels

levels

levels

Optical model

of

of

of

of

of

of

pa

LIST OF TABLES

2631
303.
3“Ar

38Ca

rameters

Page
66
67
68
69
71
72
8A

LIST OF FIGURES

Figure Page
1. Experimental electronics . . . . . . . 50
2. Particle identification spectrum . . . . 51
3. Summing circuit . . . . . . . . . . 52
A. Two dimensional TOOTSIE display. . . . . 5A
5. Triton spectra from 2ONe(p,t)18Ne and

2uMg(p,t)22Mg. . . . . . . . . . . 56
28 . 26
6. Triton spectra from Si(p,t) Si and
32S(p,t)308 . . . . . . . . . . . 57
36 3A
7. Triton spectra from Ar(p,t) Ar and
uoCa(p,t)38Ca. . . . . . . . . . . 58
8. Energy levels of 18O and 18Ne . . . . . 75
9. Energy levels of 22Ne and 22Mg . . . . . 76
10. Energy levels of 26Mg and 2681 77
11. Energy levels of 3081 and 308 . . . . . 78
12. Energy levels of 3”S and 3“Ar . . . . . 79
13. Energy levels of 38Ar and 380a . . . . . 80
1A. 0+ to 0+ ground state transitions . . . . 9O
15. Position of first peak for the ground state
transitions . . . . . . . . . . . 93
16. Position of second peak for the ground
state transitions . . . . . . . . . 9“
1?. Ratio of peak cross-sections for the ground
state transitions . . . . . . . . . 95

vi

Figure Page

18. Distorted wave calculations for pure
t)38Ca ground state

pickup in the uOCa(p,
. . . . . . . . . 99

transition.

19. Change in O as a function of configura-

tot
tion mixing . . . . . . . . . . . 101

20. Change in 01/02 as a function of configu-
ration mixing. . . . . . . . . . . 102

21. Transitions to the first 2+ states. . . . 10A

22. Ratio of peak cross-sections for the first
L=2 transitions . . . . . . . . . . 106

23. Position of second peak for the first L=2
transitions . . . . . . . . . . . 107

2A. Transitions to states in 18Ne . . . . . 108
25. Transitions to states in 22Mg . . . . . 111
26

'26. Transitions to states in Si . . . . . 112
27. Transitions to states in 308. . . . . ,. 114
28. Transitions to states in 3“Ar . . . . . 116

29. Transitions to states in 380a . . . . . 118

vii

CHAPTER I

INTRODUCTION

The two nucleon transfer reaction has been studied
for the particular case of the (p,t) reaction. The

targets studied were the even—even, N=Z nuclei in the

2sld shell. In particular, the targets were 2ONe, 2“M

328, 36Ar and uoCa, which all have Jfl=0+ ground

B,
2881,
states. These (p,t) reactions reach states in nuclei
which are two nucleons away from stability. Until
recently these nuclei had not been studied to any great
extent. The same nuclei can in general also be reached
by the (3He,n) reaction, and recently work has been done
in this area. Reports of the study of these nuclei with
the (p,t) reaction have been scattered and sparce until
now. This is most probably due to the large negative
Q-values (~-20MeV) and small cross-sections involved,
which necessitates a high energy, high intensity proton
beam of good resolution such as the Michigan State Uni-
versity Sector Focused Cyclotron is capable of producing.
The (p,t) reaction and other two nucleon transfer
reactions have been previously used to study nuclei in
the light mass region by experimenters such as Cerny and

(Ce6U, F168, Ga6A)‘

his co-workers This reaction has

also been uSed in the medium to heavy mass region by

experimenters such as Hintz and his co-workers(Ba6ua’

3365’ 8&68’ Ma66b, R867). These workers have all re-
ported that the shapes of the angular distributions of
the tritons are very much characteristic of the orbital
angular momentun transfer of the reaction.

I The two nucleon transfer reaction in general and
the (p,t) reaction in particular have some very re-
strictive selection.ru1es (see Chapter 2) which make
spin—parity assignments to the final nuclear states
quite unambiguous. This is primarily based on the fact
that the shapes of the angular distributions of the
tritons from the (p,t) reaction are to a great extent
dominated by the orbital angular momentum transfer of
the reaction. This dependence will be further investi-
gated in later chapters of this work.

Two nucleon transfer theories have been developed
which allow the (p,t) reaction to be treated by the
direct reaction distorted wave method (see Chapter 3).
It was therefore decided to study such a theory, in par—

(@165)

ticular the theory of Glendenning , and to investi-
gate the ability of this theory to predict the observed
angular distributions. In particular, the dependence of
such a theory on the initial and final state wave func-

tions and the bound state wave functions has been

studied in Chapter 7.

CHAPTER 2

SELECTION RULES FOR (p,t)

The two nucleon transfer reaction in general and
the two neutron pickup reaction in particular have some

very special and restrictive selection rules. These

rules have been discussed in detail e1sewhere<Hi6u’

G165, Ba6ua, G162), and only those applying to (p,t)

in particular will be discussed here. Let us denote the

angular momentum and isospin of the target as JA and TA

respectively. The final nucleus will be denoted by JB
and TB' The orbital angular momentum, spin and total
angular momentum of the transferred neutrons will be

denoted by 2 and £ The transferred

1’ 51’ 31 2’ 32’ J2'
quantum numbers will be designated by L, S, J and T. We

can then write:

+++

+++++
2 L=tl+22=A+X J=L+s , (2.1)

Here we have denoted the orbital angular momentum of the
relative and center of mass coordinates of the trans-
ferred pair by A and A respectively. This possibility

of the relative motion of the two transferred particles

is something which does not arise in single nucleon
transfer such as (p,d).
Conservation of angular momentum yields the follow—

ing restriction, or selection rule.

IJA“JBIiJi(JA+JB) V (2.2)

Unless a single step direct interaction model is assumed
for the transfer process, the quantity J may not be a
good quantum number. Along with equation (2.2), there
is also the following restriction on the parity change

during the reaction.

.9. +2
+

A"=(-l) (2.3)

We note from equation (2.3) that if both neutrons are
picked up from the same shell (21=22), then An=+1. Also,
from the Pauli Principle, J must be even. If An=—l,

then the two neutrons must come from shells of different
parity (N=(-l)2). The isospin of two neutrons must be

1, since each has isospin t=l/2 and projection T=+l/2.
Therefore, the following restriction on isospin is

imposed.

ITA-TB|:1:(TA+TB) (2.A)

In this work, we will be concerned with even-

even N=Z targets so that in all cases studied J =0,

A

TA=+1, and TA=O' Equation (2.2) and (2.4) then leads

to the following:

J =J T =1 (2.5)

A neutron seniority selection rule can also be
defined. Neutron seniority is related to the number of
neutrons not coupled in pairs to zero angular momentum.
Since the pickup of two neutrons can break at most two

pairs, the following restriction holds.

Avn=0,i2 (2.6)

Seniority is not necessarily a good quantum number and
thus this selection rule may not be applicable to the
reaction as a whole although it must apply to the
individual components of the wave functions which are
responsible for the process.

There are also certain approximate selection rules
that arise based on some specific properties of the
triton. The two neutrons bound in the triton are

)(Bl62)

mostly (~95% in a state of relative spatial sym-

metry (Aeven) with S=0. Then according to equation

(2.1), J=L, and equations (2.2), (2.3) and (2.5) become:

IJA-JBI1L5(JA+JB) (2.6)

Aw=(-1)A (2.7)

JB=L for JA=O (2.8)

We will find later that the shape of the angular
distribution is very much dominated by the orbital angular
momentum transfer L and therefore equation (2.8) makes
the spin assignment of the final state unique for the
(p,t) reactions as opposed to the case of (p,d) where
J=Lil/2

If it is further assumed that the neutrons are in
a relative s—state (A=0) in the triton, then L=A and we
get the following approximate selection rule from equa-

tion (2.7).
An=(-1)L=(-1)J (2.9)

We note that the approximate selection rule of
equation (2.9) restricts the nuclear states that can be

excited by the (p,t) reaction.

CHAPTER 3

TWO NEUTRON PICKUP AND THE

DISTORTED WAVE METHOD

3.1 The Distorted Wave Method

 

The details of the distorted wave method of cal-
culating direct reaction processes have been discussed
by Satchler and others(Sa6u’ 8&62). Only those parts of
the theory pertinent to applying it to the particular

case of two nucleon pickup will be discussed here.

We denote the general reaction as follows:
A(a,b)B (3.1)

We denote the spins and projections.of the particles by

M M m and m". Following the pro-

JA’ JB’ a’ Sb’ A’ B’ a b
(Sa6A)

cedure of Satchler , the differential cross-section

S

for such a reaction can be written as follows:

u u k M M m m ITIZ
do _ a b b B A b a (3.1.2)

55" 2 '—
(2nn f ka (2JA+1)(2sa+l)

 

The reduced masses are denoted by u and k denotes the

asymptotic relative momenta. The quantity T is called

the transition amplitude and is defined in equation
(3.1.3).

+ +

T=<JBMB,sbmb,kbIVIJAMA,sama,ka> (3.1.3)

V is the interaction which causes the reaction, i.e.
carries the system from one elastic scattering state to
another. In the distorted wave formalism of reference

V Sa6u, T can be written as follows:

-> ->- (_)* + *-
TDw = g 2' draA drbB x ' (kb,rb) (3.1.A)
m m
a
m!

b mb
b
<J M s m' IVIJ M s m' > x(+) (R F )
B B b b A'A a a m' m a’ a

a a

denotes the Jacobian of the transformation from the

+
individual coordinates to the relative coordinate raA
+
and rbB'

It is convenient to expand the matrix element of
equation (3.1.“) in terms corresponding to particular
angular momentum transfer. We define the transferred
quantities as follows:

+++
J=JB-JA, S=sa-sb, J=L+S (3.1.5)

We write the expansion as follows including the appro-

priate Clebsh-Gordan coupling coefficients.

(a <JBMB,s b mbIVIJAMA,sama> s gcyn (3.1.6)

IJ, M —M >

= z <JA J MA, MB —MA [J M ><L s M, ma-m B A

LSJ B B b

S

b‘mb L
<sa sb mafmbls’ ma-mb>(fl) i ALSJ(Bb, Aa)

+

+
fLSJ,M(rbB’ raA) ; where M = MB + mb - MA - ma

The product ALSJfLSJ,M is often called the form factor for
the reaction. We substitute this expansion into equation
(3.1.“) and define a reduced amplitude B as in equation

(13) of reference Sa6u.

_ 1/2
TDw - g (2J+1) <JAJMA, MB— MAIJBM B> (3.1.7)
LMm m
z A B b a + +
LS LSJ SJ (kb,ka)

Taking the absolute square of TDw and summing over the
projections indicated in equation (3.1.2), along with

symmetry and completeness relations for the Clebsh-Gordan

coefficients from reference R067, we get:

2-
lele — 2 (21B

JMmbma LS

LMm m 2
+1)|2: ALSJ BSJ b aI (3.1.8)

Substitution into equation (3.1.2) gives the follow-

ing expression for the differential cross-section.

10

 

do _ “a“b kb (2JB+1) LMm m 2
dB ‘ " 2 2 k Z '2 ALBJ BSJ b al
(2Tb ) a (2J +1)(2s +1) JMm m LS
A a b a
(3.1-9)

If only one L and S are important, this reduces to the

 

following:
2
£2 = 2JB+1 z [ALSJI o (e) (3 1 10)
d9 2JA+1 J (2sa+1) LSJ ' '

Where we have defined the reduced cross-section as

follows:

A u k
_ a b b LMm m 2
°LSJ(9)"“'““2'2 k" X I 8SJ 0 3'
(2Nh ) a Mmbma

(3.1.11)

It is this reduced cross-section that is calculated

by a distorted wave computer code such as JULIE<Ba62).

The method of calculating the BEJ'S has previously been

(Sa6A, Ba62)

described and will not be discussed here.

The BgJ's are actually calculated in a zero range approxi-

mation where f ) is replaced by a purely

-> +
LSJ,M(rbB’raA

, *
radial function FLSJ’ a spherical harmonic Y M

L and a

. + ->
three dimensional 6—function in rbB and raA'

3.2 The Two Neutron Pickup
Matrix Element

 

 

We now must evaluate the matrix element 0’); of equa-

tion (3.1.6) explicitly for the two neutron pickup

ll

reaction. The method of dealing with this matrix ele-
ment for the case of two nucleon transfer has been

developed by several workers(G165a L15“, Ba6Ab, He6Aa,

Ab66s L166: Br67). We will follow the method of

(C165)

Glendenning along with some of the details, ex—

tensions and notation of Jaffe and Gerace<Ja68).
We denote the pair of transferred neutrons as x

and write:
b=a+x A=B+x (3.2.1)

The interaction responsible for the reaction is assumed
to be the interaction of a with x. We assume the
interaction is central and is a function of the separa—
tion between the center of mass of a and the center

of mass of x.

V=V(rax) ' (3.2.2)

Following the procedure of reference (Ba62), we
write the matrix element more explicitly as follows:

* at. ->
GI? = ng dga dgx 1pJBMI3(EB) wsbmb(rax’ga’€x) V(r’ax)

m a

1P (5 Jr” ,5 > A (g) (3.2.3)
JAMA B xB x 3a a

The E's denote the internal coordinates (spin and spatial

if appropriate) of the respective particles. We assume

l2

V(rax) does not effect wJAMA and wJBMB so that we can

consider the integral over dEB separately.

(M -M ) +
A B
GAB (pr’gx)

* ->

(3.2.A)

G is then expanded in terms of normalized two

AB

particle eigenfunctions of scme potential well. In
particular, we choose the product wave functions of the
two neutrons to be transferred denoted by coordinates and

-+ -> + -> TFF.
spins r r O, and 02 (”Gde).

1B’ 2B’
(MA-MB)
GAB = €38 bYaYBLSJ<JBJ,MB MA-MBIJBMA>
LSJ ‘
¢yayBLSJ (rlB’r2B’Ol’O2) (3'2'5)
(M —M ) i i -M M -M +M
A B A B
T = Z [A (r )¢ (r )1 x * *
YOYBLSJ M Yoga 1B YBQB 2B L 3 (01,02)
<LS, —M MA-MB+M J MA-MB> (3.2.6)

 

The brackets [ ] denote vector coupling of the orbital
parts of the two neutron wave functions, and XS is the
coupled spin part. The Clebsh—Gordan coefficient assures
the prOper coupling to a specific total angular momentum

J,M. We note here that T is an L-S coupled two particle

13

wave function. The sum over a and 8 implies a sum over
all configurations of the two neutrons needetho describe
this overlap GAB’
Since the interaction is assumed to depend on the
center of mass coordinate of the two neutrons, we must
perform a transformation to the coordinates of the pro-
duct wave function T. This is most easily carried out
with harmonic oscillator wave functions where the trans-
formation coefficients are calculable in closed form.
We expand the O's in terms of harmonic oscillator wave

+
function Onl(a,r) where a is the usual oscillator

strength parameter.

u)

(r) = au 0 (O,;) 1 (3.2.7)

I
d) p Y 119.

72

Equation (3.2.6) becomes:

(M “M ) .+ ”M
A B u v *
T = Z Z a a [O (a,r )O (O,r )]
YaYBLSJ M uv ya YB uia 1B vRB 2B L
Xs (8r02) <L s, -M, MA-MB+M|J MA—MB>

(3.2.8)

The well known Moshinsky-Talmi transformation can

now be applied to the oscillator wave functions to trans-

form them to relative and center of mass coordinates<M059’

Br60, La6o)

1U

T = z a“ a“ z <nA, NA, Llpt vt L>

, 3
M, uv YO YB nANA a B
+
9
[Onl(a/2’ r12) 9NA(‘“’

_. .4
MA MB+L

X (3 3 ) <LS -M M —M +MIJ M -M >
S 1’ 2 LL, A. A B A B

(3.2.9)

NA and nA are the principle and orbital angular momentum
quantum numbers associated with the center of mass and
relative coordinates respectively. The following re-

striction on these quantum numbers holds:

2(n+N)+A+A=2(p+v)+2a+tB (3.2.10)

The expression for the matrix elements of equation

(3.2.3) now can be written as follows:
9W] = Jdga dEX 0E8 bY Y LSJ <JB J, MB, MA—MBIJA MA>
LSJ “ B

2 a“ a“ z <nX, NA, Lluta, v28, L>
M, uv YO YB nANA

i i —M MA—MB+M
[OnA(a/2’ r'12) G)N/\(20" pr)] L Xs (01:02)

P * +
<LS, -M, MA—MB + AIJ MA-MB> wsbmb(rax, Ea, EX)
V(r ) w (E ) (3.2.11)

ax sm a
38.

l5

Explicitly, the remaining wave functions can be

written as follows for the specific case of (p,t).

_ a * .
ws m (Ea) - x (Ga) , proton spin wave function

a a 82:1 (3.2.12)
MA‘MB+M + +
XS (01,02) - Z <sls2mlm2ls MA—MB+M>
m m
1 2
m1 + m2 +
x (01) x (02) (3.2.13)
81 S2 '

The wave function of the triton is assumed for simpli-
city to be a Gaussian as suggested in references G165

and Ja68.

2 2 2 2
—n (rl2 + r p + r )

W = N e 2 p x (spin function)
Sbmb

(3.2.1u)

This Gaussian wave function can be easily separated, in
terms of harmonic oscillator functions, into the relative
coordinates of neutrons l and 2, and the separation be-

tween the proton and the center of mass of l and 2

(particle x)(Gl65).

16

_ 2 + 2 +
wsbmb(rax’ ga’ Ex) ‘ G’oo(3"‘ ’ r‘12) 900(u” ’ rax)
<8 8' m m'ls m > mb(; ) 2
p p b b XS ,p . .
. 9 m1 m2

<s' s' m' n' IS'm'> l (3 ) mé (3 ) (3 2 15)
1 2 1 12 . X81 1 X35 2 ° '
We restrict S' to be zero as was discussed in Chapter 2.
The relative orbital angular momentum of the two neu-
trons in the triton is also zero indicated by the first
factor of equation (3.2.15) and mentioned in Chapter 2.
In order to evaluate‘hb the integral indicated in
equation (3.2.11) must be carried out. We note that
+ + -> ->
Jdfia dgx implies Jdr12 doa do do . This total integral

1 2
will involve the following integral.

2 + -> -> 6
Jeoogn ’ r’12) enx(a/2’ r'12)dr’12 ‘ A0 Qn(a’ ")

(3.2.16)

This definition of an is equivalent to the one of

(9165). Making the explicit substitution of

Glendenning
_equations (3.2.12), (3.2.13) and (3.2.15) into equation
(3.2.11), the integral can be evaluated making use of
the orthonormality of the spin wave functions and the
completeness of the Clebsh-Gordan coefficients. We note

that the total projection (-M) of the coupled oscillator

wave functions can be assigned to GNA since i=0 only

17

for OnA and thus enA can carry no projection. Also
since A=0, then A must equal L. At the same time, we
factor the oscillator wave function into two parts.

-M + “M A

2
GNL (2d, pr) = RNL (2apr) YL (pr) (3.2.17)

Evaluating the integral of equation (3.2.11) we

get:
= _ I
ayn <58 by y LSJ <JBJ MBMb MB lJAMA> Ay y L<pr)
a B a 8
LSJ
-M A I

- — 1“- 4 -: >

E YL (pr) <Ls, M MA MB+TI J PA NB

* _ 2 +
(Sa 0 ma 0 lsbmb> V<rax) GOO“4n ’ r'ax) 68,0

(3.2.18)
A' = 2: a“ av z < n0, NL, LI p2 , v2 , L>
YaYBL “V Ya Y8 nN a B
Q (a n) R (2ar2 ) (3 2 19)
n ’ NL xB ' '

Using the symmetry progerties of the Clebsh-Gordan
coefficients and some prOperties of the spherical har—
monics, equation (3:2.18) can be put into a form that
can be compared with equation (3.1.6). From this com-

parison, we can identify ALSJ fLSJ,M'

18

M* A

A f = 2 b A ) Y (r

v (r
LSJ LSJ,M GB quBLSJ YaYBL XB

V( 9*142"
rax) oo< ” ’ rax)

J -J +s -s

 

 

A B a b L+S-J
55,o('1) (-1)
1L (2sb+l ) 1/2 <2JA+1) 1/2
2S+1 2JB+1
where M = M — M - m + m

B A , a b

(3.2.20)

The zero range approximation must now be made in
order to be able to apply a zero range distorted wave
computer code such as JULIE to this theory.

* 2 * ~ *
V(rax)0OO (Mn , rax) - DO 5(rax) (3.2.21)

+
In order to evaluate 5(rax) and the Jacobian 9,
we write down the geometric relationships in analogy

with the results of reference Ba62.

 

3
b A 3
= = c (3.2.22)
(3 xtB+ b)
rax = -c SEX rbB - raA) ‘ . (3.2.23)

19

The quantities denoted by‘h] represent the masses
of the respective particles. Equation (3.2.23) then can

+
be used to evaluate 6(rax).

6G,...) flit/chm??? - r >

=(1/g)6(qz’-§-;bB - ; ) (3.2.2u)

9nA

+ -> -> _
When rax goes to zero then rx8=rbB. The form factor of
equation (3.2.20) with this zero range approximation can

then be written as follows:

J -J +s -s

 

 

' _ L+S-J A B a o L
ALSJ - DO 68,0 (-1) ('1) i
2 +1 1/2 2J +1 1/2
(: 3° j) < A > (3.2.25)
2S+l 2JB+1
M* A
f = I: b A' (r ) Y (r )
LSJ,M aB YaYBLSJ YaYBL bB L b8
-> +
urn—9 rbB - raA) (3.2.26)
A

The separation into ALSJ and fLSJ,M is an arbitrary
separation for convenience. We now identify

2 b with the F mentioned at the end

A
a8 YaYB

LSJ LSJ

'
YGYQL

20

of section 3.2 as the radial form factors which must be
read into the distorted wave code JULIE to calculate the
oLSJ(6) of equation (3.1.11). According to equation

(3.1.10), IA is needed to evaluate the cross—section.

'2
LSJ

IA I2 g D2 2sb+1 ‘2JA+1 .
LSJ 0 2S+1 2JB+1 ’

 

S=O (3.2.27)

Since 8:0, then J=L is the only allowed value.
Also for the case of (p,t) which we are considering,

sassbsl/2, and equation (3.1.10) becomes:

do 2
da(p,t) 0 LOL

According to reference Ba62, for the particular normali-

zation used in the code JULIE, equation (3.2.28) becomes:

do D3
d9( ) = 5533 GLOL(JULIE), [mb/st] (3.2.29)
Pst

3.3 The (p,t) Form Factor
The zero range form factor, FLSJ’ that must be
input into a distorted wave computer code such as JULIE

was calculated in section 3.2.

F 2 b A' L(r) (3.3.1)

(1') =
LSJ a8 YaYsLSJ YaYB

A' (r) = z a“ a" z <nO,NL,L|u£ , vi L>
L 0 Y8 nN a B,

Qn(a,n) RNL(2ar2) (3.3.2)

For reasons that will become evident in section 3.“, we

introduce the following factor.

 

 

 

F’ 7
20 sq Jo . (3-3-3)
“8 Se 38 =
L S J

f<2g2+1)(2JB¥1)(23+1)(2L+1)11/2
£3 SC! JG.

x “a 58 38 >
L S J
J

The symbol { } is the Wigner 9-J symbol for re-

coupling four angular momenta<Br62’ Sh63). We rewrite

equation (3.3.1).

F (r) = Z 18 A 2 (r) (3.3.A)
LSJ a8 YaYBJcJBJ YaYBLSJ '
F 1
kc so Jo
= ' . .
AYGYBLSJ £8 88 JB AYaYBL (3 3 5)
L S J
L. J

 

 

r- fl
-1
2'0. so JG
B = b 2 s J (3.3.6)
YaYBJaJBJ YGYBLSJ B B B

 

 

L J

These B's (or b's) contain spectrosc0pic informa-
tion (see section 3.“) and are often called parentage
factors. The A's contain the radial dependenceand are
made up of the radial wave functions of the center of
mass of the two neutrons weighted by the overlap of their
relative motion with the relative motion of the neutrons
in the triton. It is these A's that are calculated by
the computer code TWOFRM written by Dr. W. J. Gerace at
Princeton University. The code uses eigenfunctions of a
real Woods-Saxon well with a spin-orbit term. The
triton size parameter n is fixed at 0.2“2f as suggested.

by Glendenning<Gl65).

3.“ The Two Neutron Parentage Factor
The parentage factors are needed to calculate the
total form factor. They enter as weighting factors in
a sum over all possible neutron configurations in the
target nucleus from which two neutrons can be picked up
to reach a particular state in the final nuCleus.
In section 3.2, the parentage factors were intro-
duced in the expansion of the overlap of the target and

residual nuclei.

23

z b J J M M M |J (MA-MB)(+ )
‘ < - M > ¢ 1" E;
as yayB SJ B B A B A A yayBLSJ xB’ x
- (a w" < ) ' < + A

In order to solve this equation for the parentage factors,
we multiply both sides of the equation by (3.“.2).

*

v
<J J M [J M > cy. Yé L'S'J' (3.“.2)

B M

B A‘MB A A
O.

+ .
Then we integrate over dpr dix and finally sum over the

spin projections of the initial and final nuclei.

b LSJ(€B’ pr)]J

* *
YcYBLSJ - JEWJB(€B) ¢YaY

B A

(3.“.3)

wJA(€Bpr, Ex)d£B (1pr dEx

The square brackets denote vector coupling.

We can now interprete the b's as a measure of how
much the final nucleus plus the two neutrons looks like
the target nucleus. The b's are then a measure of the.
probability of picking two particular neutrons out of
the target and reaching a particularfinal state of the
residual nucleus. The "cross-section" for this component
of the reaction is thus proportional to the square of b.

In actuality there may be several possible neu-

trons which are available to be picked up in this one

2“

manner, so we must multiply the "cross-section" for this
part of the reaction by the number of ways that the neu-
trons can be picked up in this manner. If both neutrons
are picked up from a group of N identical neutrons (such
as from the same shell model orbit containing N neutrons),
then this factor is the combinatorial factor denoted by
(g). The general eXpression for a combination factor is

given in equation (3.“.“).

(I?) = (TEETH (3.“.“)

If the two neutrons are picked up from different
groups (such as from different shell model orbits), then
this factor is just 2NaNBwhere N2<NB) is the number of
neutrons in the a (B) group. The 2 comes from the possi-
bility that either neutron can come from either group and
yet the final configuration will be the same. More
formally it is an antisymmetrization factor. We denote

this statistical factor in general by g

Y Y

(:8

- N - Hiflzll . = = =
g YB - (2) ‘ 2 ’ Ya YB N NB N

= 2NaNB ; Ya # YB (3.“.5)

If the "cross-section" must be multiplied by g

then b must be multiplied by g1/2.

1/2

b = g

YaYBLSJ <1"J ’ ¢ LSJ’ JAI 1"J > (3°“°6)

YaYe B YaYB A

We have essentially rewritten equation (3.“.3) in a
simplified notation and included the statistical factor.
We see that the b

YGYB
(G165)

's are analogous to the BY

'3
LSJ LSJT

of Glendenning In order to proceed further with
the calculation of the parentage factors we must choose

a particular model with which to describe the wave func-
tions of equation (3.“.6). We choose a j-j coupled shell
model since it is quite often used and its concept is

fairly easy to grasp. Since by is a L-S coupled

.aYB
two particle wave function, we must transform it to j-j

LSJ

coupling. We make use of the Wigner 9-J coeffi-
(Br62, Sh63)

 

 

cients
IL01. SO. JG
2 - z A s j c (3.“.7)
yayBLSJ 3038 B B B YGYBJQJBJ
L s J
L J

The coefficients [ J are related to the Wigner 9-J
coefficients as in equation (3.3.3) and are real. Apply-
ing this transformation and interpreting the sum over

ja and jB as being included in the sum over a and B

of equation (3.“.1), we get the following:

26

1/2

E =2; «v.4» NM >'
YaYBJaJBJ YaYB JB YaYBJaJBJ A JA

(3.“.8)

We have used the B's as defined in equation (3.3.6).

A In order to calculate the parentage factors, we
must consider the particular shell model space used in
calculating lwA2 and IwB>. As an example we will consider
the case where the nucleons are limited to two shells
outside a closed core. This will allow the possibility
of picking up the nucleons from either the same shell or
two different shells. Therefore, such an example will
cover the essentials of this type calculation since in a
direct reaction description of two nucleon pickup these
are the only two possibilities. (So far we have avoided
the explicit introduction of isospin since we are pri-
marily concerned in this work with two identical particles
which we know are neutrons. Often shell model wave func-
tions are calculated with isospin explicitly included
(see reference 016“ for example), therefore we will now
introduce isospin.

In the example we have chosen to consider, the

wave function of the target nucleus can be written as

in equation (3.“.9).

N NAB TA
lw > a 2 CA [IY AC! JAG TAG XAa>|YB JAB TAB XAB>J Icore>
A a8 a8 a JATA

(3.“.9)

27

The first factor represents the two\active shells, the
shells outside the Closed core. Here Y represents

the particular shell (such as ld5/2’ 281/2, or 1d3/2)

and N is the number of nucleons in that shell. There is,
of course, the restriction that NAa+NAB equals the number
of nucleons outside the core. J, T, and x represent
respectively the angular momentum, isospin and any other
quantum numbers which might be needed to make the descrip-
tion of the state unique. The core is assumed to have
zero angular momentum, isospin and isospin projection,

and 1

and therefore JA’ T are the quantum numbers of

A A
the total nuclear state. The total wave function must

have definite isospin projection TA but the individual
active shells do not. The square brackets denote vector
coupling, and 58 represents a sum over all different

possible configurations of the active nucleons in these

two shells with amplitude C2 The final state can be

8'
written in\the same way.

B NBa
I‘l’B” = :2 Gas [IYa JBa TBa XBa>
NBB TB
IYB JBB TBB XBB>JJBTBI core> (3.“.10)

For two nucleon pickup when the core is not effected, we

have the following obvious restriction.

28

”Au + NA8 = N + NBB + 2 (3.“.11)

In order to proceed further, we consider two possible

cases .

Case I N N = N + 2

AB = NBB’ Aa Ba

In this case both particles came from the same
shell. In order to proceed we must decouple the
nucleons to be transferred from the target wave functions
by means of a fractional parentage expansion defined in

the following equation.

N N-l
IY J T x> = J'%'X'<Y J'T'x', yjtl}yN J T X)
N-l v t 9
[Iv J T x >let>l (3.“.12)

JTx

The coefficients < I} > are called coefficients of frac-
tional parentage (c.f.p.). c.f.p.'s such as these are
described in reference Sh63 and others. We have chosen

an unconventional brief notation for the c.f.p.'s. Apply-
ing equation (3.“.12) twice to one typical term of equa-
tion (3.“.10) and drOpping the core since we have assumed
it will overlap exactly with the core Of the residual

nucleus, we get the following:

29

N -1 N
A Aa ,‘ Ac
2 <v J . val}va JAG)

a8 J'T'X' a
JHTHXH

Iu’A>o:B = C

N -2 N -l

n Au
J : Yal}Ya

J3>

NAG-2‘
I
[{lya J >lya>}J,

I I NAB TA A
Y >] Y J > (3. .1“)
0‘ JAa 8 AB} JATA

In this case all three brackets denote vector coupling
in the order indicated, and for ease of writing, we have
suppressed many of the essential quantum numbers. We
must now recouple these wave functions in such a way

that we can identify the coupled pair to be transferred.

(Sh63)

Such a transformation will involve the Racah W;

functions, and can be written as follows in our notation.

1/2 1/2

[{I£l>|£2>}L [13>]L = Z (2L12+l) (2L23+l)

12 L23

w<2122L2 L

33 12L23)

[|£l>{|£2>|£3>} J

L

,23 L

(3.“.1“)

Applying the recoupling transformation to both angular

momentum and isospin, equation (3.“.13) becomes:

30

A
> = c z < } >< } > 2J'+1 2J"'+l
IwA a8 08 J T'x' I I [( )( )
JHT" H
J! v IT¥It
1 2
(2T'+1)(2T"'+1)] / W(J"jaJAaja;J'J"')
NAG-2
" . i 1'! H
W(T taTAata, T T ) [lYa J >
T
NAB

A
mm» 1 Iv J >}
a a J"' J”! G AB JATA

(3.“.15)

We can simplify the above expression by defining
what might be called a two particle c.f.p. similar to the

definition of reference Sh63.

N-2 2 N _
<y JlTle, y J2T2I}Y JTx> -
z <v“’1 J'T'x'. YJt|}YN JTx>
J'T'x' .
N—2 N-l

<Y JlTlxl. YJt|}Y J'T'x'>

W(Jlij; J'J2) W(TltTt; T'T2) ' (3.“.16)

We introduce this two particle c.f.p. into equation
(3014015).

31

A . .
> = C X .Ll.
lwA 08 GB J"T"X" ‘ (3 17)
Jf'iTl'l
N -2 N
AC1 H H H 2 I?! '9' A0.
<Ya J T X ’ Ya J T '”a JAaTAaXAa>
T
N A
NAa'2 J">{IY >IY >} ] IY ABJ >
[Iv a a A6
a J J J T
Ac A A

We now must recouple again to completely separate

out the coupled pair of nucleons. Another form of the

(Sh63)

recoupling transformation is needed which written

in our notation is as follows:

_ 1/2 1/2
[{Itl>|22>}I |9.3>]L - L2 (2Ll3+l) (2L13+1) f
’12 13
2 +22 +22 +L +L +L
1 2 3 12 13 .
(-1) W(£211L£3, L12 L13)
[{ltl>|23>} |22>l (3.“.18)
Ll3 L

We apply this recoupling transformation to both angular

momentum and isospin in equation (3.“.17).

32

2 <2 particle c.f.p.>
JIITIIXII .

JI I ITI I IJivTiv

N’A>0(B = 08

iv
" III
J +2(J +JAB)+JAo+J +JA

(—l)

1V+T

'I II!
T +2(T +TAB)+T A

+T
(-1> M

iv iv 1/2
[(2JAa+1)(2J +1)(2TAa+l)(2T +1)]

1: , 1V
W(J"' J JA JAB’ JAa J )

iv
II! II .
W(T T TA TAB’ TAa T _)

Z ivaII 1" III
T <T 1 T T ITA TA>

Tiv

>}
Jiv

-2

N N
Aa " A
[ma J 4.88..)

A8

III

T A
{lYa>|Ya>} l (3.“.19)

:9!
J JATA

Since isospin projection is important here (i.e.

(T"')z E T"' = -l for two protons, T"' = 0 for a pro-
ton and a neutron, and T"' = +1 fortwo neutrons), we
have included it explicitly with the proper Clebsh-
Gordan coupling coefficient.

We now must write down an explicit form for the

other wave functions needed to evaluate the overlap and

33

thus the parentage factor of equation (3.“.8). For the
case of both nucleons coming from the same shell we know
the overlap will vanish unless the two nucleons come from

the particular shell Ya'

|¢> = {IYaJata>|YaJata>}:T (3.“.20)
In equation (3.“.20) J, T and T are the transferred
quantum numbers in the pickup reaction. The only term
in the expansion of the final state wave function that
could possibly overlap with the particular part of the

target wave function which we have uncoupled can be

written as follows.

I > - CE {I NAG-2 J T X > ( “ 21)
wB aB ' a8 Ya Bo Bo Bo 3' °
N T
A8 B
IYB JBB TBB XBB>}JBTB

We combine this with equation (3.“.20).

N -2 N T
_ B Aa AB B
le $9 JA>QB T C08 [{IYG JBB>|YB JBB>}JBTB
T "TA
{lxa>|va>} l' (3.“.22)

JT JATA

3“

Again, we have suppressed some essential quantum numbers
in equation (3.“.22) for brevity.

The overlap described in equation (3.“.8) can now
be easily carried out for these two typical terms._ The

complete result is given in equation (3.“.23).

B = C

YQYBJGJBJT (3.u.23>

 

A B* (NAa(NAa'l))1/2
a8 Gas 2

N -2
(Y Am J TBa XBQ’ 7:

NAa
0 Ba J Tl}ya J T x

Ad Ad Ac>

a+2<J+JBB)+JAa+J +J

J
(-1) B B A

+T

a+2<T+TBB)+TAa+TB A

T
(-1) B

”(J JBo JA JBB5 JAa JB)

W(T TBa TA TBB; TAG TB)

<T T T

B B Tl T

T > 6(J

A A A8’ JBB) 6(TAB’ TBS)

The total parentage factor will be the sum of terms like
equation (3.“.23) for each component of lwA> that over-
laps with a component of IwB> plus two nucleons in the
same shell Ya’

Case II N =N

Au Ba+l’ NAB=NBB+1

35

.In this case the two particles came from two
different shells. Again a c.f.p. expansion can be

applied to lwA> , but this time just once to each active

shell.
N -l
_ Aa_ . , N
ij>aB ” CaB J 'TZ'X , (Ya Ja ’ Yal}YaAa JAa>
(I a a
I I I
J3 Ts Xe
N -1
(YBAB Js" 78'} YEAB JAB’
NAG-1
[{lva Ja'>lva>}
JAa
N —1 '
{'YBAB Js'>|Yo>} ‘1 (3.“.2“)
JAB JA

Some essential quantum numbers have been suppressed for
brevity. I

We now must reorder the coupling in order to
identify the coupled pair to be transferred. This can
be done by using the Wigner-9J coefficients. The form
of the recoupling transformation has already been written

down in equation (3.“.7).

36

 

 

 

 

P’ '1
I II
Ja JB' J
I = II
lwA>aB J 1 va v < I} >< I} > Ja Ja J 1'
a a a
I I I
J8 TB X8 JAo JAB JA
JITIJIIITIII L .J
[— '1
I I II
Ta TB T
t Tvvv -£ <T"T"' n {H >
a tB T"T"' _ T T ITATA
TAa TAB TA
._ _
N -1 N -l T"
[{IY A“ J '>|Y AB J '>}
a a B 8 JHTN
TIII TA
{Iva>|YB>} (3.“.25)
JIIITIII J T

A A

As in case I, we have included the isospin Clebsh-Gordan
coefficient to take care of the neCessary specific iso-
spin projection.

Now we write down the explicit form of the other
wave functions needed to evaluate the overlap and thus

the parentage factor of equation (3.“.8).

T
|¢>a8 = {lyajata>|y838t8>}JT (3.“.26)
N -l N -l T
_ B A A8 B
le>oB - CaB {'Ya JBa TBc XBa>|YB JBB TBB xBB); T
x B B

(3.“.27)

37

-l N -1 T

N
. _ B Ac A8 B
I"A'B’ ¢’ JA>c8 ' CaB [{IYa JBa>|YB JBB>}
J T
B B
1 TA
\ {lYa>|YB>} J (3.“.28)
JT JATA
The overlap can now easily be performed.
_ 1/2 .A B
BY y J J JT — (2NAQNAB) cos 0&8 <TBT TBT I TA1A>
a B a B .
N -1 N
Ac Ac
(Ya JBa TBo ch’ Yajctal}ya JAa TAa an>
N -l N
A8 AB
(78 JBB TB8 XBB’ YBJBtBI}YB JAB TAB XAB>
"' EF '1
JBa JBB JB TBo TBB TB
ja jB J to tB T (3.“.29)
JAc JAB JA TAa TAB 'TAJ

 

 

 

 

The total parentage factor will be a sum of terms like
equation (3.“.29) for each component of IwA> which over-
rlaps with a component IwB> plus one nucleon in shell Ya
and one in Y8 . We note that for the (p,t) reaction,
when the transferred particles are neutrons, T=+l, and
T81. " A
Some of the parentage factors can be expressed in

simple closed form. In particular, for some cases where

isospin is not explicitly included in the wave functions

38

and seniority (v) is the only other quantum number
necessary to describe completely the nuclear states in-

(6165) has given explicit expressions

volved, Glendenning
for the two particle c.f.p.'s.

When isospin is not included, and two identical
particles are taken from the same shell, and there is
only one active shell so that JBc'JB’ JAa'JA and the 8

components are included in the closed core, case I re-

duces to the following:

 

1/2
N (N -1)
A B* ( Ad Ad )
B = c c (3.“.30)
YGYBJQJBJ a a 2
N -2 N
Ad 2 Au
(Ya JB vB’ Ya JIJYa JA vA>

For the situation where N is even, JAso, and vA=O and

(6165).

Ad
therefore JB=J, this c.f.p. has the following value

1/2
N-2 2 N _ 2(N-2) 2J+l ), for v=2
‘Y J V: Y J'}* 00’ ‘ (‘(N-1) (23-1)(2J+i) ’ J¢o

.1/2
3 +3-N o g 3
( N_l 3+1 ), for v o, J o

(3.“.31)

Since isospin is not included in this case, N is the
number of particles in the active shell of the same type
as those that are being transferred. For this simple case

the parentage factor becomes:

39

 

B = CA CB* (NAQ(2JG+3-NAQ))1{2 v=O J=O
YcYaJaJao c a 2(2Ja+l)

 

B A 3* (NAa(NAa-2)(2J+1))l/2

' = C C ; V32, J#0
yayajajaJ a a (2ja-l)(2ja+l)—

(3.“.32)

When isospin is not included, and the two particles

are taken from different shells, case II reduces to the

following:
1/2 A B“ .
B = (2N N ) C C (3.“.33)
YaYBJcJBJ Ac AB 08 c8
N -1 N
Ad Ad
(Ya JBc vBa’ Yo Jal}yc JAa vAc)
N -1 N
A8 A8
<YB JBB vBB’ YB JB|}YB JAB vAB>
JBc JBB JB
J, is J
JAc JAB JA
u. .1

 

 

In the case of an even-even target with each shell

coupled to zero angular momentum and seniority zero

(N and NAB even, JA-O, JAG.O’ JAB-0, vAa-O, vABIO) then

Ac
8 3 I
JB-J, J30 Jo and JBB jB. The necessary c.f.p. s are then

trivial<Sh63).

“O

<yN‘1 j v, Yj I} YN 00> = 6v1 (3.“.3“)

The parentage factor for this simple case can then be

written as follows:

‘ *
l 2 A B
B = (2N / C08 C08 (3014.35)

N
YGYBJaJBJ Ac A8)

in 38 J 6(vBa,l) 6(vBB,l)

 

 

This particular Wigner 9-J coefficient can be evaluated

using relationships from reference Br62.

 

1/2
A B; (2NAGNAB(2J+1) )

BYQYBJQJBJ = Cos Cos (2ja+1)(2js+17

5(vBa.1) 5(vBB,1> (3.u.36)

We emphasize again that parentage factors for con-
figuration mixed wave functions consisting of combinations
of the above type configurations must be summed over all
combinations of components of IwA> and ( IwB>+2 nucleons )
which overlap. The relative phase of the components
(i.e. the phases of the CaB's ) of the wave functions

contributing are important since they add coherently in

“l.

calculating the B's and the B's add coherently in cal-
‘culating the sum of equation (3.3.“) to form the total

form factor.

CHAPTER “

THE EXPERIMENT

“.1 The Proton Beam

The Michigan State University Sector Focused
Cyclotron was used to provide a beam of protons of
energy “0 MeV to “5 MeV. The beam was energy analyzed
and spatially defined by two “5° bending magnets and
three pairs of slits. The beam resolution was ~“O keV,
as calculated from the measured magnetic fields and slit
apertures. The details of this transport system have
been discussed in reference Ma67.

After analysis, the beam was bent through 22 l/2°
and sent through a shielding wall to an experimental
vault and a 36" scattering chamber. Quadrupole focusing
magnets were used at apprOpriate locations along the
evacuated beam line. The magnetic fields of the analyzing
magnets were measured by N.M.R. probes and from these
measurements the proton energy was calculated. The magni-
tudes of the quadrupole fields were also calculated for
the particular beam energy used. Fine adjustments in
some of the quadrupoles were made by visual observation

of the beam spot on plastic scintilators in the beam line.

“2

“3

Particular attention was paid to the beam spot at the
target position in the scattering chamber.
“.2 The Faraday Cup and
Charge Collection

The beam exiting from the back of the scattering
chamber was stopped and collected in an aluminum Faraday
cup. The beam current was monitored, and the total
charge collected was measured with an ELCOR model A310B
current indicator and integrator.

The beam current was varied depending upon the
particular scattering angle. At forward angles the
elastic proton counting rate and the counting rate
capability of the electronics limited the usable beam
current to as little as S n.A. in some cases. At back-
ward angles the beam current was generally limited by
the cyclotron to about 500 n.A. The normal range of the

beam current was 50 n.A. to 250 n.A.

“.3 The Scattering Chamber
A 36" diameter evacuated scattering chamber was
used. The target post at the center was capable of
.supporting either a ladder for solid foil targets or a
gas cell target. The detector telescope was mounted on
a remotely movable arm and, in the case of foil targets,
a monitor counter was mounted on a relocatable stationary

arm. The position of the movable arm had a remote read

““

out which was accurate and reproducible to about O.15°.
A viewing port in the side of the chamber allowed visual
inspection of the beam spot on the plastic scintilator
at the position of the target by means of a closed cir-

cuit television system.

“.“ Targets

2ONe target was a 3" diameter gas cell with

The
1/2 mil Kapton* windows. The gas was natural neon which
is about 90.9% 20Ne. The gas pressure was maintained at
about 28 cm. of Hg and was monitored throughout the runs

with a mercury manometer.

The 21‘Mg target was a self supporting foil of

2“M

magnesium metal enriched to 99.96% g. This foil was

obtained from Union Carbide at the Oak Ridge National

Laboratory. It was reported to be 566 ug/cm2 thick, and
this thickness was used in normalizing the cross-sections
obtained. For this purpose the thickness was assumed to

be accurate to 15%.

The 288i target was a self supporting foil of

28

natural silicon metal (92.21% Si). This foil was also

obtained from Union Carbide. Its thickness was deter—
nuned by measuring the energy loss of alpha particles
from a natural source when they passed through the foil.

'The results were compared with range tables(Wi66) to

_I

a
E. I. DuPont de Nemours, Wilmington, Del.

“5

determine the thickness. The thickness was found to be
687 us/cm2 and an accuracy of 25% was assumed for normali-
zation purposes.

The 328 target was a 5" diameter gas cell with
1/2 mil Kapton windows. The gas was natural H28
(~95.0% 32S) at a pressure of abOut 21 cm. of Hg. The
pressure was monitored throughout the runs with a mercury
manometer. I

Two different 36Ar targets were used. Both were
3" gas cells filled with argon gas enriched to >99%
36Ar. The first cell was a sealed cell, with 1/2 mil
Havar windows and a pressure of “5.1:1.0 cm. of Hg,

built by R. L. Kozub<K°67)

. The thick windows (10 mg/cm2)
caused some problems in triton resolution. The second
cell had l/2 mil Kapton windows (~l.7mg/cm2) and a
pressure of 26 cm. of Hg. Better resolution was obtained
with this target and so it was used for energy calibra-
tion purposes. It also served as a check on the actual
gas pressure of the sealed cell which was over a year old.
The “00a target was a self supporting foil of
natural calcium (96.97% uoCa). This foil was prepared by
evaporating in vacuum calcium metal onto a tantalum
‘backing. Upon cooling, the calcium foil was easily re-
rnoved. The thickness was measured with alpha particles

in the same way as described for the 2881 target. Its

2

thickness was found to be 863 ug/cm and an error of i“%

“6

was assumed for normalization purposes. A thinner target
(~69O ug/cm2) was used for some runs, but all the data

was normalized to the first target.

“.5 The Detector TelescOpe
The detector telescope was made up of two silicon
surface barrier transmission mounted ORTEC counters. The
first counter was relatively thin and will be designated
the AE counter. The second was thicker and will be called
the E counter. The particular counters used depended on

36Ar, 2“Mg, and 2881

the specific experiment. In the
experiments, both triton and the helium-3 data was taken.
In order to allow the 3He's to reach the E counter the

AE counter was chosen to be 160 microns thick and was
kept at a bias of 50 to 75 volts.. The E counter was

2000 microns thick and was at “75 volts bias. In the
20Ne experiment, both deuteron and triton data was taken.
In order to step the deuterons the E counter was made up
of two 2000 micron counters at “75 volts bias. The AE
counter was 260 microns thick and at 100 volts bias. In

“00a experiments, only triton data was taken.

the 323 and
The AE counter was 260 microns thick at 100 to 125 volts
and the E counter was 1000 microns thick at 275 to “75
‘volts bias.

The experiments involving foil targets required only

one collimator in front of the detectors. These

“7

collimators were made of 50 to 90 mil tantalum located
at 8 to 11 inches from the target. Both round apertures
of 160 mil diameter and oval apertures of 75x170 mils,
100x210 mils, and 125x210 mils were used. The oval
collimators were smaller in the horizontal direction in
order to minimize kinematic broadening and yet increase
the effective solid angle.

I When gas cell targets were used, two collimators
were needed to define the volume of gas that is to be
considered the target. Brass plates on the sides of the
telescope were also needed to prevent particles scatter-
ing from the cell windows and other regions of the gas
from entering to detector system. The front collimator
nearest the gas cell was a tall brass slit with a full
width of about 125 mils. The back collimators were the
same ones previously'mentioned for foil targets and were
located from 9 to 12 1/2 inches from the center of the
cell. The front collimator was from 5 to 8 1/2 inches
ahead of the back collimator. A

The monitor counter which was used with foil tar-
gets consisted of a NaI crystal and a photomultiplier
tube. It was held as a fixed angle and a single channel
«analyzer (SCA) was set with its window about the elas-
tically scattered proton peak. In this way the output of
the SCA was preportional to the product of the beam current

and the effective target thickness. -This output was scaled

“8

and used to calculate the relative cross-section for

solid foil targets as will be described later.

 

“.6 Dead Time Corrections

The dead time of the pulse analyzing system was
taken account of in two different ways.~ When a monitor
counter was used, the output of the SCA was scaled and
also fed into channel zero of the pulse height analyzer.
The ratio of these two numbers then gave a measure of
the fraction of counts lost.

In the case where amonitor was not used, the
signal from the beam current monitoring meter was sent
to a voltage to frequency converter which gave out pulses
at a rate prOportional to beam intensity. These pulses
were then scaled and sent to channel zero of the analyzer.
In the same way as with the monitor counter, a measure of
lost counts was_obtained. This is not as good a method
as a monitor counter since the beam current meter cannot
follow microscopic time structure in the beam intensity,
but if dead times were kept small, the method was ade-
quate.

“.7 Electronics and Particle
Identification

Bombarding a target with “0 MeV protons produces

a large number of nuclear reactions.' Bebause of this,

some method must be used to identify the particular

“9

products of the reaction of interest, in this case
tritons. The method chosen for these experiments de-
pends on the difference in energy lost in the AE counter
for particles of different mass and charge but the same
kinetic energy.

36 2“ 28

The experiment with the Ar, Mg and Si targets

employed the ORTEC model “23 particle identifier which is

based on the technique developed by Goulding g£_al.(G°6u).
The total energy spectrum, gated by the particle identi-

‘ fier output, was analyzed and stored in a NUCLEAR DATA
160 pulse height analyzer. Figure 1 shows a block dia—
gram of the electronics involved. Figure 2 shows a
sample spectrum from the particle identifier.

20 “0

'Ne and Ca targets used a

The experiments with
different method of particle identification also based
on the differential energy loss. The signals from the
AE and E counters were summed (called the 2 signal) at
the detector telescope and all three pulses (AE, E and Z)
vvere passed through charge sensitive preamplifiers and
ssent to the data acquisition area. Figure 3 shows this
isumming circuit. Using an electronic setup similar to
13he previous method, a slow coincidence was required be-
tnfieen the AE and E signals. This coincidence was used
t<> gate the AE and Z signals. These two signals then
Mnant to a NORTHERN SCIENTIFIC dual “O96 analogy to digital

cOnverter (ADC). An S.D.S. Sigma-7 on-line computer and

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53

the data acquisition computer code TOOTSIE<Ba69> were
used to analyze the digital signals from the ADC.
TOOTSIE displays, on a cathode ray screen, AE versus 2.
Due to the difference in energy loss (AE signal) of
particles of different charge and mass for the same
energy (2 signal), the different particles fall into
bands on the two dimensional plot. Figure “ shows such
a two dimensional plot. This particular spectrum was
not taken during the present experiments. The code then
allows gate lines to be introduced in the form of poly-
nomial fits to designated points. These gate lines are
then used to route the 2 signal to any of four 20“8
channel spectrum. I I

In the case of the 328 experiment, two detector
telescopes, placed 10° apart on the scattering chamber
arm, were used. The particle identification system
using the on line computer was used. After the coinci-
dence and linear gates the AB and Z signals from the two
telescopes were mixed and sent to the ADC, along with a
Trouting signal taken from the coincidence modules.

“.8 Triton Energy Spectra and
Energy Resolution

In the earlier experiments (Ar, Mg, Si), the
falectronic limitation on the resolution was measured by
iantroducing a pulser signal, through a l or 2 pf. capaci-

t”Dr, into the preamps. It was found to be equivalent to

5“

 

Figure “ ——Two dimensional TOOTSIE display.

55

“5 to 65 keV full width at half maximum (FWHM). In the
later experiments when the total energy signal (2) was
taken from the summing circuit at the detector telesc0pe,
electronic contributions were reduced to 30 to “0 keV
FWHM.

The over all experimental resolution varied with
the particular target, counters, and electronic con-
20 d 32S cases the resolution
was about 90 keV FWHM. The 21‘Mg experiment had about

120 keV. The 2881 case was about l“0 keV. The 36Ar

figuration. For the Ne an

gas cell with the thick Havar windows gave 155 keV,
while the cell with Kapton windows gave 100 keV. The
uoCa experiment had about 60 keV overall resolution.
Figures 5, 6 and 7 show sample triton energy spectra for

each target.

 

 

 

 

 

 

 

 

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56

 

 

 

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58

 

 

 

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Figure 7.-Tr1ton spectra from 36Ar(p,t)3uAr and uoCa(p,t)380a.

CHAPTER 5

DATA REDUCTION

The one dimensional triton spectra stored in the
ND-l60 were dumped directly into the Sigma-7 and punched
on cards in the form of binary coded compressed data
‘decks. The data acquisition task TOOTSIE punched out
such data decks directly. The spectra were also plotted
by the Sigma-7 in the form of semi-log histogram plots.
Listings of the data were also obtained. From the plots
and listings, the first and last channels of each peak
along with the associated backgrounds were picked out by
visual inspection. This information was put on punched
cards and a simple Fortran computer code used these along
with the data decks to calculate areas of peaks, statis-
tical errors and centroids. The statistical error was
taken to be simply [(N+B)+B]l/2/N where N is the net
counts in the peak after the substaction of B background
counts. The centroid was calculated using only the tOp
twcwthirds of the peak for peaks over 25 counts high to
eliminate contributions from tails due to straggling in
1file target and other effects. :In the case of smaller

Peaks, statistics did not seem to warrant this approach

59

60

and the total peak was used to calculate the centroid.
The results were output on punched cards.

These cards, along with data on the proton energy,
scattering angles, and particle masses went into a
second simply Fortran computer code which used peaks
designated as being known to set up an energy calibra-
tion curve. The calibration peaks were usually taken
to be tritons from the (p,t) ground state transitions to
100, 12N, and 1”o as well as the first excited state of
10c at 3.3527 MeV<Pa69). If the ground state Q-value
of the reaction being studied was well known, these
triton peaks were also used as calibration points. The
C, N and 0 peaks came either from impurities in the tar-
get, or from a different target such-as Mylar, air or
carbon dioxide of a thickness comparable to the target
being studied. The calibration curves and kinematics
were used to calculate the excitation energies for the
peaks corresponding to the states in the final nucleus.

The same peak cards along with geometry and target
data and data concerning individual runs such as inte-
grated beam current, target angle, monitor counts and
cietector telescope angle were entered into another
ITortran computer code. This code calculated the center

(bf mass scattering angles and differential cross-sections.

FPhe formula used in calculating the cross-section for

61

the foil targets is given in equation (5.1), equation

(5.2) gives the formula used with gas targets.

N f sin 9 M

 

 

%%(6) = % DT 2 (2.66015 x 10-16) [mb/st]
C odx (A/R )
(5.1)
N f sin 6 T
$07”) =% W (1.658914 x 10'12) [mb/st]
C n P G

(5.2)

In the above formula,<a is the Jacobian of the transfor-
mation to center of mass coordinates, N is the number of
counts in the peak, fDT is the dead time correction factor,

6 is the scattering angle, 6 is the target angle, M is

t
the atomic mass of the target material in A.M.U., T is
the gas temperature in °K, C is the integrated beam
current in Coulombs, pdx is the target thickness in
g/cm2, (A/R2) is the detector solid angle, n is the
number of target nucleons per molecule of gas, P is the
gas pressure in cm. of Hg, and G is the G-factor in cm.
The G-factor is that defined and discussed in detail by

(8159). It is expressed as a sum of a series

Silverstein
of terms. Only the first two, which do not depend on
the shape of the angular distribution, were needed for

the present data.

62

When a monitor counter was used, the relative
cross—section was calculated by the expression given in

equation (5.3).

CLO;
:00

e) = N r /(Monitor Counts) (5.3)
. DT
rel.
An average normalization factor was then calculated by
comparison with the results of equation (5.1).
The resulting angular distribution and excitation
energies of the states observed are given in the

Appendices.

CHAPTER 6

EXPERIMENTAL RESULTS

6.1 The Ground State Transitions

 

The targets studied in these experiments were all
even-even N=Z nuclei with J"=O-+ ground states. The
residual nuclei following the (p,t) reaction are then
also even-even nuclei, and therefore, according to the

(R067), have J"=O+

independent particle shell model
ground states also. From the selection rule of equation
(2.2), the angular momentum transfer, J must be zero.
Applying the approximate selection rule of equation (2.6)
the orbital angular momentum transfer L must also be
zero. Therefore, all the ground state transitions should
show L=O character. Inspection of the data shows that
all six angular distributions are indeed similar (see
Chapter 7). In Chapter 7 it will be shown that the
general properties of the L=O transitions are predicted
by distorted wave calculations.

Directly from the experimental data several char-
acteristics of the L=O shape for the targets studied can
Ebe noted. This is a definite maximum in the range of
233 l/2° to 27 l/2°. This is truly the second maximum

Stince the distorted wave calculations of Chapter 7

63

6A

indicates a maximum at 0°. There is another definite
maximum in the range of A7 l/2° to 59 l/2°. Another
maximum is also indicated further back at 75° to 95°
but the data do not cover this maximum in detail.
6.2 The Transition to the First
Excited States

The angular distributions for the transitions to
the first excited states in all six nuclei are similar.
‘ The distorted wave calculations of Chapter 7 show that
this shape corresponds to L=2, and application of the
selection rule of equations (2.8) and (2.9) indicates
a spin-partiy assignment of 2+ for all the first excited
states. The lowest excited state being a 2+ for even-
even nuclei is typical for even-even nuclei in this
mass region.

Experimentally the L=2 angular distributions have
the following properties. They exhibit a peak or
plateau (washed out peak) in the region between 30° and

h5°, and a peak at 65° to 75°.

6.3 Other Transitions
The predominant dependence of the angular distri-
‘butions on L and the restrictive‘selection rules for
(p,t) outlined in Chapter 2 allow spin-parity assign-
ments to be made for some other excited states of the

Desidual nuclei. This dependence will be investigated

65

further in Chapter 7. Such assignments can be based
on comparison with either the experimental L=0 and L=2
shapes, or the distorted wave predictiOns.

The residual nuclei studied in these experiments
can in general only be studied by two nucleon transfer
reactions since they are all two nucleons away from
stability. Until recently these nuclei have not been
studied much at all. Except for the case of a few low
lying levels in some of the nuclei, the only previous
experiments studying these nuclei have employed the
(3He,n) and the (3He,nv) reactions. The Q-values for
these reactions are on the order of -1 MeV as opposed to
the -20 MeV for the (p,t) reactions. This small Q-
value makes the (3He,n) reaction possible with lower
energy accelerator, but high resolution in work involv-
ing neutron detection is difficult.

Tables 1 through 6 show the energy levels seen in
this experiment as\well as the J1r assignments for levels
inhere the experimental angular distributions were clear
enough to indicate the L—transfer. The distorted wave
calculations for reactions leading to these states, and
tune basis for the JTr assignments will be discussed in
(filapter 7. Also shown are the levels seen by other
Workers and their :1" assignments. Values in parentheses
r'e‘lbresent tentative assignments, double parentheses

infilicate that the assignment is extremely tentative. In

66

 

 

TABLE 1.--Rnergy levels of lee.
(this work) .(other works)
. . , . References
Energy 1“ Energy In
(MeV) ‘ (MeV) “

M.E.= (Ma65)

5.3193

i.00U7

- + ) ~ n t“ 1'
0.0 0+ 0.0 0 (Fact)t(0lt8)t(snto) (T060)
(IPLU)Jn(Gd(1)

'1 + \ .—, + / ‘.'. r )
Loou ._ 1.8.97.) 2 (}‘a\5;)t((.lt~c‘) (.w.) (-oct)
:.OlO i.0002 (iir6ii)In

- + w 'v')
3-35b ((0 )) (CLLC)
: 00:

0 1+ fi'w 1+ 0 "C; t/HO'C' r
3.3;0 (4 ) 3.3/6. 4 (sh6xa) (Fabn) (.ltu).‘.(sht:u)M
i.OlA i.OCUH (T068)

3.57VB (O) (Giét)

i 0020

+ - . A+ . .u i...aa

1.61M 3 3.tl«u 2 (Chéva) (Eats)1(fl;tc)an
:.013 1.030L (ChCE).(TCCo)
p rv’. ’ I it" :7.49 t("*.»! r;«”
'o/)l(—-‘ l 1.),‘U (,4 db-) \iCLU) (I‘.‘\CAL()?:
1.017 t 117
5. 50
i.OlA
6.326
1.018
/ 957
t.O25
c) 315
i.O2O

 

*
See text for explanation of notation.

67

22

TABLE 2.--Energy levels of. Hg.

 

(this work)

(other works)

5
References

 

Energy 1" Energy Jn
(MeV) ‘ (MeV)
M.E.= M.E.= (0e66)t
-0.u123 -O.380
t.0097 1.050

0.0. 0+ 0.0 0+ (Enb7)
l 250 >2+ 1.2us0 2+ (Sh68)Jfl(Gah7)t(0167)E(Be66)
: 008 1.0006 (Ce66)t
3.323 (4*) 3.353 (u+) (sn68)Jn(cao7)fi(neco) (Ce66)t
i 021 1.0N5
u.u17 (2*) u.38 (2+) (Sh65)EJfl(Ga67)t
i 027
5 057 (2*) 2 cu (2+,3') (Sh63)
i 031
5.313 5 cu (2+,3‘,u+) (onto)
:.03>

Ha ra+,;‘ u+) (sntt)
5.738 r 70 <(u+>\ (SL6;)
: 035
(.001 0+
t.037
6.281
i.033
6.6U5
1.0Au
6.836
:.ouu
7.252
1.0uu
7.961
:.OU9

 

* _
See text for explanation of notation.

TABLE 3.-—Energy

638

26

levels of Si.

 

 

(this work) (other works)
a
Energy J” Energy J" References
(MeV) (MeV
.M.E.= (En67)
-701ul
3.011
0.0 0+ 0.0 0+ (Adts) (Matt) (yan7)§"(vc67)
(3167) (U065) (AJCO)
1,705 2+ 1.7? 3+ (hath) (P060)F1(Ea07):(M167)
2.011 2.01 (AJCC)
2.700 (2*) 3.78 2* (Matt) (h068)u(ba67)2n(fic€7)
2.019 2.01 (M107) (Herb) (AJ60)
3.330 R,33 ( ) (”uL.\ (Poti)t‘(fic67) (H065)
2.019 2.03
<.7701 (ha6c crY),
‘.ukg)
. . .‘0' , t
.,;; (u (Lut.)
f K,_
.‘”P (You:) 0!?)
6"12‘,']
“.153 (u+) n.1t3 (q) , ate) o a),( (1'
—0011 f L' y
. ,(+ + . .‘( p . ‘1‘.
“’11-“? g ,g 351,: (+, I L ) :JL')}.YTT
-.Ulj 3 '1
ll. .31
t 013
{3. I.N
2.01?
5.562
1.028
5 960
2.022
6.381
-.020
6.786
2.029
7.150
1.015
7.“?0
2.020
7.695
1.031
7.902
1.021

 

I
See text for explanation of notation.

69

TABLE A.—-hnergy levels of 308.

 

 

 

(this work) .(other works)
~ - - F 3f21‘encre *
Energy Jfl Energy J0 fl ( ( L
(MeV) (MeV)
m.n.= M.E.= (En67)
—1U.08l -1A.063
t.01: 1.011
+ ”a r0 ’3 . '7
0.0 0+ 0.0 0 (A066) (Mann) (Shot) (,&L{)Ew
(N067) (H167) (Hons)
‘ + , , , . fr’. '(W /‘ 1 w
2 239 Q 2 210 2+ (haoo) (ShUC)F(DaU7)Efl(RCCI)
:.018 i.018 (Mi67) (r ts)

—O ’)+ ’3 0+ /' ”"\ 0 AL: .’r‘.' ‘t’ l”,"7
3.1330 L. 3.111.: c. (11300, (0110;)I‘.\i‘d(‘7)TTT\.‘.CO{)
1,01u i.025

+ - , -.,- .
3.707 ((O )) 3.07” (Iato) (QML0)V(L007)
:.025 ' :.023
H.386 (€303)
i.031
H.567 (C'(-)
i 030
14.721) ( is»
i.OC”
5.207 5.336 (CLt1?
$.022 1.015
5.306
i.025
(5.381) (Sh63)
(i.019)
5.U26
i.025
*

See text for explanation of notation.

7O

 

(this work)

(other works)

References

 

Energy .0 _Energy
(MeV) ‘ (MeV)
5.U80 (Sh68)
t.015
5.5u8 (Sh68)
1.02u
(5 (57) (Shoe)
(i 28
V 825 (shot)
i 019
5.807
1.027
0.01M (LhC?)
*.012
(6 1051) b.05”) (5311):")
(t 070) :.010
(6.223) b 833 (SHOE)
(: “530) 1.01-
Ll ((15
t 000
6.681
t 0H0
7.18r
i 035
7.570
t

 

'71

TABLE 5.--Energy levels of 3“Ar.

 

(this work) (other works)

, . R e e ces'
hnergy W Lnergy J“ ef P n

(keV) J (MeV)

 

M.E.= n.s.= (En67)
—18.370 -ld.30u
2.011 2.013

0.0 0 0.0 0 (ha66) (VCC?) ("167) (£166)

3.059 (Ha68) (mot?) (3107)E
3.286 2 3.30 (Ha68)E(xc67)

3.870 0 ?.H0 hat.)

:

b.050 “.05 (H865)

N. S (hate)

6.07M
.011

H-

6.5
.00

7)
L.
|

Q U1

1+

o

C\

.79u
2.011

7.322
2.006

7.u99
2.00“

70925
2.005

 

fl

5
See text for explanation of notation.

72

 

 

TABLE 6.a-Energy levels of 38Ca.
(this work) (other works)
i
Energy J“ - Energy Jw References
(MeV) (MeV)
M.E.= M.E.= (Sh69b)
—22.081 —22.007
t.011 1.021
, t
M.h.= (Da67)
-22.078
i.OUO
M.E.=
-22.050 (ha66)t
i.025
0.0 0+ 0.0 0+ (snspn) (0ao7)t(na66)§Tr
2.206 2* 2.20 2+ (Sh69b) (Madmfmnsam)t
2.005 2.03
. + 2
3.00 0 (Sho9b)
1.05
3.00 2+ (Sh69b)
:.03
r- .3 - 7.'/ t v'/ t
3.69) 3.72 3 (uao7) (112.06%.TTT
i.OO‘) _+. “:3
H.191
t.005
“,351 (2+) 0,301 2+ (Ch69b) (Da67)EJw
1.005 1.0u0
H.7A8
2.005
H.899 (2+) H.886 2+ (sno9o) (Da67)EJn(ha66)t
2.005 . 0M0
5.159
2.007
L— ’3 ‘ t
5.210 (0a67)
i.0140

 

*
See text for explanation of notation.

73

 

(this work) (other works)

. References*
_ Energy . Jn hnergy J"

(MeV) (MeV)

 

5.2614
2.005

5.U27
.006

I+

5.598,
.007

5.698'
.010

H-

1+

.810

H-U’l
O

0
U1

!+ C‘\
O H
0 LA)
C\ C‘\

O'\
N
(I;
O

(O )

H-
C
O
CO

H- C‘\
O \N
O \O
\1 (D

H- O\
O \J
|'—‘ O
O l\)

'4»
o
H r
o

 

71:

order to limit these tables to a convenient size, only
the more recent references are given. The excitation
energy listed is usually taken from the reference with
the smallest quoted error. Correspondence with levels
seen in the present work is made whenever possible. This
correspondence, of course, may not always be correct.
All references are to (3He,n) or (3He,ny) work unless
followed by a t, in which case the (p,t) reaction was
used. The subscripts J, n, or E indicate that the spin,
parity, or energy assignment was taken from that refer-
ence if it is not the only one. The mass excess (M.E.)
quoted are sometimes averages of several experiments

(Ma65)

taken from the compilations of Mattauch et a1. and

Endt and Van der Leun<En67).
A consolidation of the results of the present ex—
periments and the results of other workers are given in
Figures 8 through 13. These level diagrams are not
necessarily complete since there is some ambiguity as to
which states seen by different experimenters are the
same. These diagrams do give a general idea of the ex—

tent to which the level structures of these six nuclei

are known. Also given in these figures are the low

lying levels of the mirror nuclei. The levels of 18O
are taken from F. D. Lee, et a1.(L867) and the references
therein. The levels of 22Ne are taken from references

26

La62, A359, Pe6U, and Bu67. The levels of Mg, 3081,

75

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

EXCITATION a ENERGY (MeV)

 

 

 

 

 

 

 

:2
5
\I"
\I- '
2(3)
3'"
/
________::gg
\(2‘3')
'(-) |‘
2+ .
21 4(6)
\
\4‘.
((0 ))
2+ 2+
0‘ ' 0+
l8 l8
solo noNes
Figure 8.--Energy levels of 180 and 18Ne.

 

76

 

 

 

 

 

EXCITATION ENERGY (MeV)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6
CT’ -4
(53332:;
0.2.3.4, (22:33) -._4
——-— - 2‘” (2+)
. A,
4* (4‘)
.—J
2+ 2*
A
' '0‘ . 4' .—
22 0 . 22M 0
IONeIZ 12 9:0
22 22

Figure 9.--Energy levels of~. Ne and

Mg.

EXCITATION ENERGY (MeV )

77

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

‘ (2*)

—§'(4‘2 . (2‘) ‘

 

 

 

 

 

(O)

 

 

2+

 

 

26 °‘ 2. °‘
I2M9I4 I4S'I2

 

 

 

Figure lO.--Energy levels of 26Mg and 2631.

 

78

 

EXCITATION ENERGY (MeV)

 

 

 

 

 

 

 

 

 

 

 

 

-— fi : I I
\2‘
F

. ‘ 26

53'

:21

u. \3'

“‘2‘

'— ' ’0’
=\ '9’,

\2‘

2O

_. 0*

30 .
I 43' I6

 

 

 

 

 

 

 

 

 

 

 

 

 

 

30
nesm

 

Figure ll.--Energy levels of 3031 and 30$.

 

EXCITATION ENERGY (Mev)

79

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

24"

 

_ *

34
Issue

 

 

 

 

 

 

 

 

 

 

 

 

 

06

 

 

 

26

 

00
IB '16

 

Figure 12.-"Energylevels of 3&8 and 3“Ar.

 

80

 

EXCITATION ENERGY (MeV )

 

 

 

 

 

 

(0’)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

09
2* 2"
0‘ 0‘
38 38
IaA'zo zocom

 

Figure l3.--Energy levels of 38Ar and 380a.

 

81
3M 38
S, and Ar are taken from the compilation of P. M.

Endt and C. Van der Leun<En67).

CHAPTER 7

DISTORTED WAVE CALCULATIONS

7.1 The Optical Model

 

The Distorted Wave Method of calculating direct
reactions uses an optical model to describe the elastic
scattering in the entrance and exit channels. The
optical model takes the form of a potential which, when
put into the Schroedinger Equation, produces the dis-
torted waves (wave functions) which represent the
scattering. The form of the particular optical model
potential that was used in the code JULIE is given in

equation (7.1.1).

 

l d 1
U (r) = V (r) - V - (w —uw )
OM c o 1 + ex 0 D dx' 1 + ex'
h 2 l. d l * *
mnc s r dr 1 + ex

The first term is the Coulomb potential of a uniformly
A1/3

c . The parameter x,

charged sphere of radius Rc=rO

x' and x" in the rest of the terms have the form

1/3)/a.

+ +
x=(r-rOA For spin 1/2 particles 0 = 2s.

82

83

A search of recent literature was made to obtain
optical model parameters for protons and tritons in the
apprOpriate energy range. This corresponds to about
No MeV for the protons and 20 MeV for the tritons. There
is, of course, no elastic scattering data for tritons on
the nuclei which are the final state of the (p,t) reac-
tions studied since they are all unstable. Also there
is in general little triton elastic scattering at all.

It was therefore decided to consider the possibility of

3

using He parameters.

Since elastic scattering data were not available
for the precise nuclei and energies studied in this
experiment, it was decided that it would be best to use
optical model parameters which were consistent and
relatively constant over the range of nuclear masses
studied. Several sets of recent parameters were found,
and the ones considered here are listed in Table 7. The
geometries are average geometries over a range of A. The
potential depths are essentially averages over the range
of A pertinent to this work, the exact determinations of
which will be discussed later in this section. Variation
of potential depths with A given in the literature was
small and not smooth as a function of A, and therefore
no variation was used. When considering triton and 3He
Optical parameters, it is found that ambiguities

t(Ka68a, F169)

exis For example the elastic data can be

8H

.mcofiuwasonO Ham ca com: wumm Hmcfim
- *

 

 

Ammamv mm.a 5H0.0 00:.H 005.0 0H.H 0 0 5H 00H seepage >
A50mmv mm.a H:0.0 mz.a 050.0 :N.H 0 0 m.0m 0mH counts >H
Ammsmv m.H 5.0 0H.H 5.0 0:.H 5.0 0H.H m.5 0 5 0: 20p0p0 HHH
Am0emv m.H 5.0 0H.H 5.0 :0.H 5.0 0H.H 0.5 m.0 0 m: *couoem HH
A50smv mm.a :m5.0 :00.H m0.0 5m.H 05.0 0H.H 20.0 5.0 5.m 0.0: cowosm H
00:0 ““0 Adv gov A00 A00 Adv Adv A>mzv A>mzv A>mzv A>mzv mH0fipnmm
upmmmm oo.H :m mg .m mg m o.H m> Q3 03 o> pom

 

.mpmpwempwo HmUoE Hmoapaoul.5 mqm<e

85

fit for any reasonable value of ro by proper choice of

V0 and other parameters. A study of the literature indi-
cates that reaction data is usually best fit with
parameters where rO is in the range of 1.15 f to 1.25 f
andVO is in the corresponding range of 170 MeV to

150 MeV<F169). The choice of optical parameters to be
considered for this work was therefore limited to this
range.

In order to determine which set of parameters to
use, distorted wave calculations were made for the six
L=0 ground state to ground state transitions. The form
factor used was based on the pickup of a pair of neu-
trons from the last shell in the simplest J-J coupled
shell model configuration. For example for
2ONe(p,t)l8Ne(G.S.), the pickup of two (15/2 neutrons
coupled to zero angular momentum was assumed, for
uoCa(p,t)38Ca(G.S.) two d3/2 neutrons were used, and so
on. These should be the dominant terms derived from a
more extensive shell model wave function of these nuclei.
As will be seen later in this chapter, the shape of the
angular distribution is most greatly dominated by the L
transfer and secondly, and much more weakly, by the major
pickup configuration. From the three sets of proton
parameters and two sets of triton parameters, various

combinations were tried, and the pair which gave the

best average fit to the experimental shape for all six

86

'cases, as determined by visual inspection, was chosen.
This was sets II and V of Table 7. Small variations in
the real and imaginary well depths were made within
reasonable limits as determined by the scatter in the
values of the well depths given in the literature for
fits to actual elastic scattering data. Again the best
average fit to all six cases was used to determine the
final value of the depths given in Table 7. The strongest
dependence of the shape of the angular distribution was
found to be on the imaginary well depths of both the
protons and the tritons. All distorted wave calculations
shown in this work are based on this final set of optical

model parameters.

7.2 The Ground State Transitions

As was mentioned in Section 7.1, distorted wave
calculations were made for the six 0* to 0+ ground state
transitions assuming the pickup of'a pair of neutrons
coupled to angular momentum zero from the last shell
in the simple J-J coupled shell model picture. Each
individual proton shell and neutron shell was assumed
coupled to zero angular momentum. In calculating the
form factor described in Chapter 3, the wave functions
of the individual neutrons were taken to be those of a
particle bound in a Woods-Saxon well of the form given

in equation (7.2.1).

87

 

l h. 2
U(r)=-V ————-+( )
1 + ex mn°

1. d
v-——(
C s r dr 1 + e

(7.2.1)

In the above expression, 3 = 2; for these spin 1/2
particles. The parameter x is equal to (r-roAl/3)/a,
where A was the mass of the target minus the mass of one
neutron. The values of rO and a were chosen to be 1.25 f
and 0.65 f as suggested by Bayman g£_al.(Ba68). VS was
taken as 6 MeV, which is typical of single nucleon spin-
orbit strengths. The real well depth Vo was adjusted so
the individual neutrons were bound by one-half the two
neutron separation energy as suggested in references

Dr66 and Ba68. These Woods—Saxon wave functions were
then expanded in terms of a series of harmonic oscillator

wave functions whose strength parameters a was chosen by

the code TWOFRM to maximize the convergence of this

expansion and was usually about 0.3 f-2. This expansion
had the form of equation (7.2.2).
+ U —>

¢Y£(r).= :0 aY Ou£(a, r) (7.2.2)

1.1

* +
Multiplying both sides of equation (7.2.2) by Gu,£(a, r)

+ .
and integrating over dr, and expression for the coeffi-

cients can be found.

“a

aY J 0:2(0, r) ¢Y£<?) a; (7.2.3)

88

Both wave functions have the same angular dependence, and

this can be integrated out.

2 dr (7.2.u)

u _ 2
aY — I Rul(ar ).UY£(r) r

The ax's were calculated by numerically evaluating the
integral of equation (7.2.U).
Since these are orthonormal wave functions, the

following restriction exists for the ag's.

Ia$l2 = 1 (7.2.5)

IIMS

u 0

The sum over 11 was cut off at u=u , where “ma was

max X

determined by the following criterion. The first re-

quirement is given in equation (7.2.6).

1.1
zmax la“|2 1 0.9996 (7.2.6)
u=0 Y

The final cutoff was determined by minimizing the

quantity Q defined in equation (7.2.7).

U -
Zmax au R £(ar2)|2 r2 dr (7.2.7)
=0 Y H

QEJIu (r)-
, W n

Formally Q should go to zero as u gets infinitely

max
large, but because of numerical problems in evaluating

the integral of equation (7.2.“), Q does not go to zero,

89

but reaches a minimum. Typically “max was between 5 and

8, Q was about 14x10"6 and

IJ

maxla
0

ll Mt:

$|2 was about 0.99998.

This corresponded to fitting the Woods-Saxon wave func-
tions to better than 2% out to a radius of about
2.6 A1/3f and better than 10% out to about 3.2 Al/3f.
Since the well had a size parameter of 1.25 Al/3f, this
corresponds to fitting to 2% out to twice the nuclear
radius and to 10% to 2 1/2 times the nuclear radius. At
twice the nuclear radius, the form factor has already
dropped off by 3 or A orders of magnitude.

According to the results of Chapter 3, the
theoretical differential cross-section should be prOpor-

tional to the cross-section calculated with JULIE.

do _ K 7'
awtheory - 56TH 0(JULIE) . (7.2.8)

The proportionality factor, K should be constant
for all the (p,t) reactions studied if the proper wave
functions and thus parentage factors are used, along
with the proper distorted waves. Since the simple wave
functions described above are most probably not ade-
quate, the factor K cannot be expected to be constant,
and it is not. Figure 14 shows the data for these six 0+
to 0+ ground state transitions along with the distorted

wave calculations based on these simplest of wave

9Q

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380 0 3.2 .50
8. 8. 0! ON. 00. OO 8 0? ON 0 OO— 8. 0?. ON. 8. 00 O@ O¢ ON 0
_ . J _ a a _ _ 3.0. _ . _ _ _ a _ q n.o_

 

   

       

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as .3 as

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d

as .. a...

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«29 2.3 .28

 

 

 

 

 

 

vp/op

(JS/qw)

91

functions. The value of K was chosen to give the best
average fit to the two maxima of the data, and the value
(of K is given in the figure.

Another method of calculating (p,t) angular dis-
tributions using distorted waves is to treat the reaction
as a transfer of a rigid cluster; In this method the
two neutrons are treated as if they were an elementary
particle of spin zero and mass 2 with no internal struc-
ture present in the target nucleus as such. With this
picture, the calculation can be carried out exactly the
same way as single nucleon transfer such as (p,d). In
distorted wave calculations of (p,d), the form factor is
usually taken to be the bound state wave function of the
neutron in a Woods-Saxon well. Such a cluster transfer
calculation was carried out using the wave function of
a mass 2 particle with quantum numbers L=O, S=O and J=O.
The principle quantum number, which is somewhat arbi-
trary for such a calculation, waschosen to be 3. This
choice is based on the fact that in the expression for
the form factor (equation 3.3.2), in the more detailed
model of Chapter 3, the dominant term is the one
corresponding to N=3. Calculations were also made with.
N=1 and 2 and little difference in shape was observed.
The cluster was assumed to be bound in a well of the form

given in equation (7.2.9)

92

(r - 1.25 Al/3 f)/O.65 f

X

(7.2.9)

In this expression A was taken to be the mass of the
residual nucleus, and V0 was chosen to reproduce the
experimental two neutron separation energy. The calcula-
tions show that the general shape of the angular distri-
butions are reproduced but not nearly as well as with
the more detailed calculations.

In order to compare in a systematic way, the data
with these calculations, it can be noted that over the
range that the distributions were observed, there are
two peaks in the cross-section. The first one at about
25° will be denoted by 61 and 01 where 61 is the center
of mass angle at which the peak occurs and 01 is the
cross-section at this peak. The second peak at about
55° can be denoted by 82 and 02. Figures 15, 16 and 17
show the value of 61, 62, and 01/02 respectively as a
function of target mass number (A) for the data and the
two methods of distorted wave calculations. Of interest
is the fact that the general trend of 01/02 is repro-
duced fairly well by the two nucleon transfer theory,
but that the cluster model does not reproduce it as

well. The same single set of optical model parameters

was used throughout.

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96

In some cases the second maximum in the calcula-
tion does not appear as a peak, but as a plateau, in
which case 62 was defined as the angle where the slope
of the cross-section with 6 was a minimum. The data is
represented as bars which denote approximate limits on
the quantities as determined from the data. They were
arrived at by sketching the reasonable limits of smooth
curves through the actual data points of the angular

distributions.

7.3 Dependence on the Bound State Well

 

The dependence of the calculations on the bound
state well of the individual transferred neutrons was
also investigated. The 32S(p,t)3OS(G.S.) transition was
chosen for an example. The values of ro and a of the
Woods-Saxon well were varied to investigate their
effects. The overall shape was found to vary only
slightly. The parameters 61 and 62 were not effected at
all with any reasonable variation of rO or a. The inte-
grated cross-section (integrated over the angular range

of the calculation, 0° to 111°) denoted as was

Otot
found to vary greatly and 01/02 was also found to vary.

When a was varied by 0.10 f from 0.65 f, 01/02 changed

by about 6% and o by A0 to 50%. When r was in-

tot o

creased by 0.10 f from 1.25 f, ol/o2 changed by 20% and

otot by 115%. This large variation in the magnitude of

97

the cross—section with small variations in a and r0
indicates the necessity of good values of parameters in
order to calculate magnitudes of cross—section reliably.
Variations in 01/02 (the shape) were observed but were
small. Therefore, we again conclude that the shape is
mostly dominated by the L-transfer.

In the selection of optical model parameters, as
mentioned in Section 7.1, several sets were tried.
Although not investigated in detail, variations in

01/02 and o of the same order of magnitude as men-

tot
tioned in the discussion of bound state parameters were

1 and 62. This

indicates the need of good optical parameters in order

observed, along with some variations in 6

to carry out distorted wave calculations which are

meaningful in detail.

7.U Dependence on Configuration Mixing

 

The exact calculation of an angular distribution
by the two nucleon transfer distorted wave theory ing_
volves the use of complete wave functions for the
initial and final states. Since these are not well
known in general, the complete calculation cannot be made.
Even if they were known, the calculation of the appro-
priate parentage factors as described even in the

simplest general case of Section 3.h are very involved

98

and include sums over single particle c.f.p.'s, Clebsh—
Gordan coefficients, Racah coefficients, and 9-J symbols.
In order to study the dependence of the distorted
wave calculations on the inclusion of configuration mixed
wave functions, the case of uoCa(p,t)38Ca(G.S.) will be
considered. This is possibly the simplest case since
uoCa is doubly magic nucleus whose ground state to 0th
order might be considered a doubly closed shell. The
38Ca ground state might then be considered as a closed
proton shell and a mixture of a term with two neutron

holes coupled to zero in the 1d shell, and a term

3/2
with the neutron holes in the 25
38

1/2 shell. Therefore,
the (p,t) reaction to the Ca ground state could go by

either the pickup of two d neutrons coupled to zero,

3/2
or two 51/2 neutrons coupled to zero, and in general by
some mixture of the two. Figure 18 shows the distorted
wave calculations for these two extreme cases. The
shapes are quite similar. In the case of (d3/2)2 pickup,
01/02 equals 6.0 and in the case of (81/2)2 pickup

01/02 equals 6.8. The maxima (91 and 62) fall at
essentially the same angles in both cases. The magni—
tudes are quite different and are in the ratio of about
1.00/0.67. In order to study the effects of mixing

these two possible components, calculations were made

as a function of percentage mixture of (81/2)2 pickup

)2

with the (d , both in phase and out of phase.

3/2

99

 

0
IC) 1 ‘T f 1 l I l l
2
— (dm1mPICKUP
---- (s, ”150 PICKUP

10’3

do/dn. (ARBITRARY UN ITS)
5.

 

 

-4 1 1 1 1 1 1 1 L

 

 

o 20 40 so so ICC 120 140 l60
ecm (deg)

Figure l8.—-Distorted wave calculations for pure pickup

in the uoCa(p,t)38

transition.

Ca ground state

|80

100

Figure 19 shows the percentage change in the integrated
cross-section (Otot) with respect to pure (d3/2)2 pickup
as a function of this admixture. It should be noted that
certain admixtures can enhance the cross-section by a
factor of about 1.7 and if out of phase, the cross-
section can drop to essentially zero. Therefore,
admixtures and their relative phases can have a very
drastic effect on the magnitude of the cross-section
predicted. It also should be noted that this effect is
strongest for very small admixtures, that is the greatest

rate of change of o as a function of admixture occurs

tot
at small admixtures. Figure 20 shows the effect on
01/02 (shape) as a function of in phase admixture. An
effect is noted, but it is not as drastic as the effect
on the magnitude. Looking at Figure 18, the effect on
shape is almost not detectable on the semi-log plot of
differential cross-section versus angle.

According to the above analysis, it can be con-
cluded that the two nucleon transfer reaction is very
sensitive to the actual wave functions only in magnitude,
the shapes being mostly dominated by the L-transfer and
the distorted waves. It can also be concluded that the
magnitude of the cross-section is so strongly dependent

on small admixtures that the wave functions would have

to be known in great detail and to great accuracy in

PERCENT CHANGE IN om,

lOl

 

 

 

 

 

'00 T I T l l T 1 1 1
80 t- "
60 — F
40 — _.
20 ._ ‘
IO 1 1 1 1 1 1 1 1
1
-4o - -
-60 _.
-80 5 _
400 1 1 1 1 1 1 1 1 1

-|OO -80 ~60 ~40 -20 O 20 4O 60 80 I00

OUT OF PHASE‘--’ IN PHASE

PERCENT MIXTURE 0F (5.,2)2
ADDED TO (d ) PICKUP

Figure 19.--Change in o as a function of

3/2

tot
configuration mixing.

102

.wCfixHE coapmuswamcoo mo coapocsm w mm m0\ao cfi mwcmnolu.om muswflm

anxoa gene 00. 02:0 2. 00004
01.2.3 “.0 wastes. _5zm0mma

00.00 00 05 00 00 0e 00 00 0. o
_ _1 _ _ 4 _ a _ _

 

I I l l 1
1 1 1 l 1
(_D a) to «a- N O

, 1
E

 

20/‘9 NI 2191117113 1113011311

 

1
S

103

mfim~to calculate magnitudes of cross-sections that
are at all meaningful.

7.5 The Transitions to the First Excited
2+ States

All six nuclei studied show a fairly well popu-
lated first excited state which is well‘isolated from
any other nearby excited states that are pOpulated.

fins state is either known or expected to be a Jfl=2+

state. Two nucleon transfer distorted wave calculations

were made for these states also. Again the wave func-

tions for the initial and final states were assumed to be

the very simplest. In particular, calculations were

made assuming the pickup of a pair of neutrons from the

same shell coupled to J=2. This was the (15/2 shell for

Ne, Mg, and Si and the (13/2 shell for Ar and Ca. In

the case of 32S(P,t)3OS(lSt2+), this is not possible

since if there are two neutrons in the 231/2 shell they

rmist be coupled to zero. The 328 ground state might be

expected to contain admixtures of particles in the (13/2

as well as the 81/2 shell. Therefore, for this simple

calxnalation of the L=2 shape, a pickup of one 81/2 par-

ticle and one d3/2 particle was assumed. The experi-

mental distributions along with the results of the above

calculations are shown in Figure 21. The calculations

are arbitrarily normalized to the first and second

peaks of the data.

104

Om.

.mmpmum +m pmpflm one ow.mc00pflmcmma(1.am magmam

.000. .50

8. o... 00. oo. 00 8 0.. 00 o

 

 

_ d _ _ _

              

>0... 08.0.00
80» 2 a. 80..

>02 0.00.0.0“.
a: 2 a. 58

>02 08.0.00
000 .3. 000

_

_

_

.e

0_

To.

 

 

09

303.80
Om. 03 ON_ 8. 0m

 

(15/111111 vP/op

 

_ a _ _ _

                

>0... 005.. new

.000 .8. .000

>0... 000.70.“.

a .Q
INN 3 V OZVN

>0... 0.00. «J
020. e e 0200

0_

To.

 

 

lip/op

(JS/QUJ)

flan-ibis

‘1.

 

 

 

105

Again a 61, 62, and 01/02 can be defined as was
done for the L=O shape. In the case of the L=2 transi-
tion, the first peak (at 01) is not usually well defined
by the data. Figures 22 and 23 show 01/02 and 02 versus
target mass number for the data and these calculations.
The Optical model parameters and the bound state well
geometries were the same as for the L=0 ground state
calculations. The depth of the bound state well was

chosen so that the individual neutrons would be bound by

an energy 6 defined in equation (7.5.1).

5 = 1/2(|B.E.(2n)l + EX) (7.5.1)

In this expression B.E.(2n) is the separation energy of
the last two neutrons, and Ex is the excitation energy

of the excited state.

18Ne

7.6 Transitions to States in

Two nucleon transfer distorted wave calculations

were made for those transitions where the experimental
angular distributions were clear enough to indicate L-
transfer and for transitions to states where J7T assign-
ments have been made by other workers. The results are
shown in Figure 2A along with the configuration of the
two picked up neutrons assumed for purposes of calculation.

This assumed configuration has little meaning since it

has been shown that shapes have only a small dependence

 

 

 

3.1.1.. .. .

1. 1,- 1.. 0...

 

106

 

I

 

.5904? no 4
0.0 mm mm _ 00 .00 ON
. _ a. 0 . .

2950150130 m><>> 8.50.55 101

H

 

H

 

5202.00.06 H
. .

 

 

 

.mcofipfimcmpp mud pmnfim on» now mQOHpommlmmOQo xmmm mo oapmm11.mm mhswflm

N.c\..e

107

.COfipHmcmhp m "A 00.00% mew .009 0:009 @200 mm .Ho coapflmom11.mm @03me

0.»

mm

Poms. mo 4

 

 

H

 

00 00 , .00 00
0 .

H H

 

20.53808 0><3 00:50.0 .101
502.0000 H

 

. . . ..

(mb/sr)

do/da

108

 

IO" T l l l

 

  
  

 

I f I l
1* 1
10'2 + + + -
+
+ + 1
+ E03390 MeV
d
( 111/2)”4
I0" H.
(d 12
51/2”2
_2 ( Ex=3.6l4 MeV
IO —
2
(P
we’ve
Ex=4.576 MeV
(p .s )
l03 1 1 1 1 1 ( 1 I’21 V2 '1"

 

(D 20 4O 60 80

9cm (deg)

IOO l20 I40 ISO |80

18

Figure 2A.——Transitions to states in Ne.

 

 

HJfJWII A

 

109

on the configuration and are dominated by the L-transfer.
The calculations are arbitrarily normalized to make com-
parisons of shape with data easier.

The experimental shape of the distribution to the
state at 3.390 MeV is not well reproduced by the L=H
calculation which is shown with it in Figure 2A. The
L=A assignment is best verified by comparison with the
experimental distribution to the 3.323 MeV state in

22Mg which is most probably a 9+ by comparison with the

level structure of its mirror nucleus 22Ne (see Figure
9). This is the basis for the tentative 1+ assignment
to this level at 3.390 MeV in 18Ne.

The level at 3.619 MeV has been previously identi-
fied as a 2+ (see Table 1). The present data is very
well fit by the L=O shape. Figure 2A shows both an L=0
and an L=2 calculation for comparison. The present
experiment therefore calls for an O+ assignment to the

18Ne.

state at 3.61A MeV in
The general features of the angular distribution

to the level at 9.576 MeV are well reproduced by an

L=l calculation as shown in Figure 29. This state is

therefore assigned a J1T value of 1'.

7.7 Transitions to States in 22Mg

The state at 3.323 MeV in 22Mg is tentatively

assigned J"=u+ although the shape is not well reproduced

 

0. .44.!)15,‘ If!)il..lbp(1liitl
, 1. b . . 1w
d b

 

110

by the L=U calculation shown in Figure 25. This assign-
ment is primarily based on a comparison with the known
level structure of 22Ne, the mirror nucleus to 22Mg

(see Figure 9).

The levels at “.917 MeV and 5.507 MeV exhibit the
features of an L=2 transition. Comparison with the
transition to the known 2+ at 1.250 MeV verifies this
(see Figure 21). These two states are therefore tenta-
tively assigned J"=2+.

The state at 5.738 MeV has previously been very
tentatively assumed to be a 0+ (see Table 2). An L=0
calculation is shown with the data in Figure 25, but
there is very little similarity at all. No attempt has
been made to make a further assignment to this state.

The level at 6.061 MeV is assigned a J1T value of

+

0 . The angular distribution to this state is quite

well represented by the L=0 calculation shown in Figure
25.

7.8 Transitions to States in 2681

The angular distributions to the state at 2.790 MeV
in 26Si is not complete enough to make a definite J"r
assignment. It does exhibit some of the features of an
L=2 distribution (see Figure 26) and so a tentative 2+
assignment is made. This is in agreement with the pre-

vious assignment (see Table 3).

lll

.wz

:0 000000

 

 

          

mm
300.500
00. 03 ON. 00. Om 8 O¢ ON 0
. _ H 0 _ _ _ 0 $3
1 odwusav 00.
>02 .80 saw

- 0!
AV TO- /
D.
_ . H 0
>0: 0050.00 .0 w.
./
1 5
00.11

1 _.o.

— b H H _ — h _ O-

 

 

ou mQOHpHmcmDBII.mm onsmfim

 

       

300.500
Om. ow. 0! ON. OO. 00 00 O¢ ON 0
0 . . . _ H _ _ no.
0
.1. .. O_
N “AQGUV N1
>02 h_¢.¢ uxm
0
o
T 0:486
0. 3 00.
>02 nNn.nuxm
1 1 ..o_
h 1% . . .1 e 5 . o.

 

 

 

uP/DP

(15,1101

do/da (mb/sr)

112

 

 

 

Figure 26.—-Transitions to states in

 

1 LI.

@339 MeV
2

EZ,‘8 4. I83 MeV

Q4 .457 MeV
(65,2)L.2

. (15,2): .40

 

 

L L
20 4O

60 80 IOO

9cm (deg)

IIZO I40 ISO

2631.

I80

113

The level at 3.339 MeV is very weakly excited.
Its angular distribution is not inconsistent with the
previous tentative J=O assignment<RO68), but no defi—
nite assignment can be made.

The shape of the distribution to the state at

2681 is quite well reproduced by an L=2

“.183 MeV in
calculation but the state is only weakly excited and so
only a tentative 2+ assignment can be made.

Figure 26 shows both an L=O and an L=2 calculation
for the state at u.u57 MeV. The L=O shape appears to

give the better fit, but no definite assignment is made.

7.9 Transitions to States in 308
The shape of the angular distribution to the state

0S is fairly well reproduced by an

at 3.u38 MeV in 3
L=2 calculation as shown in Figure 27. This state is
therefore assigned a J1T value of 2+. ‘

The state at 3.707 MeV is only weakly excited. The
angles at which it was excited enough to extract a cross-
section correspond to the maxima of the L=O ground state

transition distribution to 308 (see Figure 1“). Only a

very tentative assignment of 0+ can be made.

7.10 Transitions to States in 3“Ar

The 36Ar(p,t)3uAr has not previously been reported.
The angular distribution to the state at 3.288 MeV in

3“Ar exhibits very definite L=2 character. Comparison

do/da (mb/sr)

llh

 

  
 

'04 +I T I I I I I I
.o
04‘ o
+ EXP-3.438 MeV
,,
IO‘Z— I (d5,2)2
. L82
i * '
I03—
I04r ‘
Ex=3.707 MeV
2
(d )
5/2 ”0
Us I I . I J I J I I

 

 

20
Gem (deg)

Figure 27.-—Transitions to states in 308.

 

'40 so 80 I00 I20 I40 |60 l80

115

with the L=2 calculation shown in Figure 28 and compari-
son with the distribution to the first 2+ state in 3“Ar
(see Figure 21) both demonstrate this. A J"T value of 2+
is therefore assigned to this state.

During many of the runs the level at 3.879 MeV
and “.050 MeV were not resolved. When it was possible
to resolve them, it was obvious that the state at 3.879
MeV was more strongly excited by far. Figure 28 shows
’ the angular distribution to the sum of these two states,
along with an L=0 distorted wave calculation. The shape
of the distribution is very well reproduced by this cal-
culation and so an 0+ assignment can be made for the

3“Ar.

state at 3.879 MeV in
The states at 5.909 MeV and 6.07“ MeV were also

often not resolved, and the sum of the distributions

to these two states is shown in Figure 28. When these

two states were resolved, it was n0t possible to say that

one was much more strongly excited than the other. The

total angular distribution does exhibit some L=2

character, and therefore possibly one or both states

are 2+'s, but no definite assignment can be made.

7.ll Transitions to States in 38Ca

A level in 38Ca at 3.72 MeV has been reported by

(Ha66)

Hardy et a1. using the (p,t) reaction. This level

was assigned a JTr value of 3-. Recently Shapiro

ll6

 

 

 

 

 

- '04 I I I T I I I I
o.
o . _ E-x=3.288 MeV
IO‘ _3’ . , (dB/Z'SIIZ)L.2 *—
9
o o
O
, O.
0&- 9 1
9 x
0
§ 6
. ' 0
Ex =3.879 MeV
‘. ¢ ' (AND 4.050 MeV)
'02? (SI/2 _.
' ' : LFC)
+ 9 Ex=5.909 MeV
’ ‘ AND 6074 MeV
* (C'zI/z'sI/z)._,2
'0' I. ' .l l l l . J I
O 20 4O 60 80 IOO IZO I40 |60 I80
ecm (deg)

Figure 28.-~Transitions to states in 3“Ar.

117

et al.(Sh69b) have reported a 2+ level at 3.69 MeV in
38Ca seen by the (3He,n) reaction. In the present ex-

periment, a level at 3.695 MeV was observed. The angular

distribution to this state is shown in Figure 29. An

L=3 distorted wave calculation is shown along with it,

but an L=2 shape fits it nearly as well. If the two

states previously reported are actually the same, the

present experiment does not resolve the discrepancy.

The 0+ state at 3.06 MeV reported by Shapiro e£_§l.(8h69b)

was not observed at all in the present experiment.
The angular distribution to the states at “.381

MeV and “.899 MeV both resemble an L=2 shape (see Figure

29) but the distributions are not complete enough to make

a definite assignment.

8Ca was only weakly

The level at 6.280 MeV in 3
excited by the (p,t) reaction. The distribution to this
state resembles an L=0 transition as shown in Figure 29.
The transition is too weak, and the distribution is not

complete enough to make any more than a tentative 0+

assignment to this level.

d0/da (mb/sr)

 

 
    
   

Ex=3.695 MIN 4

(f .d )
7/2 3.12“.3

6,: 4.38l MeV
(dale '5 I/2’U2

 
  
   
  

 

 

 

IO'2 ‘1
E;4.899; MeV »
d .
( 3/2 SI/2)L,2

IO'3— 8x26.2280 MeV _.
(SI/2)L'O

'04 l I l l L ' l l l

O 20 4O 6O 80 IOO l20 I40 I60 I80

ecm (deg)

Figure 29.—~Transitions to states in 380a.

CHAPTER 8

SUMMARY AND CONCLUSIONS

In this work it has been found that the (p,t)
reaction is useful as a method of studying the energy
levels of nuclei. It is particularly useful in studying
nuclei two nucleons away from stability. The energies
of the tritons from the (p,t) reaction have been used to

22 2681, 308, 34

Mg, Ar,

locate levels in the nuclei 18Ne,
and 38Ca and to assign values of excitation energy to
them. The shapes of the angular distribution were found
to be dominantly characterized by the angular momentum
transfer, and this quality was used to make spin-parity
assignments to some of the nuclear levels observed.

The two nucleon transfer theory of Glendenning<gl65>
and the distorted wave method was also studied. It was
found that the shapes of the experimental angular dis-
tributions were fairly well reproduced, and this was used
as mentioned above to make angular momentum transfer and
spin-parity assignments. The magnitudes of the predicted
cross-sections were found to be influenced very greatly
by the optical model parameters, the bound state
parameters, and, most importantly, the presence of small

admixtures in the shell model wave functions of the

119

120

initial and final nuclear states. It is concluded that
these strong dependences make the prediction of magni-
tudes of cross-section for the (p,t) reaction extremely
difficult. The detailed calculation of parentage fac-
tors from very accurate shell model wave functions would
be needed along with well determined distorted wave and
bound state parameters. Such detailed calculations and
studies would involve such an extensive project that the
present understanding of the two nucleon transfer process

might not warrant it.

 

‘fifilI
‘1

 

APPENDICES

121

APPENDIX A

 

20Ne(p,t)18Ne EXPERIMENTAL DATA 3

Abs. Norm. Error 2.8%

Angle Error 0.15 deg.
Proton Energy AA.965 MeV
Ground State Q-Value -20.02l8 MeV A

i0.00A8 MeV

123

NEECIPoT)\E18 kEEOIP:T)NE18

Ex' 00300 VEV EX0 3.390 ”EV
+/' 00014 MEV

ANG(CM) SIG“A(CM) sass? AVG(CM) SIGVAICM) ERQBR

 

(DEG) (VB/SQ) It) (DEG) (Ma/SR) (X)
11'6 “063 5'1 402 1107 1065 E“? 3308
1702 1028 E'l 708 170# 1094 EOE 2207
19.5 2.77 E-1 5.9 19-6 2061 E-2 20.6
2209 5033 E'I 305 2301 2073 E'E 1700
2805 70C9 E-1 207 2808 3004 E'Z 1408
3u.c 3.42 E-1 A.“ 34.3 1.71 5.2 20.8
3905 9029 E-E #08 3909 1087 E'B 1205 ‘
“500 ?012 5'2 908 #50“ 1029 E'Z 1300 f
5005 3050 E'B 607 5009 1042 E'Z 1003 g.
5600 F080 E'2 501 5605 1002 E'? 130“ g;
51.3 5090 5‘2 “02 6108 803 E'3 1204 I
6607 3069 E-E 503 6702 604“ E-3 1501
7606 9066 E'3 1003 7502 606“ 5'3 1300
8205 50C? 5'3 1408 83°C 7061 {‘3 1200
9000 8016 E'3 1200 9307 302 E'3 1108
9707 6073 5.3 1106 9803 5060 {-3 1“02
10501 “028 5'3 1508 10507 #027 E-3 1706
5'3 230“ 11300 3023 ['3 1505

11204 1045

NE20(P0T)¥E18 NEEOIPIT)NE18

Ex- 3.614 “EV
*/- 0.013 MEv

EXI 10894 "FV
+/' 00010 ”CV

ANG(CM) SIGMAICMI [sass ANGICMI SIGMAICMI ERF99

(DEG) (VB/SQ) Ix) (DES) (MS/3R) I2)
1107 3014 5'1, 503 1107 1'21 E'l 8'8
1703 2000 5'1 602 170“ 6096 E‘Z 1101
1906 1075 E'1 705 1907 6057 {-2 1205
2300 9063 5'2 805 2301 5064 E-2 110“
2807 “034 5'2 1108 2808 7028 [-2 809
3402 7080 E-E 902 3403 5056 E-2 1102
3907 9025 5'2 #08 3909 2068 [-2 909
4503 7026 E'Z 407 4505 1038 E'E 1206
5007 3005 E-2 609 51-0 707 E-3 1503
5603 1081 E'E 904 5605 804 E'3 1504
6106 3027 E'E 507 6109 90k E-3 1202
67.0 3.89 E-a 5.2 67-3 7.12 5-3 13o8
7409 2096 E-E 505 7502 6029 E-3 1303
8208 1060 E-E 708 8301 3066 E-3 1801
9004 906 5.3 1106 9007 “01“ 5‘3 1907
9800 7091 5'3 1104 9804 4027 E'3 160“

10509 7052 5'3 1109 10508 2069 5'3 2206

11207 6098 E'3 1002 113-0 1'63 5'3 2400

124

NEEOIP;T)\E18 REZOIPITINEIS

Ext #0576 VEV
+/- 00017 MEV

EX! 60326 MEV
*/' 00013 MEV

ANG(CM) SIGMA(CV) F9309 AV3(CM) SIG“A(CM) ERRBQ
(DEG) (VB/5Q) (%) (DEG) (MB/SR) (Z)
1108 7030 E'2 1200 1706' 2036 2'2 2105
1704 2095 E'2 1807 1909 2082 E-2 2106
1907 3068 2'2 1800 2303 2034 E'2 2103
2302 3005 E'2 1604 29‘1 2'08 5‘2 18.1
2809 3045 2'2 1308 3406 1036 E'2 3107
340“ 5088 2'2 1106 4003 1005 E'2 1805
“C00 4009 E-E 704 “509 602 E'3 2202
#506 1053 5'2 1201 51'“ 1'21 5'2 1206'
5101 706 2'3 1505 5700 1012 E'2 1307
5607 603 E'3 1806 62'“ 602 E'3 1706
6200 1006 5'2 1200 7508 4082 E-3 1707
670“ 909 2.3 1106 8307 (+087 ['3 17010
750“ 308“ 2'3 1809 9103 4090 E'3 19.9
8303 2060 E'3 2206 9900 2079 2'3 2502
90'9 2040 E'3 3005
9806 2053 5'3 2400
10600 1025 E'3 3604
11302 1032 2'3 2903
NE20(pIT)NE18 NE23(PIT)N518
EX' 50150 ”EV Ext 70957 MEV
+/' 00014 MEV +/' 00025 MEV
ANG(CM) SIGMA(CV) F9909 AN3(CM) SIGMA(CM) ERRBR
(DEG) (MB/$8) (Z) (DEG) (MB/SR) (X)
1108 901 2'2 [1102 4005 906 5'3 2306
17.5 1.146E-1 8.4 05.2 1.00 5-2 17-8
1908 1002 E'l 1000 5108 409 5'3 2605
2302 10112E-1 802 5703 703 E'3 2107
2900 10103E'1 703 68-2 605 E-3 1806
3405 6082 E'2 1003 7602 603 2‘3 1700
4001 5039 E'2 604
“507 “001 5'2 608
5102 4061 ['2 505
5608 3023 E'2 609
6202 1095 E'2 802
6705 1010 2'2 1009
7505 1009 E'2 908
8309 901 5'3 1105
9100 606 E'S 1503
9807 3001 5'3 2008
10601 1037 E-3 3604
11303 2018 E'3 2307

 

 

125

kE20(P,T)\E13

Ex“ 90215 ”EV
*/' 00020 VEV

ANG(CM> SIGMA(CP) gages
(DEG) (”B/SQ) (z)

 

1200 6059 E-B 14.5
1708 5088 E02 1a.“
2301 501“ E-E 1706 .
2306 3030 E“? 190? g
29.5 301‘) E-E 180C 1
3501 2051 E-P ?208 ?
4008 1-94 E-e 17.0 E
4604 1.55 E-a 1605 E
52.0 1076 E02 1200 5
5707 1025 E-E 1607
63'1 902 E‘3 1607 L
6806 508 E-3 2103 _
7606 50“ [-3 19.5
84.6 6'2 E'3 1803
9202 400 E-B 2508

APPENDIX B

21*1\ag;(p,t:)22Mg EXPERIMENTAL DATA

Abs. Norm. Error 6.9%

Angle Error 0.15 deg.

Proton Energy “1.875 MeV

Ground State Q-Value -2l.1820 MeV
£0.0096 MeV

126

127

M324(P;T)“022 VG?4(P,T)MG?2

Ex8 00000 "EV EX: 3.323 MeV
+/~ 00021 MEV

ANG(CM) SIGVA(CV) EDRBQ ANG(CM) SIGNAtCV) ERR?“

(DEG) (VB/S?) (Z) (DEG) (MB/5R) (X)
1805 1.428E-1 2.3 1807 9.9 E-3 1207
24.0 4.344E-1 1.4 24.3 1.15 [-2 10.2
2905 4023 E-l 303 2908 103642'2 608
3500 '20306E-1 105 3504 105805-2 509
4005 7068 E'2 207 4309 1049 E'E 1406
“509 3016 E'2 309 4604 1016 E-2 1000
5103 60“1 5'2 205 5108 9045 E-3 609
5608 6097 6'2 204 5703 9065 E. 609
6200 6006 E“? 205 6206 8069 E-3 608
6703 30048E-2 208 6709 6088 E-3 602
7206 1.673E-2 3.8 73.2 4.41 E-3 801
8209 9000 E'3 401 8305 4006 E'3 605
9301 100655'2 305 9307 3090 E-3 602

10300 7046 E-3 309 10307 2021 E'3 708

M624(P,T)VG?2 M624(P.T)M322

Ex: 4.41} HEV
*/' 00027 MEV

EX! 10250 ”EV
+/- 00008 MEV

ANG(CM) SIGNA(CN) E990? AVG(CM) SIGNA(C”) ERQQR

(DEG) (MB/SQ) (X) (DEG) (MB/SR) (X)
1806 6030 E'2 403 1808 3044 E-2 600
24.1 4.69 E-2 4.1 24.4 2.00 E-2 7.3
2906 4029 5’2 305 2909 1059 E-E 1702
3501 3085 E'2 306 3505 10192E-2 701
“006 3064 E'2 309 41°C 1091 E'E 505
4601 20C5 5.2 409 4605 10760E'2 503
5105 10380E-2 505 5200 10571E-2 504
5700 10235E'2 600 5705 8005 E-3 706
6206 106035'2 409 6208 407 E-3 2404
6705 106265'2 309 6801 3096 E-3 900
7208 106935-2 308 7304 6014 E-3 609
8301 9049 E03 402 8308 4079 [-3 509
9303 4065 E'B 505 9400 2020 5'3 806

10303 5079 E'3 405 10309 1077 E-3 900

128

M324(°0T)”822 MGE4(P.T)H622

Ex: 5.057 va EX: 5.733 “Ev
+/- 00031 ”EV *I- 00335 MEV

ANG(CM) SIGNA(C”) EQRBQ AKG(CM) SIG”A(CM) ERFOR

(DEG) (”B/SR) (X) (DEG) (MB/SR) (Z)
1808 2029 E-Z 704 1809 2057 [-2 1000
2404 1089 E-E 704 2405 1046 E-E 902
30'0 1013 E-E 2102 3001 7095 E-3 1100
3506 101C6E-2 704 3507 #045 E-3 1302
“1'1 10‘515-2 60“ #102 3070 5'3 1501
4606 10C34E'2 702 4608 3043 {-3 1507
5201 6073 E-3 900 5202 1073 E-3 2500
5706 5040 5-3 906 5708 3053 E-3 1400
6209 “028 E-3 1309 6301 “013 E-3 1107
68.3 “053 E-3 802 6804 5019 E'3 709
7305 5016 E-3 13°C 7307 5012 E-3 800
84.0 2.82 E-3 7.9 84.1 2.28 E-3 9.4
9402 1065 E-3 1003 9“03 1060 E‘3 1008

104.1 1083 E-3 Q01 10“03 1093 [-3 803

M824(P:T)VS?2 VGP4<P.T)M622

EX: 60061 MEV
*/- C0037 MEV

EX: 50313 "EV
+/- 00032 MEV

ANG(CH) SIGNA(CN) EQRBQ ANG<CN> SIGMA<CM) ERQSR

(DEG) (MB/SQ) (2) (DEC) (MB/SR) (X)
1808 1079 E-E 807 1809 4009 EOE 506

2404 1012 E-E 1001 2405 3095 [-2 609

30.0 1.57 E-a 18.2 30.1 4.34 9-2 12.5

3506 103515-2 606 3507 2035 E-Z #08

“102 90““ E-3 804 4103 2005 E02 50“

4607 6089 E-3 905 #608 10327E-2 605

52-2 4.66 E-3 11.0 52-3 108076-2 5.2

5707 5022 E-3 1005 5709 10715E'2 502

'6300 3052 5-3 1201 6302 104715-2 504
6803 3039 E-3 905 6805 9008 [-3 506

10402 1014 E-3 1108 73-8 6022 E-3 701
84.2 5009 [-3 509

940“ 5099 E-3 501

104'“ 2080 E-3 702

129

M324(P0T)“622 “GP“<P'T)”G??

EX: 60281 VEV
+/- 00033 “EV

6X: 60835 MEV
+/- O03## MEV

ANG(CM) SIGMA(CM) E2289 AV3(CV) SIGMA(CM) ERPSR
(DEG) (MB/5:) (X) (DEG) {MB/SR) (Z)
1809 1010 E-Z 1206 3303 5008 E-3 1800
2405 908 E-3 1209 3509 3049 5-3 1502
33'2 1000 ['2 IIIO 41'“ 6'09 5'3 1200
3508 101085-2 705 #700 3052 3'3 170“
4103 9-C9 E03 10.0 52.5 5.43 5-3 13.0
4609 706“ E-3 909
5204 6055 2'3 908
5709 7091 E'3 808
6302 7012 E'3 807
6806 6012 5-3 704
7309 6031 E-3 703
8403 #019 5'3 700
9405 3060 E-3 702

10405 3011 E'3 608

“524(p1T)V322 MG?h(plT)MG22

Ex“ 6'6“5 vEV
+/- 0004“ ”CV

EX! 70257 MEV
+/- C0044 MEV

ANG(CM) SIGPA(CM) 50:09 AVG(CN> SIGMA(CM> ERQSP
(DEG) (”B/8:) (z) (DEG) (MB/SR) (2)
3002 2052 2'3 7505 190C 1097 2'2 806
#104 2.68 E-3 19.5 2407 3035 E-B 6.7
“609 3.41 E-B P0.0 3303 2095 3'2 1000
5205 2039 E-3 1806 3509 2059 E-2 “07
9406 1.54 E-3 1108 4105 1069 E-E 605
#701 709“ E-3 1100

5206 8039 [-3 80%

5802 8.39 E-3 10.0

6305 9.15 E-3 1000

68.9 5.52 [-3 8.2

7402 5024 E-3 805

8406 4021 E-3 609

9408 3'91 2'3 609

10408 3.52 E-3 6.5

130

V32“(P:T)“SE

EX= 70961 ”EV
+/' O0C’09 VFV

ANG(CM) SIG“A(C”> FwaQ
(DEG) (”B/SR) %)
1901 1039 ['2 1301
2408 7.9 E-3 1302
300“ 6012 E-3 150C
3601 9036 E'3 806
Q1'7 7050 E'3 1009
#703 5009 E-B 1308
5208 6055 E-3 1002
5804 1030 E‘B 1603
6307 3058 E-B 18.0
6901 2036 5'3 1600
7&0“ 30C7 5'3 1600
8409 10R5 {'3 1205
9501 1058 E-3 1300
135.0 1.99 E-3 0.2

APPENDIX C

2881(p,t)2681 EXPERIMENTAL DATA

Abs. Norm. Error 11 %

Angle Error 0.15 deg.

Proton Energy “2.06 MeV

Ground State Q—Value -22.0lO MeV
i0.0ll MeV

131

132

SI?R(¢.T)SI?6 SI?8(P.T)8126

EXI 0.390 NEV EX: 2.793 ”EV

*/- 00012 MEV

ANG(CM) SIGVA(:”) EQQQD ANG(CN) SIG”A(CM) ERRSR

(DEG) (VB/SQ) (Z) (DEG) (MB/SR) (Z)
1900 1016 6'2 EP03 1“°1 6067 5'2 805
18.3 1.?67E-1 4.7 18.4 5.13 E-E 9-7
2200 305“ E'l 305 2201 600“ E-? 903
2703 Q0883E-1 109 2705 4051 E'a 600
3208 20395E°1 109 3301 2029 E-2 606
3801 807“ ['2 206 3804 2018 5'2 507
“3'6 3083 E-E #00 5406 1053 E-e 701
48.8 6.3? E02 13.0 59-9 1.1953-2 8.2
5402 8038 E'E 209 6503 10156E'2 306
5905 6032 E'Z 303 73°“ 1058 5'2 708
6Q09 F06? E'2 1?00 7507 10330E-2 403
70.0 1071 E'E 70“ 8600 6052 E-B 502
7503 10?1lE-2 “04 96.1 7022 E‘3 501
8505 103156-2 305
9506 Q09“ E'3 #03

5128(PaT)SI?6 SIER(POT)5126

Ex: 10795 ”EV EX: 30339 “EV

+/' 00011 vEV */' 00019 ”EV

ANG(CM) SICVA(CN) Ease? ANG(CM’ SIG“A<C”) ERRBQ

(DEG) (VB/SR) (DEG) (Md/SQ) (X)
1401 10?C5E'1 601 2705 909 E'3 1408
2704 7039 5.2 #05 3301 5080 E‘3 15.3
3209 40Q9 E'a 40“ ““00 90“ E.“ 5C0?
3803 “000 ['2 400 6504 1053 E-3 1203
4308 “032 5‘2 400 7508 702 E'“ 25'“
“900 3029 E'E 50“ 8601 #03 E-Q 2709
5405 2021 E'E 506 9602 508 E‘“ 2105
5907 10054E'2 809
6501 8063 E'3 402
7003 1051 E'Z 800
7506 104825'2 “01
8508 7031 5'3 #08
9509 306“ 5'3 703

133

8128(PaT)8I?6 8128<PaT38126

EX: 40183 “EV EX: 40821 "EV
+/‘ 00011 "EV */' 00013 “EV

ANG(CM) QIGVA(CM) EQRBQ AVG(CN) SIGMA(CV) ERQSR

(DEG) (MB/SQ) (X) (DEG) (MB/SR) (X)
1492 7.7 E-3 33.8 22.2 1.5? E-E 20.7
P202 508 E'B “909 33.3 907 E'3 110“
3302 3051 6-3 2205 4402 7086 E-3 1005
“401 50CC E'3 1305 4905 6066 E-3 1“07
4904 “062 E-3 2301 5h09 5029 E'3 1209
5408 5061 E-B 120# 6002 20Q7 [-3 2707
6301 4.83 E-3 14.7 65-7 2-70 E-3 9-1
6505 3.05 E-3 802 86o“ 2-03 5'3 10°3
76-0 2.77 E-3 1306 96-5 1.76 2-3 11.4
8603 2061 6'3 909
9604 2057 ['3 902

5128(P:T33126 8128(P.T)8126

EX! 40457 WEV
+/- 00C13 MEV

EX! 50229 MEV
+/' 00012 "EV

ANG(CM) SIGVA(CM) EQRBQ ANG(CV) SIS“A(CM) ERRSR

(DEG) ("a/SQ) (Z) (DEG) (Ma/SR) ‘ (X)
1402 1096 E-E 1705 1302 2003 E“? 1906
2202 1.43 E-2 2902 04.3 10065E-2 9.0
2706 1095 {'2 130“ 55'0 5051 E'3 1306
3303 1092 E'2 704 6507 2064 E-3 907
3806 102C E'E 809 760? 2065 E'3 1206
4402 IOISSE'E 801 9606 20QS ['3 907
4904 606 5-3 1503
5408 7042 5'3 1305
6002 60?“ 5'3 1300
6506 5022 E'3 508
7601 5024 5'3 703
8603 208? E'3 804
9604 ?057 E'3 901

134

$128(°.T>8126 5128(P1T)8126
EX= 50562 MEV

EX! 60381 HEV
+/' 00028 MEV

*/- 00023 MEV

ANG<CM) SIGPA<CM) EQRBQ ANG<CM) SIG”A(CM) ERRBR
(DEG) (MR/52> (Z) (DEG) (MB/SR) (x)
40.3 8.16 E-3 1005 33-5 6.# E-3 1601
55.0 4.26 E-3 15.9 #005 6.60 E-3 1108
6508 1064 £03 1401 6600 3001 E-3 906
75.3 1.62 5.3 17.5 76-5 3.07 E-3 10-1
9607 1094 E03 1101 8608 2004 E-3 1106
96.9 1.43 {-3 13.8
8128<P.T)3126 8128(P.118126
Ex: 5.960 MEV EX- 6.785 MEV
+/- 0.022 MEV +/- 0.029 MEV
ANG1CH) SIGMA(CN) ERRBP ANG(CM) SIGMA(C”) ERRSR
(DEG) (VB/SR) (Z) (DEG) (MS/5R) (X)
3304 2037 E'3 3309 220“ 502 6'3 5103
5501 5.12 E-3 13.9 5503 3026 E-3 1809
6509 1097 E'3 1301 6601 109“ [-3 1306
7604 1096 E'B 1401 7606 4019 E03 807
8607 1.16 E-3 15.5 86-9 3.09 E-3 9.7
- 97-0 1.79 E-3 13.5

 

 

513%(F1T)SI?6

EX:

70150 ”EV

+/- 00015 "EV

AN3(CM)
(DEG)

3306
“406
7607
8701

8122(plT)3136

EX! 70Q76 MEV
+/- 00020 NEV

ANG(CM)
(DEG)

1&04
2205
3306
“#07
5505
6603
7608
9703

SIC"A(C“)

(”B/SR)

,

(”H'U’fl
0001000)
mmmm

HMO-4".)

SIGVA(CV)

(MB/SR)

8.7
8.1
906
10?“
6038
3065
“025
3.83

mmmmmmmm
I I I I I
wwwwmwww

mmww

Enweq
(%)

3408
3901
2307
16010

EQRSQ
(Z)

3403
3504
1503
805
1‘005
905
1000
901

5128(PIT15126

Ex‘ 70695 MEV
+/- 0.031 MEV

A\G(CN)
(DEG)

1“0b
2205
3307
9703

8123(90T)8126

EX! 70903 VEV
+/' 00021 MEV

ANG(CM)
(0E0)

1‘00“
2206
3307
8703
970“

2081

SIGMA(CM)

(MB/59)

1091 E-2
1054 E'Z
1033 E-E
108R) E‘B

SIG”A(CM)

(VB/s?)

1083
1'11
1057
1081

mmmmm
I I I I I
wwmmm

ERRSR
(X)

1507
2100
1207

901

1503

136

3128(p;T)3126

EX: 7.476 ”EV AND
EX' 70695 MEV AND
EX: 709C2 REV
ANG‘CM) SIGMA(CM) F9359
(DEG) (VB/S?) (%)
1404 5.59 5*? 1201
2205 #03? E-E 1306
33-7 3.31 E“? 701
4407 3023 E'a 502
5505 2036 E'E 301
660# 9.28 5‘3 602
7609 10118E'2 601
8702 7.60 E-3 792
9703 7955 5'3 609

APPENDIX D

328(p,t)3os EXPERIMENTAL DATA

Abs. Norm. Error I 2.3%
Angle Error 0.15 deg.
Proton Energy 39.915 MeV
Ground State Q-Value -19.593 MeV
$0.012 MeV

137

138

832(PaT)S?G S3?(P:T)S3O

EX= O-OCO ”EV fix: 3.433 ng

*/' 0-01“ "EV

S32(P;T)SBC

532(P-T)33O

ANG<CM> SIGVA<CW) EQQBQ ANG(CM) SIS”A(CM) EQQfiQ
(DEG) (VB/8:) (z) (DEG) (MB/3R) (z)
1601 903 E-E 1101 16-3 7-72 E-E 12-8
18-8 3.21 E-l 4-9 19-6 6095 {-2 1105
2105 5099‘E-1 208 2106 5098 E-Z 908
2608 7032 E'l 108 2701 “027 {-2 707
27.0 5.51 E-1 1.8 29.7 3-11 E-2 19-6
29-5 5.36 E-l #05 32-4 2043 E-Z 7-3
32-2 1.740E-1 2.0 37-7 1.72 2-2 15.6
37.5 7.67 E-a 6.4 37-8 2-51 [-2 11-7
“208 5-CO 5'2 503 43-1 3009 E-E 700
48-2 10031E‘1 2-7 #8-6 2-41 E-E 509
53-6 1-121E-1 2-5 53-9 1-24 5'2 8-2
5807 7039 5-2 3-4 59-1 604? E-3 1301
63-9 3-70 5'? 5-7 6“-3 6079 E-3 1202
6902 10C“ E-E 1103 69-6 1004 E-E 1108
79-4 1056 E-E 9-2 73-8 4033 E'3 19-1

EY' 20239 MEV
+/' 00018 MEV

Ex: 3.707 MEV
*/' 0-025 ”EV

ANG(CM) SIGMA(CH) gage? ANG(CM) SIGNA(CM) ERRBR

(DEG) (“R/S?) (Z) (DEG) (MB/SR) (X)
1602 1-“1 E-l 9-0 27-0 3-8 E'3 44-0
18.9 1.C87E-1 8.8 «3.1 2006 5-3 37.2
2106 80R8 E-E 8-1 53-9 1-08 E-3 39-2
2700 5.99 E-E 6-7 59-1 1-19 {-3 #0-2
29-6 4.92 E-z 15.5 6a.3 7.9 g-a 44.5
3203 “055 E'E 707
3707 307“ E'Z 907
4300 3078 E-E 602
48-# 2-61 E'Z 507
53-8 1-060E-2 903
5809 8053 E-3 1105
6302 8036 5'3 10-8
6904 1017 ['2 1006
79-6 5-2“ 5'3 1608

139

S32(P1T)S3O

EX: S-2C7 “EV
+/" 00022 MEV

ANG(CM) SIGHA(CM) ERR
(DEG) (NB/SR) (x

54-2 8.14 E-3 13.7
6406 7089 E.3 11

532(PIT)S3C

£x= 5.305 vav
+/- o-oes vev

AN3(CM) SIGHA(CM) gage?
(DEG) (MB/S?) (2)

54-2 6.14 {-3 12.7
6406 5.14 E-3 14.2

S32(P:T)330

EX= 5.207 MEV AND

EX! 5'306 MEV

EX’ 50426 MEV
+/- 0-025 MEv

ANath)
(DEG)

1604
1901
2108
27-2
2703
32-6
43-3
48-8
SQOE
590“
6“-7
69-9
83-2

SIG”A{CM)

ANG1CM) SIGMA(CM,
(DEG) (MB/3R)
19-1 2-71 E-E
21.8 30100 EOE
27'1 3-52 E'E
2703 4.32 g-a
32-6 3002 E-2
43-3 2051 5*?
4808 1084 [-2
54-2 1043 E-E
59-3 1032 E-E
5“°6 1030 5-2
69'9 1-31 E'E
8C'1 702 [-3
S3?(PJT)S3O

(MB/3R)

3-59
2087
2056
3004
1068
1006
900

7062
3006
9-6

7008
5-51
“-42

E-E
E'E

mmmmmmmmmmm
wwwwwwwmmmm

EQRBQ

(X)

E998?

(%)

2101
2103
1509
10-7
1306
11-6
15-6
1108
10-9
1008
1203
1609
19-2

 

1M0

S?2(P0T)833 532(P-T)830
EX- 5.207 “EV AND Ex: 60108 MEV
EX- 5-306 HEV AND 0/- 0.029 MEV
EX! 50“26 MEV
ANG(CM) SIGVA(CM) £996? AMG<CM> SIGWAth) EQQSQ
(DEG) (VB/SR) (Z) (DEG) (MB/SR) (X)
3709 ?095 E‘B 1Q02 3207 3016 ['3 2707

38-0 3.79 E-a 10.1

 

532(POT153O S32(P:T)S3O
EX“ 50897 VEV EX! 60223 MEV
*/' 00027 MEV */- 00033 MEV
ANG(CM) SIGMA(C") FQRBQ ANG(CM) SIGMA(C”) ERRBR
(DEG) (NB/SQ) (x) (DEG) (MB/3R) (X)
2702 5'3 43-3 32-7 3035 E-3 28-7

408
2704 601 E'3 3003
3207 603 5'3 1803

141

832(POT)S?3 532(P-T)S30

EX' 6-415 “EV
+/- 0.040 VEV

EX‘ 70185 “EV
+/- 0-035 MEV

AN3(CV) SIG”A(CM) EQQSQ AN3(CM) SIG“A(CM) ERQBR

 

(DEG) (VB/SR) (Z) (DEG) (MB/SR) (X)
21.3 6.4 E-3 46.1 1605 1.62 {-2 36-6
27.3 1.21 E-a 19.9 21.9 6.6 E-3 39.5
3207 705 E-3 15.8 2703 1009 E-E 19.0 $
49.0 3.91 E-3 19.7 32-8 8-9 5-3 14-9 1
5#04 5051 E-3 1R0“ 3801 701 [-3 2909 ‘
59.5 6.62 E-3 13.9 43.6 7.3 5-3 18-3 ,
6408 3053 E-3 19-5 #901 5-53 E-3 15-8 7
54.5 4.50 2-3 17.8 f
59-7 3065 E-3 1908 ;
.65-0 3045 E-3 21-0 1
fi;
832(P1T)S3C 832(P0T)330

EX' 60861 VEV
+/- o-ouo VEV

EX! 70570 MEV
+/"' 00045 MEV

ANG<CM) SIGMA(CV) EQQBR ANG(CM) SIGMAch) ERRBR

(DEG) (“B/SQ) (X) (DEG) (MB/5R) (Z)
16'Q 2-86 E’E 2205 270“ 907 [-3 22-1
19.2 1095 5'? 2505 27-5 1030 E-a 17-4
21.9 1-96 E-E 3009
270“ 3-11 5'? 9-7
3208 2076 ['2 609
38-1 1054 E-E 1508
“3-5 1030 E'a 12-2
“900 1-06 E'E 10-7
54-# 8017 5'3 11'“

5906 70C9 5'3 1207
6“-9 6-28 E-3 1305
8005 507 E'B 1807

APPENDIX,E

36Ar(p,t)3uAr EXPERIMENTAL DATA

Abs. Norm. Error
Angle Error
Proton Energy

Ground State Q-Value

142

u.8%
0.15 deg.
39.9 MeV

-19.523 MeV
$0.011 MeV

 

mJlfl- :0.‘

1&3

A0361F.T)AQ34 AR36(P0T)AQ34

Ex: 0.003 va EX: 30288 "EV
+/O 00014 MEV

ANGtCM) SIGVA(CV) EQRBR ANG(CM) SIGMA1CM) ERRBR

(DEG) (VB/SR) (z) (DEG) (Ma/SR) (X)
1506 101356‘1 308 1507 7001 E'2 500
21.4 0.88CE-1 1.8 21-5 7018 E-2 5'6
2602 u.SuCE-1 1.3 26.0 4.26 5-2 «.7
32'0 109C5E'1 107 3202 2042 E-E 503
3609 “003 ['2 208 3701 202308-2 309
42.5 6.00 {-2 2.7 42.9 2.42 g-e «.4
“704 10C65E'1 108 #707 107OQE-2 407
5206 9095 5'2 205 5209 802“ E'3 809
53.1 8.68 E-a 1.4 53.0 9.97 E-3 5.1
5709 “057 E'E 204 5802 10078E-2 505
6305 10615E'2 302 6309 10460E‘2 305
7308 101435-2 300 7402 8088 E-3 307
8400 10528E'2 205 840“ 2076 E-B 605
9307 “072 E'3 501 9“01 2080 E-3 702

AQ36(PIT)A?3# AR36(P.T)AQ3u

EXI 2009B “EV
+/- 00011 MEV

EX! 30879 MEV
+/- 0.015 MEV

ANG‘CM) SIGVA(CM) £9050 ANG(CM) SIGMA¢CM> EERBR

(DEG) (VB/SQ) (X) (DEG) (MB/SR) (X)
1506 70“? E'E “08 2500 6006 E“? 1208
2105 508“ E'Z 707 2707 5021 E'Z 308
2604 3090 E'E #09 3702 6020 E'3 804
3202 2067 E-2 #09 4901 1042 E'Z 908
3700 2020CE'2 309 5303 5099 E‘3 707
“208 2070 EVE “02
“706 10660E'2 407
5208 7029 5'3 906
5303 8077 E'3 504
5801 3.55 E-3 10.0
6308 6019 E'3 506
7#01 “086 5'3 “09
8‘03 1035 E'3 1002
9400 2081 E-B 701

144

A936(P,T)AR?Q AR36(P:T)ARB“

Ex: 4.050 VFV EX: 4.523 MEV

+/' 0.014 ”EV */- 0001“ MEV
ANG‘CM) SIGVA(CV) EQRGQ AMG(:M) SIGMA(CM) ERRBQ
(DEG) (VB/SR) 1%) (DEG) (MB/SR) (X)
2500 704 E-3 4908 1507 2029 E-E 906
2707 1050 5'2 705 2605 2000 E-E 704
3702 5028 5'3 907 2708 2017 E-2 603
4901 902 5'3 1301 4708 8079 E-3 702
5803 1025 5'3 2101 4902 709 [-3 1306

 

A936(P0T)AQ34 AR36(P,T)AR33

Ex. 3.879 MEV AND Ex- 4.651 MEV

EX. “0050 ”EV ?/- 00014 MEV
ANG(CM) SIGNA(CH) EQRBP ANG(CM) SIGMAth) ERRBR
(DEG) (VB/SQ) (X) (DEG) (Ma/SR) (Z)
1507 “029 5'2 70“ 1507 1'68 E‘a 1209
2106 5093 E'Z 605 2605 1085 5'2 802
2605 5051 E'E “01 ‘2708 1076 E’Z 703
3203 2032 E“? 504 #709 7070 E-3 707
3702' 10150E'2 601 4902 300 [-3 1308
“209 10578E'2 601
4708 10832E-2 406
5300 1062 5'2 603
5304 10397E'2 #01
5803 7023 5'3 701
6400 3070 5'3 804
7#03 3010 E'3 706
8405 2032 E-S 709
9402 1011 ['3 1206

145

A0361P,T>AR34 AR36(P.T)AE34

EX: 4.5P2 YEV AND
EX= 4.651 ”EV

EX= 40985 HEV
+/" 00014 MEV

ANG<CM) SIGNA(CM) tease AVS(CN) 515vA(cM, geese

(DEG) (NR/SQ) (2) (DEG) (Ma/SR) (X)
1507 3097fiE-E 703 2708 1078 E-2 609
2106 4001 5'2 802 4902 703 E-3 1407
2605 3095 E'E 501 5301 5083 5'3 1008
32-3 2-99 E-E 4.6 58.4 3.49 2-3 13.9
3702 206275'2 306
4300 2006 E‘E 500
4708 106485.? 409
5301 102C7E‘2 703
5305 10194E-E 405
5804 10C0CE‘2 508
640C 10C37E'E 404
7404 6031 5-3 406
8406 4078 ['3 500
9403 4004 5'3 600

 

A936(P;T)APRQ ARBétplT)ARB4

Ex: 40867 VEv
+/- 0-014 MEV

Ex. 4.857 MEV AND
EX! 40985 MEV

ANG(CM) SIGMAccva £990? ANG<CM) SIGMA1CN) ERRBQ

(DEG) (Va/$9) (Z) (DEG) (MB/SR) (X)
2708 904 E'3 1906 2106 1088 E‘E 1306
49.2 3.93 E-3 25.3 25.5 2.09 E~2 7-1
5804 1016 E'3 2101 3204 1068 E”? 604
3702 10098E-2 509
43°C 8030 E-3 807
4709 7096 E'3 704
5301 5080 E'3 1008
5306 6011 E'3 608
5804 4066 E-3 901
8407 1017 5'3 1109
9404 700 [-4 1603

146

An361P.T)AQ34 0936(PIT)A93“

EX“ 503C7 “EV
+/' 00013 “EV

Ex. 60974 qfiV
*/' C0011 “EV

ANG(CV) SIG%A(CM) EQRBD AVG(CM) SIG“A(CM) ERRCQ

(DEG) (MB/SR) (X) (DEG) (MB/SR) 1%)
21.6 ‘6.9 E-3 26.1 27.9 2.09 5-2 6.6
2606 1018 E-E 1007 4302 6075 E-3 002
3204 10052E'2 ~ 809 4904 604 E-3 1601
3703 8.94 E-3 7.0 53.3 4.07 5-3 14.1
4301 7037 £03 906 5806 2019 E-3 1504
4709 6011 5'3 806 9406 1005 5'3 1309
5302 5053 5'3 1105
5306 4095 E'3 801
5805 4054 5'3 908
6402 4049 5.3 704
7405 2011 E-3 1002
8407 2018 5'3 803
9404 1033 5'3 1106

Pt...-

 

AQ361PJT)AR34 A936(P:T)AQ34

EX: 50909 “EV
+/- 00012 MEV

EX: 50909 MEV AND
EX' 60074 MEV

ANG(CM) SIGMA(CW) EQRBQ ANG(CM) SIGMA(CM) E3939

(DEG) (VB/S?) (Z) (DEG) (MB/SQ) (Z)
2709 5.90 E-3 15-7 15-8 4.06 {-2 7.4
4301 4.88 E-3 1203 2107 3019 [-2 1000
4903 1055 E-3 4401 2606 2058 E-Z 607
5302 ?060 5'3 1801 3204 1054 E-Z 704
~5806 10C] 5'3 2700 3704 10298E'2 508
9405 706 ['4 1706 4302 101605-2 706
4800 100105“? 5'9

5303 6069 E'3 1006

5307 5059 5'3 804

5806 3019 E'3 1302

6402 4069 5'3 801

7406 3047 {-3 801

8403 1093 E-3 1007

9406 1080 E-3 1007

147

A936(p.T)A924 A?35(P.T)A934

EX' 60525 vEV
+/' O0CC9 VFV

EX0 7.322 Mgv
*/' C0006 MEV

ANGtCM) SIC“A(CV) EQQSQ A\G(CH) 513MA(CM) ERQSQ
(DES) ("R/SR) (X) (EEG) (ME/SQ) (X)
2606 407 6'3 23906 508 207‘, 2.2 906
32.5 “012 E'3 9900 2108 1083 E-e 1308'
3704 E035 E'3 1702 2607 2043 E-Z 703
4801 2063 E'3 1603 3705 10286E-2 600
5303 2017 5.3 P103 4303 10113E'2 800
5807 1025 E'B ?505 4905 704 E’3 1407

A035(P.T)AR34 ARBétP.T>A934

EX. 7.499 MEV
+/- 00004 MEV

EX= 60794 KEV
+/- 00011 VEV

ANG(CM> SIGVA(:v) EQRSQ ANG(CM) SIGMA(CM) ERRSR

(DEG) ("Q/SQ) (X) (DEG) (MB/SR) (X)
1508 1039 5'2 1609 1509 806 E-3 2306
2107 703 5'3 9800 2108 1057 E“? 1503
2607 10C1 E'? 1208 2607 1067 E-E 808
32.5 5.74 E03 15.6 37.5 102“8€*2 602
3704 6041 E'3 1000 4303 8015 [-3 1301
4302 6013 5'3 1204 4306 907 E-3 1205
4801 4026 E'3 1206
5304 4.56 ['3 1500
5308 3062 5'3 1301
5807 P097 E-3 1408
6404 4080 [-3 806
7408 ?013 E'3 1107
8500 1034 ['3 1405
9407 1052 E93 1100

148

A936(P.T)A:34 AR35<P.T)AR34

EX0 70322 ”EV AND Ex: 7.925 MEV
Ex- 70499 "EV +/- 0.005 Nev

ANG(CM) SIC"A(CV) E9989 ANG(CN) SIG“A(CM) ERRSR

(DEG) (VB/$3) (z) (DEG) (“B/SR) (X)
15.9 3.62 E-a 8.8 15.9 1.94 E-2 1209
2108 3038'E-2 1300 2108 2006 [-2 1402
2607 4010 5'2 503 2608 2012 E-2 709
3206 2060 ['2 509 3206 1047 E-2 901
3705 2053 E-E 402 3706 105536-2 506
4303 _1092 E-2 602 4304 9003 E-3 1004
4802 10530E'? 508 4803 9056 E-3 709
5305 1021 E-E 807 5306 5057 [-3 1501
5309 103135-2 607 5400 3094 E'3 1301
5808 R091 E-3 704 5809 4072 E-3 1008
6405 R052 E-3 506 6406 5001 E'3 802
7409 5071 E-3 602 7500 3087 E-3 809
8501 4086 E'3 601 8502 3014 6'3 900
9408 3091 E-3 704 9500 2034 ['3 1105

 

APPENDIX F

uoCa(p,t)380a EXPERIMENTAL DATA

Abs. Norm. Error 7.3%
Angle Error 0.15 deg,
Proton Energy 40.1“ MeV
Ground State Q-Value -20.u28 MeV
$0.010 MeV

149

150

CA“C(PIT):A38 CA40(P:T)CA33

Ex: o-ooo va Ex- 3.695 MEv
¢/- 0.005 MEV

AN3(CM) SIGMA(CN) EDRQQ AN51CM) SIGMA(CM) ERPBR

(DEG) (“B/SQ) (z) (DEG) (MB/SQ) (X)
1600 105245'1 308 1601 3015 ['2 906
2103 30383E'1 205 2105 3045 E-Z 708
2606 30C17E’1 205 2608 3005 E'2 1000
3201 9081 E-E 40C 3203 2021 E“? 700
3701 30C9 E-E 403 3704 1089 E-Z 507
4204 7.59 E02 406 4207 1042 E-Z 110-
4706 10CP9E-1 200 4709 10036E-2 801
5209 7037 ['2 207 5302 7052 5‘3 900
5801 30CCEE‘2 P06 5805 6032 E-3 602
6303 8094 E'3 402 6306 5082 E-3 506
7305 10449E'2 403 7309 4068 E'3 802

CA4O(90T)CA38 CAkO(P:T)CA38

EX! 20206 YFV
+/' 00005 “EV

Ex- 0.191 “EV
0/- 00005 MEV

ANG<CM) SIGMA(CM) [PRBQ ANG(CM) SIGMA(CM) ERROR

(DEG) (MB/SQ) (X) (DEG) (MB/SR, (X)
1601 R087 E'Z 502 2105 400 E-3 3508
2104 7067 E-B 501 3204 2018 E-3 3500
2607 6002 E-E 500 5303 1023 E-3 2604
3202 3040 5'2 500 58-5 509 E-4 2802
3703 3038 E'E -401 6307 707 E-4 2204
4206 3040 E'E 6'9
4708 2077 5‘2 500
5301 10512E'2 600
5803 10317E'2 401
6305 10655 '2 301
'7307 8087 E'3 507

151

CA4O(P:T)CA38 CA00<P0T)CA38

Ex' 40381 MEV
4/. 00005 MEV

Ext «.899 Mgv
0/- 00005 MEV

ANG<CM) SIGMA<CM> ERRBR ANsccr) SIGMA(CM) ERRBQ
(DEG) ("B/SR) (X) (DEG) (MB/SR) (X)
1601 3081 E'2 805 1602 3082 E'E 804
2105 3031 E'E 802 2105 4043 E“? 700
2609 ?047 E02 1200 2609 2078 E-2 800
3204 1091 5'2 1000 3204 2003 E-E 700
3704 1064 [-2 603 3705 1069 E-E 601
4800 10219E'2 700 4208 1022 E“? 1207
5303 8015 E-3 807 4801 9080 E-3 800
5805 6077 [-3 601 5304 7010 E'3 902
6307 7091 E-3 407 5806 6070 E'3 601
7400 5019 E-3 707 6308 8084 E-3 405
7401 5026 [-3 706
CA40(P:T)CA38 CA40(P:T)C038
EX' 40748 MEV EXI 50159 MEV
0/- 00005 MEV +/- 0.007 MEv
ANG(CM) SIGMA(CM) ERROR ANG<CM> SIGMA(CM) ERRBP
(DEG) (VB/53) (X) (DEG) (MB/SR) (X)
1602 1085 E'E 1303 1602 506 E'3 3100
2105 1004 E-B 1702 2105 306 5'3 3405
2609 709 E'3 3000 2609 303 {'3 3500
3204 6049 5'3 1500 3204 3042 E03 1900
3705 3099 E'3 1403 3705 2093 E-3 1805
5303 2099 E'3 1503 4801 1046 5'3 2705
5806 1093 5'3 1209 5806 1033 E'3 1607
6308 1054 [-3 1300 6308 806 5'4 1908
7401 1025 E03 1902 7401 609 E04 2608

3} 4 ,0 i»

CA4C(F0T)CA38

EX' 50264 WEV

+/- 00085 VEV

ANG(CM)
(DEC)

1602
2105
2609
3205
3705
4801
5806
6308
7402

CA4C(p0T)CA38

5x-
+/-

ANG(CM)
(DEG)

SIGMA(CM3
(VB/SP)

902
1005
80C
6047
4071
1078
1028
20C

608

mmmmmmmmm

:mwwwwwmw

50427 “EV
00006 “EV

SIGMA(CM)
(VB/SR)

3205 P026 5'3

3705

1022 5'3

152

[0039
(X)

2009
1704
1800
1500
1300
2400
1700
1102
3004

A“)

u)
C) \l

CA40(PIT)CA38

Ex. 5.598 Mgv
+/- 00007 ”EV

ANG(CM) SIG”A(CV)
(DEG (MB/SR)
15°? 1000 5'2
3205 3095 E-3
3795 3.99 E'3
4801 2073 E'3
5304 1067 E-3
6309 1061 [-3
7402 1054 E-S

CA00<90T)CA38

Ex. 5.693 HEV

+/- 00005 "EV

50930
(X)

2105
1900
1504
1906
2108
1308
1708

ANG(CM) SIGfiAtcfi) E930E

(DEG)

3205
3706
4802
5305

(MB/SR)

3051 E‘3
2061 5'3
2010 E'3
3044 [-3

(X)

2500
1907
2206
1409

153

CA40<P.Y>CA38 CA«O(P0T)CA38

EX! S0810 ”2V
+/- 00005 NFV

EX: 60280 MEV
+/- 00308 MEV

ANG(CM) SIGVA(CV) F3359 AVGKCM) SIGHA(CM) E3239

(DEG) (MB/SQ) (z) (DEG) (MB/SR) (X)
1602 1031 E'2 1705 1602 1004 {-2 2003
2106 1048 E-2 1504 2106 101R [-2 1707
27.0 908 E'3 1500 27.0 702 {'3 2600
32'5 606 E'3 1700 3205 504 [-3 2000
3706 7067 2'3 100? 3706 2090 2'3 1909
“209 4015 E-3 2309 4802 7020 E-3 1009
4802 4012 2'3 1501 5305 3065 E-3 1506
5395 40GB E'3 1301 5508. 1087 2‘3 1“'9
5807 2076 2'3 1006 7403 808 E.“ 2603
7402 1068 E-3 1606

CA4C(P0T)CA38 CA00(PIT)CA38

EXs 60136 MEV
+/- 00006 MEV

Ex. 6.593 MEV
+/- 00007 MEV

ANG<CM) SIG”A(CM) £0009 ANG(CM) SIG“A(CM) 5:930

(DEG) (VB/SQ) (X) (DEG) (Mg/SR) (X)
1602 407 E-3 3500 2700 1007 E'2 1500
2106 406 E'3 3307 3206 5075 E-3 1500
3205 2097 E-3 2400 3706 3028 E-3 1709
3706 3062 2'3 1608 4300 400 E-3 2608
“892 1077 E-3 2409 4803 5019 [-3 1301
5305 1009 E-3 3307 5306 5083 E-3 1101
7403 706 E-4 3003 5808 2032 E-3 1200

6400 1086 E'3 1400

CA40(PIT)CA38

EX‘ 60702 VEV
+/' 0'010 MEV

ANG(CM) SIGMA(CM) E3289

(DEG) (MB/SQ) (X)
1602 2060 2'2 309
3206 608 E-3 1505
3707 60C8 2-3 1202
4300 400 2'3 2509
4803 5002 5'3 1400
5306 3010 E'3 1507
5809 2045 5'3 1107
6401 1089 5'3 1302

CA40(P)T)CA38

EXa 60768 WEV
+/- 00015 MEV

ANG(CM) $10MA<CM) £005:
(DEG) (MB/SP) (x)

21.6 20
40

1
3206 7

154

CA4O(PIT)CA38

EX: 6.801 MEV
+/- 0.012 MEv

ANG(CM) SIGMA(CM)
(DEG) (MB/SR)
3707 1018 5'3
4803 1015 E'3
5306 1066 E-3
CA40(PIT)CA38

EX: 70208 MEV
+/- 00015 MEV

ANG(CM)
(DEG)

7405

SIGMA(CM)

(MB/3R)

105

f"
V

E-3

ERRBR
(X)

3400

3502
2508

ERRBR
(z)

1906

155

CA40<P:T)CA38 CA40(P0T)CA38
EX! 70800 "EV EXI 80535 MEV
+/- 00012 MEV */- 00010 MEV
ANG(CM) SXGMA(CM) EQRBR ANG(CN) SIGVA(CM) ERRBR
(DEG) (MB/SR) (z) (DEG) (MB/SR) (X)
1603 1048 E-E 1904 1604 1018 E-2 2309
2107 1002 5'? 1807 21°8 606 E‘3 29.5
3708 #029 5'3 1605 2702 506 E'3 3701
5308 “035 5'3 1502 3208 “03 E-3 2508
5900 P099 E-3 1306 3709 4028 E-3 1906
6‘03 P004 5'3 1406 “806 3083 ['3 1908
5309 3006 E-3 1808
6“°“ 1060 E-3 1901

LIST OF REFERENCES

156

udifiu,

m.

Ab66

Ad67

'Ad68

AJ59

A360

Ar68

Ba62

Ba6ua

Ba6Ab

Ba65

Ba68

Ba69
Be66

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