PHYSICAL REVIEW CVOLUME 44, NUMBER 6DECEMBER 1991
Dynamical pola iza ion po en ial dne o he exci a ion o collec i e s a es
M. V. And es, '"F.Ca a a, '''and E.G. Lanza' '
"'Depa amen o de Fisica A omica yNuclea , Uni Ue sidad de Se illa, 41080Se illa, Spain
''Dipa imen o di Fisica dell'Uni e si a, 95129Ca ania, I aly
''Is i u o Nazional'e di Fisica Nuclea e, Sezione di Ca ania, 95129Ca ania, I aly
(Recei ed 21 Ma ch 1991)
Wi hin he Feshbach o malism we calcula e he nucleus-nucleus dynamical pola iza ion po en ial
a ising om he coupling o he elas ic channel o he collec i e ib a ional s a es, desc ibed by he
andom-phase app oxima ion. Calcula ions o he sys ems '0+ Ca and Ca+ Ca show he impo -
ance o he high-lying s a es. They gi e he main con ibu ion o he eal pa o he pola iza ion po en-
ial a e e y inciden ene gy, while hey domina e he imagina y pa only a e y high ene gy. A low
ene gy he abso p ion is gi en by he low-lying s a es. The eal and imagina y pa s o he po en ial a e
shown o obey adispe sion ela ion. Calcula ions o elas ic c oss sec ion gi e agood desc ip ion o he
expe imen al da a.
I. INTRODUCTION
The elas ic sca e ing o nuclei is well desc ibed by he
op ical po en ial. I s eal pa is usually cons uc ed by
means o he double- olding model [1]. The imagina y
pa desc ibes he depopula ion o he elas ic channel due
o i s coupling o he nonelas ic ones. This coupling gen-
e a es also aco ec ion o he eal pa . Recen ly, a
enewed in e es in his so-called dynamical pola ized po-
en ial has been aised by he expe imen al e idence [2] o
as ong ene gy dependence o bo h eal and imagina y
pa s o he op ical po en ial a ene gies close o he
Coulomb ba ie , which is known as h eshold anomaly.
Adispe sion ela ion has been used [3,4] o ela e he
s ong inc ease o he eal pa o he dec ease o he
imagina y one. Thus i is in e es ing o s udy he beha -
io o he pola iza ion po en ial wi h he ene gy wi hin a
mic oscopic app oach.
Semiclassical models ha e been p oposed in o de o
calcula e he pola iza ion po en ial by including low-
lying collec i e ib a ional s a es and one nucleon
ans e [5], only one nucleon ans e [6], and bo h low-
and high-lying collec i e ib a ional s a es [7]. Al hough
he po en ials so ob ained gi e agood desc ip ion o he
expe imen al elas ic c oss sec ion, ad awback o hese
models is ha he pola iza ion po en ial is no cons uc -
ed di ec ly bu only by making app oxima ions on quan i-
ies exp essed as in eg als along classical ajec o ies.
The pola iza ion po en ial can be calcula ed in acom-
ple e quan um way h ough he use o he Feshbach o -
malism [8]. This app oach has been exploi ed by Vinh
Mau by using he closu e app oxima ion [9]. The main
ad an age o his me hod is i s simplici y. One can ob-
ain he pola iza ion po en ial wi hou de ailed in o ma-
ion on he s uc u e o he nonelas ic channels. A he
same ime, one limi a ion o he model is i s inabili y o
desc ibe how speci ic channels con ibu e o he o al po-
la iza ion po en ial.
In his pape we calcula e, ollowing in spi i he ap-
p oach o Vinh Mau, he pola iza ion po en ial aking ex-
plici ly in o accoun he e ec s o he exci a ions o he
collec i e ib a ional s a es, desc ibed wi hin he
andom-phase app oxima ion (RPA). An analysis o he
eal and imagina y pa s o he pola iza ion po en ial in
e ms o he single collec i e modes shows he impo -
ance o he gian quad upole esonance (GQR) s a es.
They gi e as ong con ibu ion o he imagina y pa
only a high inciden ene gies, while he eal pa is dom-
ina ed by he high-lying s a es a all ene gies.
We show ha he ene gy dependence o he eal and
imagina y pa s o he pola iza ion po en ial ollows he
end o he expe imen al da a and sa is ies adispe sion
ela ion. Finally, calcula ions o elas ic c oss sec ion pe -
o med wi h ou mic oscopic op ical po en ial gi e a
good desc ip ion o he expe imen al da a. P elimina y
esul s o his model ha e been p esen ed elsewhe e [10].
II. THE MODEL
An elegan and anspa en way o ake in o accoun
he coupling o he elas ic channel o he nonelas ic ones
is h ough he use o he Feshbach o malism [8]. In his
app oach he e ec i e hea y-ion in e ac ion is w i en as
(R R')=(oo I.(R)loo )S(R R')
+g(oolu(R)lK, K)
EC lE2
&G««, (R,R')(Ic,Ic,lU(R') loo)
=VF(R)+hV(R, R') .
In he i s e m he nucleon-nucleon in e ac ion U(R) is
double olded wi h he g ound-s a e densi ies o he wo
nuclei. In he second e m he sum is o e all he none-
las ic channels, and U(R) is double olded wi h he ansi-
ion densi ies o he wo nuclei: i desc ibes he coupling
o he elas ic channel o he nonelas ic ones. This e m is
he so-called dynamical pola iza ion po en ial. I s physi-
2709 1991 The Ame ican Physical Socie y
2710 M. V. ANDRES, F.CATARA, AND E. G. LANZA
cal meaning is anspa en om Eq. (1): he in e ac ion
ac ing a he dis ance R' akes he sys em in o one o he
elimina ed channels; hen i is p opaga ed a ano he dis-
ance Rwhe e he in e ac ion, ac ing again, b ings he
sys em back in o he elas ic channel. Then he pola iza-
ion po en ial is nonlocal, and i one o mo e o he elim-
ina ed channels is open i is also complex and i s abso p-
i e pa desc ibes he loss o Aux om he elas ic chan-
nel. The coupling also make b,Vene gy dependen be-
cause o he appea ance o he ene gy in he p opaga o
Gx. x(R,R'):
Vk«)V (R')
Gx x(R,R')= dkEE~ —
E —
i k—
/2p+i i
whe e E, is he cen e -o -mass inciden ene gy, EK1
and EK a e he exci a ion ene gies o nucleus 1and 2, e-
2
spec i ely, and pis he educed mass. The yk(R)'s a e
he ela i e mo ion wa e unc ions o he colliding nuclei.
The calcula ion o Eq. (1) is a e y di icul ask be-
cause o he p esence o GK K.We calcula e he p opa-
12
ga o in he WKB app oxima ion, which has been shown
o be agood app oxima ion o a-nucleus sca e ing [11]
a ene gies g ea e han he Coulomb ba ie . In his ap-
p oxima ion we ha e
a e e ms o he o m
a , =y&oolU(R)ll~, o&G~,(R,R')&I~,olU(R')loo& .
K1%0
(8)
The closu e app oxima ion model o Vinh Mau con-
sis s o eplacing he exci a ion ene gies EK and EK by
12
a e age alues E& and E2, espec i ely. In his case he
p opaga o Gdoes no depend on he pa icula exci a-
ion and i can be aken ou o he summa ion o Eq. (8);
hen aclosu e ela ion o e he s a es E, and K2 can be
used. This app oxima ion is e y good a e y high in-
ciden ene gies, while a low ene gies he alues o M(p)
[Eq. (3)] can be qui e di 'e en depending on he alues o
EK and EK .Fu he mo e, he e6'ec i e nucleon-nucleon
12
in e ac ion is app oxima ed by asepa able o ce.
We wan o calcula e he pola iza ion po en ial a ising
om he exci a ion o he collec i e ib a ional s a es and
o s udy he ela i e impo ance o low-lying and high-
lying ones. Thus we do no make use o he abo e ap-
p oxima ions and we explici ly sum o e he ele an
s a es, whose ene gies and ansi ion densi ies a e calcu-
la ed mic oscopically wi hin he RPA.
Wi hin he double- olding app oach he o m ac o s
can be w i en as
F~,(R)—
=&ool U(R) lI~ &o &
pexp[i'. x. (p)s]
12
G~ ~(p,s)=- 2~%2 S
wi h
2p
M~, x,(p) =,[E. Ex, Ex, —
~c(p—
)—
1'c(p)]
whe e
p=-,'(R+R'),
The local op ical po en ial VL (p) is gi en by
(2)
(3)
(4)
= d d 2px. O( , )U(l , —
z+Rl)poo( z) .
The explici exp ession o he adial pa o he ansi ion
densi y pox ( &) is
J'& +i/2
p,o(") g( ) (l —lg —,
'lL o)
4m
X(Xpq' —
Y„h' )8 ( )Rh ( ), (10)
ph ph p
1.5
&L, (p) =&p(p)+ &VL(p),
while Vc is he Coulomb po en ial be ween he wo nu-
clei. Ap ocedu e o de ine he local pola iza ion po en-
ial b,VI will be gi en la e on in his pape .
In o de o make mo e anspa en he con ibu ion o
he single e ms in Eq. (1), we can sepa a e he summa-
ion o e K,,E2 in o h ee pa s,
X=X+X+X
K1 K2 Kl =0 K2%0 K1%0 K2 0K1XO K2+
which gi e ise o h ee di 'e en con ibu ions o b,V,
AV=AV2+hV, +AV)2,
0.0
—
0.50
L= 1
l
0.5
cos 8
I
0.5
cos 8
L=3
I
0.5
cos 8
'i
/I.
I'
I
espec i ely. In he i s e m nucleus 1s ays in i s
g ound s a e while nucleus 2is exci ed, ice e sa in he
second e m, while in he hi d one bo h nuclei a e exci -
ed. This las e m is expec ed o gi e asmall con ibu ion
and will be neglec ed. Then wha we ha e o calcula e
FIG. 1. The quan i y Al, de ined in Eq. (11),as unc ion o
cosO o h ee alues o he angula momen um L. In each sec-
o he h ee cu es co espond o a ixed alue o p= 8 m and
o he ollowing alues o s: 1 m (solid line), 4 m (dashed line),
and 8 m (do -dashed line). The esul s co esponding o odd
alues o La e mul iplied by —
1.
DYNAMICAL POLARIZATION POTENTIAL DUE TO THE. ..2711
whe e R~ (Rh )is he pa icle (hole) wa e unc ion; X~&'
and Yza e he o wa d and backwa d RPA ampli udes
K)
co esponding o he mode K& whose mul ipola i y is L].
The calcula ion o b,V,z(p, s) is epo ed in he Appen-
dix. Exp essions o i and o EV(p, s) a e gi en by Eqs.
(AS) and (A9), espec i ely. Thei angula pa can be
easily wo ked ou i we assume ha he po en ial b,Vis
weakly dependen on he angle 0be ween pand s. This
has been shown o be ue o nucleon-nucleus sca e ing
[12]. The angula pa o b.V, can be w i en as
A (p, s,x)=(+p +—,
's +psx Qp +„'s —
psx—
)pi/2 1/2
2L+1 2L+1
[2(L—
A, )+1]'~ [2(L —
A, ')+ I]'
A, +A,
L
()i.
X,X=0
A, +A,
Xg(—
) W(A, ,L—
A, ,A.',LA, ',L —
)(L —
A, 0L—
I,'Ol 0)(A, 0A,'Ol 0)P (x) .
whe e x=cos8 and 8is he angle be ween pand s. Nume ical calcula ions o Eq. (11) show ha he a ia ion o A
wi h 0may be e y di e en depending on he alues o pand s. In pa icula , we show in Fig. 1 he beha io o Az,
o p= 8 m and s=1,4,8 m, as unc ion o cosO o h ee alues o L. We see ha a apid a ia ion wi h cosO is ound
o p=s =8 m, while o small alues o s he quan i y A~ is almos independen o cosO. This ange o alues o la ge
pand small sis he physically in e es ing domain, whe e he localiza ion p ocedu e ha we will desc ibe la e on also
makes sense. So in his ange o alues hV is weakly dependen on he angle 8. Choosing he alue cos8=1, which
co esponds o ake he maximum con ibu ion, we ha e
A (p,s,x) =( '—
—
'')
x=1 A. =O
2L+1
2P2A, (A, 0LOlL —
10)(—
)~
2L +1
A.'=0 (A,'0LOlL —
~' 0)=(—
)c .(12)
Then he po en ials AV&z and b.Vi can be w i en as
and
EV,~(p,s)—6gG»» (p, s)L E qgA»» (p,s)%»» (p, s)(L, 0L~ Ol J0—
)
»1212
l~ 2J(13)
EVi(p, s)= 5XG», o(P s)+»,o(P s)+» o(p~ —
s)L i,
16m g1
(14)
whe e %»» (p,s) is de ined in Eq. (A7) o he Appendix.
The po en ial calcula ed wi h jus he desc ibed p o-
cedu e is nonlocal. Bu , since i is always p e e able o
use local po en ials, a leas o he calcula ion o physical
obse ables, we use as anda d p ocedu e [13] o ob ain a
local po en ial om anonlocal one. This p ocedu e is
applicable whene e he ange o nonlocali y o b,Vis
small wi h espec o i s adius. Thus, we ha e
I has been shown [9] ha he hypo hesis o e he ange
o nonlocali y o b,Vis alid, a leas o la ge alues o
p. Indeed, in Fig. 2we show he eal and imagina y pa s
o Eq. (14) as unc ions o s o wo ixed alues o p. Fo
la ge alues o pi is ound ha he ange o nonlocali y
is abou 1 m. This alue is consis en wi h p e ious ones
gi en in he li e a u e [9,14]. Thus he hypo hesis o in-
dependence o b,Von he angle be ween pand s, dis-
cussed be o e, is a1so jus i ied.
AV(p) =4m jo(ks )RehV(p, s)s ds, (15) III. RESULTS AND DISCUSSION
8'(p) =4m jo(ks) Imb, V.(p,s)s ds,
whe e
p[&c.m. VF(p) Vc(p)] .
2p
g2
We ha e done calcula ions o he sys ems 'O+ Ca
and Ca+ Ca a se e al inciden ene gies. The le els
used in he calcula ions a e epo ed in Table I. They
ha e been ob ained wi h asel -consis en RPA code using
an SGII o ce [15]. We ha e included all he s a es which
exhaus a leas 10% o he ene gy-weigh ed sum ule
2712 M. V. ANDRES, F.CATARA, AND E.G. LANZA
30
I
20- 150 +40C
El,b=104 MeV
10-
—
10-
—
200s( m)
10-
leO +4oc
Es ab =104 MeV
-100s( m)
FICx. 2. Real and imagina y pa s o he nonlocal pola iza-
ion po en ial AV, (p,s), o he sys em '0+ Ca a E&,b=104
MeV, as a unc ion o s o wo ixed alues o p: 7 m (dashed
line) and 8 m (solid line).
(EWSR). The RPA calcula ions ha e been done wi h a
la ge numbe o pa icle-hole con igu a ions (-350 o
he 3o "Ca) in o de o ge agood desc ip ion o he
low-lying s a es [16]. Bo h ene gies and ansi ion densi-
ies compa e well wi h he expe imen al da a. The
e ec i e M3Y nucleon-nucleon in e ac ion has been used
Nuclei
1
1
2+
2+
2+
3
3
3
E* (MeV)
19.096
19.996
19.945
20.620
21.280
7.220
32.870
34.610
%EWSR
27
21.4
44
13
12.6
8.8
8.4
8.9
"Ca
1
1
1
1
2+
3
3
16.302
16.765
17.372
17.883
18.296
18.632
16.741
4.830
31.281
9.8
15.2
10.6
10
10.6
16
80
15
13~5
TABLE I. P ope ies o he RPA s a es used in he calcula ions.
in o de o cons uc he double- olding po en ial V~ o
Eq. (1) and he o m ac o s o Eq. (9).
In o de o s udy he in e play be ween he low-lying
and high-lying s a es we ha e made an analysis in e ms
o he single ib a ional collec i e s a es. In he le pa
o Fig. 3we show he eal and imagina y pa s o he po-
la iza ion po en ial o he sys em Ca+ Ca a
E&,b=240 MeV. Each line co esponds o he con ibu-
ion due o di e en mul ipola i ies as indica ed in he
igu e; he o al po en ial is gi en by he solid line. The
ela i e con ibu ion o he single collec i e s a es can be
be e seen in he igh pa o Fig. 3, whe e o he same
sys em we show he eal and imagina y pa s o he pola -
iza ion plo ed in pe cen age o he o al po en ial as
unc ion o he ela i e dis ance R. Each line co e-
sponds o he ela i e con ibu ion o he di e en mul-
ipola i ies, as indica ed in he igu e. We no e ha he
abso p i e pa is gi en almos comple ely by he low-
lying 3s a e. Con e sely, he con ibu ion o low-lying
and GQR s a es o he eal pa b, Vis o he same magni-
ude. The high-lying 3s a es and he gian dipole eso-
nance s a es gi e essen ially ze o con ibu ion. Fo he
dipole iso ec o s a e his con ibu ion comes only
h ough he small mix u e o T=O componen s due o
he Coulomb in e ac ion. This esul is no peculia o
he pa icula sys em: in gene al, a low ene gy he ab-
so p i e pa o he op ical po en ial is gi en by he low-
lying s a es, while bo h low and GQR s a es con ibu e o
he eal pa . In ac , also o he sys em '0+ Ca a
E»b =104 MeV (see uppe pa o Fig. 4) asimila beha -
io is ound.
In o de o in es iga e he pola iza ion po en ial a en-
e gies much highe han he Coulomb ba ie we ha e
calcula ed 6Vand 8' o he sys em '0+ Ca a se e al
inciden ene gies. In Fig. 4we show he ela i e con i-
bu ion o he single modes o he eal (le ) and imagina y
( igh ) pa o he pola iza ion po en ial as a unc ion o
Rand o h ee di e en alues o he inciden ene gy.
Again, he mul ipola i ies a e indica ed o each cu e.
We no e wo s iking ea u es. (1) The eal pa is dom-
ina ed by he GQR s a es which gi e, a high ene gy,
80% o he po en ial. The con ibu ion o he low-lying
3s a e, compa able in he pe iphe al egion o he 2+
s a es, is dec easing when he ene gy inc eases. The
high-lying 3s a es and he GDR s a es beha e in an op-
posi e ashion. Howe e , hey ne e each impo an
alues. (2) Con e sely, he beha io o he abso p i e
pa changes d as ically, going om 104 o 640 MeV. In
ac , as we ha e seen be o e, a low ene gy he imagina y
po en ial is essen ially due o he 3low-lying s a es,
while as he ene gy inc eases he dominan abso p ion
comes om he GQR o bo h nuclei. This beha io can
be unde s ood in semiclassical e ms [7); highe ene gies
co espond o sho e in e ac ion imes; hence, by he un-
ce ain y ela ions, he p obabili y o exci a ion o high-
lying s a es is highe . This mechanism does no wo k in
he case o he eal pa because he p ocess in ol ed in
he o ma ion o 6Vis a i ual one.
Adeepe analysis shows ha he majo pa o he o-
al 3con ibu ion comes om he low-lying s a e o he
Ca. This can be seen in Fig. 5, whe e o he sys em
DYNAMICAL POLARIZATION POTENTIAL DUE TO THE. ..2713
0+ Ca a E]b=104 MeV we show how he pola iza-
ion po en ial due only o he low-lying s a es is dis ibu -
ed be ween he wo pa ne s o he eac ion. We see ha
he hea ie pa ne is esponsible o mos o he e ec .
This end does no change a highe ene gies; he only
di e ence is ha , o ins ance a EI,b=640 MeV, he wo
cu es ge close o la ge R. On he con a y, o he
high-lying s a es i is he ligh e pa ne which gi es he
g ea e con ibu ion, al hough he di e ence is no as big
as in he p e ious case. As an example, we show in Fig. 6
he pola iza ion po en ial due o he GQR s a es o he
case o '0+ Ca a E),b=640 MeV.
In o de o check he ange o alidi y o he p e ious
esul s wi h espec o he ela i e impo ance o GQR
and low-lying s a es, we ha e done calcula ions o labo-
a o y ene gies om 10 o 320 MeV o he sys em
'0+ Ca a a ixed dis ance R=9 m. The esul s a e
epo ed in Fig. 7, whe e he eal and imagina y pa s o
he pola iza ion po en ial a e plo ed as unc ions o he
ene gy. Thei ene gy dependence esembles e y much
he expe imen al indings o Re . [2]. The dashed line is a
esul o acalcula ion done including only he low-lying
3s a es o he wo nuclei; he solid line has been ob-
ained by using all he s a es o Table I. As was expec ed,
a low ene gy he abso p i e pa is gi en only by he 3
s a es, he con ibu ion o he high-lying s a es inc easing
wi h he ene gy un il eaching amaximum a ound 650
MeV (no shown in he igu e) and hen dec easing e y
slowly o 0.
Su p isingly enough, o he eal pa we ha e acom-
ple ely di e en in e play be ween low- and high-lying
s a es: he la e a e gi ing abig con ibu ion e en a
e y low ene gy. The di e en beha io depends on he
physical p ocess gi ing ise o he eal and imagina y
pa s o he pola iza ion po en ial. The la e is due o
eal exci a ion o he nuclei, and hus i anishes a low
inciden ene gy when he channels a e closed. The eal
pa is due o i ual exci a ion o he nuclei, so i is
p esen a all ene gies. P e ious calcula ions o he pola -
iza ion po en ial ha e no aken in o accoun he GQR
s a es, missing in his case a leas 50%%uo o he e ec .
The ene gy dependence o he pola iza ion po en ial,
shown in Fig. 7, has a wo old o igin. The dynamic ene -
gy dependence is due o he appea ance o he cen e -o -
mass ene gy in he de ini ion o he p opaga o . The
second one can be hough o as aspu ious one [17]be-
cause i is due o he localiza ion p ocedu e desc ibed in
Eqs. (15) and (16) and i comes h ough he local momen-
u n k.
The ene gy dependence o he eal and imagina y pa s
o he pola iza ion po en ial a e go e ned by he dispe -
sion ela ion. This ela ion is deduced om he gene al
p inciple o causali y, so e e y pola iza ion po en ial
should obey i . The possible iola ion o he dispe sion
ela ion due o he ex a ene gy dependence in oduced
by he localiza ion p ocedu e has been ound o be small
[9,18,19].
In o de o check i in ou case we use he linea
Ca +Ca, EI g~240 MeV
0.0100
-2.580 -2+
-5.0K4 60
40
-7.5
/
/
-10.08R( m) 10
20- 3ll
1
0910 11
R( m)
0100
60-
-10 40-
20
-15
8R( m)
I
10 08R( m)
I
10
FIG. 3. Real and imagina y pa s o he local pola iza ion po en ial o Ca+ Ca (E& b=240 MeV) as a unc ion o he ela i e
dis ance R. Each line co esponds o he con ibu ion o di Fe en mul ipola i ies as indica ed in he igu e. On he le pa he o al
po en ial is gi en by he solid line. On he igh pa he po en ials a e plo ed in pe cen age o he o al one.

2714 M. V. ANDRES, F. CATARA, AND E. G. LANZA
schema ic model gi en by Mahaux e al. [4]. In pa icu-
la , we use he model o Eq. (3.1S) o Re . [4], whe e we
ha e changed he misp in ed plus sign o he las e m o
aminus. In his model he imagina y pa is segmen ed
in o h ee pa s, and his makes i possible o ind an alge-
b aic exp ession o he co esponding eal pa h ough
he sub ac ed dispe sion ela ion. By using Eq. (3.1S) o
Re . [4] we ha e hen calcula ed he eal pa o he po-
la iza ion po en ial o bo h o he imagina y pa s o
Fig. 7; namely, he one due o only he 3low-lying
s a es and he one calcula ed wi h all he s a es o Table
I. In bo h cases he schema ic po en ial has been no mal-
ized o he calcula ed one a E ,b=320 MeV. The
squa es in Fig. 8a e he esul o his calcula ion. The
ag eemen wi h he cu es calcula ed by means o Eq.
(1S) is e y good. This implies ha he localiza ion p o-
cedu e used he e is agood one in he sense ha i does
no iola e he dispe sion ela ion, a leas a he dis ance
conside ed in he calcula ions.
Op ical model analysis [1,20,21] on elas ic sca e ing
da a is a ailable o he sys em '0+ Ca. Ou imagi-
na y po en ial shows abeha io in quali a i e ag eemen
wi h he empi ical one, namely, i dec eases in he
E&,„&80MeV domain and goes o 0a E&,b-40 MeV.
In he same ene gy egion we ha e aco esponding in-
c ease o 6V. Un o una ely, he empi ical alues show
aconside able sca e which makes aquan i a i e co n-
pa ison ha d. As poin ed ou by he au ho s o Re . [4],a
consis en eanalysis o all he da a would be in e es ing.
In o de o ha e a u he es o ou mic oscopic op i-
cal po en ial we ha e calcula ed he elas ic c oss sec ion
o he sys ems '0+ Ca and Ca+ Ca a a ious en-
100 100
80 -2+
60-
40-
80-
60'3 g,
E) b~104 MeV
1
03hl
~as+~~~,~, ,~1»
789
R( m)
I
8
R( m)
100
+
80- 320 MeU E&b ~320 McV
60-
40-
I
20 -3
3hl
P'
0~y
&4O
20-
0
~~~~
~W~~~
10 789
R( m)
100
80 =
100
80-
60-
40 "
E~b ~640 McU 60 =2
0
RID ~640 MeU
~~MM~M~~~
20-3 D
'3 hl
le
0I
89
R( m)
20-
78
R(~m~
~K
'g pso he Pola iza ion po en ial o '~+ Ca d -
plo ed in pe cen age o he o al po en ial and as a unc ion o he ela i e dis ance R.
TENTIAL DUE TOLARIZA TION PO™&T
DYNAMICAL P
4o( aR,=9
0+i
2715
100)
104 MeV. &
pep +"Ca, E)b =
I04
80-
4DCa
6p
40 ~0.1-
on&y
$0
06
I
8
R( m)
I
910 p.0
0.15 yes
100
80 Ca
4D
60 .
40-
o.&o
I005-
0.00
800
El b{MeV)
oh' 3
I
300
80-
10
I 9
78
6R( m)
he pola iza ion p
no en-
lo di
a s oeeo
E«04 MeV
lab
o1b }1 lo
he po en ial gi en on y
nuclei.
he pola iza ion po
7. Real adim gi p. he inciden en
a s oe
FIG. 7. unc ioil oaeal-
l16O+40Ca plo ed as shed cu es ~e e
ial o R=9 m. The das
3s a es o
a ixed alue «.l. he low-lying
o a
dbincluding on y acalcula ion
cua i,h, 1,
olid lines a e
uclei. eso '
done wi a
h ll he s a es oa
=640 MeV, 8GQR
0+ Ca, Ei~b =, R
I100
80-
60- ieO
40 ."Ca
di6'e en ial c oss
g' .Fi .9a e epo ed he elas ic i
=104 MeV (uppe
e gies. In ig. +Ca a Ei,b=
he cases 0+ =
'h he da a is e y goo
h h l
he ola iza ion po en ia
1
a e no a all exhaus i e. n
06
100
I
8
R( m) 10 0.4Ca R= 9 y
0+ a,
80
60
40-
20-
4DC
0.1—
~i
010
786R( m)
=640 MeV. He e h p
he ola iza-
as Fig. 5a Elab =
FIG. 6. Same as 'g.
GQR s a es.
ion po
o en ial is due y
o'onl o he
300
I
100 200
0E|,b(Me )
Fi .7. The squa es co e-
he lowe pa o Fig. .co e
Fddispe sion e ai
spon o
d o asub ac e i
2716 M. V. ANDRES, F.CATARA, AND E. G. LANZA
1O0
-„. 1O-'
j10
16p +40C
104 MeV
0.8
I
I
0.4
'0+ Ca, R= 9 m
l
20
I
40 60
(deg ees)
I.I.
80 100 180 0.0
104
10~
102
E
Cl
b10'
40C +40C
E...=1.43 MeV
0.4
0.2
0.0
/
I
I
I
I
I
Il. . . .I
100 800
Eg b(Me )
I
300
100 40 I
50 60 70
8,I(deg ees) 80
FIG. 9. Elas ic sca e ing angula dis ibu ion o 'O+ Ca
and Ca+ Ca. The la e is plo ed in absolu e uni s. The
da a a e om Re s. [21]and [22], espec i e1y.
side ed he ans e channels which a e hough o be im-
po an a leas o asymme ic sys ems. This may be he
easons why we ha e s ong oscilla ions in he elas ic
c oss sec ion a backwa d angles o he '0+ Ca case.
In ac , as is shown in Re . [5], he imagina y po en ial
due o he ans e channels ha ing alonge ange may
smea ou he oscilla ions.
Ano he poin o be in es iga ed is how much he e-
sul s depend on he alues o he exci a ion ene gies o
he collec i e s a es used in he calcula ions. In o he
wo ds, is he use o an a e age exci a ion ene gy in he
p opaga o and hen he use o he closu e app oxima ion
jus i ied, o is i impo an o use he p ope ene gies o
low- and high-lying s a es'? To answe his ques ion we
ha e done acalcula ion including aH he s a es o Table I,
whe e he ene gies ha e been a bi a ily pu equal o a
Axed alue. In pa icula , we ha e chosen E160 6.5
MeV and E4, =5.0MeV (Re [9]). Th.e esul s a e e-
4'Ca
po ed in Fig. 10 as dashed lines, while he solid lines e-
sul om he RPA ene gies epo ed in Table I. The be-
ha io wi h he ene gy o he imagina y po en ial is qui e
di Fe en in he wo cases. In he calcula ion co espond-
ing o he solid line he e ec o he high-lying s a es is
e iden : he po en ial goes up as he inciden ene gy in-
c eases. The dashed line esembles he one in Fig. 7,
FIG. 10. Same as Fig. 7. The dashed cu es e e o acalcu-
la ion pe o med wi h cons an exci a ion ene gy E&6 =6.5
0
MeV and E4O =5.0MeV, while he do -dashed ones co e-
Ca
spond o he alues EI6 =20 MeV and E4o =17 MeV (see
ex ). The solid lines a e he same as he ones plo ed in Fig. 7.
which was calcula ed wi h only he wo low-lying s a es.
Thus all he s a es now gi e an equally impo an con i-
bu ion, a leas a low ene gies. The magniAca ion ac o
be ween he wo cu es is no equal o he numbe o
s a es used in he calcula ion because we ha e kep he
RPA o m ac o s in bo h cases. Fo comple eness we
ha e done acalcula ion wi h he a e age ene gies close o
he ene gies o he gian quad upole esonance s a es, i.e.,
E,6=20 MeV and E4o =17 MeV. The esul s (do -
dashed line in Fig. 10) show an enla gemen o he ene gy
scale: he apid inc ease and hen dec ease o he eal
pa o he pola iza ion po en ial is now sp ead ou o e a
much la ge ange o inciden ene gy. As expec ed, he
imagina y pa is impo an a high ene gies and anishes
a E„b—
100 MeV (see also lowe pa o Fig. 7). The eal
pa s a e go e ned by he dispe sion ela ion: di e en
beha io s o he abso p ion will p oduce di e en beha -
io s o he eal pola iza ion po en ial. In pa icula , ac-
co ding o he simple o m o he linea schema ic model
o Re . [4] [Eq. (3.17)], he posi ions o he maxima o he
eal pa s a e de e mined by he ene gy alue co espond-
ing o hal he in e al whe e he imagina y po en ial
goes apidly o 0. The ac ha he eal po en ial co e-
sponding o he do -dashed line has amaximum nea 200
MeV conA ms his simple model.
We ha e shown, a leas o he wo ex eme cases
p esen ed in Fig. 10, ha he use o a e age exci a ion en-
e gies in he p opaga o o Eq. (2) gi es ise o pola iza-
DYNAMICAL POLARIZATION POTENTIAL DUE TO THE. . . 2717
ion po en ials which di e in shape and magni ude om
he po en ial calcula ed wi h he p ope ene gies o bo h
low- and high-lying s a es. Since hese esul s a e no a
all exhaus i e, his p oblem should be u he in es iga -
ed.
IV. CONCLUSIONS
Wi hin he Feshbach o malism we ha e calcula ed he
dynamical pola iza ion po en ial a ising om he cou-
pling o he elas ic channel o he collec i e ib a ional
s a es. The la e we e cons uc ed wi hin sel -consis en
RPA wi h he SGII o ce. Bo h ene gies and ansi ion
densi ies compa e well wi h he a ailable expe imen al
da a. The p opaga o was calcula ed in he WKB ap-
p oxima ion, which is e y good a inciden ene gies
g ea e han he Coulomb ba ie . The ba e nucleus-
nucleus po en ial has been cons uc ed by double olding
he e ec i e M3Y in e ac ion wi h he HF densi ies o
he wo nuclei. We ha e no made use o he closu e ap-
p oxima ion bu we ha e summed o e a ini e numbe o
ele an s a es, each one wi h i s own ene gy. The locali-
za ion p ocedu e we ha e used does no des oy he ene -
gy dependence o he pola iza ion po en ial: The eal and
imagina y pa s sa is y he dispe sion ela ion.
We ha e done an analysis o he eal and imagina y
pa s o he pola iza ion po en ial in e ms o he ela i e
con ibu ions o he single collec i e s a es o he sys-
ems '0+ Ca and Ca+ Ca a se e al inciden ene -
gies. As one should expec , a low inciden ene gies he
main con ibu ion o he imagina y pa o he pola iza-
ion po en ial comes om he 3low-lying s a es. As he
ene gy is inc eased he ole played by he 3s a es is ak-
en o e by he GQR s a es which gi e he main e ec .
Con e sely, he eal pa is domina ed by he high-lying
s a es in all he ene gy ange in es iga ed, and hey con-
ibu e up o 80% o b, V. This no el esul shows ha
he GR s a es canno be dis ega ded in he cons uc ion
o adynamical pola iza ion po en ial.
We ha e also shown ha he use o he a e age exci a-
ion ene gies, in he wo ex eme cases ea ed he e, gi es
ise o apola iza ion po en ial which di e s app eciably
bo h in shape and magni ude om he one calcula ed
wi h he p ope ene gies.
The ene gy dependence o he pola iza ion po en ial
has been checked by means o he dispe sion ela ion. In
pa icula , we ha e used he linea schema ic model o
Re . [4]. The esul is e y good. Calcula ions o elas ic
c oss sec ion gi e agood desc ip ion o expe imen al
da a: hese calcula ions ha e been done wi h no adjus -
able pa ame e s. ACKNOW%'LED
GMENTS
We wish o hank M. A. Naga ajan, N. Van Giai, and
N. Vinh Mau o use ul discussion and sugges ions. We
also hank N. Van Giai o making his RPA code a ail-
able o us.
APPENDIX
In his Appendix we wan o calcula e he quan i y b,V,2which is de ined as
AV,2(R,R')= gFK K(R)GK K(R,R')FK K(R'),
K1K2
whe e FK K(R) is gi en by
12
FK K(R)=(00~u(R)~K, Kz) = d , d 2pK o( , )u(~ , —
2+R~)poK ( z)
(Al)
/dp pu(p)pK o(p)poK (p)i ~1L2(
4~'" 12
Xgj~(PR)(L, M, L, M, ~J M, +M, )(L, 0Lz O~J 0)YJM +M (R),
J
(A2)
whe e we ha e used he ac ha
pK, o( i)=pK, o( , )YI. M( i)
and he de ini ion o he Bessel-Fou ie ans o m
pK 0(p)=4~ " i«iJI.,(p i) K,o( i ).
(A.3)
(A4)
p('s) g(J—
AMpA p~J M) YJ iM
—
„(p)Yq(s) .
We ha e deno ed wi h jl and Y& he Bessel and sphe ical ha monic unc ions, espec i ely. The ca e o e he spin
a iables deno es J=VZJ+ I, and 8'(p) is he Fou ie ans o m o u. Acco ding o Eq. (4) we can w i e R=p+ —,s;
hen in his case we ha e [Eq. (6.50)] o Re . [23]:
I/2
2I. +1
RYJM(R)=&4m. g
Q=o