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Michigan Stats
University

4:"

chSiS

 

This is to certify that the

thesis entitled
ANALYSIS OF NUCLEON-NUCLEUS
SCATTERING WITH THE WKB METHOD

presented by

Hyang Key Lee

has been accepted towards fulfillment
of the requirements for

fl {

7 Major professor

do Date ‘3" ‘9 Pig

0-169

 

 

 

 

ABSTRACT

ANALYSIS OF NUCLEON-NUCLEUS SCATTERING
WITH THE WKB METHOD

by Hyang Key Lee

The WKB method together with the impulse approximation
is applied to the inelastic scattering of 156 MeV nucleons
from the nuclei 012, 015 and Ca“°. The optical potential
employed includes a spin-orbit part. In general, the agree-
ment between prediction and experiment for the cross-sections
is impressive. The predictions by the WKB compare favorably
with the results of the DWIA, although the WKB appears to
predict slightly larger cross-sections than the DNIA.

For normal parity states, the effect of the spin-
orbit potential on the polarization is as eXpected. In the
cases where spin-flip is important, the polarization is
sensitive to the spin-orbit term. For Cano, the imaginary
part of the spin-orbit potential contributes little to the
polarization.

The WKB method is applied also to the cases where the
incident energy of the projectile is #0 MeV. In place of the
impulse approximation, a two-body potential of the Yukawa
type is assumed. The agreement between data and prediction
is qualitative only. The WKB is clearly inadequate at this

low energy.

ANALYSIS OF NUCLEON-NUCLEUS SCATTERING
WITH THE RED METHOD

By

Hyang Key Lee

A THESIS

Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics and.Astronomy
1966

PLEASE NOTE: Thesis is tightly bound. Some print
lost in spine. Filmed as received.

UNIVERSITY MICROFILMS , INC.

ACKNOWLEDGMENTS

The author wishes to express deep gratitude to his
parents and brother, who made it possible for him to study
in this country by personal sacrifices on their part.

Sincere appreciation is felt for the patient guidance
and understanding provided by Professor Hugh McManus through-
out this work.

The author gives special thanks to his wife for her
unceasing encouragement and personal endeavors in preparing

this manuscript.

11

on

"I
h

'0!
‘

IOU

TABLE OF CONTENTS

ACKNOWLEDGM EN T8 0 O O O O O O O O O O O O O O O 0

LI ST OF ILLUSTRATIONS O O O O O O O O O O O O O 0

LIST OF TABLES O O O O O O O O O O O O O O O O 0

LIST OF APPENDICES O O O O O O O O O O O O O O 0

Chapter
I.
II.

III.
IV.

V.

VI.

INTBOWCTION O O O O O O O 0 O O O O O O
DISTOBTED WAVE THEORY . . . . . . . . .
Cross Sections and Polarizations . . .
Optical Potential; Perturbation Theory
Impulse Approximation . . . . . . . .
"KB APPROXIMATION O O O O O O O O O O 0

FORM FACTORS . . . e e e .

Nuclear States 0 e e e

Isospin Dependence . . . . . . . . . .
FomFaCtorSeeeeeeeeeeeee
Final State for Spin S = O . . . . .
Final State for Spin 3 = l, J = L .
Final State for Spin 3 = l, J = L 1

DETAILED CALCULATIONS O O O O O O O O 0

Reduction of the Scattering Amplitudes
Scattering Amplitudes for S = O . .
Scattering Amplitudes for S = l . .

Numerical Calculations . . . . . . . .

DISCUSSION 0 O O O O O O O O O O O O O O

012 e e e e e e e e e e e e e
016 e e e e e e e e e e e
Calm e e e e e e e e e

111

Page
11

viii
ix

N00 0\ l-‘

Page

VII. THE WKB METHOD AT #0 MEV . . . . . . . . . . . 52
VIII. CONCLUSION . . . . . . . . . . . . . . . . . . 60
LIST OF REFERENCES . . . . . . . . . . . . . . . . . . 95

iv

Figure

1.
2.
3.
h.
5.
6.
7.
8.
9.
10.
11.
12.
13.
lh.
15.
16.
1?.
18.
19.
20.
21.
22.

LIST OF ILLUSTRATIONS

Cross-Section
Polarization
Cross-Section
Polarization
Cross-Section
Polarization
Cross-Section
Polarization
Cross-Section
Polarization
Cross-Section
Polarization
Cross-Section
Polarization
Cross-Section
Polarization
Cross-Section
Polarization
Cross-Section
Polarization
Cross-Section

Polarization

for

for

for

for

for

for

for

for

for

for

for

h.“3 mev level of C12

9.7 MeV level of C12

12.7 MeV level of 012

15.1 MeV level of 012

16.1 MeV level of 012

18.2 MeV level of 012

19.3 MeV level of 012

21 MeV level or C12
I N I I I

22.3 mev level of 012

21.2 MeV level of 012

19.5 MeV level of c12

Page
98
99

100
101
102
103
10h
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119

23.
24.
25.
26.
27.
28.
29.
3o.
31.
32.
33.
34.
35.
36.
37.
38.
39.
no.
41.
42.
43.

45.
46.
47.
48.

Cross-Section for
Cross-Section for
Polarization ”
Cross-Section for
Cross-Section for
Polarization ”
Cross-Section for
Polarization ”
Cross-Section for
Polarization '
Cross-Section for
Polarization '
Cross-Section for

Cross-Section for

23.2 MeV
19.2 MeV
n n
24.2 MeV
6.15 MeV

11.5 MeV

13o]. MGV

15.3 MeV
11 n
180? MeV
20.2 MeV

level

level

level

level

level

level

level

level

level

of C
of C
n n
of 012
of 016

of O16

of O16

of 016
II I!

of O16
of 016

Polarization for 19.1 MeV level of 015

Cross-Section for 13.5 MeV level

Cross-Section for 16.6 MeV level

Polarization for 17 MeV level of

Cross-Section for 17.3 nev level

Polarization '

Cross-Section for 17.6 Mev level

Polarization '

of O16

of 016

016

of 016

of 016

Cross-Section for 20 MeV level of 016

Polarization '

Cross-Section for 20.2 MeV level of 016

Polarization '

Cross-Section for 3.7 MeV level of Ca“0

vi

Page
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146

Page
50. Polarization for 3.7 MeV level of Ca”0 . . . . 147
51. Polarization " " ” " ” ” . . . . 148
52. Cross-Section for 4.4 MeV level of Ca“0 . . . 149
53. Polarization ” ” " ' ” " . . . . 150
54. Cross-Section for 7.15 MeV level of C340 . . . 151
55. Polarization " ” " ” ” " . . . . 152

56. Cross-Section £8r 7.73 MeV and 8.34 MeV
levels Of Ca 0 e e e e e e e e e e e e e e 153

57. Polarization for 8.34 MeV level of Ca”0 . . . 154
58. Cross-Section for 8 MeV level of Ca’+0 . . . . 155
59. Cross-Section for 12 MeV level of Cauo . . . . 156

60. Cross-Section for 4.43 MeV level of 012
With 40 MeV e e e e e e e e e e e e e e e e 157

61. Cross-Section for 9.6 MeV level of 612
With 40 MeV e e e e e e e e e e e e e e e e 158

62. Cross-Section for 6 MeV level of 016
With “0 Nev e e e e e e e e e e e e e e e e 159

vii

LIST OF TABLES

Table Page
1. Optical Parameters . . . . . . . . . . . . . . 40
2. Central Absorption . . . . . . . . . . . . . . 42
3. Spin-orbit Integration . . . . . . . . . . . . 42
4. Effective Form Factors for C12 2*, T = 0 . . . 57
5. Parameters for the Distorted Wave . . . . . . . 57

O . . . 59
7. Effective Form Factors for 016 3', T = 0 . . . 59

6. Effective Form Factors for 012 3', T

80 Identification Of the Final States 0 e e e e e 63

viii

LIST OF APPENDICES

Appendix Page

I. . . . . . . . . . . . . . . . . . . . . . . . 64
II. ....................... 69
III. . . . . . . . . . . . . . . . . . . . . . . . 71
IV. . . . . . . . . . . . . . . . . . . . . . . . 73
V. . . . . . . . . . . . . . . . . . . . . . . . 76
VI. . . . . . . . . . . . . . . . . . . . . . . . 89
VII. . . . . . . . . . . . . . . . . . . . . . . . 91

ix

CHAPTER I
INTRODUCTION

The theory of the scattering of high energy nucleons
by the nucleus as developed by Kern-n, hohanus and Thaler (1)
(hereafter referred to as EMT), often called the distorted
wave impulse approximation (or DWIA), has been very useful
in the study of the nuclear reactions. In DNIA the incoming
nucleon is considered to be scattered by an essentially free
nucleon in the target nucleus while traversing an average
(or Optical) potential due to the presence of other nucleons
in the nucleus before and after the collision. The calcula-
tion of the wave function (distorted wave) of the beam
nucleon (as the optical potential is of complicated functional
form) has to be carried out numerically using a high-speed
computer. Since this procedure is long and complicated, it
is difficult to keep the essential physics in sight. There-
fore it is hoped that the calculation of the distorted wave
can be simplified by making further approximations and.there-
by bringing out the essential features of the nuclear re-
actions. The Eikonal or VKB method is used for such purpose.

(Several authors have applied the HEB method to nuclear

reactions with high energy beam particles above 100 nev.
Squires (2) used WEB to calculate the cross section and

-1-

-2-
polarization of the inelastically scattered nucleons leaving
the target 912 in 2+ state. Squires' optical potential was
of Gaussian form and had no spin-orbit potential. The
particle-hole interaction was not taken into account in the
nuclear wave functions. Agreement with the existing data
was good if an effective charge correction was made to the
cross-section. Making use of HEB, Sanderson was reasonably
successful in reproducing the broad peak in the cross sections
observed at the excitation energy near 20 Kev for C12. (3)
The optical potential Sanderson used was purely absorptive
and had no spin-orbit part. Sanderson's nuclear wave functions
were linear combinations of the particle-hole states.
D. Jackson used WEB to study the scattering of a nucleon
‘with incoming energies of 40 Rev; 90 MeV'and 150 Rev from
L16, exciting the target to 2.18 HeV’above the ground state.
(4) The theoretical prediction was compared with the experi-
mental crcss section for a beam energy of 40 wev. The
experimental cross-section was greater than the theoretical
prediction by a factor of 1.5 and peaked at smaller angles.
This is not surprising as DHIA.and.HKB are high energy
approxhaations. However, the polarization agreed with the
experiment reasonably well. This is because the polarization
for excited states of normal parity is essentially given by
the Born approximation. D. Jackson also used a complex
optical potential of the Gaussian form. She showed that the

effect of the real part of the optical potential was very
small at energies above 90 nev. T. Erickson calculated the

-3-

cross section for incoming nucleons of 180 nev exeiting the
016 nucleus to 2‘ state at 12.5 nev. He obtained good agree-
ment with the experiment for the cross section although he
was less successful with the polarization. The cptical po-
tential used was purely absorptive and central. The nuclear
wave functions Erickson used were combinations of particle-
hole states. (5)

There is now much more experimental information about
the inelastic scattering of high energy protons (around 150

16, and c340. DeBouard et a1. (6) at Orsay

new from (:12, 0
measured the polarization of inelastically scattered nucleons
from 012. Jacmart et a1. (7) at Orsay published results on
the inelastic cross sections of the light nuclei. More re-
cently, Hasselgren et a1. (8) measured a large number of
inelastic cross sections again from the light nuclei. Boos
(9) measured a few scattering cross sections from Geno.

Looking at existing theoretical calculations by the
VIP method, one is impressed with its reasonable success at
high energy. The present investigation applies the WEB method
to nuclear reactions involving an inelastically scattered
nucleon with an incident energy of 156 HeV'and.many excited
states of the nuclei, 012, 016, and c340. The optical potential
used has a.flood-Saxen form and a spin-orbit part as given by
Haybron and Satchler from analysis of elastic scattering. (10)
The nuclear wave functions are taken from calculations of

Gillet et a1. (11) Cross sections and polarizations are com-

pared with experiment and also with the results of DUIA

-4-
calculations. (19) Particular attention is given to the
effect of the spin-orbit potential on the polarization as,
it is suspected that for some states, the spin-orbit term
may be very important. The same calculations are done for
the inelastic scattering of incident nucleons of 40 nev from
012 and 016. ‘As the use of the'HKB method is not entirely
Justified at this low incident energy, one would be grati-
fied to get qualitative agreement with the experiments.

In Chapter II, the distorted wave formalism is out-
lined. Expressions for the scattering cross-sections and
polarizations are given. The definition of the cptical
potential is also given, along with the definition of the
scattering amplitude. Then follows the discussion of the
impulse approximations.

In Chapter III, the distorted wave is calculated by
the WEB method.

Chapter IV deals with the nuclear state functions in
the particle-hole formalism. Form Factors are defined for
spins S = O and S - 1.

Further reduction of the Samatrix is carried out in
the first part of Chapter‘V. .A typical integral that is in-
volved is carried out. The most general expressions for the
scattering amplitudes are given both for spin S a O and S = 1.
In the second part of Chapter V, the data for the optical
potentials are given. The results of the numerical calcula-
tions of the distorted waves are shown.

In Chapter VI, the results are discussed and compared

-5-
with recent experiments and also with DWIA calculations.
WEB is applied to the incoming energy of 40 HeV’in
Chapter VII. The main features of the calculations are the
same as for 156 HeV except that the two-body interaction is
of the Iukawa type. The numerical work involved has been
carried out again with a computer.

The conclusions are drawn in Chapter VII.

CHAPTER II
DI STORTED WAVE THEORY

Cross Sections and Polarizations
Assume that the Hamiltonian H of the system of a nu-

cleon and a nucleus can be separated into two parts:

H=HO+V X‘
where Ho =- Hn «Ivfiand V . 12:1 WEFj-rza ).

The symbols have the following meanings:
'fi is the kinetic energy operator of the incident
nucleon.
fin is the Hamiltonian of the target nucleus.
7", 5"" are the position vector and the spin operator,
respectively, of the incident nucleon.
E, f are the position vector and the spin operator
of the ith nucleon in the nucleus.
X is the total number of the nucleons in the target.
V (R, 5’33, if ) is the interaction cperatcr between

th nucleon in the target.

the incident nucleon and i
Let En have the iegenfunctions (<h, (A , -- he") with
corresponding eigenvalues (to, £19 nan"). Assume (Q, to)
represent the ground state eigenfunction and energy, which we

shall take as zero. Let 1‘0 be the wave number of the incident
-6-

-7-
particle when it is far away from the target. Win the mass

of the nucleon.

a???”

no Q8“)% UV é“ (a, '"fi 3 $1, 5'0)

a?”
E WA- “.1 4)“ (R, W}: 3 S,--- S»)

 

(2.1)
:02

where m: a .2.“— +En and (sin-an) denote the spins of the
nucleons in the target; U,,is the spin wave function of

incident nucleons with projection 1"

For the scattering of a nucleon of energy E by a
ground state nucleus,

(so + v) 42 mg) -_- E‘Mi’d) (2.2)
with the boundary conditions of incoming plane wave and out-
going spherical wave (g represents the variables of the

target). We can rewrite (2.2) as:

-.

-' Ke'?

+(?o$)= 1:13—94. My‘h(§)+ E1fl+:ev\{>£af)

 

(2.3)

corresponding to outgoing spherical wave.

The asymptotic form is

4?? -‘ K V" :K‘r
e' e' gsne
+£PI§) .3 (E94 4:1“) + 50—? Q“ U + SAC“ \" 41“”) “v

 

(2.4)

where kn2 = k02 - Zmnh and summation over n is to be carried

-8-
out for all possible excited states of the nucleus. The

coefficients 8 and Sn respectively, are called the elastic

o,
and inelastic scattering amplitudes and are to be calculated
from (2.3) and (2.4).

One gets in the center-of-mass system of the nucleon-

nucleus,
2
sn . -(211) gaié 6%) (n, kn) le°,o) (2.5)
1

1:0 and kn are the wave numbers of the free particles before
and after the collision. (6) represents the ground state of
the target nucleus. Since we are concerned with closed shell
nuclei, the ground states have the total angular momentum
J I O and isotopic spin T a O. n is a set of radial quantum
number, J, T, and projection of J,)a, for the excited state.
Since we will be concerned with a specific final state one
at a time, only )Awill be kept in writing Sn as 89A).

89*) is an operator in the spin space of the beam and
hence can be expanded as

39.) - 8°91.) x 1 4» 3‘3“»); 4» s’gufl'y + 82y"; (2.7)
where 1, Vi, Cy'and. {2 are the unit and Pauli matrices. The
cross section for the nucleon with any spin state after the
scattering of an initially unpclarized nucleon against the
target nucleus is

Ar 1'
1?: "" 1’ §§.§|<W‘ “M W).

A 1
= Z; Z? \ $"(,»)\
" r (2.8)
The polarization of an initially unpolarized particle
after the scattering is

. 21.1: St W SH]
r p I (It 3%.”, SW]

2 E I We (3°79 3’90) + 1.15%») SW»)
£1154») s m}

(2.9)
where He and Im.mean the real and imaginary parts, respectively.

The scattering plane is taken to be the xz plane.

thical Potential; Perturbation Theogz
Our Job is now the calculation of the T matrix as de-

fined in (2.6). Because the interaction potential V is
not well known and whatever is known at present about V has
a complicated functional form, it is not possible to calcu-
late T completely. The approach taken in this paper was
developed in Reference 1.

The interaction potential V is split into two parts;
one that is responsible for the elastic scattering and the
other for the inelastic scattering. .As the elastic scattering

is generally'much.more pronounced than the inelastic

scattering, the part responsible for the inelastic scattering
is treated as a perturbation.

-10-
Let P and Q‘be the projection operators to the ground

state and the excited states respectively.

 

 

P + Q - 1 (2.10)
Operating with P on (2.6) from the left,

PT - PV + we (M)!!! (2.11)
where G I 1
Likewise with Q.

or - cv + eve (Pi-Q)!“ (2.12)
From (2.11) and (2.12), one deduces

PT - PV (1+6 1-3m QV) (1+GPT) (2.13)
Define To I PT, (2.14)

u - rv 4- ms 141% ov (2.15)

(2.13) can be rewritten as
TO I U +UGTo . (2.16)

Since we start out with the nuclei in the ground
states, To is that part of the T matrix which is responsible
for the elastic scattering. Then 0 is that part of the
interaction potential responsible for the elastic scattering.
He call 0 the optical potential. 0 in general is, as de-
fined by (2.15), neither local ncr hermitian.

Let V I U + 6. Then T can be written in To and 9
which is assumed to be small,

'1 - so + (1+TOG) 6 (I‘D-GT)

To + (1+TOG) G (1+GTO). (2.17)

The matrix element of T between the initial state \kgd7' and

-11...
the final state \k, n) is
(x, nl'rlxo, c» a <K, nl (ld-TOG)B (1+GOTH KO, 0) (2.18)
(1+GTO) Inc, 0) is with (2.16)
(1+oro)|x°,0)- no.0) + Momma-venom.--) no.0) (2.19)
That (2.19) is the solution of the equation

9*
(so 4» U))(: {m - g. x“. 1,. (i) (2.20)

with the incident plane wave and outgoing spherical wave can

be seen by rewriting (2.20) as

+ Eo-H.+:€- uxkofi

 

 

in)"
12.7 59.?
.. + Q U
W (U+ U6 + ) (2.19%
(2.21)

To simplify (2.20), we assume that U is local. Multiplying
from the left side of (2.20) by (Hg) and integrating over
the variables of ;

(_21‘:V1 + U°.(r)) X13: ,_ E. n?)
(2.22)
where U " 84¢“) U (”0“
co 5 . I

Likewise one can find (1+To G)... I Ln) by solving
.. ‘ v) _ t)
(3,, Y1 1’ UN) ) Xx E“x¢
u“ - X 4; (0 own“); (2.23)

 

-12-

2
where Ek Ifin 4- gm— andT means 'adjoint". The reason we

have Uml in place of U00 is because if we have our \n) as
the initial state, then the elastic scattering will leave
the target nucleus at ln). Then P and Q of (2.10) will be
the projection operators on the state (a; and all other
states respectively. we approximate Uml by U00. As long as
our projectile does not disturb the nucleus too much, this
certainly will be reasonable. In practice, U00 is found from
the analysis of the elastic nucleon-nucleus scattering and

is in general complex and has a central as well as spin-

orbit part. The parameters for the U are listed in Chapter

00

V. Hereafter, U 0 will be referred to as U.

o
Combining (2.23), (2.20) and (2.18), we get

<Km|TlK.,o7 --<<L,x£’l 0| xi", 4))

(2.24)

=<£’x5)\ V ‘ X3; 450)

(2.25) is possible as (4“\U)¢¢)I 0.

(2.25)

In lse A roximation

 

The objective of this section is to approximate V
with nucleon-nucleon scattering amplitudes. To reduce the
complexity of (2.25), consider a term, suppressing the spins

and using f in place of n,

<4,- E’\ v one) X3? 4).)

-13-

: n-
)‘EUIFJ XS) (1') vote) Xfi’ 44:33) 473ml?"
(2.26)

denotes the coordinates ofn-l nucleons.

Using the Fourier transforms

X: 38‘3”») Keck.) 35?"? A}?

etc. (2027)
and also changing the variables
2

integration over H yields
'k ... , * lie-E1" .F' 6) 4. 1.4"")
“-L ((4834.42))(180Wine-t ) WWW-“I‘M“ r' 3‘”

*‘ ._. 4 ‘9'
2 S ¢1*(),K’E*E“) X8“) (K!) V\\<u‘lx?0‘" *0“) K“) “J dad“ J’-

_., .-, (2.28)
But («A v \ h) = V; (1‘, i“ P“, P" 1)
where 7:1- (Sufi, (33" ‘2' 1 Ta (2.29)
Experiences in two-body scattering show that V is insensi-
tive to the energy. So we assume that V I V (q'). Further-
more, assume

V (q') - V (q) (2.30)
where q = )‘i-‘fo‘ , the momentum change of the incident particle.

-14-
Substituting (2.30) into (2.28) and using the inverse

Fourier transform of (2.27), (2.25) is written now

<f.)<\ leno) = (.1058 4:03) fitmé [we 3:72)]

:x:r QQ‘QLE3) (A; (AFQ (d?!

(2.31)
Approximate the Fourier transform of‘V as
2“ 1a

(2.32)

where t1(q) is the scattering amplitude of two free nucleons
as listed in Reference 1.

Charge independence, invariance against time reversal
and space inversion, and finally rotational invariance re-
strict the general form of two-body scattering to the
following expression (15)

t I A + Brn'crm + own-H111) + Er-q + INS-0;;

 

q q
(2033)
where (€L‘£.‘$) are unit vectors as shown below,
.9
333°. 23°“. 3.51“"
\“k-ko) (1‘03 I (203) (2.34)
i
A
r
V a
\

-15-
The coefficients A's are the operators in the isotopic

spin space as shown below:

A =- 2<3A1+A0) + tel-AD) (51,- (2.35)

with corresponding expressions for B, C, E and F. '[ refers
to the incoming nucleon and t; to the nucleon in the target.

A0 and A are the coefficients for the singlet and triplet

1
states.
(2.31) can be written as

<1. '4 Tl Kw") 7‘65.“ i“; W’- Sfifi‘ag) X???)

1"
\ 11‘ the ScV—eflxs (w) {(223) 4?." an;

(2.37)
(2.37) is still an operator in the spin space of the incident
particle. ‘8‘”: are given by Gillet et a1. So the remaining
task is to calculate2X:(t),which is done in the next
chapter.

CHAPTER III
WEB.APPROXIMATION

The differential equation (2.22) is, in the center of

the mass of the nucleon-nucleus system,

.-:1: "'* Q9 3 Q,
[ 1' U<(r) '1’ Ustr)0'~L] KC?) E°xK¢ 1?)

(3-1)

with the incident plane wave of the wave number ko and the
outgoing spherical wave. U0 and Us are the scalar functions
of Wood-Saxon type which vary very slowly and have short
m8”. In practice U0 >? Us. 3”: is a spin-orbit opera-
tor acting on the incoming particle. (’13 the reduced.mass
and 1% I c I 1.
Using Green's function, one can write
55°? 4K. P-F'
XS " €79); " 2‘31; 8%) l [ U‘CF)+U3(H)?':1XS:G’)a-rw
(3.2)

where U», the spin wave function of the beam, is suppressed.

Vith.thauley and.Brown (12) and Glauber (13), we write

:Z-?
X "” - ' (1') C
k° )( (2"54 (3.3)

X'(r) is an operator in the spin space and has to be deter-

mined by (3.2) and (3.3). Assume that the potential is very
-16-

-17-
msooth and varies very little within a distance d and the
incident energy is sufficiently high so that kod )) 1. Then
it is reasonable to expect that x' (‘1‘)-U varies very little
in d. we shall drop the terms of the order (53:3).

From (3.3) and ($2), one gets
: ew- ) . ,-
X'(?) '3 I " S e " r[Uc('J) +U U‘(yb)r? e-z .F'X; (?) 91?,

 

41‘ [r 4"]
(3.4)
One gets from (3.4)
eKo'IRWI Ea, (r1?
0 -'n
X‘V)‘ ‘ - _%S_e___ lr-V'I OUeCr'W‘F'M'

 

.‘KJFZ-W) __, a
__ 42’ e ‘o. KO. (r-v9 . VI ' A
4,. ”,4," e. Uswfifi «02(th

(3.5)
Changing the variable 35- - 19.2" in the right-hand side of
(3.5) and using di’I r2 dr 01d}
21f . c
- ( «.m- , ..
X4” 2! - it 8 g g e ’1) uc((?-1="|)X_ (. 4"") "”1448"

411 r)!
O -I 0

 

” 2“ .W.
‘i LJ’I a-fin . _ I]; '-II II; I'
4.“ S S g§7_ US (|r~r 0717!?) KX‘PP)Y' Jr 414?
o -n o

(3.6)
Consider the second term of (3.6). Integration by

-18-
parts in )l yields

f8 Jail-h, x(' ‘9”) Ar" 1-{1R'.Sew U cufl- 4-”9X ””11
° = --t

«3(- ,
+§2°Sc r ! n)-§L;((ch)°)14V"

(3.?)
Because of the rapidly varying phase, the second and third
terms are (HE-:3) compared to the first. (13) Hence they are
dropped.

One could apply the same argument to the second term
of (3.6). One gets

X1?) = ‘ + 7&8 WHEN-W) KHZ-WNW]
i=1

+ if: 8 USU-P4”) XXV-V”) r. (F'J-"QXK dr']
0 ’11"
(3.8)
Now taking our coordinate system such that k0 is

along the z direction, TI cosy is measured from the z

direction.

X'Um = . + i—S-g u‘(k,z-z~))('(?,z-e") ole"
+ LE- 8 Us ()3, 3‘2” 6’. (PX?) XXI.) E‘t”)0)2"

° (3-9)

-19-
where b .J x2472. Hereafter, whenever convenient, we shall
use the cylindrical coordinate system as defined by r’ - (x,y,z)
- (VJ). If the particle is incident along 2, then b is the

impact parameter.

2
X = ' "if-8 [ 1145,29 + (1,“,291afx'xilx (1,2948

-v0

(3.10)

One can rewrite (3.10) as

e _. .
o -:‘_E S [ uctm') + 11,1529 r3375] alt
X (L2) = 6 Kb -70

 

(3.11)
Combining (3.11) and (3.3). one finds the solution to (3.1)
8
7...; _, :5. S [ uccu') + u,c1,zua’.'¢x'l??l de'
Xt") : Q. ° —N
k. £2,197;
(3.12)
To calculate xx“), one must solve
2. {-1 t)
[ ‘11: + Life) + u:<r>?o?]X. 35X:
(3.13)

with the incoming plane wave of the wave number k and the

incoming spherical wave. The procedure is exactly the some

as before except this time il- -1 term survives.

-20..
3E5, + ii. a.
At” ._. .3. K 8 [ “5 (5,29 + uf(s,z')?.t’x?14%'
@1054- '3

(3.1a)
The scattering matrix (2.5) is now, with (2.37), (3.12) and
(3.14):

so» ”(131%) 81531;.) [:14 pupae-29]

{n
K,“ W £3,911,112.“

(3.15)

CHAPTER IV
FORM FACTORS

Nuclear States
The nuclear wave functions (bf and O. that appear in
(3.15) will be described in the language of the second
quantization. Let a1 and a be the creation and annihilation
operators of the fermions, i.e. the nucleons. a is the ad-

Joint of a) .

aJatd'atat-s s,t

a8 at + at a8 = O

a; a; + a; a; =- 0 ((4.1)
Let lo) be the "vacuum“ state (closed shell state). Then

33,10) represents a state of the closed shell plus a

particle with the angularwmomentum J and its projection.m.
(-1)J+naJ-mW57 is a closed shell state minus a particle
with the angular momentum J and the magnetic quantum -m.
The effect of this missing particle on the shell is that as
a whole the shell now has quantum numbers (3, m), which we
call the "hole” state. The phase (_1)1flm is necessary to
give the hole state the same rotational transformation
property as the particle state. (16, 17)

We confine ourselves to nuclear reactions in which a
particle makes the transition from a closed shell state to
a higher state. The particle-hole states were used as the

-21-

-22-

basis to calculate the nuclear wave function by the pertur-
bation technique, the perturbation being the interaction
between the particle and the hole. Then the nuclear state
is a linear combination of the particle-hole states.

‘A particle-hole state q“ with a total angular momentum
J, its projection )& and.the total isotopic spin T is given
below. (5)

2 .Al ’ 1‘ R” S» 3“
‘9‘ £1,1uL5). . ,0. M.
" Lawn: I!” 5" 3n 's’ H" ‘1'“)
L S T

I I

. 1 0‘,
* I. <1» 1.1,>\u,>\nlL. m H) ” (1" . ,
N)“ I , xul'xu; “V QAUI'A”, “J ‘0)

1 \
, ‘ sum,
* i. “'0. 5», mu, m" ‘3,“‘2 )k'" 0t: arm. \0)
mm; “W” "’

13+ ;
x ' ' -\ 1.
CE 1.: 1W, [v.52 \ T, o) U atfiitfl‘ottéfl'; \ o)

(11.2)
where primed and unprimed variables refer to the hole and the
particle respectively and f. =- {3171 '5". ‘QN’ “N and I N are
the total angular momentum, the orbital angular’momentum, the
spin and the isotopic spun of the particle respectively.
nu is the radial quantum number. The corresponding primed

quantities have the similar*meanings for the hole.

-23-

L and 141 are the sum of the orbital angular momenta
and its projection of the particle-hole state, while 8 and
H2 refer to the total spin and its projection. The quantity{ }
is Wigner's ‘(j symbol and <\ ? is the Olebsh-Gordan co-
efficient. (18)

The nuclear wave functions ()0 and 4“ are given as
in - to?
Q‘ = E. (1‘90" Cf"

(11.3)
For the meaning of x and I, see the last paragraph of this
Chapters

*-
The factor Xi) E’t‘ X: is to be replaced by

EX K SEXE’*t X13: £4,415 a: (1,4)

4,4!

(Mb)
where t is any one of ti's and in, and Q”, are the eigen-
functions of a single particle in the harmonic oscillator
well.

The scattering matrix (3.15) now becomes

(9* Q1
3 (1*) = 21%. Q???) 2;,me ail (M ,3.) Pka ’ch. L“)

t
(1,, 01,, \o)\ I.) (11.5)

where [to and {1}) are the initial and final isotopic spin
states of the beam particle.

-2h-

Isotopic Spin Dependence
The two-body scattering amplitude has the most general

expression in the isotopic spin space,

t :- ta + t' {-11 (M6)
1', I? t8 and t' are defined in (2.35). As we exclude the
charge exchange scattering from our considerations and our
targets have T . a before the scattering, the projection of
the isospin after the scattering again will be zero. Sur-
viving terms will either be t8 or tv depending on the final
states of the nucleus.

When T a O, the operator t becomes {éts'

When i - 1, ~ - t - J'étv. (11.?)

Form Factors
Carrying out the algebra involving the creation and
annihilation operators, (#.5) becomes

’9" " 2E ( ... gums“.

(“'08)
where Q
{— :6,» Lg $5.53» L<LSHM|T)
Z ‘ ,, 'I I
L5 lN’uL' ‘9'.) S”! 3N "1,"; 1 F
L. S. T
r x‘ s‘+n~‘

'55:) a» 1.. Nam. «02: “g...” "<s.,s.‘,m..,m.:\s,n.>

*
* Ln...2~,>\..l<s.,m.1x§’ J: x3: \s:,-m.:> mix, a)

(4.9)

where ‘1‘", Luke» and (“'2’“) 9"”) are now the eigen-
functions in the harmonic oscillator well.

Final state S 8 0. L 8 J
with (2.33), one may write

3.) u

A A .L . 1 i
T» = l j . L " 1. J" (,1)‘. ' '
E NJ.‘ )1 a 1i Nfi“ Z. (IMO My 0‘ TV»)? tr") (‘0’ (a, AN,“ I b)"‘)
o I

L:

'<n.,z.,,\..\ X§’*[A * cr1x:\n;, may.)

4 a) A k“ 12? 3,, (£1.10)
= 2.)“ J“ Li ‘ J_ ‘ ‘ &+1\
" 2— M Z' ’49) (10,23).”)..‘(lbp/IA)
I 0 I M’
, at (u
i CV" .11 )CKr
8K '1'» 9N:Q’Q)XS [A 31‘“ Rn" ’39» A»?
(n.11)
where R's are the radial eigenfunctions.
Raking use of the identities,
M it
“,3,“ a EU {5"} MC“ (u 12)
11‘
Ye;
21:: 4T: ((99. ’M"W| L ’M><’e""‘ °) °'L' °>\‘:M= 1m. «tum.
(ll.l3)

and the tonal properties of the Clebsh-Gordan coefficients,

one can deduce from (lull)

a , A, .
T; = 2 ~' 3‘ i”! 2” J2: J“ {J
W ca: 12: at: (cm 1‘); O, 9‘ L: °> (A)
I o ‘J'

. * * 9*
SKN,9.1YL7; Xk [A " “Exit: fi’NAJr-LAFA‘D‘

(n.1u)

Define the fom factor FJOJ as

1’30" " 2 EWY’N fflfflf'w‘“ I" j“ 3"
M {:5 '

1'
K (2,1,6, 0; ‘3‘ 3-,”) Kym)!» RN) 31‘
)

3.1}
I
I

[2, (1L“)(zlgi|)(z(,;+l)] >4.

(b.15)

(a
(hie, 0; 0‘19) g 9"

3'

0 (0"- N)-
‘i".

walnut-i

= so

(n.16)
is called Vigner's 3-j symbol
as given in reference 18. (£1.15) becomes

where quantity (

A .
‘1; = 21. ' £11, Q» J:5. 3.. 4”;
a? u :- 5; 0., 1.}, o,.\ L, .> W
3' o 'J'

. * * 9*
)KMWY‘V’ X“ [A * “517C: fimwfl’m

(n.1n)

Define the form factor FJOJ as

w - 2 1(me in; me» a. 1. 3..
M W ‘9» 2. J“

*-
x <2», e, o, .1 1,0) We”, RM. 2.:
I

in}
T
l

Lumuczwvueéwl’é

(n.15)

(a
(43,“, 0; 0‘1"») 9.:

I

O (~4- N"
i".

I e
1.4.. +111“):

.1 (.4)

(n.16)
is called Higner's 3-j symbol
as given in reference 18. (huls) becomes

where quantity (

-27-

OJ:- L¥+ )JWJ” 2 NJ
I"J ‘1— (‘1')“6 3%,: (2"; (ARI):

”)1” “Q ('
(n.17)
(MB) is simplified as

JoJ * "'
s(/u. s-o) .. 2({(%))’ Y1» XE) [Ai 031%: “iv-ML
(4.18)

PinalBtateS-l J-L
Algebras involving the spins of the particle-hole

 

states are more tedious than the 8 - 0 case. Writing out
explicitly the spin operators of t,
o x y 2
t-t1+to'x+to~}+trz (M19)

with t° =1 4» fly
13"!
’ - c +-s§y
t anz

where (1's are the spin operators for the particle-hole and
6"s for the incident nucleons. The coordinate system is
defined in Chapter II.
A typical calculation goes as follows:
dfim‘c 5‘ l M)($
£3“: :1, a, New. S t an M1...) “S, PM)

aw

_ N i 1
_ ML MEI-4) (SI) S" I MI. “I. ‘ SI ”3-)(suj m”‘ M‘ S" I an.)
U, u ‘b

= g, [gnu + Shy-1)

(b.20)

-28..

where i =- 41, 02* . “—ggii‘i} £3 6;; «'03::
Using (n.20) and the like, (M9) becomes

A A t g ‘Qfl S“ 3”
Tu U‘) ’5')“ "'LS‘“ \ Z, (L, 3,91» (441,»)

" fl

943.;
L- S T

(4.21)

where

0‘7 = ’t C (5mm + Shine!) + E (‘Mt4vsnudfi

+ $450,,“ + 5m») + 5F 5a,. 6;

((4.22)
Remembering the identities (14.12) and (b.13), (“.21) can be
simplified as

.__ o 4‘, A A ‘9' (N '5» Ju
IA») c4313" ("3‘1”); H) "J1 9,; 3": 3,3} l: <L,,l mm] 1'1“"
L s ‘I "'M‘

“ <1 ' o 0 a yr *
u’ 2”, 0’ “I, > X Rnfllefilfi Ct > £2 R191,“ 473’

(n.23)

,m :

am:

the a

With

-29-

Define the form factor

1: {5. 4 A A A ' e
’3 " 35‘ T. WW). 3,, 3,; $1.. 3.; W2" ‘v‘ 3» h
N 111 Se‘ 3,;
T S I

1* <Q~,£J,o,0\ L.°> Res, Km:

(4.24)
making use of the properties of 3-j and 9-j symbols listed in
the appendix of Reference 18, one can reduce (n.24) to

1: 4| ‘0 . . ‘
FJ = E (K+Y)“ Ede” J“ 3‘ )0! j“ I R“. a
N W J. J .4 11.3.,

7- a

 

1112.;

(ii-25)
with (4.23) and (11.25), (4.8) can be written
L." = , ‘=
3( 1,5 1 ») 24%. ('4)£m «Amman,»
1
‘SFI‘ Y:XS’ <t7 XS r AVAIL
(2.26)

nggl gpppe s - 1, 1 2 L31

.All the calculations are exactly the same as J - L
and 8 a 1 except that we have two values of L for a given
.7.

Define the form.factors by

Fm - fiécxwu'

-30-

1‘ {I A Al I: A
“uh: 51» 1e H) <1“, (Leno) L,» L

 

{a Si in *
* l 2' Sf; 3'1} Rmhflfl Kin“; (N.
L. S T
(n.27)
For L = J-l, one gets
)9. It)”
.J-1.1.J._. r LL) um?It2.4mm)+(ej-3;)(z,,;+u u
*" " IIUIHHII—UVQ.
)w )a: I "'
U
( 2t “'1': o) K'NJ» RMLJLI
((4.28)

For L =- J+l,

FJ+1,1,J_ J} E (X Y)“ (cum 3+1). ”111‘ 1.. j 1..“ L‘lue “”11!“th )u)(23.1|)+'l'il]
[wum (21w 1V2.

. (33:- L‘. 0‘) Kn'ihfla

I
3..

(n.29)
“Li/4’8"” I 2J1£(%) L2,“ "‘(L, 5 H1, Ha.) I M)
+9
x ) FL“ ‘11:.)(5) <t> X( “n"
(b.30)

.Although we have assumed that the ground state of the
nucleus as the Hartree-Fock ground state, in (h.3) allowances

for the correlations in the ground states were made by the

coefficients Y. Gillet et a1. (11) have allowed the

-31-

ground state of the nucleus to have the excitations of the
particle-hole pairs. Then a nucleus can be excited in two
ways: first, by creating a particle-hole pair and secondly,
by destroying a particle-hole pair that is present in the
ground state. X is the amplitude of the first mechanism and
I is the amplitude of the second mechanism. These are given
in References 11 and 12. X and I satisfy the normalization

condition
2 -
Z (x 22)" - 1

where summation is over all the particle-hole states that
contribute to our’nuclear state function. In evaluating the
form factors, the corrections to the phase of (Xe!) should
be made by multiplying (-l)JN . (19) All the relevant form
factors are evaluated and listed in.Lppendix I along with the

radial wave functions.

CHAPTER V
DETAILED CALCULATIONS

Reduction of the Scattering Amplitudes

Scatterin Am litude for S I 0

 

For the final state of the nucleus with L = J, the
total scattering amplitude will be the sum of (M18) and
(#.26).' First let us consider (hula).

with (3.12) and (3.1h)

B(L=J, 3-0.1“) ' Zflfl‘ij) 8 FM ‘1’1b flaw-KW
o \‘O

LH

-'" ?u H) "IUJH’
-58! c sr'(l‘x

3455 S2016 U‘O'JPXEHJZ'
e '3 ° "°

[A 1 C63]

(5.1)
To evaluate the exponents with the optical potential,

we use the coordinate system defined in Chapter II in which

i; is along the 2 component of vector '1'. we further make
the following approximations:

i. 2 i; “mi '22 (5.2)
The basis of the first part of (5.2) is that the deflection
of the particle after scattering is roughly approximated by
é‘m‘m‘? and t (d). The second approximation of (5.2) is
-32-

-33-
based on the fact that the energy loss of the beam is small
compared to its incident energy.

Let

I
K}
Q. ' ‘3ng u,n., g')dz‘
-o~

- b t N ‘
Q‘ "$733 0.0.20 A:

'% UCLI.)d2‘
F‘C 5’8» c 3

we
Q .3 Qt + Q =21;
2. IE, Usflei') d2'

‘9‘

(5.3)
Then (5.1) reduced to
x a"?
SHIP-T. 8'09)” ' 2E. 6%)! \(LH FJOJ e 1 Nb)
6:0‘0—‘(tm [A + ca'y] é:e.c-.C$XK.) a? (5.1;)

A A - ..a cy-e
where b, to are the unit vectors of b,'§b, and.q a k-ko.

Let

é 6'; e, . G; (,0; (Qfiaz) ‘3 «Rm 4?“ (Qfl‘Qz)
+ i- [Aa mac.) - at. (61.-62.)] [6; r...)
+§ {-cosmaa.) + CoSLQer)] a}, [431m]

(5-5)

-34-
Considering Q1, Q2, and Q1 + Q2 to be small, (5.5) becomes

4ch- ".Q°— .
e MW e ‘m=fi-:tha.ma+$[(¢*°m~(Qr°‘=’]tfi.r~]

Manama 4:71-61 Ema-m]

(5. 7)

)l Us 42‘ is dropped for convenience. (See
30 2

:ob2

 

where Q1" Q2 -

Appendix II)
With (5. ‘7), (5.1+) becomes *
3(L=J, sac, ) =- 211:) M)??? PP YIM‘W) FJwtawr’w?

4&3- (€72) é:- (gtp 17-? ['(b) I,JoJ Qct)‘fi-:.(0,Q)¢ Firm} .5?

-zlégk’zvia L 8 53'? ("5)pr Q (L) Yr; 0;“ [A + ‘63] d?

(5.8)

where 6;: r A...”- QC”?
Expanding 8(W, 8-0,”) in the spin-space, as 8(L-J,
8-0,») - 8°(L-J’, 3-0,“) 1+s‘(L-J, s=o,u).—x + syn: , s=o,,u)o-

y
+sz(L-J. M. )r
F 2 (5.9)

He get

8°(L-J'. 3.0,») . 21?; (71C) Ae-‘S 1.?[700 YT: (,0?) PJ’OJ’ d?
+28IE-(MC8631rP(H {”QCL) Him?) cos? dr
s‘u-J. 3-0.») - -2 Pr (T’gxMSé-fi F1113” 01m ‘(wimemQ dr

shy-J, s=o,») . 2E; (‘15.ch '5'? ['er TI}. (94) a?

-35-
+ 215—34 (7’5") g e-WVFCB) FJOJQ fling cosc) d?

sz(L=-J, 3-0,») =- o (5.10)
The integral over (b is carried out below:

1(- _ x~ -»
flush?) — YT,» (9,0) 6 +

17'? ‘1)”an 60¢

Using the formula (20)
‘ 8 m : 2 ca}:

73"” E, ) = 2
L o cab” 0‘» I‘M“ (5.11)

(2)

J being the Bessel function of order m, we have

M
gl‘é.7'?e1.fl4 A§ = 2W ‘2‘)“ 1")“ (1"431‘9)
° (5.12)

a.“ . . -.
) 945’? 6‘») “44¢ = W (*1 w NJMI‘M'MMH iw‘Iya-Jr‘m”)

0

S1“ .39.? -'»§ (5013)
-. ‘ Tn d. a g “in“ " I .:. ' I“)
o A ) Q “1"“ Jun?” n0) ‘J thflmw]

(5.11:)
Let us look at the integrand of (5.8). Pi” is Q'drz
multiplied by a polynomial in r of even or odd parity as
given in Appendix I. We shall approximate P and Q by the

functions of the following type. (See next section)

2 2
(5.15)
2 2
Q“) " 6°33" “MC“ 4' 05b3) -31) +(DFb + Dsb3)e'sb]
(5.16)

Then all the integral can be carried out analytically. The
integral of (5.8) along Z is the following type: (21)

 

N
-at‘ . .-« (an-0
2“€. =‘3 11
g o\£- 2"“ a.“ E
o

(5.17)
A typical integral in (5.8) along b is given below: (22)

9° 1. -|
0

I.

1! £5 1.-1— ” -. 1
“1* 1X" (”'9’”) e. W (F. [é'éfl 3m); .11.

 

)

2v Ldfp)»; PUT”) 4 (0H?)
(5.18)
where
(F. (“In 2) = 9i (3),. i“
“=0 7". (f)n
m°.i Q‘Lfi AWOL) "" UL+n-n
(5331 (f), =§U+§0 (gym)

(5-19)

-37-
and. 'F(1+m) = 10
,F. is called the generalized hypergeometric function and in
our case always a polynomial of the finite degrees.

Scattering Amplitude for S =- 1

Following exactly the same procedure as before, we

have, using 0;,- (0‘“ sin+- dices“,

 

”za‘f‘“ -{ Qtrm

3. c; e. = [6“ ~§Q4InQJ
-:Qtrh ‘:Q\o'm
e “S ’— = L ‘3 t WW4]
~JQea-M JAG"
rt 2. m -= 6‘E

(5.20)
substituting (5.20) into (4.26), we have after integrating

over Q ,
8°(L.S=1.}\) ° kgk?) 31c ”FHKIHE" + JM\)<L"|’H‘IH*‘T1)‘)M£'H“

8111‘”)mewvgmvct)ear-Am» + M QMEJM‘me‘bfi'mm'TI/O

1: gYh*nI(o’O) F”: [,(b) Q (5) [LNMH‘ S\Hd§($b) ~AU1H) I|HH)(1B)1 Hardin 0‘9

- M )1. “am + Sum) <t,l,n..mn.,»)("nattoF‘nrcm(L)
3

"‘1"1

[;‘“'“‘IW\-i\») 1A‘H“‘] (1)91 ruminants

(«.41

(5.21)

-38-

s‘(L,s-1, L . ”If -
,0 LE (7) -Z'WEFEIM‘( Jm.“c)nu,)<LJ,N1.M:IT,/A) i'H“

LA]

1-
. g YHHSOP) F P(b)]'|m(-tt)r"o\r-Ain9cle

+ 2c“ 2; (5&7. + 5,4,”) <u.|,n.,M.| 3-,») S FMIY * (9 0) [‘(b)
b,m ’

QM.) [1 W‘NJ—unmu)’%‘° " ’1 "W I

[New] r‘amma alt)

(5.22)

8y(L,8-1, —‘- . 22f - '
, )8) 2 ($3) )r ) " z)“ B Z. “Muir Sm») (LI‘IH‘IM‘UM’Q

0"

. \J
)F“ 11.: (0,0) [’(uT. H.101”) Mr- anode

Nil

4: o.
gym... + 5'40.) W) H» "41») SF“ This»)

(1961(5) """"‘ ‘
P [1. I‘Mfié‘lb TA‘NPfi'I‘H‘lF1DJrLAr’AInDJD

(5.23)

82(L,8‘19 )‘L O * a ' ‘
J“ ml); (‘14!) 5F 2'“ ($5M 5m,» “WWW!
, L\]' 1‘
SF YM‘M’ ['(U Ilmkib) r‘olr' 4;.er

A11 1: (5.2“)
he integrals involved in (5.21) --- (5.21)) may be carried

-39-
out with (5.17) and (5.18).
The total scattering amplitude for a normal parity
(L I‘J) state with any spin (0 and 1) is given by the swm
of 8(L IJ, 8 = 0,}5) and 8(L =J', S = 1,»).

8(L-J,})) - 8°(La1',)k) -'1+s‘(L=J,p)a;+sy(L-=J,1)ry+sz(WW0;

(5.25)
where
s°(L-J,),) s°(L=J, 3:0,») 4- s°(L=o, saw)
sflwvy) =- sI(L=J, 3:0,}1) + sI(L-J, 3:1,»)
SUD-.13») . slaw, 3-0,») + ‘Sy(L=J, 3-1,“)
82(L=J,p) - sz(L=J, 8:1,») (5.26)
The total scattering amplitude for the abnormal parity

states (JfiL, 8:1) is the sum of amplitudes for J = L-l and
J ; le. One may expand this scattering amplitude as in
(5-25):

3°91) - s° (Law-1, 3:1,») + S°(L=J+l, 3:1,”)

519,) - s: (LBJ-1, s=1, + arm-0+1, 3:21,,»

an») - 3! (150-1, 3-1,») + s’(L=-J+1, s=1,,.)

389“) - sz (Lea-.1, 3-1, ) + sz(L-J+1, 3-1,”) (5.27)
where the terms on the right-hand sides are exactly the same
as the corresponding terms (5.21) --- (5.2b) except that

L i J and the form factors are different.

numerical Calculations
The optical potentials used in this investigation are
of the form, (10)

-80-

U") "’6‘” — V010)" -£ (w- 41w, $3 (n e") “‘

+ Ft'c‘ (WMAWI) f. isrLH-e‘)" E]:

&

(5.28)

where x - (r-r°A&)/a, x' - (r-ro'AK)/a'.
we shall ignore V§(r), the Coulomb interactions, and
-V(ex+1)'1, the real part of the central interaction. The
effect of the Coulomb scattering is significant only at the
forward angles in the elastic scattering. Furthermore, in the
inelastic scattering, the momentum transfer is never zero and
hence the effect of Coulomb interaction is still smaller where
the inelastic scattering is important, i.e. at the angles
210°. The real central part contributes to the phase of
the distorted wave. As we will be squaring the amplitudes
for any Observable quantity, this is not going to be important
at the high energy. This was borne out in References 8 and 23.

The relevant parameters for the optical potentials at
the incident energy of 182 uev are given below for 012 and
Cauo. The parameters for O16 are assumed to be the same as
those of C12 except the atomic number.A. H is the mass of

W meson. (10)
TABLE 1.--Optica1 Parameters

re a W wD re a' wR wI

 

012 .902f. 0.852 15.6 0 1.186 0.556 8.12 - .10
ce“° 1.012 0.588 16.8 0 1.368 0.582 8.06 -2.0

-81-
The unit for the length is fermi and the energy is in MeV.
From (5.3), (5.15) and (5.16), we have

’0
__(_<_e S 1)", A1‘
5 «1‘- - *-
F(\»)=€ Q?” ”Q “ =1- can on We. - (cutoub‘ks

(5.29)

618.) - :Jé(m)(wnnw1)g +§~.(.+e‘)" 12‘

3 FER. kanwflf (CF L 1- 85 P) 3‘ + (DFH 059) is}, I

(5.30)
The integration along 2' was carried out numerically
with Simpson's rule. (A typical program for the numerical
integration is listed in Appendix III. The results were
fitted to the function on the right-hand.side of (5.29).
Take ]‘(bn) for N different points of b. Then adjust the
parameters CH, CN, A, B, DH and DN such that

N _ L . ‘
0'" Z. [ ['(bn) " If (.CH‘i- 00998 Ab+(DH‘D”b‘)Q 3512.
“‘3'

becomes very small. This was done again with the computer
and a typical program is listed in Appendix IV.
The exact same procedure was repeated for Q(b). In

80

the case of Ca , the curve for Q(b) was fitted by three

different sets of the parameters on the right-hand side of

-82-
(5.30). The quality of each fitting is measured by 6' . As
we shall see, each fitting gives different polarizations for
0e40, indicating that perhaps the polarization depends sensi-
tively on the shape of the Q(b) (i.e., spin-orbit potential).
Final values for the parameters in (5.29) and (5.30) are

given below.

TABLE 2.--Central Absorption

 

 

r (b2)

 

Ihten- Nb.of
val pt». c— CH CN UM DN A. B

 

012 0~8 80 .17110'3 1.613 -.086 -l.078 -.127 .17 .39
JP

016 . ~ .86x10 3.158 -.035 -2.586 -.269 .2 .35

ca"0 o~10 50 .33::10"'2 38.89 -.517 -33.75 -.911 .0991 .183

 

TABLE 3.--Spin-Orbit Integration

 

 

Q(b)

 

lunch Nb.of
val Ins. CF CS DP D8 R s

 

(:12 0~7 35 .76n0 .783 «1006 ..315 .520 .3 .39
015 - ~ .33x10"2 -3887 -11.58 389.2 .568 .5 .85
on“ v - .3910”? .152.5 6.89 152.8 7.02 .29 .38
I ~ - .79x10 .1868 29.01 1869 27.11 .29 .32
n - - .20x10'1 -9117 23.33 978.9 30.69 .33 .39

-83-

Using the two-body scattering amplitudes given in
References l and 35, we now are ready to evaluate the in-
elastic scattering amplitudes. All the integrations in Z
and b can be carried out analytically with the aid of (5.17)
and (5.18). In Appendix V, the outline of the actual calcu-

lations of the scattering amplitude for O16 1', T 8 1 state
is followed by the computer program.

CHAPTEB VI

DISCUSSION

C12

8.8'Mev

Figure 1 illustrates the comparison between the experi-
ments at Orsay (7) and Uppsala (8), and the theoretical
predictions for 2+’T = 0 state at 8.8 mev. The theoreti-
cal prediction reproduces in both shape and magnitude the
data obtained by the Uppsala group. The agreement between
theory and the Orsay experiment is equally good in shape,
although not as good in magnitude. Also shown in Figure l
is the result of the DHIA calculation. (19) The WKB and
DHIA show good agreement.

The polarization predictions obtained by theory con-
firm the Uppsala results (27) and also the Orsay data (36)
at large angles. (Figure 2) The disorepancy between
theoretical prediction and the Orsay results at the small
angles is serious as the addition of spin-orbit interaction
makes the situation worse. Again, agreement between the

WKB and DHIA is good.

202 “av

Figure 3 shows the comparison between the cross sec-
tions measured by the Orsay (7) and.Uppsala (8) groups and
-88-

-85-

the theoretical calculations for 3‘, T s 0 state at 12.8 MeV.
The'HKB calculation shows good agreement with the Orsay
result, but predicts larger cross sections than were ob-
tained in the Uppsala experiment. Agreement between the WKB
and DUIA.results are reasonable although DWIA favors the
Uppsala data.

The theoretical prediction of the polarizations matches
the result of BET (27) as shown in Figure 8. .Although.not
shown in Figure 8, the DWIA calculation (19) gives practically

the same polarizations as the “KB.

12.2 Rev
Figure 5 illustrates the comparison between the theoreti-

cal cross-sections (1+, T I O at 18»MeV) and.the experimental
results. Although the angular dependence shows reasonable
agreement, the experimental cross-sections are too small by
a factor of 10. In Figure 6, the theoretical prediction of

the polarizations is given.

15.2 MeV
The angular distribution predicted for the state 1+,

T = l at 16.6 MeV is in good agreement with the experiment,
but the magnitude of the cross-sections is too large.

(Figure 7) Electron scattering and B decay agree with the
form factor given by the intermediate coupling. This gives

a transition strength equal to one-third of the Gillet vector
and would give fair agreement with the curve here. The

polarization is too large when spin-orbit is taken into

-86-
account but too small when it is ignored. Because of the
paucity of the data and the large associated errors, we can
only claim the consistency between the theory and the experi-
ment. (Figure 8)

16.1 MeV

The theoretical curve of the cross-sections for the
final state 2*, T - l at 16.3 MeV’matches the Orsay result
(7) both in angular dependence and in magnitude, and is con-
sistent with the Uppsala data (8) as shown in Figure 9.
Also, the polarization curve is consistent with the Orsay
measurements (6). (Figure 10) This conclusion is in dis-
agreement with.Reference 28 which stated that the observed
polarization was a mixture of 2", r =- 1 and 2*, T = 0 at
16.? Rev. The reason for such a disagreement is that the
spin-orbit potential was ignored in the calculations con-
tained in Reference 28.

18.2'Mev
Figure 11 gives the theoretical predictions of the
cross-sections for 2', T a 0 at 16 heV’and the experimental
ponuts. Two sets of the data do not seem to agree with
each other at the small angles. The agreement between the
theoretical result and the Uppsala experiment is very good.
The polarization measurements are crude and the number
of points measured few (Figure 12). Our calculation with the
spin-orbit potential is consistent with the experiment.

-87-

12.3 MeV

Figures 13 and 18 show the theoretical calculations
of the cross-sections and polarizations for 2‘, T = l at
19.3 Kev along with the experimental points. We note that
the change in polarization due to the L-S term is signifi-
cant. We can claim only the consistency of the theory with
the experiment.

21.0 Nev

Figure 15 compares the Orsay data with the theoreti-
cal prediction for the state of 3', T = l at 23.5 MeV. Two
results agree with each other in shape but not as well in
magnitude. The polarization prediction is given in Figure 16.

variations of the polarization due to L°S are large.

22.3_hev

The theoretical level of l‘, T = l at 21.5 nev is
compared with the experimental results of the Orsay group
in Figure 17. The agreement between the two results is very
good. In Figure 18, the polarization prediction is shown
along with the Orsay data. The Born approximation is con-
sistent with the experiments but the introduction of the L-S
term changes the curve a great deal.

Figures 19 and 20 represent the theoretical predictions
of the cross-sections and polarization for the final state of
2', T a 0 at 21.2 Mev. We note that the L-S term changes the
polarization curve extensively. The cross-sections and

polarizations computed for the state 3', T a 0 at 19.5 Mev

-88-
are given in Figures 21 and 22. The change in polarizations
due to the L°S term is the ”standard" type.
Figure 23 shows the calculations for 2', T = 1 at
23.2 Kev. In Figures 28 and 25, the predictions for the
cross-sections and polarizations are given for the state 2*,
T = 0 at 19.2 MeV while Figure 26 shows the theoretical

cross-sections of 1‘, T = l at 28.2 Nov.
016

6.15 MeV

The observed cross-section is larger than the theoreti-

cal prediction for 3‘, T I 0 at 6.25 MeV by 0.6r».1.0 mb.

(See Figure 27) Since the experiment did not resolve 0+
level at 6.05 MeV, the discrepancy is certainly reasonable.
The general shape and angular dependence of the predictions
are in good agreement with the experiment. (Also, our calcu-
lations compare favorably with the DWIA results. The po-
larization with L-s is shown. There is reasonable agreement

between the theory and the experiment. (Figure 28)

11.5 Mev

Figure 29 compares the Uppsala experiment with the
theoretical prediction for 2', T a 0 at 10.5 MeV. nAlthough
the theory predicts the cross-sections about 50$ smaller
than the observed values, the angular distribution matches
the data. In view of the fact that T. Erickson gets slightly
larger cross-sections by using different form factors, (5)

we may obtain better agreement by using a better form factor.

-89-
Polarization prediction is given in Figure 30.

13.1 Mev

The prediction of the cross-sections for 2‘, T = 1 at
13.0 Kev and the experimental data show good agreement.
(Figure 31) The effect of the L-S potential on the polariza-
tion is shown in Figure 32. No data are available for com-

parison.

15s: HeV
The Uppsala results are compared with the level 2‘,

T s 0 at 18.6 MeV in Figure 33. The theoretical curve is
too low, but the angular distribution looks good. The pre-

dicted polarizations are small as seen in Figure 3“.

18oz ”av
The experimental data and the prediction of the cross-

sections for the state 3', T = 0 at 15.1 wev are given in
Figure 35. Agreement between them is good in shape, but

not in magnitude. ‘Also shown is the result of the DWIA
calculations. The WXB result gives a factor of 3 larger

than that of DWIA. The polarization curve is practically the

same as Figure 28.

20.2 HOV

Figure 36 compares the measurements of the Uppsala
group with the prediction for 2', T a 1 at 19.1 Nev. Two
results agree with each other in angular distributions in-
cluding the plateau at 25°. The height of the predicted

 

WW“

-50-
curve is again too low. The polarization prediction is given
in Figure 37.

Figures 38 through 88 show the cross-sections and
polarizations for various final states. No data for these
states are available to us. The results for 3", T = 1 states
are not shown as the maximum cross-sections are around 0.2 mb

at the angle of 25° for both states at 12.7 MeV and 28.1 Mev.

Cal"O

 

2'2 HOV
Figure 89 shows the results of the experiments (9, 30)

and the theoretical predictions for 3' at 3.88 hev. Curve I
is calculated on the assumption that the excited state is
purely T I 0 while Curve II takes into account both the

T I 0 and T I 1 states. Two sets of experimental points do
not seem to agree with each other. Roos's results show the
peak at a smaller angle than the results of the Harwell
group. Our Curve I agrees reasonably well with the Harwell
results and the DHIA calculations. However, Curve II over-
estimates the magnitudes of the cross sections. This is
probably due to the fact that the addition of T = 1 changes
the form factors.

We have varied the imaginary part of the spin-orbit
potential in the calculation of the polarizations. The re-
sults are given in Figure 50. We conclude that the imaginary
part of the spin-orbit potential is not necessary in our opti-
cal potential as it does not affect the cross-section either.

In Figure 51, as we improve the fitting to Q(b) (see Chapter v

-51-
for the definition (“Mend Q(b)), the wild interference
pattern disappears. Finally we get the curve in Figure 50
if we include 8 I 1 to our wave functions. Such interference
was also noted in Reference 19. So we note that the polari-
zations are very sensitive to the changes in the shape of

the spin-orbit potential.

8.8 Rev

The experimental cross-sections are lower than the
theoretical calculations for 5' at 8.38 MeV. (Figure 52)
The angular dependence of the curve seems to be consistent
with the experiment. In Figure 53, polarization predictions
are illustrated.

Figures 58-59 show other excited states of 3’ and

5'. No data are available to us for comparison.

CHAPTEB VII
THE WKB METHOD AT 80 MEV

We wish to apply the WKB method to a few nuclear re-
actions with the incident particle energy of 80 Nov. At this
energy, we do not expect any good quantitative answer, as
the validity of the WKB requires the incoming particle energy
to be above 100 MeV. However, if we could reproduce the
rough features of the experiments, the WKB method should
help us gain insights into the nuclear reaction mechanism at
this energy.

We shall make the drastic approximations. The spin-
orbit part of the cptical potential will be ignored as will
be the Coulomb part. However, unlike the situation at
156 MeV, the real central part of the optical potential will
be retained. The two-nucleon potential will be assumed to be
independent of the spins and isotopic spins, which amounts to
making B, C, E, and F of the two nucleon scattering ampli-
tudes zero in the limit of the impulse approximations. Hence
the final states have zero spin and zero isotopic spin.
Furthermore, the two-nucleon interaction potential will be
assumed to be of the Yukawa type. We shall examine the final
states 2*, r a 0 at 8.8 MeV and 3', T = 0 at 9.6 hev of
C12 and 3-, T a 0 at 6.15 hev of 016.

-52-

-53-
Everything is the same as at 156 MeV except for the
fact that since the two-body interaction potential is used,

the two-body scattering amplitude cannot be used.

89):) I {E'L 4X38Xx 90,) < I (3)) Z VWIPHEP F:fiX£’(P)AF’Ar
(7.1)
where'T’refers to the position of the incoming nucleon and
33 to 3th nucleon in the target nucleus. Xéz), and Xé') are
distorted waves with outgoing and incoming boundary conditions
in a central optical potential only. L and I: are the
nuclear state functions of initial and final states. V(?,‘?J)
is the two-body interaction potential of Yukawa type. In the
following equation, symbols have the same meanings as in

Chapter V:

(w 58 '
390 a “FEET—3N Z, ( XVI) fl 2)»);‘1‘2‘ S" J3» Eiflvofluoh. ll“)

L. 0 1-
:I Z, <8”, (~)Xu,)\p\ L,M> e” n 3)“)
le&
*- _, 3 ..,
8 Mam." 20")? Ag?) erfg (P) K (it) XSWW’W'
U) N a) U ‘A
.2
2. e: "5 1
V = 173-— SW?) +£V _g'(*.w1 may be expanded using
date: N
-————_ - 09
ITF-P'l '1 go (118)} WK)?» Wm) B (3%")
fl
. *(A‘f'(r
gfl‘ZMp-QR HIGHS “‘11 ) ') (7.3)

[so

-58-
where 3( If V3.) and h+( ‘66) are the spherical Bessel function
and Hankel function of the first kind, respectively, and are
given by

‘ flier
3t m ._.. “F I 31 25"-

w c.
Rt ‘" ‘ (R: * :82) .3—

r

4
Re ‘\’ 3 3&2 2:; A- M r‘S

z‘sl (1-5)!

(7.8)
The arguments of J( :fr‘ ) and h+( “("3 ). r, and r<,
refer to larger and smaller of lengths (r, r'). (25)
Using the identities (8.12) and (8.13), the ortho-
gonality of IQ , m and the definition of the form factors
given in Chapter IV, we get

t 0]" ’

391) = 2a (35‘. 4‘;\X:’*Y L“ *‘I‘M aft (fr) 803! “VHF: r 4,-
~ (>0 Q) ' ‘ 3.] L '
‘8' Va )QUQr) 8'. £{ (If?) F (Y0) r] A?
‘1’ LU1&Q(CH 831LW6.9F103” .

* ”I 3. W“ 8" i)" W) F‘" 9‘4"] 1.5"") “-
P

(7.6)

-55-
Distorted Wave

 

As before, we have
90

”1?- )5. S udz‘
-”

962%: = e‘" w

(7.7)

12 are in the

where the real and imaginary parts of U for C

notation of (5.28), (28)

UB - -V(l +e‘)‘1 ; v = 38.5 MeV. r0 = 1.22r a = 0.67r

HI = -W(l +ex')-1 ; w = 8.9 MeV. r5 = 1.8f a'= 0.7:
(7.8)
Integral along Z is carried out numerically and the results
are fitted to suitable functions of b, the impact parameter.

The methods employed are exactly the same as in Chapter V.

_ 85" d.
e za_wUIz

GAMCA(b) =
= 1.0 - (CM 4' CNb2)e‘Ab2 _ (DM + DNb2)e-Bb2 (7.9)
K.
FEE“) = agURdz'
= 38-5 (0“ + CWDZM’A"DZ + 38.5 (UM + DNb2)e'Bb2
(7.10)
1:-” 1:” 3 e—z‘i-‘f- (Oos (FEE) - £sin(FRE)) x GAMCA
° (7.11)

where b = r sine.

Determination of V3, VI 9 and f '

 

(A rough estimate for the constants of the Iukawa po-
tential is made by calculating the free two-particle
scattering amplitudes in the Born approximation in the two-

body center of mass system and comparing the result with

-56-
the coefficient A of the two-body scattering amplitude. Be-
membering that the interaction is complex,
Mq) - AB 4- iAI(q)

-fr __'
=-.(£21L)_§m(k\vfi ° +1v1e 9’ ‘k?

 

 

 

 

1‘ r
m V V
8‘?q_2+—1€Z - in)" q 2.5.2 (7.12)
From EMT, we have values of A(q)
Ea _ i;
q = 0 .78 fermi q = 0.0 .86 fermi
a 1.1 .39 fermi = 1.8? .83 fermi

Equating'x(q) to A's for two values of q's, we get

ya . - .217 r1 vI - - .78
r - 101 f f. a 1067

Effective Form Factor
For a given state, one can evaluate numerically the

‘111al.'11::).‘l:iezH Y n I

A ‘ . 3°

FRJOJ a Re Ltd} ‘1‘“ hi) FIOI'.‘ 1.". 4,3“ (:cfl 8te(;frc)F P L‘V"
' r

(6.13)
A Q) .I I'. ,. It} . ,1 3°12.
1'1"” - ’8. W) 30 w r) F 91-1.. + W“ {81er "(284“
.1

as the functions of r and the results are fitted to the linear
combinations of Gaussians multiplied by polynomials. ‘Again

the numerical methods are the same as those in Chapter V.

s (p) - 2&[§r)fltflm XkeCFRB-i Si-LFR91[VK$:°3+WI’]¥‘1]

-57-
1350.0)“ (“1") . GAMCA(b)r2dr sinfi «10 (6.15)

91§_2+. T a 0 at 8.8 MeV

J2(1fr) = t [?l_+___ ____}'[. f 1‘ ...S’ 1‘} + 53;. [e"r+efr]
(6.16)

héfiufr) =1 [—1.+ yr +P3‘T] .4" (6.17)

2
FJ°J(r') -.A1’r'2e '°(r' (See Appendix I)
The results of integrations over r' are fitted to the functions
of the form

2 2

11 - rzch .“AL r 4.03 e‘AH r ) (6.18)
“ 202 ’ 202

The constants CL, CB, AL, and.AM for F3 and PI are given

below:

rsts h.--Effective Form Factors for 012 2*, r - 0

 

 

CA as AL AM
A

33202 -.116 -.057 .35 .20
‘9 202 -.065 -.018 .35 .21

I

 

For FRE(b), and GAHCA(b), the coefficients are as follows:

ILBLE 5.--Parameters for the Distorted Wave

 

 

CH ON (A UK DN B

 

FRE(b) .129 -.0031 .12 -.032 -.00#8 .54
GLMCA(b) 1.000 -.0153 .12 -.0580 -.050# .29

 

-58-
Jz(r) is evaluated using the polynomial approximations to
JO and.J1 (see Appendix 6) and the relation

(r) g JN(r) 2N

r
The Fortran program for IBM 1620 is given in Appendix 7.

 

The result is plotted in Figure 60 along with the experimental
points. (29) It is gratifying to note that, despite drastic
approximations and validity of the WKB in doubt at this
energy region, the curve shows a qualitative agreement with
the experiments. The dotted line is the result of local,
impulse, spin-independent approximation, i.e., exactly the
same as at 156 Nov except that B, C, E, P of t(q) were set
to zero.
012 3‘. T . 0 at 9.7 adv

.All the necessary informations are contained in C12 2',

T =- 0 case except :3“? r) and h“;(i fr) and the effective

form factors:

- L
33(1)”) = 1)) F (€19)ng -f". -(+( 6%:— ))
h(+ ) 8 C 15 |s 2:)
(ifr) [__?" + ———--W_)3 H + 0’0" V

A (A
33303 and F1303 are fitted to

 

2 2
(CA r3 e‘ALr + on :3 e'AMr ).

The constants are listed in the following table.

-59-

TABLE 6.--Effective Form Factors for 012 3', T = 0

 

 

 

ca 03 AL AM
A
33303 -.110 -.031 .35 .2
[$1303 —.063 -.oo86 .32 .2

 

With 33 from equation (6.19), one can integrate along Z
analytically but along b, the numerical integration is per-
formed. The result is shown in Figure 61 along with the
data. (29) Again the results look reasonable.

016 3’-gg - 0 at 6.15 ncv
Calculation is identical with 012 3‘, T = 0 except

that the effective form factors have different constants.

TABLE 7.--Effective Form Factors for 016 3‘, T = 0

 

 

 

oi 03 ‘AL AM
A
33303 -.0888 -.0698 .32 .20
A
21303 -.0595 -.0182 .30 .20

 

The results are shown in Figure 62. There are no

experimental data available to us.

CHAPTER'VIII

CONCLUSION

1 6 MeV
The predictions of the cross-sections by the WKB

method show reasonable agreement with the experiments. Our
predictions of the cross-sections also compare very favorably
‘with the DHIA results. The effects of the spin-orbit terms
on the cross-sections are found to be small. Also, we have
found that the Born approximations give essentially the same
angular distributions as the distorted wave predictions ex-
cept that the former gives cross-sections which are a few
tunes larger than those given by the distorted wave predic-
tions, a conclusion reached by the DWIA. (19) The WEB
appears to predict consistently larger values of the cross-
sections than the DHIA does. The reason for this is not
evident as the precise comparison between the “KB and the
DHIA is not possible because of different approximations
made in.algebraic work. He may change our magnitude a little
by correcting for the notion of the center of the mass of
the nucleus as in our calculations we have assumed that each
nucleon is in an orbit of a central potential. Also, we be-
lieve that better nuclear wave functions would also change
the magnitudes. On the whole, the WEB method as a theoretical
-60-

-61-
tool for the prediction of the nuclear cross-sections is
comparable to the DHIA.

For the final states in which the contributions of
the 8 - 1 states to the cross sections are small, the
polarizations predicted by the WEB agree with the results
of the DHIA. The predictions by both the WEB and the DWIA
agree with the experiments except at the small angles where
the theoretical values are too large. (Figure 2) This
feature seems to be common among the calculations that use
the impulse approximations. Addition of the L's term in the
optical potential makes the situation worse.

For the final states where the S 8 1 contributions to
the cross-sections are important, our predictions of the
polarizations are quite irregular.

In some cases like 1*, T = l of 012 at 151 HeV’(Figure
8) the L08 term merely enhances the Born approximations. The
spinporbit term changes the curve in the right direction.

In other cases like 2", r = 0 of c:12 at 18.2 uev (Figure 12)
the changes brought about by the L's term are quite radical.
We cannot say that the L°S term improves our prediction over
the Born approximations, but we can say that the L's term
should not be ignored in the calculations of the polariza-‘
tions as Perrain and.Vinh4Mau did. (Also we have found that,
at least for 3' of 0a“, the contributions of the imaginary
part of the spin-orbit potential to the polarizations are
negligible. Also for 3" ,of Ca“, we have found that the wild
interference pattern observed in the DWIA calculations (19)

-62-
disappear as we improve the fitting of the numerical integral
along Z of U3(L°S) to a Gaussian function multiplied by the
polynomials. This is another indication that the polariza-
tions are sensitive to the spin-orbit potential.

.As much as the polarizations are very sensitive to
the L°8 term, the sensible thing to do is to take the polari-
zation data of the inelastic scattering and adjust the L-S
part of the optical potential to fit the data.

On the basis of our calculations, we have made an
attempt to identify some of the states observed in the exp
periments. The results are given in Table 8. However, it
should be remarked that as many as six peaks are observed

near the excitation energy of 20 nev of C12

, our identifica-
tions may change when further studies are done on these

peaks.

#0 uev

Despite the drastic approximations involved, we re-
produced roughly the experimental situations. When we used
the Iukawa two-body potential in place of the impulse
approxhnations, we improved the predictions considerably.
Our calculations show that although the WEB in its present
form is inadequate at 80 nev, it may be possible to improve
on the "KB at this energy.

-53-

TABLE 8.--Identification of the Final States

 

I
r

L

 

 

 

 

 

 

 

 

Exp. State Theoret. State Comment
(Gillet et a1.)
C12
“.4 Nov . 2*, T = 0 at 8.8 Mev
9.7 ' . 3‘, T = 0 ' 12.8 "
12.7 " . 1*, T - 0 ' 18.0 " . . . Poor
15.1 ~ . 1*, T = 1 ~ 16.6 -
16.1 n . 2*, T = 1 v 16.3 "
18.2 ” . 2', T = 0 ' 16.0 ”
19.3 ” . 2‘, T = l ' 19.3 ” . . . Poor
21.0 ~ . 3', T = 1 ' 23.5 "
22.3 ' . 1', T = 1 ' 21.5 "
016
6.15 ' . 3', T a 0 ' 6.25 ”
11.5 ' . 2', T 8 0 ' 10.5 ”
13.1 ' . 2', T = 1 ” 13.0 "
15.3 ” . 2', T 8 0 ' 10.6 ”
18.7 ” . 3', T a 0 ” 15.1 ”
20.2 ' . 2', T 8 0 ” 19.1 '
Ca“0
3.? ' 3‘ at 3.88 '
4.8 ” 5' ' 8.38 '

 

APPENDIX I

The radial wave functions used in evaluating the form

factors are: (33)

a 4A 34 2 +1. 15 Q _a(\~l

“H

 

&
+4 8' r{ 2¥fi5 1e-:?”
= u' oL41 ——- -
32/“ (Trgfitx [ 2. «r]
334 a: K'yfl'xa/4 re[ QWU‘Q')” (24+S')°‘rl+dl)‘4 8.:
(2&1; H

The values of O( for C12, ()16 and Calm are taken from

the analysis of the electron scattering experiments: (38)

012 ..... .37 fm'z
016 ..... .311
Cauo ----- .22

Using the definitions of the form factors (8.17),
(8.28), and (8.29), we get the following results:

 

C12 1‘, T . 1

 

p101 . -0.2250(2 c-Krz [Tl r + 22 0Lr_3_7
F111 - .159o(2 e" (121-El r + 112 K12]

 

 

E Y1 12 H]. 32
1707 “CV 8.59 -2e7 3.28 -2017
21.5 .678 -2. 77 .682 1.38

_‘

-68-

'0 .-l

-65-

 

012 1+

 

 

2" 2
F011 = 0.159 K(3/2)c" Kr [21. + mxr

 

 

 

 

 

+ mcfr"

 

 

 

 

 

 

 

9211 - 0.159 K(5/2)e" '6er r2 + YBeriV
T = 0
E YL _3QL_ IN IA YB
18.0 MeV -0.885 2.38 0.53 -1.58 0.167
T a 1
16.6 MeV -0.137 2.6 0.336 -1.068 0.083
012 2-
F112 =- 0.159t(2 e" °“PRAYl r + Y20(r_3_7
F312 - 0.159713 1%" “1‘2 r3
T = 1
E n .2. 423..
13.2 HeV -5.0 2.36 -0.256
19.3 -2.89 3.29 0.685
23.2 -0.312 1.51 -1.085
2.2
15.6 MeV 2.86 0 0.78
16 -5.08 8.09 0.398
21.2 -0.188 -0.97

1.57

 

c:—

ufa -

-66-

 

012 2+

 

2
F202 = -0.225°k(5/2)e"<1‘ [h r2 + Y20( r37
F212 - 0.1590L(5/2)e-°’~1‘2[§1 r2 + 112 0337

 

 

 

 

 

 

 

 

 

 

 

 

 

T a 0
E 11 12 El 32
8.8 MeV 5.52 -0.183 -3.536 0.665
19.2 -2.011 1.60 0.195 0.535
T = 1
16.3 ucv 1.98 -0.027 -2.68 -0.083
c12 3-
F303 = FORM 0(3 r3e" °(r2
F313 = FACTORxB r3 e" '02
T a 0
E -EQ§!_ 229228
12.8 Mev -0.585 0.126
19.5 0.2 -0.318
T‘s 1
18.8 MeV -0.306 -0.067
0.39 -0.318

23.5

   

0 ° 1-, T = 1

 

F101

 

 

2
.. -0.225 of c'rxrz [h r + 130(127

Bl HZ

 

F111 = 0.159002 e" 0(r [fil r + 320L127
E 11 12
1305 HOV -3e32 2e175

8.56 -2.85

e.—

:

an;

and \

-67-

 

o16 2'

 

 

 

2

F112 a 0.1590(2 e"°(r [Y1 r + 220(r_3_7
2

F312 = 0.15901 313.12?“r Y3

 

 

 

 

 

 

 

 

T = 0
E ._.-11.. .__;!§_ ._-Zl__
10.5 MeV 0.0056 1.72 -0.369
18.6 0.0056 -0.989 1.78
16.6 -0.068 2.225 1.78
17.3 -5.68 3.72 -0.0219
T = 1
13.0 ncv 0.38 1.72 -0.875
17.6 . -l.l9 0.578 1.96
19.1 2.88 0.08 1.08
20.2 -8.88 8.38 0.62
016 3-
12303 :- -0.387,<3 r3 c-sz n
F313 = +0.2680(3 r3 c" 4’2 1:2
T - 0
E Y1 (._Z§__
6.25 ucv -1.93 .839
15.1 1.815 .519
20.1 - .621 -l.268
T = 1
12.7 nev -0.858 1.18
18.5 0.822 -0.226

28.1 -1.06 -1.18

 

=
I.

.3.

2

9;

-68..

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Ca80 3- #fi
.303 = - roe-2 [a .3 + 12.1.27
F313 = fl 1‘;_°(r2 Ail r3 + 32er7
811'
T - 0
3 Y1 12 31 32
3.88 Mev 8.95 -3.83 -1.775 1.309
7.15 -2.795 1.123 2.828 -0.866
7.73 1.59 -0.872 -2.19 0.88
8.38 -0.1 -0.385 1.81 -0.85
T a 1
3.88 MeV -0.07 0.088 -0.1 0.025
7.15 0.399 -o.058 -0.885 -0.505
7.73 0.223 0.098 -0.118 -0.357
8.38 0.378 -0.078 -0.92 0.1
08180 5-
1,505 . _ $16.2 x8 15 a. air-2
('2' 1!
F515 = 9,569 of r5 e" “1'2 12
81?
T a 0 T = 1
___ .s 11 12 II 12
8.38 MeV 2.163 -1.92 -0.058 -0.118
8.0 -0.232 -0.288 -o.822 2.657
12.2 0.787 0.212 -0.19 0.303
13.8 0.22 0.0728 0.628 -0.989
13.2 0.98 2.888 -0.161 -0.253
§Q«5 -0.13 -0.355 -1.138 -2.82

 

APPENDIX II

The basis of this approximation is the experimental
observation that the polarization of an initially unpolarized
nucleon and the asymmetry (to be defined below) of a 100%
polarized nucleon are very nearly the same. (31,32) We
shall take the xz plane as the scattering plane.

Asymmetry is defined as follows:

Assume that the incident nucleon has a polarization vector

E and the target has zero spin.

The cross-section at an angle 9 is (15)

i - + “
3.3;”) =;§;1<smsw +3108” 5 w 3)\

(111.1)
1f the incoming nucleon is 100% polarized along the y direction
(“f-.12“ 2 (I; when scattered to the left (L),
an.
a; 'P; = 4-3- when scattered to the right (B).

-69-

(:2

-70-

Then asymmetry A is defined as

8'8) - 3300 = yrlswfajy’f]
TM + Nu.) at [5123?“ (311.2)

With the definition of P in Chapter II, we have

 

 

[A'F] 58150» 9091 -- 3T.- [ 51m; 5’90]
- QT. [ $8,065 500]

(AII.3)
Using My) = 809A) + ngph’x + 3’9”"), + 32900-2
we have
UV?) ELUWSWU '-' 8 in (3‘90 558;»)
(AII.8)
If we rewrite (5.6), following Reference 37
40,0" -:Q,0’ .
e "c; e. " = q- («.1610 -2[Q+1°1m,m]
(311.5)
where
Q : i U‘da'
1E, m
a t
13 .. LES thew
ZE§ _£

It can be shown that 1" a 0 makes the right-hand side of

(AII.8) either very small or zero.

APPENDIX III

Bead the integrand.\( (the cptical potential) into
the memory as a subfunction. First fix the value of b.
For a given interval in Z, do the integration twice:
first, by dividing the interval into two sub-intervals we
have the result Fl; secondly, by dividing the interval into
8 sub-intervals we get F2. Compare the results F1 and F2.
If \Fl-FZ‘ (0.01, go to the next interval. Repeat the
procedure until Z a 16.0. The program evaluates the integral
for 0 S b s 16.0 with an increment of 0.2

This program is written for CDC-3600 at Michigan

State University.

-71-

-72-

* 151238 LEE,H
PROGRAM INELAS
8:000
105 Z=0.0
SUM:'=OeO
107 DH=0.01
108 x1=z+DH/2.0
x2=z+D3
X3=Z+3.0*DH/2.0
x8=z+2. 0*DH
110 Fi=D3*(1(B, Z)+8. 0*Y(B,X2)+Y(B, x8))/3. 0
F2=DH*(Y(B, Z)+8. 0*(Y(B, x1)+1(3, X3))+2. 0*Y(B, x2)+Y(B,X8)V6.0
120 IF(ABSF(Fl-F2)-0. 01 ) 130, 130, 125
125 DH=DH/2. 0
30 T0 108
130 sun=sun+32
Z=X8
IF(Z-16. 0) 107,180,180
180 PRINT 18, 3 .31, 32, SUN
185 FORMAT(/ F20.6
Bas+0.2
13(3-16. 0) 105,150,150
150 CONTINUE
END
FUNCTION 1(3, Z)
'3313m .9/(1. 0+EXPP((SQRTF(B*B+Z*Z)- 3. 00)/0. 556))
END
END

APPENDIX IV

Read the results of the integration performed by the
program in Appendix III as the elements of the matrix E.
Evaluate the exponential function Q using the elements of
the E matrix as the exponents.

Consider

THF(N) = 1.0 - (x +Yb2)e-A b2 .. (z + v b2)e-D b2
<r= {3mm .. mmnz

Make (r as small as possible by varying x,wr , A, Z, V and
D. First fix A and D. Then solve the equations:

30- =0
3!
or =0
av
or =0
3!
QO' =0
JV

These four equations are solved to get X,1’, Z, and V. Then
'we evaluate 6'. Repeat this procedure by giving different
values of‘A and D. TBF corresponds to [’(b) in the text while
1,1’, Z, V,.A, and D correspond to CM, CN, DH, DN, A and B.
This program is written for CDC-3600 at Michigan State

University; 73

-78-

* 151238 Lee H

10
11
15

25

30

3303333 FITTIM
DIMENSION 0(50), ,E(50). QI(5O).B(50), FL(5).FM(5) FN(5).

1AF(50),AT(50 ,BT(50), THF (50)

0 IMAGINABY PART CENTRAL FITTING
DO 10 K=1,85

vex

B(K)=0. 2*(U-l. o)

CONTINUE

READ 11, (QI(N),N=1, 80
FORMAT(8F10. 5)
DO 15 N=l, 8O
Q(N)=EXPF(-0. 0176fiQI(N))
A=O. 18
DO 800 1.1,5
AaA+O. 02
D=0.09
DO 750 J=1, 20
DO 2 M=1, 8
FL(M§=0. O
M=0 0
FN M)=0;o

CONTINUE
D=D+0. 02
DO 30 N=1, 80
(AF(N)=EXPF(=A*B(N)**2)-EXPF(-D*B(N)**2)
.AT(N)=E(N)**2*EXPF(.A*E(N)**2
BT(N)=B(N)**2*EXPF(-D*B(N)**2
E(N)=Q(N)-1. 0+(1 0-Q(1))*EXPF(-D*B(N)**2)
PL(1)=FL(1)+E(N)!AF(N)
FL(2 =PL(2)+AF(N)**2
$2 =FL 2 +33 N;*AT(N

=FL +3? N *BT(N

Eil)=FM(l +E(N)*AT(N)

mz)=FM(2 +AF(N)*AT(N)
FM(B):FM a

of 3:8
FN(2 :FN(2

 

tAT N)**2
+3T N)*BT(N)
1+E(N)*BT(N)
+AF(N *BT(N

”(2 +ATN N*BT(N
8):FN

+DT N **2

CONTINUE
DET=FL(2)*FM(3)*FN(8)+FL(3)*FM(8)*FN(2)+FL(8)*FN(3)*FM(2)
1-FL(8)*FM(3)*FN(2)-FM(8)*FN(3)*FL(2)-FN(8)*EM(2)*FL(3)
DEX=FL(1)*FM(3)*FN(8)+FL(3)*FM(8)*FN(1)+FL(8)*FN(3)*FM(1)
l-FL(8)*FM(3)*FN(1)-FH(8)*FN(3)*FL(1)-FN(8)*FM(1)*FL(3)
DEI=FL(2)*1=N(1)*3N(8)+3L(1)*FM(8)*FN(2)+FL(8)*FN(1)*FM(2)
l—FL(8)*FM(1)*FN( 2)-FM( 8)*FN((1)*FL(2)-FN(8)*FM(2)*FL(1)

1-318517213521133(3132313 3313333337

=-DEX /DET
YB-DEY/DET
Vi-DEV/DET

 

 

 

 

-75-

z=1.0-x-Q(1)
EQA=FL(1)+X*FL(2)+Y*FL(3)+V*FL(8)
EQB=FM(1)+X*FM(2)+Y*FM(3)+V*FM(8
EQC=FN(1)+X*FN(2)+Y*FN(3)+V*FN(8
PRINT 301,x,y,z,v.A,n
301 FORMAT(5X,1HX,E15.5 ,2X,1HY,E15.5,2X,1HZ,E15.5,
12x,13v,E15.5,2x,1EA,E15.5,2x,IHD,E15.5)
PRINT 302,EQA EQB,EQC
302 F02NAT<3E15.55
SIGMA=0.0
DO 00 N=1,8O
THF N)=1.0-(X+Y*B(N)**2)*EXPF(-A*B(N)**2)
1-(Z+V‘B(N)**2)“EXPF(-D*B(N)**2)
500 SIGMA=SIGMA+(Q(N)-THF(N))**2
PBINT16 (Q(N) THF(N),N=1,80)
16 FOENAT(8FIO.6S
PRINT520,SIGMA
520 FORMAT(1OX,SHSIGMA,E15.8)
750 CONTINUE
800 CONTINUE
END
END

87.74001 87.61321 87.23718 86.60882 15.717 5 88.55089 83.09150 81.32066
39.2026 36.77877 18.99509 30.890 58 27.517 5 23.36597 20.36385 16.85980
13.602 2 10.71003 8.2 023 6.23781 8.68869 3. 1680 2.89008 1.801833
1.2968 .92962 .6 #37 .87371 .3371? .23965 .17018 .12070
.08555 .06060 .08291 .03036 .02188 .01519 .01073 .00758

APPENDIX V

A brief outline of the calculations of the scattering
amplitude for the final state 1‘, T a 1 of 016 13 given
below.

From the consideration of the parity, L = l and S = O,
or 10

Let CON - 2Q(§;A
S()\al, 8:0) 8 CON 8 3.41.3? [‘(p) ‘(H'ew)+) FHHU') [A + €03]

- - . .. .3." ’F
‘83-’- L '018)'W' 8 e“ r ['(V') Yule,» PM.» Q(b)[°'§,fl C

- L CON'G'OWW'SJV'WQMN1’: (Mb) 6;. [A + J31

(AV-1)

Consider

SI 8 CON S €26??? PU?) \("9" [Y1 r +Y2, T5] EQWDLAfl‘

*. 1
- 2 n con 8 {1(0) ‘0. (9,6)[Vf1r +‘f2. r’} (24".): (1L)
WW" 5;“; AB

(AV-2)
Integration over Z yields

- 1. L‘- 4 2
:00». "if; S [‘(biefl’ [ $1. + 34:75 + {%‘]I‘HU°\L
-76-

“=l

-77-
Define

8'24“” L" V 1W“) AL -.-. FUZJAIU)

0 (AV-3)
go‘s-WW NJ. (NM = FM W”
O
1 (AV-8)
8" (WW L510,» 4b = F86 mu)
0
(AV-5)
Likewise we define FB2(B,U) --- and FBZ(Z,U) --- when A is

replaced by B and zero respectively. The variable 1 is re-
placed by U.
Define also
CF32 . PBZ(Z,U)-CM'FBZ(A,U)-DM'FB2(B,U)
- CN°FB8(A,U)-DN°FB8(B,U) (AV-6)
CF38 =- PB8(Z,U)-CM'FB8(A,U)-DM'FB8(B,U)
.. CN-FB6(A,U)-DN-F36(E,U) (AV-7)
Then making use of |fl(b2) from the text,

31 . CON.2U R 1{[% + 22 01122 + 12 crank
0L ’1
=- Ismu M R

The third term in (AV-1)”is, after integrating2 over Z,
311 - -iCON']l’ (0. 098) “’8 (b2)[_K_Y1 + 22 b2
R Y' m.)

(AV-8)

4- Iéfie ez'“ 0(6) L 1 o~ (.1200) + J (cm)

4.0313200) ] [A + Cwy] db
(AV-9)

“I
J'-

a).

3‘1

no

1‘” a

a“...

-78-

Define p 2
01323 = SV(b2)Q(b)b2 e'db J2(qb)db (AV-10)
0
w 2
cans = 3V(62)O(b)bl‘ e" “Lb J2(qb)db (AV-ll)
o
2
GDZS = fiwzmwmz e‘ *b Jo(qb)db (AV-12)
o
2
0043 = gubzmum“ e' ““0 Jo(qb)db (AV-13)
a _ e e e . 2; Y2
F'BDP CON 5; 1t 0 098 [(0323 + GDstfi + 5175-)
+ (GB8S + GD8S) XL] (AV-l8)
{3L
FBDM a -CON- . -o.0 8 01323 - CD23 3.1— + Y2
u g 9 < >( a m“)
+ (GB8S - anus) 2.2. (AV-15)
{87.
Then
SII acrx'onBDP [A + ocy] + 1 G‘y-FBDN-w [A + 0 try]
(217-16)

We can follow the same procedure to evaluate the
second term of (AV-1).
89581, 8:0) =- id 'SPAl + iC°W'FBDM

+0"! A-w-FBDP + rue-3211 + now-FBDN)
5' (AV-17)

For spin 8 = 1, we can show that the only non-zero tem

is the coefficient of 0'2.

321412 a -21! {'21 F CON-1% ( 1,1,1,0\ 1,1) SYT(9,¢)2111(1-)
['(b)Jl(-qb)r2dr sin 9 d9

=- - . - El 32 >CFBZ+§-2.CFB8
2“ CON J38 F ((532007:- {a- (117-18)

-79-
Then the addition of (AV-17) and (AV-l8) gives the scattering
amplitude when the projection quantum number IA = 1. For
’18 -l, we get similar results.
We have to repeat the whOle process to get the
scattering amplitude for )1 = 0.
The program shown is for l“, T a l of 016. The pro-
grams for the other states considered have similar structures.
Read in the momentum transfers at Q(N). Then the real
and imaginary parts of the two-body scattering amplitudes A,
B, C, E and F are stored at AR, AI, BB, BI, CR, CI, EB, EI,
PR and PI. The parameters for the distorted waves are read
in by the statements 153 and 139. The statement 123 reads
in ~(1, ‘{ 2, El and 82 for the form factor. NE and HI are the
real and imaginary parts of the spin-orbit coupling constants.
The results of the integrations performed by (5.18) are
stored as the subfunctions FA8, --- FB8, --- T85, --- T09.
Only these functions appear for all the states considered,
although not all of them are used for a given state. The
real parts of the scattering amplitudes are given the
following'names:
Be@°(l4 =31)] - SAVRl
Re[§x( #:117 =- SAXRl
Rafi“ )1 -1)_7 - 32131
BelEzgfit=lI7 = SAZRl
3918094 =0)_7 =- sumo
3.57 ( [1:017 - sumo

Similar'names are given for the imaginary parts. The DO

hogs
fiffe
appro

CI‘OSE

-80-
loops are necessary to evaluate the cross-sections for the
different values of the q's and also to consider the Born
approximations by making CM=DM=CN=DN=O and WB:WI=0. The
cross-section to be plotted is obtained by 10 x TCBOS(mb).

-81-
238 LEE H
PROGRAM SCATER
DIMENSION Q(50)9 ARH(50)9AIH(50)9CRH(50)9CIH(50)9
1WRM(5)9 WIM(5) 9CME‘5)9 CNE‘519 DME(5)9 DNE(5)9AE(5)9 BE(5) 9
28RH(50)9BIH(50)9ERH(50)9EIH(50)9FRH(50)9FIH(50)
COMMON P
O 1- T=1
READ 1009 (0(N)9 N=1939)
FORMAT(8F1006)
READ 165.(ARH(N).AIH(N).CRH(N).CIH(N).N=1.39)
FORMAT(4F1006) ' I
READ 1679(BRH(N)9BIH(N)9ERH(N)9EIH(N)9FRH(N)9FIH(N)9N=1939)
FORMAT(6F10.6)
2:000
P=O.311
SQP=$QRTF(P)
CON=1095
pEK3-0098
READ1519 (WRM(N)9 N=192)
READ 1519 (WIM(N)9 N=192)
FORMAT(2F1005)
READ 1609CF9CS9DF90$9R9$
FORMAT(6F1005)
READ 1109(CME(N)9CNE(N)9DME(N)9 DNE(N)9AE(N)OBE(N)9N=192)
FORMAT(6F10.5) ’
DO 951 JF=192
READ 1019 Y19 Y29 R19 R2 1
FORMAT(4F10.5)
YA=-.225*P**2*Y1
YB=-.225*P**3*Y2
RA=.159*P**2*R1
RB=0159*P**3#R2
PRINT 1249 Y19 Y29 R19 R2
F0RMATt/5X.4F15.3)
DO 950 L=192
CM=CME(L)
CN=CNE(L)
DM=DME(L)
DN=DNE(L)
A=AE(L)
8=BE(L)
pRINT IBSQCMQCN’DM’DN’A’B
FORMAT(6F10.5) '
DO 910 J=192
WR=WRM(J)
WI=WIM(J)
PRINT 1709CF9CSoDFoD$9R9$9WR9W1
FORMAT(/8F15.5)
DO 900 I=1939
AR=ARH(I)/4.0
AI=AIH(I)/4.0
C1=CIH(1)/400
CR=CRH(1)/4OO
BR=BRH(I)/400
BI=BIH(1)/400
ER=ERH(I)/4.o'
EI=EIH(I)/4e0

~82-
FR=FRH( I )/4.0
FI -FIH(I)/4.0
U= 0(1)
6841: CF*T25(R92. U)+DF*T25(S. Z.U)-CM*CF*T25(A. R.U)
l-CM#DF*T25(A. s. U)-DM*CF*T25(B. R, U)- -DM*DF*T25(B.S .U)
0843=cs*T27(R.Z.U)+Ds*727(s.z.U)-(CN*CF+CM*cs)*T27(A.R.U)
l-(CN*DF+DS*CM)*T27(AoSoU)-(DN*CF+DM*CS)*T27(B.R.U)
2-(DN*DF+DM*DS)*T27(89$.LH
GBAS=-CN*CS*T29(A.R9U)-CN*DS*T29(A.$oU)-DN*CS*T29(B.R.U)
1-DN*Ds*129(B.s.U)
GBZI=CF*T23(R,Z9U)+DF*T23(SoZoU)-CF*CM*T23(A.R.U)
l-CF*DM*T23(B.R.U)-CM*DF*T23(A.S.U)-DM*DF*T23(B.S.U)
6823=CS*T25(Raon)+DS*T25($929U)-(CM*CS+CF*CN)*T25(A.R.U)
l-(CS*DM +CF*DN)*T25(89R9U)-(DS*CM+DF*CN)*T25(A.S.U)
2-(DS*DM+DF*DN)*T25(B.SoU)
6825: -CN*cs*T27(A.R. U)-cs DN*T27(B. R.U)
l-CN*D$#T27(A. s. U)-DN*Ds*T27(B.s. U)
0021- —CF*T03(R.Z.U)+DF*103(S. 2. U)- CM*CF*T03(A. R. U)- CM*DF*T03(A. s. U)
1 -DM*CF*T03(B. R9U)-DM*DF*T03(B. s.U)
GDAl=CF$TOS(RoZ.U)+DF#TOS(SaloU)-CM*CF*TOS(A.R.U)
l-CM*DF*TOS(A959U)-DM*CF*TOS(BoR9U)—DM*DF*TOS(B.$.U)
6023=C$*T05(R92. U)+DS*TOS(S.Z. U)-(CM#CS+CF*CN)*T05(AoRoU)
1-(DM*C$+CF*DN)*TOS(B. R9U)-(CM*DS+DF*CN)*T05(A. s. U)
2-(DM*DS+DF*DN)*TOS(B. s. U)
GD¢3=CS$TO7(R.Z9U)+DS#TO7(S.Z.U)-(CM$CS+CF*CN)*TO7(A.R.U)
1-(DM*cs+CF*DN)*T07(B.R.U)-(CM*Ds+DF*CN)*T07(A.s.U)
2-(DM*D$+DF*DN)*TO7(B9S9U)
GDZS=-CN*CS* T07(A9R9U)-CN*DS*TO7(A9$9U)
1" DN*CS*TO7( B9R9U)‘DN*D$*T07(B9S9U)
0045=-CN*cs*T09(A.R. U)-CN*DS*TO9(A. S.U)
1-DN*CS*TO9(B. R. U)—DN*DS*T09(B.S. U)
6845: 0841+6843+6845
GBZS=GBZI+GBZ3+GBZS
GDZS=GDZI+GDZB+GDZS
604$=GD4I+GD43+GD45
CF84=F84(z.U)-CM*F84(A.U)-DM*F84(B.U)-CN*F86(A.U)-DN*F86(B.U)
CF82=F8212.U)-CM*F82(A.U)-Dm*F82(B.U)-CN#Fsa(A.U)-DN*F84(B.U)
AWM=AR*WR-AI#WI
BWM=BR*WR-BI*WI
CWM=CR*WR-CI*WI
EWM=ER*WR-EI#WI
FWM=FR*wR-F1*w1
AWP:AR*WI+AI*WR
BWP=BR*WI+BI*WR
CWP=CR*WI+CI*WR
EWP=ER*WI+EI*WR
FWP=FR*WI+FI*WR
SFA1=CON*(CF82*(YA/SQP+YB/(2.0*P*SQP))+YB/$OP*CFB4)*3.84
FBDP= CON*PEK*((GBZS+GDZS)*(YA/$QP+YO/(2.0*P*SQP))
1+(684$+604$)#YB/SQP)*1.92
FBDM=CON$PEK*((GBZS-GDZS)*(YA/SQP+YB/(2.0*P*SQP))
l+(GB4$-GD4S)*YB/SQP)*1.92
SAVR1=-AI*5FA1-CWP*FBDM
SAVII=AR*$FA1+CWM* FBDM
SAXR1=AwM*Fstg~
SAX11=AWP*FBDP
SAYR1=-CI*SFA1-AWP*FBDM

 
 

 

-83-
SAYIl=CR*SFA1+AWM*FBDM
OCB=(RA/SQP+RB/(250*P*SQP))*CFBZ+RB/SQP*CFB4
AMIZ=CDN*QCB*3.84
SAZR1=-FI*AMIZ
SAZII=FR*AM12
GBDP=(RA/SQP+RB/(2. 0*P*SQP))*(GBZS+GDZS)
l+RB/SOP*(GB4S+604$)
GBDM=(RA/$QP+R8/(2.0*P*$QP))*(GBZS-GDZS)
l+RB/SQP*(GB4S-GD4S)
AMA=-5.45*CON*QCB
AMB=2.72*CON*PEK*GBDP
AMC=2.72*CON*PEK*GBDM
SAVRO=CR$AMA+EWM*AMB-BWM*AMC
SAVIO=CI*AMA+EWP*AMB-BWP*AMC
SAYRO=BR*AMA-CWM*AMC
SAYIO= BI*AMA-CWP*AMC
SCRO= SAVR1**2+SAVI1**2+SAXR1**2+SAXI1**2+$AYR1**2+SAYII**2
CRO$1= SCRO+SAZR1**2+SA211**2 .
CROSO=SAVRO**2+SAVIO**2+SAYRO**2+SAYIO**2
TCROS=2.0*CRO$1+CROSO
SCROS=2.0*SCRO
PRINT 519. U. SCROS..CROSI. CRoso. TCROS
FORMAT(5F15.5) ‘ _
RRINT 521. OCB.SAZR1.5AZI1.SAVRo.SAVIo.5AYRo.5AYIo
FORMAT(7EISo4)
CONTINUE
CONTINUE
CONTINUE
CONTINUE
GO TO 139
END
FUNCTION FA4(x.U)
COMMON R
DEN=1 oO/(P+X)
v= U*U/(4.0*(P+X))
EVF=EXRF(-V)
FA4=U**3*EVF*DEN**4/16.0
RETURN
END
FUNCTION FA5(x.U)
COMMON P
OEN=1.0/(R+x)
v= U*U/(4. 0*(P+X))
EVF=EXRF(-V)
FA6= U**3*EVF*DEN**5 *O.25*(1.0-O. 25*V)
RETURN
END
FUNCTION FA8(x. U)
COMMON P
OEN=1.0/(R+x)
v= U*U/(4.0*(P+X))
EVF=EXRF(-V)
FA8=1.25*U**3*OEN**6*EVF*(1.0-o.5*v+o.os*v*V)
RETURN
ENO
FUNCTION F82(x.U)
COMMON P

 

(

DEN=loO/(p+X)

V= U#U/(4o0*(P+X))
EVF=EXPF(-V)
F82=U*EVF*DEN**2*0025
RETURN

END C

FUNCTION FB4(X9U)

COMMON P

OEN=1.0/(R+x) .

V= U*U/(4o0*(P+X))
EVF=EXPE(-V)
FB¢=U*DEN**3*EVF*O.5*(1.0-005*V)
RETURN

ENO _

FUNCTION F86(X9U)

COMMON P

DEN=loO/(P+X)

V: U*U/(4.0#(P+X))
EVF=EXPF(-V)
F86=1.5*U*DEN**4*EVF*(1.0-V+V*V/6.0) (
RETURN

END

FUNCTION FBB(X9U)

COMMON P

DEN=1.0/(P+X) _

v: U*U/(4.0*(P+X))
EVF=EXPF(-V) .
FBB=6.0*U*DEN**5*EVF*(1.0-1.5*V+O.5*V*V-V**3/24o0)
RETURN .

END

FUNCTION T45(X9Y9U)
COMMON P

OEN=1.0/(P+X+Y)
V=U*U/(4.0*(P+X+Y))
EVF=EXPF(“V) '
T45=U**4#EVF*DEN**5/32.0
RETURN

END

FUNCTION T47(X9Y9U’
COMMON P

DEN=1.0/(P+X+Y)
V=U*U/(4.0*(P+X+Y))
EVF=EXPF(-V)
T47=0.156*U#*4*DEN**6*EVF#(l.0-O.2*V)
RETURN . '

END

FUNCTION T49(X9Y9U)
COMMON P

DEN=loO/(P+X+Y)
V=U*U/(4.0*(P+X+Y))
EVF=EXPF(-V)
T49=0094*U**4*DEN#*7*EVF*( l 00-004*V+000333*V*V)
RETURN '
END

FUNCTION T411(X9Y9U)
COMMON P

DEN=1.0/(P+X+Y)

-35-
V=U*U/(4.0*(P+X+Y)) V
EVF=EXPF(-V)
T411=6.55*U** 4*DEN**8*EVF*(1.0-0.6*V+0.1*V*V—0.00476*V**3)
RETURN

END

FUNCTION T25(X,Y.U)

COMMON P ‘

DEN=1.0/(P+X+Y)

v=U*U/(4.o*(R+x+Y)T

EVF- -EXPF(-V)

T25: 0. 375*U*U*DEN**4*EVF*(1.0 -0.333*V )

RETURN

END

FUNCTION_T27(X9Y’U)

COMMON P

DEN=1.0/(P+X+Y)

‘V=U*U/(4.0*(P+X+Y))

EVF=EXPF(-V)

T27: 1.5*U*U*DEN**5*EVF*(1.0-0. 6667* V+Oo 0835*V*V)
RETURN \ T

END

FUNCTION T29(X9Y9U)

COMMON P
COEN=I.0/(P+X+Y)

v=U*U/(4.0*(P+X+Y))

EVF= EXPFC-V)

T29=7.5*U*U*DEN**6*EVF*(1.0-V+0. 25*V*V-O. 016667*V**3)
RETURN ‘

END

FUNCTION T211(X,Y9U)

COMMON P. _

'DEN=Io0/(P+X+Y) .

V=U*U/(4.0*(P+X+¥))

EVF=EXPF(-V)
T211-—45.0*U*U*DEN**7*EVF*(1.0-1.333*V+0o 5*V*V+0 00278#V*#4
1- o. 0667*V**3) . I
RETURN

END

FUNCTION T23(X9Y9U)

COMMON P

DEN=1.0/(P+X+Y)

V=U*U/(4.0*(P+X+Y))_

EVF=EXPF(-V)

T23: O.125*U#U*DEN**3*EVF

RETURN

END

FUNCTION T03(X¢Y9U)

COMMON P

DEN=1.0/(P+X+Y)

V=U*U/(4.0#(P+X+Y))

EVF=EXPF(-V)

TO3=0.5*DEN#*2 *EVF*(1.0-V)

RETURN

END

FUNCTION T05‘X9Y 9U)!

COMMON P

DEN=1o0/(P+X+Y)

 

I ‘ I ' ‘ ' '
.(~ .1. I

i ’
I ‘1 l h -
I I . . ‘l. ..J . A

-86-
v=U*U/(4.o*(P+x+Y))
EVF=EXPF(-V)
105=EVF*DEN**3*(1.0-2.04V+0.5*V*V)
RETURN
END
FUNCTION TO7(X9Y9U)
COMMON P
0EN=1.0/(P+X+Y) '
v=u*U/(4.o*(P+x+Y)).
EVF=EXPF(-V)
T07 =3.0*DEN**4*EVF*(1.0-3.0*V+1.5*V*V-0.16667*V** 3)
RETURN . <
END
HflmTION T09(X9Y¢U)
COMMON P ‘
DEN=100/(p+X+Y)
V:U*U/(4.o*(P+x+Y))
EVF=EXPF(-V)
109:12.0*DEN**5*EVF*(1.0-4.o*v+3.0*V*v-o.6667*V**3+0.0417*v**4)
RETURN '
END
HmmTION T011(X9Y9U)
COMMON P
[EN=100/(P+X+Y’
V=U#U/(4.0*(P+X+Y))
EVF=EXPF(-v) .
1011=60.0*DEN**6*EVF*(1.0-5.0*v+5.0*V*v-1.667*V**3+0.208*v*#4

 

10.00832*v*#5)
RETURN
END
ENO
0.192 0.288 0.382 0.479 0.572 0.665 0.760
0.940 1.03 1.12 1.200 1.290 1.38 1.450
1.620 1.690 1.770 1.840 1.910 1.980 2.040
2.160 2.220 2.280 2.330 2.360 2. 30 2.47
2.5550 2.580 2.620 2.640 2.670 2.690
a -.4523 -.0256 -.0029
' -.4411 -.0505 -.0050
5 -.4228 -.0741 -.0055
’ ‘03980 “00957 ’00037
5 ”03672 “.1149 .0008
s -.3314 -.1131 .0082
3 -.2915 .-.1442 .0187
“02485 ”01538 .032
-.2035 -.1598 .0479
‘01576 -01623 00657
-.1120 -.1613 .0851
-.0675 -.1570 .1054
”00252 “01498 .1262
-00142 -01399 01470
00498 “.1279 01676
.0812 -.1135 .1877
.1078- -.0978 .2075
5 .1293 -.0810 .2271
3 .1454 -.0632 .2466
3 .1559 -.0448 .2663

84
26
J7
92
178
mg
95

.1608
.1599
.1533
.1410
.1230
.0994
.0704
.0362
-.0030
”.0469

”.09500

”.1467

”.2014‘

”.2580
-.3157
-.3734
”.4298
-.4836
”.5336
”.0721
”.0704
-.0676
”.0640
”.0597
”.0548
”.0498
”.0448

.‘.0401

”.0360
-.0326
”.0302
-.0288
”.0286
”.0297
”.0319
-.0352
”.0395
-.0450
-.0511
”.0580
”.0656
”@0737
-.0823
”.0915
“.1014
-.1118
-.1231
”.1351
4.1481
-.1620
-.1769
”.1927
”.2092
”.2262
”.2435
-.2607
”.2774

”.0260
”.0069
.0121
.0310
.0498
.0681
.0858
.1027
.1186
.1331

‘.1458

.1564
.1644
.1695
.1714
.1696
.1640
.1545
.1412
”1.005
”.9390
”.8339
”.6951
”.5299

,”.3472

”.1562
”.0341
.2150
.3794
.5216
.6376

.7254

.7850
.8178
.8266
.8151
.7874
.7478

1.7004

.6486
.5952
.5421
.4904
.4407
.3929
.3465
.3009
.2555
.2100
.1641
.1181
.7235
.0275
”.0157
”0.563

-.0937.

”.1271

.2864
.3071
.3284
.3502
.3722
.3939
.4147
.4338
.4503
.4633
.4721
.4757
.4735
.4650
.4500
.4283
.4000
.3653
.3248
”.0706
-.0646
”.0547
”.0414
-.0250
”.0061
.0146
.0366
.0590

0.0811

.1023
.1217
.1389'
.1531
.1639
.1709
.1737
.1721
.1661
.1554
.1402
.1206
.0966
.0686
.0368
.0014
”.0371
”.0784

”.1222

”.1679
-.2150
-.2631
”.3115
”.3596
”.4068
‘.4525
”.4958
-.5361

-87-

-1.519
-1.492
”1.447
”1.388
”1.317
”1.238

-1.153.

-1.065
-.9772
”.8908
”.8068
”.7255
-.6465
-.5688
-.4914
”.4130
”.3328
-.2503
”.1659
”.0803
.0049
.0874
.1646
.2337
.2921
.3373
.3674
.3814
.3790
.3609
.3285
.2841
.2304
.1703
.1069
.0432
-.0182
”.0752

-.0986
-.0954
”.0901
-.0828
”.0735
-.0623
-.0495

'-.0351

”.0194
”.0027
.0149
.0329
.0511
.0690
.0862
.1025
.1173
.1304
.1412
.1494
.1548
.1570
.1557
.1508
.1422
.1298
.1137

2.0939
”.0706
-.0441

.0148

'”.0168
"-.0503

”.0849
-.1201
”.1551
-.1892
-.2215

 

”.2931
4.31
-.11
-11.58-
0.0
-.03459
2.175
-2.57

'5.1562

349.2
0.0
-2.5859
4.56
-4.64

-.5727

-5.642
0.0
-.26942
”2.85
3.32

-88-
".1262
.5

'0.0
.2

”.2514

.45
0.0
.35

APPENDIX VI

To evaluate J2 and J3 for the calculations involved

in 40 Mev, we have used (26)

J (3‘) + Jnfl‘X) . gr; Jn(X) (AVI.1)

n-1

and for -3 < X6 3
x 2 x 4
J00!) = 1 -2.2499997(3) + 1.2656208(3)
- .3163866(§)5 + .044447993E 8
- .0039444(§)1° + .00021(§ 12 (AVI.2)

Jim

 

x - 1 -.56549985<§>2 + .21093573<§)"
- .03954289(§)6 + .00443319(§)8
- .00031761(§)1°‘* .00001109(§)12 (471.3)
and for 3 g: x.<,9o
J00!) - i?- fo ' Case,
to = .79788456 - .00000077(%) --.00552740(§)2

-.00009512(g)3 + .00137237(%)h

-.00072805(;)5 T .00014476cg)6

6°: 1 -.78539816 «041663970;

-.000039546%)2 + .0026573(;)3
-39-

-90-
-.00054125(%)“ -.00029333(;)5 + .00013558(3x)6

(AVI.4)
J1 " £1. °°8 91
‘5 2
r1 . .79788456 + .00000156(%) + .01659667(;)
+ .00017105(;)3 .. .00249511Q)”
+ .00113653(%)5 .. .0002033(%)6
91 = -2.35619449 + .2499612(§>
+ .00005650(«3)2 .. .00637879(%)3
1|,
4- .00074348(%) 4» .00079824(;-)5
- .00029166(3.)6
x (AVI.5)

One must be careful in using the above approximation
for x<n where n is the order of the Bessel functions as the
round-off errors may become large. we have evaluated J2
and J3 and compared the results with the values of J2 and
J3 listed in a standard table. (26) The discrepancies exist
for the small values of X where 32 and J3 are very near
zero. For these points, we have used the values of J2 and

J3 listed in the tables.

APPENDIX VII

The program listed here calculates the cross-sections
of 2", T =- 0 at 4.4 MeV of 612 with the incident energy of
40 MeV. The effective form factors are read in by CA, CB,
AL, AM. VB, VI, BER and RH]: are the real and imaginary
parameters of the two-body potential (v3, 3 , VD: U. The
integral over the impact parameter b is carried out by
Simpson's rule in the statements 14 and 2?. The functions
appearing in the integrands are stored as the sub-functions.
GJZMZ represents the result of the integration over 4) and!
of the effective form factor multiplied by e‘iq‘r YL: when
M = 2. GJ2MO has similar meaning for H = O. T316 1 GAMCA
represents the distortion while BESis the Bessel functions.
The cross-section is given by 0308. This program was written

for IBM-1620 at Duquesne University.

-91-

.-92..

DIMENSION 551M2‘393195JMO(3.3). CAO(2). C8012).AMO<2).
ISAD(2 ) , SBD(2)9BLD(2).BMD(2)9ALD(2)
CDMMDN A9B.CAQCB9'5A9589AL9AMgBL98M, _
READ4 9 ( CAD(N)9N=19"2).(CBD(N).N=1921.(SAD(N)9N=1.2)9(SBD(N).N=1.2)
81:10.6)
4 :gigAgi ( ALD(N).N,=1.2).(AMD(N).N=1.2)9(BLD(N).N=1,2),(BMD(N),N=1.2)
5 FORMAT ( 81:10.6)
READ 6 9VR9V19RHR9RHI
6 FORMAT ( 41:10.5)
EBA=EXPF(”.1)
CFSO=CO$F(.O) ' _ "
SID=SINF(.O) '
soR=soirzTr=14.0)
A=-.225*5.52*.084
=.0
=.0
DO 711 L.==1.14
DO 611“ K=192
CA=CAD( K )
CB=CBD(K)
SA=SAD(K)
SB=SBD(K)
AL=ALD(K)
AM=AMD ( K )
BL=BLD(K)
BM=BMD(K)
DO‘601 12.1.2
=.0 '
DH=.4
SJM2(I,K)=.O
0 x1=z+oH
X2=Z+2.*DH

3.114211 .K ) =(YI(2.1.0)+4.*v11x1.1.0)+YI(x2.1.0) )*DH/3.+$JM2(I.K)
=x2

’F‘ZT6-l 10.21.21
I 2:.0

DH=04T

SJMO(I ’K)=.0
25~ X1=Z+DH

X2=z+2°*DI—1
:7 - . .

fjfi’g‘I’K) =(YS(Z.I.0)+4.0*Y$(X1.I.Q)+YS(X2.I.0))*DH/3..+$JMO(I.K)
#01 CONTINU
’H CONTINU

SVSXR2=V R *RHR*5JM2( 1 . 1 ).-v1 *RHI *SJMZ ( 2 .2)

Svlzfo*RHR*SJMO(191)’VI*RHI*SJMO(2.2)
SVIO:VR*RHR*SJMZ(2.1)+VI*RHI*SJM2(1.2)
c605: R“‘RI—uzzatsaMmz.1)+v1=1<R1-11::<s.1M0(1.2)
ewe: SB ===1.9181442412.045VR2**2+sv1244242.+$VR0442+$V104421
32 FOR 82. 5.1M211.1).5JM211.2).5JM2(2.1).SJM212.2)
MM.(‘tho 6)
PUNCH -
FORM... 3.sJMoc1.1).s.1M0(1.2).SJM0(2.1).SJM0(2.2)
1-(‘H=20.6)

 

25.30.30

~93-

PUNCH 51. QQCRDS’ SVR295V1295VR095N10

FORMAT(6E12.5)

030+.2

CALL EX IT

END
:MJFREQGAMCAQ GJ2M0.GJ2M2. BEFI.BEZE

FUNCTION YS(X.I.Q)

R=X*Q

IF(R-.00001}15.16.16

BES=1.0

GO TO 18

BES=BEZE(R)

CONTINUE

FE=FRE(X)

IF(I-1)21.21.23

TRIG=COSF(FE) 7 6

GO TO 31

TRIG=SINF(FE)

YS=-44.*TRIG*BES*GAMCA(X)*GJ2M0(X)

RETURN '

END

FUNCTION YI(X.1.Q)

R=X*Q

IF(R-.00001)15.16.16

8E5=.0

GO TO 18 .

BES=BEFI(R)*2./R-BEZE(R)

CONTINUE ’

FE=FRE(X)

IF(I-1) 21.21.23

TRIG=COSF(FE)

GO TO 31

TRIG=SINF(FE)

CONTINUE

YI=54. *TRIG*GAMCAJXJ$OUMK3*BES*GJ2M2(X)

RETURN

END

FUNCTION BEZE(X)

S=X/3.0

R=S**2

V=1.0/S

T=1.0/R ’

IF(3.0-X) 15.15.20

BEZE=1.0-2.2499997*R+1.2656208*R**2-.3163866*R**3+.0444479*R**4
1-.0039444*R**5+.00021*R**6

GO TO 22

FZE=.79788456 -.0055274*T-.00009512*V*T+.00137237*T**2
1-.00072805*T*T*V+.00014476*T#*3 -.00000077*V

ANG= X-.78539816-.04166397*V +‘ .00262573*T*V
1‘.00054125*T**2 -.00029333*T*T*V+.00013558*T**3-.00003954*T

BEZE=FZE*COSF(ANG)/SQRTF(X)

CONTINUE

RETURN

END

FUNCTION BEFI(X)

s=X/3.0

R=S*S

-9u-
V=100/S
T=100/R
IF(390-X) 15915920
EFI=95-956249985*R+921093573*R**2-903954289*R**3+.00443319*R**4
1-900031761*R**5+900001109*R**6
BEFI=X*EFI

GO TO 21

FI=979788456 +901659667*T+900017105*V*T-900249511*T**2
1+900113653*T*T*V-900020033*T**3+900000156*V

ANG=X~2935619449 . +900005650*T-900637879*V*T+

1.00074348*T*T+.00079824*V#T*T-.00029166#T**3 +.12499612*V
BEFI= FI*COSF(ANG)/SGRTF(X)

CONTINUE

RETURN

ENO

FUNCTION GJZMZKX) (

COMMON AoBoCAQCB9 SA9$B9 AL9 AM9 BLOBM

Y=X*X
GJ2M2=A#(CA*EXPF(-AL*Y)/SQRTF(AL)+CB*EXPF(-AM*Y)ISQRTF(AM) )
1+B*(SA*EXPF(-BL*Y)/SQRTF(BL)+SB*EXPF(-BM*Y)/$QRTF(BM))
RETURN

END

FUNCTION GJ2M0(X)

COMMON A.B.CA.CB. SA9$B9AL.AM9BL9BM

Y=x*x '

GJ2M0= A*(CA*EXPF(-AL*Y)*(X- Y*X*AL)/(AL*SORTF(AL) )
1+CB*EXPF(-AM*Y)*(X-Y*X#AM)/(AM*SQRTF(AM)))
2+B*(SA*EXPF(-BL*Y)*(X-Y*X*BL )/(BL*SQRTF(BL))
3+SB*EXPF(-BM*Y)*(X-Y#X*BM)/(BM*SQRTF(BM)))

RETURN

END

FUNCTION FREtD)

CM=.129144

CN=-0003138

DM=-.0316928

ON=-.004752

=012

=.51

F21=(CM+CN*D*D)#EXPF(-A*D*D)+(DM+DN*D*D)*EXPF(-B*D*D)
FRE= 34.5*F21

RETURN . J

END '

FUNCTION GAMCA(X)

CM=1.00094

CN=-.0153128

DM=-.579689

ON=-.05035

=012

8:029
GAMCA=190-(CM+CN*X*X)*EXPF(-A*X*X)-(DM+DN*X*X)*EXPF(-B*X*X)
RETURN

END

15518 -.O755 -.057468 -.01687 .34524 .22570 -.73135 -.34119
.34 .2 .2 .57 .60 .19 .215
97 -046 101 106

 

1.

2.
3.

5.
6.

7.

8.

9.

10.

11.

12.

139

1b.

15.

LIST OF REFERENCES
A. Kerman, H. McManus, and R.M. Thaler. Ann. Phys. §,
551. (1959).
E. Squires. Nuclear Physics 6, 50“, (1958).
E. Sanderson. Nuclear Physics 15, 55?, (1962).
D. Jackson. Nuclear Physics 25, 199, (1962).
T. Erickson. Nuclear Physics jfi, #97. (196”).

X. DeBourd, J. Fos, B. Geofferson, N. Marty, C. Holland,
and B. Tatischeff. Le J. De Physique 25, 1059, (1963).

J. Jacmart, J. Garron, R. Riou, and C. Ruhla. “Pro-
ceedings at the Conference on Direct Interactions and
Nuclear Reaction Mechanism,” Padua, 1962. (Gordon and
Breach Science Publishers, New York, 1093, (1963).

D. Easselgren, P. Renber , 0. Sunberg, and G. Tibell.
Nuclear Physics 62, 81, I1965).

P. Boos, Massachusetts Institute of Technology Theses,
Unpublished, (196%).

?. gitchler, and R. Haybron. Physics Letters 11, 313,
19 .

V. Gillet, and N. VinhPMau. Nuclear Physics #, 321,
(1964). V. Gillet and E. Sanderson. Nuclear hysics
h, #72, (196k).

G.P. McCauley and G.E. Brown. Proceedings Royal So-
ciety. Z1. 893. (1958).

B.J. Glauber. Lectures in Theoretical Ph 3103 Vol. 1,
Intersoience PubIIshers 50., New York, 31%, (1959).

D. Jackon. Nuclear S ectrosco with Direct Reactions
1;. Unpublisfiea, Ifgonne Nationai fab. (1965).

H. Goldberger and K. Watson. gellision Theory, John

Wiley 00., New Ibrk, (1964).

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16.

17.
18.

19.
20.

21.

22.

23.

24.
25.

26.

27.

28.

29.

30.

31.

32.

33-

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G. Brown. The Unified Theor of Nuclear Models. John
Wiley 00., New York, 33, (1965).

D. Brink and G. Satchler. Nuovo Cimento g, 599, (1959).
D. Brink and G. Satchler. An 1ar Momentum, Oxford
University Press, London, #3, (1962).

R. Raybron and H. McManus. Preprint.

P. Morse and H. Feshbach. Methods of Theoretical Physics
1;, 1321, McGraw Hill 60., New York.

See any Mathematical Tables, for example, Standard
Mathematical Tables, 11th Ed. Chemical RuEEer Co.,
eve and. 9 7.

G. Watson. Theory of the Bessel Functions, Cambridge
University Press, 394, 2nd ed.

1(2..R8BS Elton and D. Jaokon. Nuclear Physics 21, 209,
19 2 .

M. Fricke and G. Satchler. Phys. Rev. 112, 567, (1965).

A. Messiah. Quantum Mechanics I, North Holland Pub.
00., Amsterdam, 9 .

F.W.J. Olver. Bessel Functions of Integer Order,
Handbook of Mathematical Functions. National Bureau
3?'§£andards,‘§§5. “"

P. Hillman .A. Johanson and H. Tyren. Nuclear Physics E,

M. gerrain, and N. Vinh-Mau. Physics Letters 13, 236,
19 5 .

Preliminary data from Minnesota group, private communi-
cations to Hugh McManus, Professor, Michigan State
University.

L. Elton, R. Shaw, and A. Swift. P.L.A. Pro ess Re ort,

Rutherford High Energy Lab., HarweII, (I963).

Th. ASJ. Marie and H. Tyren. Nuclear Physics E, 662,
1957 .

E. glphone, A. Johansson, and G. Tibell. Nuclear Physics
_. 72.

E.G. Mayer, and J. Jonson. Elementar Theor of Nuclear
Shell Structure. John Wiley, New York, 236, (I933).

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35-
36.

37-

-97-
V. Gillet and M. Melkanoff. Phy. Rev. 132, 1190, (1969).
H. McManus and D. Slanina, private communications.

Marty, Private communications to Hugh McManus, Professor,
Michigan State University.

Hooton, D.J. and Ascheroft, N.W. Proceedings Physics
Society. £1. 193. (1963).

 

-99-
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