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Date

  
   
  

  

L I B R A R Y
Michigan State
University

   

This is to certifg that the
thesis entitled
THE OPTICAL POTENTIAL IN
PROTON—NUCLEUS SCATTERING

presented bg
Donald A. Slanina
has been accepted towards fulfillment

of the requirements for

PhoDo degree in PhYSiCS

   

Major professor

May I“, 1969

/

 

ABSTRACT
THE OPTICAL POTENTIAL IN PROTON-NUCLEUS SCATTERING

BY

Donald A. Slanina

The optical potential for 40 MeV protons is calculated

12 40 58Ni 120Sn 208P

for the spin zero nuclei C, Ca, , and b.

r
The real central part of the potential is calculated to

first order in the nucleon-nucleon effective interaction
which is taken to be the G-matrix used in studies of the
bound state properties of finite nuclei. The impulse
approximation is used for an estimate of the spin orbit
potential. The imaginary part of the optical potential is
calculated from a perturbation treatment of the channels

open for inelastic scattering. The energy dependence,
isobaric dependence, effect of possible proton-neutron
density differences, and antisymmetrization effects are
considered for the real part of the central potential. Cross
sections are calculated for the scattering of 20 and 40

MeV protons on 12C and 40Ca using the theoretical optical

potential and compared to cross sections obtained from

empirical optical potentials.

THE OPTICAL POTENTIAL IN PROTON—NUCLEUS SCATTERING

By_ {
. ;;

Donald A: Slanina

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

1969

657355
7’ 3‘67

II.

III.

IV.

V.

TABLE OF CONTENTS

Introduction.........................
General Theory........ ........ .......

l. OptiCal Potential.Series

2. Antisymmetrization.Scattering

3. Imaginary Optical Potential
Effective Interactions...............

l. Impulse Approximation,

2. G—Matrix Effective Interactions
Nucleon Density......................

DiSCUSSionOOOOOOOOIOOOOOOOOOOOOOOOOCO

ReferenceSOOOO ..... OQOOOOOOQOOOOOOOOOOOOOOO

\

AppendixAOOOOOOOOOOOIOOOOOOOOOOOOOOOOOOOOO

Appendix B000...OOOOOOOOOOOOCOOCOOOOOO......

AppendixCOOQOOOOOQOOOOOOOOOOIOOOOOOOOO....

AppendixDOOOOOOOOOOOOOO0.0060000000000000.

ii

.20

.25
.28
~36
.39
.43
.46

.50

Table

Table

Table

Table

Table

Table

Table

Table

LIST OF TABLES

l: The impulse approximation parameters fit to one

and two Yukawas.,..

.............. ..... ................52

2: The charge and proton point densities,............53

4O

3: Hartree—Fock parameters for. Ca oscillator

constant = a = 2.08f.; energy/nucleon = -7.47 MeV....54

4: UR and r2 for only the direct term of the real

central potential.....................................55
5: Energy dependence (VB) and isobaric dependence
(VI) of WC and SG for the direct term and the total

potential. .............. ., ......... .. ................. 57

6: UR and <r2> for WG and SG for the direct (D),

exchange (E), density difference (p) and total (T)

potentials. .... .......... . ..... . .......... . ........ ..58
7: <r2> and.Woods~Saxon-parameters for proton,

neutron and matter distribution. ...... ................59
8: Spin orbit parameters. ...... ............... ......60

iii

LIST OF FIGURES

Figure l: The charge and proton point distribution for
40Ca compared to the distributions obtained by pure

oscillator functions and Hartree—Fock functions.,,,,,61

Figure 2: The theoretical real central potentials of IA, KB,
and KK with no antisymmetrization and no density
difference are compared to the empirical potentials

of Fricke et al and Greenlees et al... ............... 62

Figure 3: The theoretical real central potentials of WG and
SG with no antisymmetrization and no density difference
are compared to the empirical potentials of Fricke

et al and Greenlees et al............................63

Figure 4: The theoretical real central potential of WG
where the effect of antisymmetrization and density
difference are included.................. ...... ......64

Figure 5: The variation of the strength of the real central

potential with energy for 12C and 40Ca using the WG

effective interaction................................65
Figure 6: The real central potential of WG for 40Ca at the
lab energies of 20 and 40 MeV........................66

iv

Figure 7: The theoretical real central potentials of WG
and SG with antisymmetrization and density difference
are Compared to the empirical potentials of Fricke

et al and Greenlees et al.... ...... ........ ....... ...67

Figure 8: The real spin orbit potential of IA is compared
to the empirical potentials of Fricke et al and
Greenlees et al. ....................... . ....... ......68

Figure 9:- The imaginary central for 40Ca at 20 and 40 MeV

are compared to the empirical potentials of Gray et
al and Fricke et al.......... ..... ...................69

Figure 10: The theoretical optical potentials for 12C at

20 and 40 MeV are compared to the empirical potentials
of Cameron and van Oers. ................. . ........... 70

Figure ll: The cross section for 12C at 20 and 40 MeV for

the theoretical optical potential compared to the
potential of Cameron and van Oers. ................... 71

Figure 12: The cross section for 40Ca at 20 and 40 MeV

for the theoretical optical potential compared to

the potential of Gray et al and Fricke et al. The
crosses denote the theoretical potential with the

imaginary term being replaced by its' empirical

counterpart. ........................... . ............. 72

I . INTRODUCTION

The optical potential is calculated for 40 MeV

protons and the spin zero nuclei 12C, 40Ca 58Ni 120Sn

208P

I I I

and b. To first order in the nucleon-nucleon effective
interaction, t(r), the optical potential V(r) is written
as a folded integral of t(r) and the nucleon point density

or matter density pm(r)l

V(r) = Aft(l£'-£l)pm(£')d_r_' (l)

The effective interaction t(r) must be a continuation
of the effective interaction G(r)2, derived from free
nucleon-nucleon scattering data and used in calculating
the bound state prOperties of nuclei. Such an interaction
is state dependent and is different in different relative
angular momentum states. To calculate the folded integral
easily, it is necessary to have the effective interaction
expressed in configuration space and the only angular
momentum projection for which this is easy to accomplish
is the separation of the interaction into parts acting in
even and odd relative states. However, the interaction is
strongest in s-states and, for hard core potentials like
the Hamada—Johnston3, can be approximated in this and other
even states using a Scott—Moszkowski4 separation distance

method giving a configuration space interaction that

vanishes inside the separation distance. In this approxi—
mation there is an effective central force in triplet.even
states arising.from the tensor part of nucleon-nucleon
force. Thus, the strong part of the force is given mainly
as an effective central interaction in configuration.space.
Estimates for the interaction were taken from Kuo and

Brown (KB)5, Kallio and Kolltveit (KK)6, and a density
dependent interaction designed to mock up the state depend—
ence of the G-matrix from Green7. Two forms of Green's
density dependent interaction are used; WG having a weaker
density dependence than SG. As the last three interactions
act only in relative s-states, the further approximation
that they are the same in all relative even states is made.
Furthermore, since the interaction in-relative odd states
has little effect on the binding energy of nuclei8,.and is
not normally given in configuration space, it is neglected
here and t(r) will be set to zero in odd states except for
the two-nucleon spin orbit potential. Here the impulse
approximation was used to estimate the effective interaction.
Motivation for this arises from the successful spin orbit
splitting calculations of Elliott et a19 where the inter-
action was expressed in terms of free nucleon-nucleon phase
shifts. With these approximations, the effective inter-
action is a central force acting-only in relative even
states, similar to Serber force, with a separation distance
of approximately 1 f, tOgether with a tensor force, negelcted

here, and a two body spin-orbit force.

The importance of using a G-matrix effective.inter-
action is illustrated by the calculation of the realrcentral

40Ca and 40 MeV protons using-a

optical potential for
Serber type interaction which acts only in relative even
states and fits low energy scattering lengths and effective
rangeslo. This interaction gave a much larger strength
and range for the Optical potential than is empirically
observed or calculated using the G-matrix.

The matter density, assuming no proton-neutron
density difference, was obtained by unfolding the finite
electro-magnetic size of the proton from the empirical

charge density. The charge densities of Acker et al11

120Sn, and 208Pb while Hofstader's

values were used for 12C and 58Ni. The matter and charge

were used for 40Ca, 12

densities are related by
och(r) = fop(I£'-£l)pm(£')d£' (2)

where ch, p, and m refer to charge, proton, and matter.
The matter density is assumed to have the same algebraic
form as the charge density. The matter parameters were
obtained by matching the empirical charge densities second
and fourth radial moments to those calculated using equation 2.

The calculated optical potentials were compared to

13

the empirical potentials of Fricke et al for 40 MeV protons

and Greenlees and Pylel4 for 16O, rescaling the numbers to

12C. With no antisymmetrization and no proton-neutron
density difference, the real central potentials closely

resembled those obtained by empirical analysis. The major

difference occured in the energy dependence.

 

This difference was accounted for by including
antisymmetrization in the scattering process. .Antisymmet-
rization accounted for 80% of the energy variation given
in Fricke et all3. The potential due to antisymmetrization
is non-local and its' local equivalent was estimated using
the method of Perey and Saxonl6.

In order to correlate some of the Optical model
parameters in the empirical analysis, Greenlees et all4
define a nucleon point density which is independent of
the proton density. They then assume a two nucleon inter—
action with strength and range as parameters, and search
on these parameters for best fit. This analysis leads to
a large neutron skin, ie. a large difference in
(<r2>p—<r2>n)l/2. If this large proton-neutron density
difference is used in Equation 1 with a realistic G—matrix
effective interaction, the strength and range of the
resultant real central potentials overestimate the empirical
potentials. Thus, much of the effect attributed to proton-
neutron density difference by Greenlees is included in the
present G-matrix effective interaction. In contrast to
Greenlees, the isobaric analogue state calculations of
Nolen et all7 given a small proton—neutron density difference.
On the theoretical side, the Hartree Fock wave functions
of Tarbutton and Davies18 give a small neutron skin. Their
values of the mean squared radii, renormalized so that the

calculated and empirical proton mean squared radii were

equal, were used to estimate the neutron skin for 40Ca and

208Pb. The difference between neutron and proton.distri-

208Pb was about half

bution radius obtained this way for
the value obtained by Greenlees and was almost duplicated
by a harmonic oscillator shell model calculation. For
this, the oscillator constant, taken to be the same-for
protons and neutrons, was fixed by the mean squared radius
of the empirical proton point distribution. The inclusion
of this small neutron skin for the four nuclei brought the
calculated strength and range of the real central potentials
closer to the observed values.

Since the effective interaction used is hermitian
the leading term for the imaginary part of the optical
potential comes from that part of the second order term in
the effective interaction which involves inelastic scatter-
ing on the energy shell. It was assumed that, in the sum
over intermediate states, only the low lying collective
states excited by inelastic scattering are important.
The method of Perey and Saxon16 was again used to estimate
the equivalent local potential from the resulting non-local
and angular dependent potential. The examples considered

are for 20 and 40 MeV protons on 12C and 40C

a. The cross
sections obtained by using the theoretical optical
potential, using the weak Green effective interaction (WG)
to estimate the real central part, were calculated and
compared to the cross sections obtained by using the
empirical optical potentials. The comparison was relatively

good in the sense that the general shape of the cross

sections are the same.

Chapter II contains a general derivation.of.the.
optical potential series and the algebra associated with
the antisymmetric part and imaginary part of the potential.
The effective interactions used are presented in Chapter III
and the nucleon point density is given in Chapter IV.

The results are discussed in Chapter V.

II. GENERAL THEORY

1. Optical Potential Series:

 

The optical potential reduces the nuclear many
body scattering problem to the equivalent problem of one
particle scattering in a complex potential well. The
nuclear T-matrix is reordered so that the variables of
the target nucleus are assimilated in an effective
potential, the optical potential. The assumptions used
in this section are that the nucleon-nucleus potential
may be written as a sum of nucleon—nucleon interactions
and that antisymmetrization of the incident nucleon with
the target nucleons may be neglected. This point will
be returned to later.

The Hamiltonian for the system is1

H=H +K+v=H +v (l)
n 0

where Hn is the nuclear Hamiltonian, K is the kinetic energy
of the incident nucleon, and V is the nucleon-nucleus
interaction
V(r)=E(lr-zi|)
l
and

= 2
HnUm(zl,...,zA) EmUm(zl,...,zA) ()

where Zi contains the spacial, spin, and isospin coordinates
of the ith nucleon in the nucleus.

The nuclear T-matrix is

T = V+VGT (3)

where G=(E—H +is)—l. For elastic ground state scattering

O

we wish to obtain an integral equation for T of the form

T = W+WGTe (4)

el 1

where G contains no excited states Of the nucleus, W is

the optical potential and Te is diagonal with respect to

l

the nuclear ground state. Let n=0 be the nuclear ground

state and define the projection operators

P = |0><0| ; Q =% O|n><n|
with

P+Q=l
and

TelzPT

If we operate with P and Q on Equation 3, we obtain

PT=PV+PVG (P+Q) T

QT=QV+QVG(P+Q)T (5)
or

PT=[PV+PVG(l-QVG)—1QV][1+GPT]

and the Optical potential is

w: pv+pvc(1—st)”lov (6)

to first order: W=Ul=PV

to second order: W=U1+U2=PV+PVGQV

If U1 is written in coordinate representation,
Ul(r) = Aft(|£'-_r_l)p(r')d£' (7)

where A is the number of nucleons in the nucleus, t(r).
is the effective nucleon—nucleon interaction, assumed.to
be local, and p(r) is the density of point nucleons in

the nucleus.

2. Antisymmetrization Scattering

 

Including antisymmetrization in the collision
process has been considered in detail by Levinzo, the
leading term of which replaces the effective interaction
t(r) by an operator tas(r) which antisymmetrizes the

incident particle with one Of the nucleons in the nucleus.

tas(r)

or

= tD(r)+tE(r)

where pt, pS, and p2 exchange the isospin, spin, and
spacial coordinates of the two nucleons involved in the
collision and D and E refer to the direct and exchange
due to antisymmetrization parts of the amplitude. The
effective interactions used to estimate the exchange are
the density dependent interactions of Green7, WG and SG,
acting only in relative even states. If SE and TE label

the singlet even and triplet even parts Of the potential,

lO

tas(r) = t(r)[l+p21 (9)

where

] N-Z

t(r)=%€IV +v 'IEAA

SE TE V

v (10)

SE- TE]
and N and Z are the number of neutrons and protons in
the target nucleus.

In coordinate representation, the first order

potential is

<£'|Ull£>=A<o,£'Itasl£,o>=UD+UE

where D and E label the direct and exchange parts Of U1.
In what follows it will be assumed that t(r) is a local

potential in coordinate space, ie.
I
l .. __ |_. '—
<£1£2|t|£2£1>‘t(|£1 r2|)5(-r-1 £1>5<52 £2) (11)
As before, the direct term is
UD(r)=AIE(|£'-£l)o(£')d£' (12)

where p(£') is the ground state density. The exchange

term is

_ A _
U (r r£>=AId£ d£l[:_2d£ilt<l£-£l

E _ l)<0|£i><£1l0>5(£l_£1)

5(£i‘£> (13)

and

O(£,£')=I<OI£><£'l0>d£2...d£A

The exchange potential is non-local and the equivalent
local potential VE(r) is estimated by using a method similar

16

to that used by Perey and Saxon and is defined from the

Schroedinger equation.

ll

vE(r)w(£) = AIE<l£—£'|>o(£,£')w(;')d£' (14)
where w(r) corresponds to the distorted wave for elastic

scattering by the real direct well and

V21“;

V
II

-k§w(£>
where k: is the local wave number in the real direct
potential well. {See Appendix A for an outline Of the
same method based on the integral representation of
elastic scattering.}

The method starts by taking the Fourier transform
Of that part of the function that depends on §é£'-£,
and then expanding the Fourier transform in a Taylor
series about some wave number k0.

E<s> = (2w>"3fdg¢‘i2'§ t<p2> (15)

2

0t +...]

= (2n)"3fdpe"i2°§[t(k§)+(p2—k

_ 2 _ 2 2.
_ [t(kO) (Vs+k0t +...]5(§)

where t'=gE§| 2 2. k3 is a free parameter and its' value
d p=k
is determined by flaking (v:+k§) as small as possible.

Keeping only the first two terms in Equation 15

v (r)w<r> = A[{t(k§)—<v§+k§)t'}p<£,5+g)w(£+§)15:0

Since W: Operates on p(r,r'), the density is estimated by

p(£,£'> = 4§m ¢,(r>¢,(r')y*,m(f)ylm<f') (16)

12

where the sum over 2 goes over all of the occupied 2
subshells in the nucleus, and ¢£(r) is the radial harmonic
oscillator wave function for the ith subshell with the
oscillator constant being determined by the empirical

mean squared radius of the density distribution. The
v2p<r r'>| =42 ¢ <r>Y* <f)v2[¢ <r'>Y (fUJ
_'_ 5:0 gm 5L 2m ft 5m s=0

—2 12 2
=q ;,(22+1)¢,<r>
Consider the term

<v§+k§)p(r,r')w<r'>|s=0=< S—kfi+§2>o<r,r>w<r)+[1p<r,r'>-2w<r')13=0

The gradientp gradient w part of the above will be neglected.
If the &2 term and the gradient gradient can be neglected,
the above equation corresponds to a local density approxi—
mation. Perey and Saruis20 calculate a term of this type,
retaining the gradient gradient part, and Obtain a small
correction to the local density approximation. Thus, the
present calculation should give the effect of exchange on
the collision process to within 20% or so. The equivalent

local exchange potential is

_ A 2 _ 2 -2_ 2 , 2 _
vE<r> - ;[t(k0) k0+q k,)t 1E<22+1)¢,<r) (17)
2 . . 2 -2
and k0 IS the maXimum Of zero or kg—q .

The most important part of the preceeding develop-
ment is that t(p2) be smooth enough to be approximated

by the first two terms in the Taylor series expansion.

13

If this were not the case, higher order terms in (v:+k§)
would have to be retained and the complexity of the
problem increases by orders of magnitude.

The IA is used to calculate the spin orbit potential
and contains the effects of exchange as noted by Takeda.
and WatsonZl. Hence, the above procedure will not be used

for the spin orbit potential.

3. Imaginary Optical Potential

 

In Operator form, the first correction to the Optical
potential of Equation 7 is

U2=PVGQV

and in coordinate representation is

 

 

(3' 107-15. 32.4.0 (2">fl3fda<£'o|V|n£">d£"Gn(meiB' (Ii—"‘3" "<15: "
X <.r_"'n|VlO_r_> (18)
where
Gn(P) =[fi::2 - nJh:fiz + ie]“l

n labels the excited states of the nucleus, and the matrix
elements of V are related to the form factors used in the
calculation of the inelastic scattering amplitudes. Since
the interaction depends on spin and isospin, the integration
over 2 includes a sum over spin projections. Let V be a
local real interaction and let a', b, and a refer to the

spin-isospin state Of the incident proton. Then

14

(210'3 l 2

2 m 'm m Id2<a|olV|nb>Gn(p) (19)
(

a b a
is _r_' -£>

x e <bn|V|oa>

The 1/2 arises from the average over final spin projections.
Equation 19 will be used to estimate the imaginary
part of the optical potential. Since V is real, the
imaginary term comes from the on energy shell part of.
Gn(p); ie. those inelastic states which can be energetically
excited. The calculational model neglects all other
intermediate states such as pick up. The real part of
the second order term corresponds to the term used to
estimate the effects of core correlations on the binding

energy and its' effect is small8. The imaginary potential

 

is then
. 2 2 2 2
. ___1L__ 12' <_r_'-£> 41.1.; _ 31.2.
W(£’£)_ 3g+0 H'm mfdpe 5(2m En 2m ) (20)
2(2fl) a b a.
x <a'olVInb><bn|V|oa>
If k2=k2—2mEn, then
n hz 2 2
,fi k 2 2
5(..__.n -12.) = i5(p-k)
2m 2m /h2 n
P

Let §;£'—r and integrate equation 20 over p. The result

 

 

is
sin(k 8) ~ ~..
W(r',r)=- m 2 Z Z, n E FLSJn Tn(r)FLSJn Tn(r,)
— 4W“ n+0 mambmamn s Sm
Lislml
* A, l l _ _ l l _ . _
X YLM(f)YL'M'(r )<§ 2ma mblsma mb><2 Ema' mbIS ma' mb>
(21)

C‘ _ < I ' I _ >
x <Lsta mlenMn> L s M ma, mbIJnMn

 

15

where the F's are the form factors for the inelastic
scattering from the ground state to an excited state n
and are defined in Appendix B. Thus, the imaginary
potential is both non—local and angular dependent.

The major contribution to Equation 21 arises when
the spin transferred is zero, S=S'=O. This removes the

Clebsch—Gordon coefficients from Equation 21 and

sin(k s)
W(r,r')=- m2 2' s n FJ(r)FJ(r')ZY
4Wfi J M

where the prime indicates that the contribution of the

*
JM

 

 

(by (55')

JM
ground state is to be omitted and FJ(r)=FJOJ’T(r)

The procedure of Obtaining the equivalent local
potential is more complicated than it was for the exchange
potential: sin(kns)/s does not have a useful Fourier
transform which can be expanded in a Taylor series. TO
obtain a suitable Fourier transform, a function of s, f(s),
should be taken out Of FJ(r)FJ(r') such that f(s) sin(kns)/s
does have a Fourier transform. The same effect should be

produced by multiplying Equation 22 by

-a52 +2152
1 = e e
and let
—as2
g(s) = e sin(kns)/s
2 (23)
GJ(r,s) = eas FJ(r')

Here a is a free parameter and its' value is

determined from the condition that

16

2 2 d 2
o 2 2_ 2
dq q —k0

over the interesting range of q2 where g(q2) is the Fourier

g(q2)=g(kg)+(q

transform of g(s). The value of a used is a=l.0 as it
gave a g(qz) that was approximately linear.

The local equivalent potential was then found by
using the Perey-Saxonl6 method previously outlined. Here
k5 will be the lab energy wave number and V(r) will

correspond to the distorted wave corresponding to elastic

scattering by the real potential well.

_2

’H

The equations needed for the potential are

lab

v2w<r>=—k:w(r>

ki=Z%-[E
”H

v2[6J(r,s)YLM<e'>JS=O=[6aFJ<r)+§J<r>JYLM<f>

1ab"VREAL(r)]

 

where
“J _ l a 2 d J _J(J+1) J
F (fl-[73?]? fiF (r) ——-2——F (1')]
r r
and
2J+l *
= Z A
4n MYJM(f)YJM(r)

Also the gradient gradient term will again be neglected,
giving a local equivalent imaginary Optical potential of

w (r) = ———E§—§ §'(2J+1)XJ(r>FJ(r) (25)
l6fl‘h

17

where

xJ(r>=[g(k2>—6a-9-2-g<q2>| 2 21FJ<r>-§J<r> (26)
dq q =kO

The question of convergence is more important
here than it was for the exchange potential mainly because
the imaginary poential is not a small effect added to
a much larger potential. In general, convergence will
be served if Equation 25 is relatively insensitive to
changes in kg.

but inelastic studies22 indicate that convergence may

For small values of r, the above holds

be a more serious problem at the nuclear surface.

TO calculate the contribution from the on energy
shell inelastic scattering states it was assumed that
the most important contributing inelastic scattering
states are the low lying collective states which include
the effects of long range correlations. The effect Of
long range correlations on the imaginary potential wa
studied by Terasawa23 and he found that pairing correlations
enhanced the potential by a factor of three. Thus, the
strongly correlated states should be the most important
and these are the strongly excited T=O states, 2+ and 3—

in 12C and the 3— and 5— in 40C

a. An energy weighted sum
rule was used to estimate the strengths Of the higher

excited states Of a given multipole.

18

The low lying collective state wave functions for

12C and 40Ca were taken from Gillet and Sanderson?4

0+, 2+, and 4+ states are important in 40Ca but were not

The

available in Reference 24. These states were then assumed
to be a sum of all energetically possible 2am particle-
hole pairs. This procedure will underestimate their
contribution to the imaginary potential because of the
importance of correlation523.

A sum rule is used to estimate the strengths of
the higher lying collective states Of a given multipole.
The energy weighted sum rule is taken from Lane25 and is
a measure of the total electromagnetic transition

strength of a given multipole J,

SJ

_ .zJ 2
A(En E0)I<n|<iri YJO(fi)IO>I (27)

and

U)
ll

2
J 'gfig-J(2J+l)fer(r)d£/fp(r)d£ (28)

where En is the energy of the nth excited state of multipole

J and p(r) is the nucleon density of the nucleus. Now,

the inelastic scattering matrix elements are very similar

to those in Equation 27 and it will be assumed that Equation

27 is a good estimation of the relative strengths Of the

inelastic scattering states corresponding to a multipole J.
Consider Equation 27 tO be rewritten as

J_ J -J
S —Sl+82

19

where Si is the transition strength of the low lying

state, n=1 in Equation 27, and Si will contain the rest Of
the transition strength of the multipole J. Then S3

will be considered as a pseudo-state which lies lfim
higher in energy than the lowest collective state. The

value of S; is Obtained using Equations 28 and 29 where

Si is calculated using the wave functions in Reference 27.

The wave function associated with Si is then considered

to be Of the same form as the low lying state but rescaled

J

by a value associated with S2.

III. EFFECTIVE INTERACTIONS

l. Impulse Approximation

The impulse approximation effective interaction
(IA) comes from solving the free nucleon-nucleon t—matrix.
It is basically a high energy approximation as it neglects
the binding of the struck nucleon. Watson and Takeda21
place the lower limit of its' application at around 100 MeV.

The impulse approximation presented here will
include off energy shell kinematics. The nucleon-nucleon
collision will conserve energy in the nucleon-nucleus
center of mass system but not in the nucleon—nucleon
center Of mass system. The ansatz used will be that the
momentum transferred, q, is the same in both systems.
This is equivalent to taking nuclear recoil into account.
Under this ansatz, the final nucleon—nucleon center of

mass momentum is

,2_ 2 A-l 2
k —k +——A q (1)

where A is the number Of nucleons in the target and k is
the initial nucleon-nucleon center of mass momentum.
For elastic scattering from spin zero nuclei, the

relevant part of the nucleon-nucleon t-matrix is

t(q) = A(q)+C(q)g'fi (2)

20

21

where A(q) and C(q) are the appropriate Wolfenstein26

parameters and n is a unit vector perpendicular to the
scattering plane. These parameters are still Operators

in iSOSpin space, ie.
A(q)=AO(q)+Al(q)11°12

The calculation uses the Hamada-Johnston potential and
the off energy shell matrix elements are calculated by

the method Sobel27

used in his bremsstrahlung calculation.
The algebra necessary tO Obtain the pseudo phase shifts

is presented in Appendix C and the expressions for the
Wolfenstein parameters in terms of the reaction matrix
elements proceeds in the standard manner26.

The amplitudes A(q) and C(q) are fitted to a sum
of two Yukawas and, in order to Obtain an idea of their
strength and range, they are fitted to a one Yukawa
potential in which the range is obtained from the mean

squared radius of the two Yukawa fit. Let t(q) stand

for either A(q) or C(q), then
V V

 

 

t(q)=41T[ : 2 + g 21 (3)
al(q +al ) a2(q +a2 )
or
_ -a r —a r
t(r)—Vle l /alr+V2e 2 /a2r
and
t(r)=V e—ar/ar

O

22

The values of the parameters of the Yukawa potentials
that are used are listed in Table l.

The IA was used to calculate the real central
potential, but its' main purpose is to estimate the spin

orbit potential for reasons mentioned in the introduction.

2. G—Matrix Effective Interactions

 

At low incident lab energies, the best estimates
for the effective interaction should arise from the
continuation in energy of the G-matrix interaction used
in bound state calculations. Like the impulse approximation
t-matrix approach, they are based on low energy free
nucleon-nucleon scattering but they also describe nucleon-
nucleon scattering in a finite nucleus. As explained in
the introduction, they act mainly in relative even states,
resembling a Serber force, and will be zero inside a
separation distance, d. Under these assumptions, proton-
proton (pp) and neutron-proton (np) parts of the inter—

action are

_l , _1 3
tpp(r)_ZVSE(r) , tnp(r)—8VSE(r)+8VTE(r)

where SE and TE refer to the singlet even and triplet even
parts of the force.

All of the G-matrix effective interactions are given
as effective central interactions. The first estimate comes

from Kuo and Brown5

23

VSE(r)=Vc£(r) r>d : =0 r<d
KB
_ _ 2 . _
VTE(r)—Vc2(r) 8vt£(r)/24O r>dt . —0 r<d

where VCR and Vti are the long range parts Of the Hamada-
Johnston potential for the central and tensor components,
respectively. For lab energy of 40 MeV, the separation

distances are
d =1.05 f ; d =l.O7 f
s t

The next estimate comes from the studies of Kallio and

Kolltveit6
v (r) =-330.8e_2'4021(r—°4) r>d : =0 r<d
SE s s
KK
__ -2.5214(r-.4) . _
VTE(r) — 475.0e r>dt . —0 r<dt

and, for 40 MeV protons, the separation distances are
d =l.O46 ; d =O.924
s t

The effective interaction of Green7, using the KK inter—
action, used the local density to account for the state

dependence of the interaction.

_ _ 2/3 KK . _
VSE(r)—Cs(l asp )VSE(r) r>dS . —O r<dS
Green
_ _ 2/3. KK . _
VTE(r)—Ct(l atp )VTE(r) r>dS . —O r<dS

24

where WG:
C =.992 a =.035
s s
Ct=l.07l at=l.454
SG:
C =l.157 a =.323
s s
Ct=l.623 at=l.845

and p is the local density. This interaction uses the

KK separation distances.

IV. NUCLEON DENSITY

The nucleon densities for the target nuclei were

Obtained from the electron-nucleus scattering results of

12 12 58

Hofstader for C and Ni and the muon-nucleus

scattering results of Acker et al11 for 40Ca, 120Sn

208P

I

and b. The proton point distribution is Obtained by

unfolding the finite electromagnetic size of the proton

from the empirical charge distribution28.

och(r)=f-op(l£'-£|)pm(£')d_1_:_' (l)

where ch, m, and p refer to the charge, proton point, and
proton densities. It was assumed that pch and pm were of
the same algebraic form. The proton density used was of

the form
p (r)=n_ a e p ; ag=0.427 (2)

and the target nuclei were of Woods—Saxon form

(r-c)/a]-l

p(r)=po[l+e (3)

Equation 1 gives relations between the radial
moments of the three distributions.

(r2> =<r2> —<r2>
m ch

-<r4> - £9,<r2> <r2> (4)

4 _ 4> _ 4
<r >m_<r <r >ch p 3 ch p

ch

25

26

These moments are, for a Woods-Saxon distribution,

<r2>=.2C2(3+7x)

<r4>=c4(3+18x+31x2)/7

_ na 2
x —(c )
The proton point parameters, cm and am, were Obtained

by solving the two equations in Equation 4. Then

TTa
_ m 2
xm—(Ee—)

S

(49y—3DX§+(42y—18)xm+9y—3=o
where
y=7<r4>m/25<r2>i

and cm and am are obtained from

_ 2_
—5<r >m/(3+7xm)

Btu B N

=xmci/fi
Of the neutron distribution is assumed to be the
same as the proton point distribution, the above parameters
are those to be used for the matter distribution.
Tarbutton and Davies18 found a small difference between
the neutron and proton densities in their Hartree—Fock
calculations and their results were closely duplicated
for 208Pb by using the harmonic oscillator shell model

picture Of the nucleus. The shell model proton density

was assumed to be spherical and of the form

o(r)=% <22+1>¢§<r>

Z
2

27

where the sum over 1 goes over all the filled A subshells
and ¢£(r) is the radial harmonic oscillator wave function
for the 2th subshell. The harmonic oscillator constant,
a, is obtained from the charge density by using Equation 4
and the neutron density is Obtained from this a and using
the lowest filled neutron states.

The values used for the density parameters are listed
in Table 2. For a further explanation Of the symbols
used and a discussion concerning the replacement of the
pure oscillator wave functions used in Equation 5 by Hartree-
Fock wave functions, see Appendix D along with Table 3 and

Figure 1.

V. DISCUSSION

The first part Of the Optical potential that will be
considered is the real central potential. Initially,
exchange scattering and a possible proton-neutron density
difference will be neglected. Under these assumptions,
the potentials Obtained are listed in Table 4 and Figures 2

and 3 with

UR=fV(r)d£ (l)
and

r2 =fr2V(r)d£/UR (2)

The theoretical potentials are compared to the 30 MeV
proton analysis of Greenlees et all4 and the 40 MeV proton
analysis Of Fricke et all3.

The agreement of the potentials between themselves
and to the empirical potentials is good considering the
calculation is a first order one and that rather rough
approximations to the G-matrix were made. The major
point is that the theoretical potentials which come the
closest to matching the empirical potentials are those
based on the G—matrix problem for finite nuclei which do

take into account, even if only approximately, nucleon—

nucleon phase shifts up to several hundred MeV and the

28

29

presence of other nucleons. The importance of using a
G-matrix based effective interaction was illustrated by
using a Yukawa force, acting only in relative even states,
taken from Preston10 to calculate the real central potential.
This interaction fits low energy nucleon-nucleon scatters
ing lengths and effective ranges but using it to calculate
the Optical potential gives a much stronger potential

than the empirical potentials. For 40 MeV protons incident

on Ca40, the Preston interaction gives a potential with.

UR=—22,400 MeV f3 and <r2>=24 f2 while the empirical

l3 3 2 2

potential of F gives UR=—15,330 MeV f and <r >=l6.43 f

and the weak Green (WG) G-matrix interaction gives a

3 2 2

potential with U =-12,910 MeV f and <r >=l5.12 f .

R
The same result occurs when the Preston interaction is

208P

used for b. The resulting Optical potential has

UR=—ll6,300 MeV f3 and <r2>=41.92 f2 compared to the

3 2 2

empirical valuesl3 of U =—79,2oo MeV f and <r >=37.19 f

R

and the theoretical values, again based on the weak Green

G-matrix interaction, Of UR=-69,OOO MeV f3 and <r2>=33.80 f2.

Two other characteristics of the real central
potential should also be considered. In the analysis Of

13

Fricke et a1 , the strength of the real central potential

varies with respect to energy and neutron excess as

N—Z
(r 0) 0+ EELAB+ I[ A ] ( )

 

 

30

where V =4l.l MeV, V =-.22 MeV, VI=26.4 MeV, and the

0 E

Coulomb term was suppressed. The theoretical direct
potentials gave almost no energy dependence but~gave-
approximately the proper isobaric dependence, see Table 5.
The WG and SG potentials are used to consider the effects
of exchange and neutron—proton density difference on

the characteristics Of the real central potential.

The effects of exchange on U <r2>, and the general

RI

shape of the WG and SG potentials is small as seen in

Table 6 and Figure 4. The inclusion of exchange slightly

increases the <r2> and makes the resultant potential

more Woods-Saxon in shape as the exchange contribution
Of exchange is in the energy dependence. Exchange
accounts for 80% of the energy variation between the lab
energies Of 30 to 40 MeV, see Table 5. The theoretical

values Of V =-.21t.01 compare favorably to the value

E
1
obtained by Fricke et all3. Visually, the change of the

strength of the potential with energy is given in Figure 5

12 40C

for C and a, and the change of shape with energy

4OC

is given in Figure 6 for a. The concave shape of

the energy dependence seen on Figure 5 also seems to be

indicated by the empirical analysis Of Cameron and van Oersl5

for 160. There also appears to be a mass effect for the

energy dependence, V decreasing with A, but this effect

E

may be beyond the resolution of this calculation.

31

The empirical analysis Of Greenlees et all4 used

the proton-neutron density difference to reduce the:
number of free parameters used in the search procedure
for the optical model potential by relating the real
central and real spin orbit mean squared radii-to a
matter distribution. In terms of the mean squared radii
for the real potentials,

real central: <r2> =<r2> +<r2>
R 2n,c m

. . 2 2 2
real spin orbit: <r > =<r > +<r >
so 2n,so m

where 2n refers to nucleon-nucleon and m refers to the
matter distribution. The analysis used

_ 2 2 _ 2
€-<r >2n,c+<r >2n’so—2.25 f

The value of E was obtained from a best fit search and

it leads to a large neutron skin. The values Of 5 obtained

from the G-matrix effective interactions are g=5¢1 f2.

Thus, the use of a reasonable G—matrix effective inter-

action absorbs a large amount Of the neutron—proton

density difference inferred by Greenlees et all4.

Another source for the estimate Of the neutron-

proton density difference is from ththheoretical Hartree-

19 4O

Fock calculation of Tarbutton and Davies for Ca and

208Pb. They Obtain a much smaller neutron skin that

Greenlees et all4 and their values are comparable to the
values Obtained in the isobaric analogue state calculations

of Nolen et all7. Since the harmonic oscillator method

32

outlined in Chapter IV gave a neutron skin similar to
40 208

the one of Tarbutton and Davies for Ca and Pb, it
was used to estimate the neutron distributions for 58Ni
and 120Sn. The mean squared radii for the various neutron

distributions are listed in Table 7. The effect Of the.
density difference on the.form of the WG potential is
illustrated in Figure 4.

The density difference and.exchange effects were
included in the WG and SG potentials and are presented
in Table 6 and Figure 7. The values Of UR and VI isolate
the WG as the best estimate of the effective interaction.
Because of this, the WG potential is used for the real
part of the theoretical optical potential for the calcula-

12C and 40Ca.

tion of the cross sections for
The IA was used to estimate the real spin orbit
potential because of the reasons presented in the intro-
duction. Since p(r) is Woods-Saxon in form and <r2>2n,so
is small, the potentials are fit to a Woods—Saxon form
whose parameters are given in Table 8 and is illustrated
in Figure 8. The theoretical representation is good in
general and eSpecially good for 58Ni and 120Sn.
Despite the assumptions used in the calculation of
the imaginary central potential, a surface type peaked
potential was obtained which agrees in form with the
observed empirical forms, see Figure 9. The major differ-

ence is that the theoretical potentials peak inside of

 

33

the empirical potentials: The same difficulty occurs
in the micrOSCOpic form factors used in inelastic scatter—
ing. Even though the convergence Of the Perey-Saxonl.6
method used is difficult to estimate, the method should
give the-gross structure of the imaginary part of the.
optical potential. The important points of the calcula—
tion are that the strongly excited low-lying collective
states are very important and give a large contribution
to the-imaginary potential: The-total contribution Of
a given multipole to the imaginary potential can be
extracted from the use of a sum rule. A case in point
is that the T=0 2+ state in 12C accounts for about 80%
of the calculated potential for 20 MeV-protons. The
importance of using collective states was noted by
Teresawa23 in his calculation of the imaginary potential.
He noted that pairing correlations increased this part
of the potential by a factor of 3. A similar effect
was noted for the T=0 3— state of 40Ca. The collective
state gave a contribution Of 1.15 MeV for 40 MeV protons
while, if the 3— state was replaced by all possible 1h
particle-hole pairs and its' contribution to the imaginary
potential is certainly-underestimated.

The total optical potential for 40Ca is illustrated
in Figures 7, 8, and 9 and is compared to the empirical

Optical potential Of Gray et al29 at 20 MeV and Fricke

et all3 at 40 MeV and the total optical potential for 12C

 

 

34

is illustrated in Figure 10 and_is compared—to the rescaled
160 parameters of Cameron and van Oersls. These optical
potentials, both empirical and-theoretical, are used to
calculate the differential cross sections for incident
protons of 20 and 40 MeV. The cross sections obtained

are compared in Figures 11 and 12. The general shape

and magnitude of the cross section based on the theoretical
Optical potential is much closer to the empirical cross
section at 20-MeV. To see how much of the discrepancy

was due to the imaginary potential, the cross sections

are also plotted for the case-where the theoretical
imaginary potential is replaced by the empirical imaginary
potential. This effect is denoted by crosses in-Figure-lZ.
The agreement is amazingly good at 40 MeV and implies

that only the real potentials are well represented by

the theoretical estimates at that energy. (At 20 MeV there
is no noticeable improvement resulting-from the interchange
Of imaginary potentials. This illustrates that the
theoretical estimate Of the imaginary term is approxi-
mately as good as the empirical estimate whereas the

neglected inelastic channels may be important at 40 MeV.

35

Even with all of the assumptions used-in the-
calculation, the-theoretical-empirical agreement of.
the Optical potential is good. The major point of this
paper then rests on the consistency of the G-matrix
effective interaction which is a good estimate for the
Optical potential effective interaction, is used in.the
bound state problem of finite.nuclei, and has its‘

foundation in free nucleon-nucleon scattering.

l.

10.

REFERENCES

A. K. Kerman, H. McManus and R. M. Thaler, Ann. Of
Phys. 8(1959)551; H. McManus. Les Mecanismes Des
Reactions Nucleaires, (Grachen/Saint-Nicolas Press,

1964 page 289).
T. T. S. Kuo, Nucl. Phys. A103(l967)71.
T. Hamada and I. D. Johnston, Nucl. Phys. 34(1962)382.

S. A. Moszkowski and B. S. Scott, Ann. of Phys. 11(1960)

65.

T. T. S. Kuo and G. E. Brown, Phys. Lett. 8(1965.54;

T. T. S. Kuo and G. E. Brown, Nucl. Phys. 82(1966)40.
A. Kallio and K. Kolltveit, Nucl. Phys. 53(1964)87.

A. M. Green, Phys. Lett. 4B(l967)384; A. Lands and

J. P. Svenne, Phys. Lett. 25B(l967)9l.
G. E. Brown and C. W. Wong, Nucl. Phys. A100(l967)24l.

J. P. Elliott, H. A. Mavromatis and E. A. Sanderson,

Phys. Lett. 4B(l967)358.

M. A. Preston, Physics of the Nucleus (Addison-Wesley

Publishing CO., 1962, page 27).

36

ll.

12.

l3.

14.

15.

16.

l7.

18.

19.

20.

21.

22.

23.

37

References (continued)

H. L. Acker, G. Backenstoss, C. Daum, J. C. Sens and

S. A. DeWit, Nucl. Phys. 81(1966)l.

R. Hofstader, Ann. Rev. Nucl. Sci. Z(l957)23l; H. F.
Ehrenberg, R. Hofstader, U. Meyer—Berkhout, D. G.

Ravenhall, and S. S. Sobottka, Phys. Rev. 113(1956)666.

M. P. Fricke, E. E. Gross, B. J. Morton and A. Zucker,

Phys. Rev. 156(1967)1207.

G. W. Greenlees, and G. J. Pyle, Phys. Rev. 149(1966)
836; G. W. Greenlees, G. J. Pyle, Y. C. Yang, Phys.

Rev. Lett. ll(l966)33.

J. M. Cameron and W. T. H. van Oers, University of

California Report, UCLA-10Pl8-10.
F. G. J. Perey and D. S. Saxon, Phys. Letto 10(1964)107.

J. A. Nolen, Jr., J. P. Schiffer and N. Williams,
Phys. Lett. 21_(1968)1.

R. M. Tarbutton and K. T. R. Davies,(to be published).

F. S. Levin, Nucl. Phys. 46(1963)275.

F. G. Perey and A. M. Saruis, Nucl. Phys. 19(1965)225.

Gyo Takeda and K. M. Watson, Phys. Rev. 91(1955)l336.

H. McManus and F. Petrovich,(to be published).

Tokuo Terasawa, Nucl. Phys. 39(1962)563.

24.

25.

26.

29.

30.

31.

32.

38

References (continued)

Vincent Gillet and E. A. Sanderson, Nucl. Phys. 54

(l964)472.
A. M. Lane, Nuclear Theory (W. A. Benjamin Inc., 1964).

H. P. Stapp, T. J. Ypsilantis and N. MetrOpOlis, Phys.

Rev. 105(1957)302.
M. I. Sobel, Phys. Rev. 138(1965)B1517.

L. R. B. Elton, Nuclear Sizes (Oxford University Press,

1961).

W. S. Gray, R. A. Kenefick and J. J. Kraushoa, Nucl.

Phys. 61(1965)542.
F. Petrovich, private communication.

D. M. Brink and G. R. Satchler, Angular Momentum

(Oxford University Press, 1962).

R. Muthukrishnan, private communication.

APPENDIX

APPENDIX A

In Chapter 2, the Schroedinger equation was used
to Obtain an equivalent local potential from the non-local
exchange potential. An alternate way to define the
equivalent local potential is by the T—matrix. This
approach will be useful for inelastic scattering.

With forces acting only in relative even states,
the exchange term in the distorted wave born approximation

is built up of components of the form:

T r2)d£ld£

=2
AS if CiffMif(rl 2

_ *
Mif(£l£2)=x *(51)¢f(£2)v(I51'£2|)¢i(£l)x(£2)

where the x's are the distorted waves discribing elastic
scattering in the final and initial channels and ¢(r) is

the wave function of the bound nucleon.

Following Perey and Saxonl6, one takes the Fourier

transform of that part Of the matrix element which is a

function Of the non-locality, 3 -£

1 2
transform in a Taylor series about the wave number k

0
V(s)=(2fi)_3feil.(£l_£2)[V(kg)+(A2—kg)gxz]dl
dk
2 2 2 dv
=[V(k0)_(vs+k0)g;2]6(£l_£2)

39

=3 and expand the Fourier

40

where
9312:“
dk2 dxfix 2 =k0

The wave number k2 is a free parameter which will be

0
determined such that the first few terms will be important.

If r

—l and s are chosen to be the independent variables,

IMif(£1.32)d§?x_*(£1)¢;(£1)V(k§)¢i(5l)x(£l)

-x— (r1)¢i (r1):k2[(k§ +v 2)¢f(rl)x(rl )1

and, if the independent variables are £2 and g,
_* * 2
[Mif(_r_l£2)d§_=x (£2)¢f(£2)V(kO)¢i(£2)x(£2)
-x(r 2)¢f(r2d)-— V2[(kO 22+v )x (3:2)¢i(1:_2)]

dk
Coming the two results gives
M f(£)=!Mif(£l’£2)d§
_ _* 'k 2
—i 9!; x'*<g)¢i(£)[<k§+v2)¢;(5)xgg>

2 dk

2 2 —* *
+[ (k0+V )x (3) chi (_r_) 1 (bf (_r_)x(§_)
also,
2<¢;x)=-<k§+k2>¢§x+2<2¢;)-<2x)

v2<x'*¢i)=-(k'2+k§)x’*¢i+2<3x'*)~(2¢i)

41

where

2.* 2 * 2 2
V ¢f=—kf¢f' V X(r)=—k X(r)l etc.
Take 1/2 Of the cross term appearing in Mif(r) and

2

integrate by parts remembering that dV/dk is a function

of r because Kg will depend on the optical potential. Then

_* *
(2x )-(V¢i)932¢fx

dk
goes to
—* 2 dV * dV * * dV dV
x [(V ¢.)—-—¢ x+(V¢.)~(V———)¢ x+(V¢-)°(V¢ )-—-x+--¢
i dk2 f i dk2 f l f de de f
(V¢i)°(VX)
and
_* dV *
¢~———(V¢ )°(Vx)
ide f
goes to

..‘k * dv .0de * _* dv *
[(Vx )(V¢ )-- -+X - (V¢.)'(V¢ )+x ¢.(V-——)°(V¢ )
f dk2 i de i f i dk2 f
.* dv 2 *
+X ¢.———%V ¢ )]X
lde f

it
Now, absorb the V2¢i and V2¢f into the non cross term of

Mif(£) (l)
_* * 2 dV
Mgf(£)=x (r)¢f(£)[v<k§>-{kO-2<k'2+k2)}g;21¢i<£)x(5)
and (2)
l l —* dV * dV dV *
M. (r)=—x [(V¢-)(V———)¢ +2(V¢-)(V¢ )-—+ .(V—-—)°(V¢ )]x
if —- 2 i dk2 f i f dk2 i dk2 f
l -* dV * l -*

dV
-¢-(V¢ )]-—-(VX)
1 f dk2

42

where

1

Mif (_r_) =M(i)f (33.) +Mif (33.)

*
SO if terms like Vx and Vx are neglected,

Mif(_r.)=X—*(£_)F(r__)x(£.) <3)
and
k3=§Real[k'2+k2]
F<3>=¢E(_r_)vmg)¢i<£3+2ilmagtk'2+k21§1’2 ¢§<r>¢im
dk -— —_
d l d * * (4)
V O V O
+EF(V¢1) (vef)+2(vg;2) [(V¢f)¢i(r)+¢f(r)<V¢i)1

The expression given in Equations 1 and 2 are similar
to those Obtained by Perey and Sarius.20 In Equation 2
the Vx terms contain a nucleon current density which
should approximately be zero for elastic scattering.

_*
Also, for elastic scattering, x (r) becomes a plane wave.

APPENDIX B

In this appendix the expression for inelastic scatter—
ing form factor, which will be used to calculate the

30

imaginary part of the potential, is presented. The process

considered is
a+A +b+B
with the matrix
<bB|v|Aa>

The interaction will neglect exchange, tensor and spin

orbit forces, and is

A
V(r)= IV(I£-£il)

i
and if ser-El
V(S)=v00(s)+vlo(s)(g’gi)+vOl(S)(Efii)+Vll(S)(2°Ei)(l°li)

Using tensor notation for spin and i-spin have

00:1 TO=1
01:3 le-T-
and
_ _ x+y s s . t t .
v(s)-gtxy( ) vst(s)o_xox(i)T_yTy(i)

To aid in the separation of nuclear and interaction

information, write

43

44

vst(s)=]vst(|£frl|)6(£'-£i)dr'

and expanding both integrands in spherical harmonics
6 '- .
(r r1)

stL(rIr')'_—T§"_'r
r

(s): -2 Y* (r)YLM(fi)fv '2

I
LM LM dr

Vst

In order to treat spin and space equally, introduce the

spherical tensor of rank J as

TfiiJ= fi'x'<LSM'x'IJMJ>YLM,Oi,
Also if
OTSJ’ MJ(r .)_ 2Ty(' )TL:JV VSTL(r'rr)6(r;::i)
then
<bB|V|Aa>=§ST (—)X+yY£M(f)<LSMxiJMJ><b|o§XTTy|a>
<B|fO$§J’MJ(r,r')r'2dr'|A>

The processes considered are restricted to those
where the i-spin projection Of the target and the spin,
i-spin, and i—spin projection of the incident nucleus
does not change. Upon using the Wigner Eckeart theorm,
31

using the phase conventions Of Brink and Satchler,

the matrix element becomes

Sa’mb 1 l
<bBlVlAa>=ESTJM(_) <2 Ema—rubls ma_mb>
<JAJMAMB-MAIJ MB><LSMma-mb|JMB-MA>
1 LSJ, T *
<2TTa O|2Ta ><T ATT ABo|T TA >F (r)YLM(f)

45

where
~LSJ,T _ , LSJ,T , .2 .
F (r)—IVSTL(r,r)F (r )r dr
and
6(r'-r.)
FLSJ'T(r')=/7‘/TZT113<a J T ||ZTT(1)TLSJ(1) 1
BBB]. r,2
lldAJATA>

where the quantum numbers used above are defined in

IAa>=IdAJAMATATA>IsamataTa>

IBb>=|dBJBMBTBTA>IsambtaTa>

and L,S,T, and J are the orbital angular momentum, spin,

i-spin, and total angular momentum tranferred to the nucleus

during the reaction.

46

APPENDIX C

26

In an ordinary phase shift calculation the M—

matrix is defined from the scattered spherical wave
ikr

wsc<r,e,¢)=M<e¢>lxinc>er

 

where wsc is the scattered wave and lXinc> is the

initial spin state. M is related to the S—matrix and

the R-matrix by
S=R+l

and

1 1/2

M(G¢)=E£, ,m,(e¢)<£'m'|R|£o> (l)

m' Q
For the spin zero case, the differential equations

we have to Solve are

2

(C: 2 ug(r)+[k2-—————£(£:l) -U(r)]u£(r)=0 (2)
r r

2

d .2 £(£+l) _

g;— 2(r)+[k --;—2-—]F2 (r)—0

where u£(r) is the actual wave function and F£(r) is the
regular bessel function with the boundry conditions

a . _£fl
F2(-)r_>°o_...> sn.n(k r —2)
(3)

n in
u£(r)r*59»51n(krm—§+6£)

47

From Equations 2 and 3, we get

sin6£=-%IOF£(k'r)U(r)u£(r)dr

This defines the one energy shell phase shift.

k'=k, the pseudo phase shift is defined

(X)

__l .
Ag— ETJOF£(k r)U(r)u£(r)dr

with
Ag =sin 62 if k'=k
and since
<£|R|2>=2i|sin6£ for k'=k
then

<£|R|£>=2i A for k'+k

Q

Once the matrix elements of R are represented in
terms of the pseudo phase shifts, the procedure to obtain
the Wolfenstein parameters is the same as that for the
regular phase shift calculation26. Equation 4 holds for

£=j(j=£+§) and s=0 or 1. Since the Hamada-Johnston3

potential has a tensor part, the wave functions for

If

£=jil are coupled. Using Blatt—Biedenharn phase shifts,

the equations that are to be solved are
2

 

.d 2 (j+l)“ — . - m o +
[—-+k —————l e—mv. (r)]u (r)———V. (rm (r)=O

dr2 r2 4’12 3 l 152 3 l

2 9

d 2 (j+l)(3+2) m + + m 0 - _
[———+k — -—-Vy(r)]u (r)——-V.(r)u (r)—O
dr2 r2 412 3 2 ,52 3 2

(5)

(4)

48

where

V§=VC'(j+2)VSL'[2(3'1)/(2j+1’1VT'(j+2)VLL
vj=vc+<j—1>VSL-[2(j—1)/(2j+1)1vT+<j-1)vLL
v3=£etj(j+1)11/2/(2j+1)}vT

and C, SL, T, LL refer to the central, spin orbit, tensor
and quadratic spin orbit parts of the Hamada-Johnston
potential.

These u's are then to be solved for numerically.

The u's are then expressed in terms of W's

+
W’+ sin(kr-(jil)g+6_)

H

Wi+ sin(kr—(jil)g+6+)

2
where
ut=(sin€,)eié_wi
1 3 1
0056,
J
:_ _cose. i6+.i
112'.( !)e W2

sine.
3

Then, the pseudo-phase shifts are defined as

+

W1)

A. =1. (Wl)+tanejh

0 <
3'1 3-1 j-1

9 (w—

+ + )
j+l l

+
A. —I (wl)+coteju

j-l j+l (6)

+
2)

Aj+l=Ij_ (w

‘ _ 0
l(W2)—cot€jIIj_l

(w-)

+ _ + + _ o
Aj+l—Ij+l(W2) tanEJIj+l 2

 

49

where
16(f)--E———IwF (k'r)V i(r)f(r)dr (7)
— — 2 2- j
l 4h k' 0

Equation 6 arises in the same way as Equation 4 did for
the £=j case. By using the W's in Equation 6, we have

for k'+k

A _l+51n6j_1

L-la H

i .
Aj+l+s1n6j+l

Then the matrix elements of R become

16‘A7_ -e15+AT ]

<°—l R '+l>=sine.coss. e
3 ' I] 3 J[ J 1 3+1

. ._ _ . is- + _ 15+ +
<3+l|R|3 l>-31nejcosej[e Aj_l e Aj+l]

2

j+l

i6- -

<j-llle-l>=coszeje Aj_l+sin ejel6+A

. . a 2 is- + 2 i6 +
< +l R +l>=31n e.e A. +cos €.e +A.
3 I I] J 3-1 3 3+1

 

APPENDIX D

The method used to obtain the proton point density,
and thus the matter density if there were no neutron skin,
from the empirical charge density is outlined in Chapter IV.
Table 2 lists the parameters for the nuclei considered

where the density is of two forms

2 2
form 1: p(r)=p0(l+c rZ/a2)e r /a (1)
form 2: p(r)=po[l+e(rnc)/a]_l (2)
The values of the harmonic oscillator constant a and the
energy‘hw, where
2 2
’, yfi c
hw——§———§ (3)
a m c
P

and mp is the proton mass, listed in Table 2 are obtained

by matching the oscillator density's mean squared radius to
the empirical proton point mean squared radius. In obtaining
the oscillator constant, the center of mass correction was

12

included for C and 40Ca but neglected for the rest of

the nuclei as it corresponds to a l/A correction.

50

 

51

The example of 40Ca from Acker et alll is used to
illustrate the forms of the various distributions and the
effect on the oscillator distribution if the pure oscillator
radial wave functions are replaced by Hartree-Fock wave
functions.

As in Chapter IV, the oscillator density is
o<r>=i <21+1>¢2<r) (4)
NE 2

but now ¢£(r) is replaced by a radial Hartree-Fock wave

function

where Rm£(r) is the radial oscillator wave function for

the 2th subshell (defined by the quantum numbers m and 2).
The 40Ca Hartree—Fock wave functions used32 are listed in
Table 3. Then, in Figure l, the charge, proton point,
proton point oscillator, and the oscillator charges (the
distribution obtained by folding the finite electromagnetic
size of the proton into the proton point oscillator

distribution) densities are given for both pure oscillator

and Hartree-Fock wave functions.

 

52

~\HHAmuv\.m_nm.

 

 

 

 

 

 

 

 

 

m~.~ m.mm HN.H oo.m omva om.m omwa om
mm.H s.mo mm.m mm.m mme os.H vvm ma
em.H o.mm em.m om.m oamm mm.~ omem mm nnmom
sm.~ m.mm 5H.H oo.m Nmma om.~ omma ow
om.a o.mm vm.m Nw.m smv os.a m¢a a:
mm.a s.mm mm.m om.m oeaw mm.m ommm mm cmoma
ov.m m.Hn vo.a oo.m mam om.m OHOH ow
om.a m.mm mm.m No.m mme os.a mmm ma
vm.a m.em wm.m om.m came vq.m ooem mm flme
mv.m m.mh Ho.a oo.m Ham om.m «mm om 1
sm.H $.40 mH.m No.m was os.a smm a:
vm.a o.mm vm.m om.m ossv ev.~ omvm mm mooe
my r>mzvo> A no. Hv A comm A>mzvm> A mean A>mzva>-
H- 4 m m H- H- #

 

 

 

 

 

H mqm<B

 

.mmzmxsw 03p cam wco ou uflm mumnmfimumm aoflumEonummm mmasmfifl one

53

 

 

 

 

 

 

 

 

 

 

 

 

HH mn.m mmw.N wm.mN me.H mmm.m N mH.om HN.N hw.m QANON
NH mm.m mmv.N me.mN moo.N HHm.m N mH.mN om.N om.m QmmON
HH mH.m omN.N om.oN Hmm.H hmv.m N om.HN HN.N mv.m cmONH
NH om.m mho.N Hm.vH Nvm.N NmN.v N mw.mH om.N mN.w Hme
HH .Nm.OH mmm.H om.HH mNo.N www.m N oo.NH HN.N bwn.m moov
NH mv.OH mom.H >>.HH Hmm.N mow.m N H¢.NH om.N vw.m moow
mN mo.bH Hmm.H mm.v Nmo.H ONN.N N Nm.m mm.H om.N UNH
NH mH.mH mmw.H mm.m omm.H mmm.H H NN.m H>.H NH.H UNH
.Hmmmu A>02v3nx Amva EANuv Em EU Snow noANuv now £00
.wwHuHmcmp unaom cououm paw mmnmno mge

N

mqm<B

 

 

 

54

 

 

 

 

 

 

 

 

 

moHo.n mHmo. memo.u Nmam. o.mHu oH
memo. vao.u Humm. NHmo.n s.nHu mN
meoo.- mmoo. «wee. mama. «.mm- mH
emoo.u mNoo.u mHmo. swam. o.mmu mH
acmo acmo acHo acoo mumnMHmwmwunmm mumpMwamwuumm
.>m2 h¢.hl n comHosc

\mmumcm “.mmo.N u m u unopmcoo HOpmHHHowo mo How mummemmmm xoomnmmuummm

ow
m mqmfifi

 

55

 

 

 

 

 

 

 

 

 

 

 

 

Hm.mH osn.HN m mv.mH omm.mH m
NN.mH omo.mN o ms.oH omHINH o
mv.q 0N.NH oqN.mN om «4H.e om.mH ome.mH om
em.m ms.mH ONo.mH oz .mN.m NH.mH on.NH oz
NH.m em.NH omm.NN mm mH.m om.¢H onm.mH mm
ms.o me.HN OH¢.HN mm es.o mo.mH omm.eH mm
mH.m mm.NH one.NH NH mH.m Nm.qH Hmo.NH NH
mN m a» mN m m»
A .HV A .HV o m I w A .HV A .HV o m I m
ANmV N Ava N Amm > 2v.o u ANmV N Ava N Amm > so a u
Hm.vH I 5A HV 42 o . I a u no
I N . mm m HH I AN v 04

 

m

.HMHpcmuom Hmuucmo Hmmu on“ MC Show uomuHU map ch0 How A Hv cam D

w mqmfie

N

 

 

.GOHflUMHO#GH w>flfiUQMMO ComHUDCIGOGHUDC @5# MO mDHUMH UGHMUUW QMOE m>H#OmMM® ®£# mH UGM

 

56

 

 

 

 

 

 

 

 

 

a mN
.H I. .H H
AN v AN v ANHV

mm wmchmp wm paw 03 How mNANHv

.4.

m pm wow noon "0

4HH H o
m um m UHMm ”m
mHH x

mH.Nm omN.mN m om.NN O4H.m4 m

OH.nm omo.mm o ms.sN ONH.om o

No.m om.4m omm.Nm om HN.4 sm.mN 044.54 mm
4N.4 om.mm omm.mm oz No.4 mm.4N onm.mm oz
NH.N 4m.Nm om4.nm MM NH.m sm.mN omm.m4 mm
o>.m mm.mm oso.Nm mm Hs.m O4.NN omm.o4 mm
ON.m 4N.Nm omH.mo 4H NN.m mo.4N onm.em NH

mN m mN m
H H . w I mm» H H . 0 I ma»
ANHVAN v AvaAN v Amm > so a u ANmVAN v ANMVAN v Amm > so a u
vm.mN I E Hv am mm.0N I E H cm
I AN mON I AN v ONH

 

 

 

Aomscflpaoov 4 mamas

 

 

57

TABLE 5
Energy dependence (VB) and isobaric dependence (VI)

of WG and SG for the direct term and the tOtal potential.

 

 

 

 

 

 

 

 

 

 

 

 

type -VE(D) -VE(T) VI(D) VI(T)

40Ca we .03 .21

se ‘ .04 .20

58 . -

N1 we .03 .20 28.63 24.47
se .03 .20 137.12 37.76
lZOSn we .03 .22 28.82 26.81
se .03 .18 36.59 37.69
208Pb we .03 .22 28.76 24.68
se i .04 .21, 36.72 36.50

 

 

 

where D and T refer to the direct interaction alone and the

total interaction.

 

58

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MHHm um monHm um
mH.nm .omth m
mh.mm oH.bm ww.mm mm.vm oommm OHmmm ommm ommNm 0m
mh.>m mm.mm Hm.om om.mm chomh omNHw Oth ommmw 03 QmNON
om.hN oeHmv m
Hm.5N mm.wN mm.mm hm.mN oommm owwmv ommw oevhe Om
vh.mN oh.mN mv.mm mm.4N oonww omHow onw onmmm 03 amONH
Hm.mH ombHN m
Ho.ON m4.mH mm.4N 0N.mH omNmN 0N4mN ommN owNmN 0m
ov.mH mm.mH mo.NN mh.mH ochHN OHHmH ommN ONomH 03 Hme
m4.mH ommmH m
mh.mH mm.mH hm.mH om.mH om4hH ommmH oomH ommmH 0m
Hm.mH mm.vH mm.hH NH.mH onmvH ommNH omnH QHmNH 03 moov
BANHV QANHV mANHv QANHV ABVMDI onmDI AmeDI AQVmDI wmmw

.mHMHpcmuom ABC Hmuog cam Adv

wosmummeo muHmcmo .Amv mmcmcoxm .AQV uomnHU map

m mqmfifi

.HOM Om UCM 03 HOW A .HV Ufim

N

mo

 

 

 

 

 

 

 

 

 

 

<r2>

and matter

distribution.

59

TABLE 7

and Woods—Saxon parameters

for proton, neutron

 

 

 

 

 

 

 

 

 

 

 

 

 

 

<r2> <r2> <r2> c a <r2>l/2—(r2>1/2
p m n n n n p
40Ca TD 11.36 11.23 11.10 3.69 .461 -.04
58Ni e 14.81 16.97 19.80 5.15 .532 .60
HO 14.81 14.97 15.12 4.32 .532 .04
120
Sn e 20.86 25.20 28.09 6.49 .450 .73
HO 20.86 21.54 22.03 5.66 .450 .13
208
Pb TD 29.54 31.66 33.04 7.11 .446 .31
e 29.54 34.11 36.97 7.55 .446 .65
HO 29.54 32.18 33.89 7.21 .446 .39
. 19
TD: Tarbutton and DaV1es an=ap

14

G: Greenlees et al

HO: Harmonic oscillator basis

 

 

60

TABLE 8

Spin orbit parameters

 

 

 

 

 

 

 

 

 

 

 

 

 

 

type VO(MeV) cso(f) aso(f) <r2>2n UR '<r2>
400a IA 5.17 3.68 .554 1.01 1320 12.37
e 5.70 3.94 .70 1920 16.08
F 6.22 3.52. .778 1680 15.78
58Ni IA 5.04 4.19 .621 1.04 1890 15.85
e 5.20 3.93 .70 1735 16.03

F 5.53 4.15 .641 2040 15.99
12°8n IA 5.19 5.42 .565 1.17 3833 22.03
e 6.20 5.78 .700 5743 26.82
F 6.11 5.21 .800 4470. 25.15
208Pb IA 5.05 6.63 .563 1.21 6600 30.75
e 5.13 6.72 .700 7230 33.90
F 5.84 6.08 .794 ‘6421. 30.88

13

Fricke et a1

14

Greenlees et al

 

61

 

 

 

 

  

 

 

 

FIGURE l
C040
l2__ CHARGE ——
- Mr) PROTON POINT ——
\ PURE OSCILLATOR —--
~\,_ HARTREE FOCK —«-
I— §
O.\.
\\\f\\.
.CXB== ————— E:i:::& ‘
'\
CHARGE \
'2\, \ .06
\\
l__ \\Q§>\ ‘—'$D‘l
.08———-— E. 02
K ..
\
06—- \
° PROTON POINT \\ -* O
o.\\
04‘— :\
\
\
.02—
l 1 l J
l 2 3 4 5 M)
Figure l. The charge and proton point distri«

. 4O . . . . .
bution for Ca compared to the d1str1butions obtained

by pure oscillator functions and Hartree-Fock functions.

 

FIGURE 2

l I l I l TI I II

ffpg FRICKE ET. AL.——
° '“"'“ GREENLEES ET. AL —-
IA- -

 

 

 

3O
— 20
- IO

 

30
~ 20
_ IO

Re Vc (r) (MeV)

 

’ 30
~20

 

 

 

ICD r(f)

Figure 2. The theoretical real central potentials of
IA, KB, and KK with no antisymmetrization and no density
difference are compared to the empirical potentials of Fricke

et a1 and Greenlees et al.

 

63

FIGURE 3

7 ‘I T7 l I If I 1 . I ‘ I
26an FRICKE ET. AL.——
60% GREENLEES ET. AL.--

_._._._._.__. wen"

SG---

 

 

  

30
-20
'1 IO

 

 

 

3O
‘20
" IO

 

30
~20
- IO

 

 

 

 

l l 1 -

8 9 IO r(f)
Tigure 3. The theoretical real central potentials of

HG and SG with no antisymmetrization and no density difference

are compared to the empirical potentials of Fricke et al and

Greenlees et al.

63

 

FIGURE 3
1

TI 11"

263% FRICKE ET. AI..—
GREENLEES ET. AL.--
we----

se--—

  

 

 

 

 

 

 

 

 

 

l l -

9 IO r(f)
Tigure 3. The theoretical real central potentials of

VG and BC with no antisymmetrization and no density difference

are compared to the empirical potentials of Fricke et al and

Greenlees et a1.

64

 

60 ”ZOBPD
50 ~ “ \ \

FIGURE 4
I

I I I l I l

WEAK GREEN '

DIRECT
DIRECT + EXCHANGE ---
DIRECT (p t p)

 

 

 

30

 

~20

 

 

 

 

 

 

C) l J. 1 l, l 1 AJ 1 l
O I 2 3 4 5 6 7 8 9 IO r(f)
Figure 4. The theoretical real central potential

of WC where the effect of antisymmetrization and density

difference are included. .

 

65

.cOHuomnmucH O>Huommmm

DB map mchD mu use 0 How.>muwcm nuH3 HOHpcmuom HmHucwo

ow NH
Hmmn map mo cumcmuum man mo coHumHum> 639 .m OHDmHm

 

His; 93“. om oh om On 04 0... ON 0.
H

H H H H H H H

I Inn

0 UN. ,/Il'

:3); 3n: U> mm

 

 

 

m meoE

 

.>02 04 6am ON 46 mmflmumcm 66H.6:u

pm 6004 How 03 we HOHucmuom Hmuacmo Home one .m wusmHm

E. m

 

66

 

 

 

 

II >229»
I >m_>_ON

0004 :32“— So> mm
ZMMKO x<M>>

 

_ _ _ _ _ _

 

m mmDoE

 

67

FIGURE' 7
I

 

I I I T I
FRICKE ET. AL.

208 GREENLEES ET.AL-.--
WG

I

   
  
    

 

 

 

 

 

 

 

 

Figure 7. The theoretical real central potentials
of WC and BC with anitsymmetrization and density difference

are compared to the empirical potentials of Fricke et al
and Greenlees et a1.

 

68

 

 

 

 

 

 

 

 

O—NU‘O—NUJO—NOJA

 

 

 

FIGURE 8
7_208Pb FRICKE ETAL. ——
6- GEEENLEES ETAL.::
SEER
4
7
6
5
4
7
6
5
4
7-
6
5
4
3
2
I \
O 1 1 I l 1‘ _ I l l l
O|23456789 r(f)
Figure 8. The real spin orbit potential of IA is

compared to the empirical potentials of Fricke et al

and Greenlees et al.

 

 

 

69

 

.Hm um OxOHHm paw .Hm um wmnw mo mHMHunmuom

HMOHHHQEO msu ou Uwummeou mum >6: O4 one 0N um 60 How Hmuucmo MchHmmEH 039 .m mnsmHm

 

 

 

o
E. m n 4 .. m N _ 0
AIM]. ‘ _ _ _\IIII IIIII I
II IIIIII [Hlel _\\\\ II_
I / III / \\ \IIIIIIIIIIIN
/ / \\\I./ / m
I / / ..l
/ x // \.\\\ /I
I / /.I.\.\ 4 I?
I- //v \\. \x 11mm
II\\ x
I /// \\ I0
I /I\ IN
I 1m -
I ll >22 0? 10
I m
I II 82889: I _ _
II 3.8 9.0:...
III 60:83,: 32>: Cvo> .025
I 6.8 >80 cool4

 

 

 

m mmDoE

 

 

70

FIGURE IO

 

I2

C
20 MeV CAMERON VANOERS—

THEORETICAL
40 MeV CAMERON VANOERS—-

IHEORETICAL
E‘;.

4501
CO

\
Q‘ \
oI) \.\

‘\\

OJ
Q

./

\
\\

//

30
20
IO

 

IMAG. Vc (r)

—
C

I.—

\\

.\'\\\j\

—I\)0J-Il5(fim\lmLO—NOJAUIOI

/./
//

 

I
3

l
I 2

Figure 10.

The theoretical optical potentials for

 

5 r(f)

12C

at 20 and 40 MeV are compared to the empirical potentials of

Cameron and van Oers.

 

 

71

 

20 MeV

-———emphmal
0
'0 I‘ o Iheoreficol

 

 

 

'04 I I I I I, I I I I
20 40 60 '80 I00 I20 I40 I60 I80
BCMIdeg)

Figure 11. The cross section for 12C at 20 and 40 MeV
for the theoretical optical potential compared to the poten-
tial of Cameron and van Ores.

 

72

 

40
-| CO
IO -
O O
0 o 000
AA 0 00
O 00
A O O
o O
A O
0 AAA
A AA
A
A
A A
AA
empirical

0 theoretical

A theoretical
[empirical imaginary term ]

 

 

 

[0-4 I I I I I I I I J
20 4O 60 80 I00 IZO I40 I60 |80

90M ( deg)

Figure 12. The cross section for 40Ca at 20 and 40 MeV
for the theoretical Optical potential compared to the potential
of Gray et al and Fricke et al. The crosses denote the
theoretical potential with the imaginary term being replaced

by its empirical counterpart.