PHYSICAL REVIEW CVOLUME 39, NUMBER 1JANUARY 1989
Random-phase-app oxima ion-based dynamical pola iza ion po en ial
M. V. And es
Depa amen o de F&'sica A omica yNuclea , Uni e sidad de Se illa, 41080Se illa, Spain
F.Ca a a
Is u o di Fisica dell'Un e si a, 98100Messina, I aly
and Is i u o Nazionale di Fisica Nuclea e, Sezione di Ca ania, 95129 Ca ania, I aly
Ph. Chomaz
Di ision de Physique Theo ique, Ins i u de Physique Nucleai e, 91406 O say Cedex, F ance
E.G. Lanza
Is i u o Nazionale di Fisica Nucleai e, Sezione di Ca ania, 95129Ca ania, I aly
(Recei ed 18 July 1988)
Adynamical pola iza ion po en ial is de ined aking in o accoun he e ec s on he elas ic chan-
nel due o he exci a ion o ib a ional collec i e modes, desc ibed wi hin he andom-phase app ox-
ima ion. The p obabili y ampli udes o exci ing hese modes a e e alua ed by in eg a ion along
classical ajec o ies de e mined by he eal pa o he nucleus-nucleus po en ial wi h ene gy and
angula momen um loss. Calcula ions pe o med o he Ca+ Ca sys em show ha , a high bom-
ba ding ene gy (E/A =44 MeV), bo h he eal and he imagina y pa s o he pola iza ion po en ial
a ise mainly om he exci a ion o he high-lying modes. A ene gies nea he Coulomb ba ie , he
calcula ed elas ic c oss sec ion is in good ag eemen wi h he expe imen al da a. The inclusion o
he eal pa o he pola iza ion po en ial imp o es his ag eemen .
I. INTRODUCTION
An impo an p oblem in he s udy o pe iphe al
hea y-ion p ocesses is o connec he nucleus-nucleus po-
en ial o he unde lying nucleon-nucleon in e ac ion and
o he mic oscopic s uc u e o he wo pa ne s. In he
olding model' he eal pa o he op ical po en ial is
cons uc ed om he g ound-s a e densi y dis ibu ions
o he wo colliding nuclei. On he o he hand, e en in
he pe iphe al collision egime, p ocesses like nucleon
ans e and exci a ion o collec i e deg ees o eedom
ake place. The p oblem hen a ises o cons uc ing apo-
en ial which desc ibes he e ec s o such p ocesses on
he elas ic channel, he pola iza ion po en ial.
The pola iza ion po en ial a ising om he ans e
mode was s udied in Re . 2(see also, e e ences quo ed
he ein). The modi ica ions o he olding po en ial due
o he exci a ion o he collec i e ib a ional s a es we e
calcula ed, in he adiaba ic limi , i i Re . 3. I was ound
ha he co ec ions a e impo an and gi e as ong
enhancemen o he sub-ba ie usion c oss sec ion.
In he p esen pape we ex end he p e ious analysis by
aking in o accoun he dynamics o he collision. This
gi es ise o acomplex pola iza ion po en ial which, o-
ge he wi h he olding one, de ines an op ical po en ial
while aking in o accoun he coupling o he elas ic
channel o he ones co esponding o he exci a ion o
collec i e deg ees o eedom. The la e a e desc ibed
wi hin he andom-phase app oxima ion (RPA) while he
ela i e mo ion is ea ed classically as go e ned by he
olding po en ial plus he eal pa o he pola iza ion po-
en ial. The ene gy and angula momen um loss a e also
app oxima ely aken in o accoun .
The op ical po en ial has been used o calcula e he
elas ic c oss sec ion. We ha e s udied he sys em
Ca+ Ca a a ious ene gies. Ou esul s can be sum-
ma ized as ollows. The heo e ical c oss sec ion is in
good ag eemen wi h he exis ing expe imen al da a,
which shows ha ou model is able o gi e, wi hou any
adjus able pa ame e , a easonable desc ip ion o he op-
ical po en ial. A low ene gies, bo h he eal and imagi-
na y pa s o he pola iza ion po en ial a e needed in o -
de o ep oduce he da a. On he o he hand, a sligh ly
highe ene gy (E/A =6 MeV), once he abso p ion has
been aken in o accoun , he heo e ical esul s wi h and
wi hou he eal pa o he pola iza ion po en ial a e
equi alen ly good wi h espec o he a ailable expe i-
men al da a. A low ene gies (e.g.,a E/A ~10 MeV),
he con ibu ion o he low-lying collec i e s a es o he
pola iza ion po en ial is ound o be dominan .
In o de o analyze o which ex en he exci a ion o
he gian esonances plays a ole in he de e mina ion o
he op ical po en ial, we ha e also made acalcula ion a
E/2 =44 MeV, whe e hei e ec s a e expec ed o be
la ge . Due o he absence o expe imen al da a in his
case, we only make acompa ison be ween heo e ical e-
sul s ob ained wi h o wi hou hem. We ind ha he in-
clusion o he gian quad upole esonance (GQR) s ong-
ly modi ies he elas ic sca e ing c oss sec ion a angles
nea he g azing.
39 99 1989 The Ame ican Physical Socie y
M. V. ANDRES, F.CATARA, PH. CHOMAZ, AND E.G. LANZA 39
II. THE MODEL
The basic assump ions o ou model a e ha o g az-
ing collisions, he ela i e mo ion can be ea ed classical-
ly and he mu ual exci a ion o he wo pa ne s can be
desc ibed as due o he mean ield o one nucleus ac ing
on he o he one. In pa icula , we will concen a e on
he exci a ion o collec i e ib a ional s a es, which we
desc ibe mic oscopically wi hin he RPA. Unde he
abo e-men ioned assump ions, he in insic Hami1 onian
o he sys em is he sum o wo e ms, one o he a ge
and one o he p ojec ile, each o hem ha ing he o m
H;„,„=gEkB),
"B),+gVko( )Bk
kk
+H.c.+gV)k.( )B„"Bk
kk'
whe e he ime dependence comes in h ough he ela i e
dis ance R( ), which obeys classical equa ions o mo ion.
The eedback o he exci a ion on he ajec o y is aken
in o accoun by de ining he ene gy and angula -
momen um loss as equal o he mean alues o he ene gy
and angula momen um s o ed in he collec i e deg ees
o eedom. In addi ion o ha , a each in eg a ion s ep
in ime, we modi y he nucleus-nucleus olding po en ial
by adding o i he eal pa o he pola iza ion po en ial.
In Eq. (1), B&(B&)a e boson ope a o s co esponding o
he RPA ope a o s Qk(Q„)
whe e
N( ) =glI„( )l' .
In he amewo k o he semiclassical app oach, we hen =
de ine he complex pola iza ion po en ial as
(ol p), &=exp —
i d '[b, V( ')+iW( ')], (12)
whe e AV and 8' depend on ime h ough he ela i e
dis ance R( ) and he in eg a ion is made along aclassi-
cal ajec o y.
F om Eqs. (10) and (12) i ollows:
0V( )=P( )= Im QI„*( )I„( )
k(13)
whe e
P( ) =Im g F„*( ')d ' F„( ")d ",
I„( )=—
i Fk( ')d ',
iF.~I
F„( )=V„o( )e
and all he in eg als a e e alua ed along aclassical ajec-
o y. F om Eq. (6) i ollows ha he p obabili y ampli-
ude o he sys em o emain in i s g ound s a e is gi en
by
(()l@ &—
i(0( )—
i/2A( )
Qk'= gI&k(Ii)[I il~+I"~(Ih)[h Il~l,
ph W( )= ,'N( )= —
Re—
—QI,
*( )I„( )
k(14)
lq &=Q'lo&, Qlo&=o,
whe e lo& is he RPA g ound s a e and l&Ip), & he collec-
i e s a es we a e in e es ed in. The ampli udes Xand Y
and he ene gies Ek a e solu ions o he RPA equa ions.
The ma ix elemen s
(4)
a e calcula ed wi hin he RPA. The ope a o U
ep esen s he mean ield o he o he nucleus. In wha
ollows we ha e neglec ed he nonlinea e ms in he in-
insic Hamil onian o Eq. (1) since hey a e no impo -
an o he p esen calcula ion. Indeed, as i was shown
in Re . 4, a low-bomba ding ene gy, hei inclusion
enhances he popula ion o he collec i e high-lying
s a es which, howe e , emains much smalle han he
popula ion o he low-lying modes. Then, hese mixing
e ms a e impo an only when one is s udying aselec i e
p ocess like he inelas ic sca e ing. A high-bomba ding
ene gy hey a ec e y li le, e en he inelas ic sca e ing
c oss sec ion.
The s a e o he sys em a any ime is he p oduc o
wo cohe en s a es
[I ( )] k(B1') k
k
whe e he Ik( ) a e de ined in Eq. (8). The so-de ined
unc ions 6Vand 8'canno di ec ly be in e p e ed as he
eal and he imagina y pa s o apo en ial. Indeed, hey
depend on he whole his o y o he sys em up o he ime
. In pa icula , hey ha e di e en alues when he sys-
em eaches he same ela i e dis ance in he app oaching
and he ou going phase. They wi11 also depend on he
conside ed classical ajec o y and hen on he angula
momen um as well as on he ene gy. In he nex sec ion
we will discuss ap ocedu e o de ine alocal-pola iza ion
po en ial s a ing om Eqs. (13) and (14). Adi e en
p ocedu e is conside ed in Re . 7whe e he a en ion is
mos ly concen a ed on he ela i e impo ance o he
low-lying s a es and high-lying s a es in de e mining he
ail o he imagina y pa o he op ical po en ial a high-
inciden ene gy.
III. THE POLARIZATION POTENTIAL
The ba e nucleus-nucleus po en ial is calcula ed by
olding he M3Y e ec i e in e ac ion wi h he Ha ee-
Fock densi ies o he wo nuclei. The collec i e s a es a e
calcula ed wi hin he consis en RPA wi h he Sky me in-
e ac ion SGII. The esul s we will show la e we e ob-
ained by including he s a es epo ed in Table I, i.e.,
hose exhaus ing a leas 5% o he ene gy weigh ed sum
ules (EWSR).
In o de o discuss how o ex ac he pola iza ion po-
en ial om he quan i ies i)), V( ) and W( ), le us exam-
ine in de ail aspeci ic case. %'e conside he Ca+ Ca
lPl
DYNAMICAL
NgASED D
ROXIMATIO
pHASE-AP
ANDOM
0
39
E(MeV)
J7T
NucleI
5031
16.561
16 970
17 488
17.667
18.20&
18 399
19.219
22.166
23 554
32 716
11
24
17
8
10
14
14
17
10
3
2+
2+
2'
4+
4+
0+
0+
0+
0
3
4oCa
calcula Ion.
ic da a use
Spec oscopIc
&E~SR
TABI-E
0
-2
O
4
-6
III
1O 11
n y 3&2 pn~y 3
91O 11 12]O 11
Me& bomb s unc 1ons
jn ene gy he
44 Me gp'and Vas ugIn he
sys em d2we hw
5O and 26
In F1gs R o L=o he esul s
~]an
ob a1ne
ela i e d1 h igu e we Pl he lpw-ly' g
s P eac e Pn yj.e., he
ee sec o jgh o 2+ s a es
~ om h ee pnucl
s a ~
ll he s a p each c
GQ 'ho he (nPi h)n e a&&e . o in-
R) and ae Pa edue o
app oa p' inc ease, They con he
ach eac jn abso n jnue
yand 's a es. eaches
i ies ~he collec 1 e he sys em
ion o he ime ~ e dec ease, a
gi a ol o so ed hen h y
ease s ead1 yach an . +Van
c o closes aPP The beha ' lybe ween
di anc a dis an~~ h he in i ~) o
1s 1n .Phase 1s ga ac e is lc
collision ime
~clusjpn o
es, he '"
he collec om he gn ibu 1o
ea en main
As i 1s app di es he
ucj» an
GQR is c
a , Eq. (1
}eimagina y p
F1bu FIG. a ne as F'g'
a h1s enc-
e ac ha , apn-
~'due o he Ris la ge. Co
xc1 a ' ",. he con ' "'
nc 1o s~~ nd ou go1ng p e ec s
The unc oaching an he memp y
same R.man1 es
la'Lcula ion
jn he app 's a jpn
hjs is aamjca ca .p en 1a
lisipn. T jn ou dyna pla iza 1on p
h1ch a «e in pon
&&es a& dene g» each L)
he s1m a aje«o y'loses
(i.e.,
~~+iW) deP ha each dis ance o
i e dis an he pp en hcollec i
d ines he hexci a 1P
alueo 'np nce 0
e'nce e.a oun .e he
d he d1s a
oach. S'n ll in e al uld 1nclu.
aPP in asma i ion shou be easily
plac hsp ese P
app oach, h ep ocess-
loses
le ec s
1n phy qu an 1
he wo q
shown
+gV( )d
p(+ ~)=
(&s)
and +"w( )d'
)~(+m)=
O'
O
+.,pn~y 3
)y 3&2
I
1O 11 12
10 11
-5 g10
E.(i3)] «
o en ~al [q he
he po as un
a iza ion P c ion o
Ral pa,l o880 Me 'd he esul
he gIll .h o Je ,,07~gR), an
edis ance -
m igh o' he
ela I ~
1ding, om + a e~ ('e' e o
b ained bP, nd he h e
in u'ee 2s es e
he h ee cu 50,
s a e, an hsec o , e.L, =240, 2
s a e, e Table I. In ela mome~™
a»,„ alue. o
he s a e he ang
h ee di e en a
and 260 A.
R=R1+R~,
0
whe e
he eal
jpn conne .The
ing ha sen in he poo app o
his 1s ah ajec o y ession o i he
iza 1on so e jcexP lwe
ha e ion p
an ana ypp en ja e w1 h
In o de o he pola jza u p ocedu
a s p-dby us1ng
jmag1n lues ob a .9
lc 1on o a i h adius
col eon o ms, wWoods-Saxon
102 M. V. ANDRES, F.CATARA, PH. CHOMAZ, AND E. G. LANZA 39
R, =(1.20M,'—
0.09) m .(16) ~~~~
The deduced alues o he dep h and he di usi i y,
when all he s a es a e included, a e AVo= —
14.26 MeV,
az&=0.491 m o he eal pa and lgo= —
25.34 MeV,
a~ =0.495 m o he imagina y pa . The op ical po en-
ial is hen ob ained by adding he eal pa AV o he
olding po en ial while he imagina y pa is gi en by 'lV.
Calcula ions ha e been done a se e al ene gies, in pa -
icula , we ha e s udied he beha io o he pola iza ion
po en ial nea he Coulomb ba ie . A such ene gies,
he main con ibu ion comes om he low-lying 3s a e.
All he esul s a e collec ed in Table II, whe e o each
ene gy we epo he alues o he dep hs and di usi i ies
o AV and lH. The ene gy dependence o he eal pa o
he op ical po en ial is be e illus a ed in Fig. 3whe e
we compa e he ba e- olding po en ial plus he Coulomb
pa (dashed line), wi h he ba ie s ob ained a he ene -
gies indica ed in Table II and in he cap ion. As he in-
ciden ene gy dec eases, he co ec ion o he olding po-
en ial inc eases, eaching e en ually he adiaba ic limi
o he pola iza ion po en ial (see Re . 3).
This beha io wi h he ene gy can be ela ed wi h he
indings o Re . 10 whe e i was shown ha in o de o
ge agood i o he expe imen al elas ic c oss sec ion, i
is no su hcien o a y he pa ame e s o he abso p i e
pa o he op ical po en ial, bu i is also necessa y o
mul iply he olding po en ial by an ene gy-dependen
ac o g ea e han 1, when he ene gy app oaches he
ba ie om abo e. This has been explained, and
con i med by explici calcula ions, "as acoupled-
channels e ec . The au ho s o Re . 12 ha e in e p e ed
his ene gy dependence by using he dispe sion ela ion
be ween he eal and imagina y pa o he op ical po en-
ial. Ou esul s a e no based on aphenomenological
analysis, bu a he on an app oach which akes in o ac-
coun he collec i e ib a ional modes in acomple ely
mic oscopic way. In he nex sec ion we will see ha he
c oss sec ions ob ained, including he eal pa o he po-
la iza ion po en ial, a e in be e ag eemen wi h he ex-
pe imen al da a han he ones wi hou such co ec ion.
IV. ELASTIC SCATTERING CROSS SECTION
The op ical po en ial V ,&d+AV+i%', de ined in he
p e ious sec ion, has been used o calcula e elas ic
O
O52- /
l
I
~
~o 10
FIG. 3. Ba e double- olding po en ial plus he Coulomb e m
(dashed line) compa ed wi h he ba ie s calcula ed a se e al
cen e o mass ene gies: 880, 120, 71.8, 64.8, 60.6, and 55.45
MeV ( om abo e o below).
di e en ial c oss sec ions o he Ca+ Ca sys em and
o se e al alues o he ene gy. The case E/A =44
MeV is in e es ing since a such high ene gy he p obabil-
i y o exci ing he GQR is much highe han a lowe en-
e gies. The e 'ec s o he GQR, which ha e al eady been
shown in he p e ious sec ion o be e y impo an in he
calcula ion o he pola iza ion po en ial, show up also in
he elas ic c oss sec ion. In o de o ha e abe e insigh ,
we ha e done calcula ions wi h he op ical po en ial con-
s uc ed by including only he low-lying 3s a e and all
he s a es o Table I. F om he esul s shown in Fig. 4,
we see ha o angles la ge han he g azing one he
di e ences a e impo an . This is in ag eemen wi h he
indings o Re . 13, whe e i ws shown ha he GQR is
s ongly exci ed a his ene gy and a ound he g azing
angle.
The impo ance o he eal pa o he pola iza ion po-
en ial is illus a ed in Fig. 5whe e he esul ob ained,
wi hou including i (dashed line), is compa ed wi h he
c oss sec ion calcula ed wi h he comple e op ical po en-
ial. F om he igu es we see ha in he same angula
ange as be o e he inclusion o he I'eal pola iza ion po-
en ial s ongly enhances he c oss sec ion. This no el e-
TABLE II. Pa ame e s o he Woods-Saxon o ms o he eal and imagina y pa s o he pola iza-
ion po en ial o he Ca+ Ca sys em. The alue o Ro (Re . 9) is 8.0279 m o bo h. The o al eac-
ion c oss sec ion calcula ed wi h an op ical po en ial whose eal pa is jus V Jd is deno ed by az', he
one calcula ed by Sa chle and Lo e (Re . 1) is deno ed by o~". The las line e e s o he calcula ion
done including only he low-lying 3 s a e.
(MeV)
55.45
60.6
64.8
71.8
120
880
880
~&O
(MeV)
—
179.87
—
155.84
—
142.68
—
127.06
—
49.78
—
14.26
—
1.27
( m)
0.380
0.385
0.386
0.389
0.459
0.491
0.504
lÃo
(MeV)
—
37.40
—
33.23
—
50.36
—
92.14
—
31.22
—
25.34
—
8.54
( m)
0.424
0.441
0.411
0.371
0.443
0.495
0.475
(mb)
100
361
534
779
1642
2423
2038
old
(mb)
59
302
474
719
1583
2414
2037
SL
(mb)
671
939
1893
39 &ANDOM pH ASE-APPRQX IQN-BASED 0
I
YNAMICAL
I
~~ ~
ioo
i0-2
Ig » &~-~ ~ -
A--~
%/ x/
l
84
FI 4El .(d g)
IO
sys em a E— ' 1c o
88O
sec ion
ob ained b.oMeV. The do he OCa+ 4o
using he edashed line aCa
low-lying 3—epo en ial ceshows he
a«, wh;l ns uc ed i
pie e calcula . '»e he ull ]i 'ncluding onl h
a (on co esPonds ye
s o he co n
1O3
102
10
ns whe e h
by he old; o he op ical
men al dgone. The po en jal is g'
aa a is im o eag eemen wgi en
« he pol P o ed by he 'wi h he exp
po a izaiion o 'einclusion o &pe i
on he o 1
"po en ial. I s ."0 he eal pa
Table II o h ec ion, whose 1
pp eciable also
ege wi h ho aues a e e
oen jal
op ical p« .oecalcula d.'p ed jn
sec ion. Th' ob ained b i
e . 1by usin
in R
is e ec is y ing he clas '
a o } ss sec jon ip~a
angula pola iza ion 'o sensi i e
whe e epo en ial ao
pe imen al da leas in he
~e»s (gee F;
10+
1g. 7)
sui shows ha
o a a eno maaliza ion o h
1
ia is also e eal a
e ec able a igh en
In o de j esi h
1
ssec ion.
ea e also caacula ed he
en al clas po en ial is
ic c oss se
gyp
eima in
o e o i
We ha e he ec ion nea h eexpe i-
dmb ba ie .
ee o which d
u'ea jo wi h mo el is able
h Th
eines co es a e e-
spon o calcula-
L
E
10
10
IO'—
10
l0 2=
lO ~
0
Ecm= 880 MeV
6
cm. (d g)
Kl
U
I
8IO
10 I
50 60
c.m. (deg)
80 90
FICx. 5.
~.Same as Fi
~.ig. 4, bu in edashed line is h
is e
gyp
e eal o1
ep esen ed aken in o apo a iza ion co ec-
iine. ain ecalcula ion
FIG. 6. C
Compa ison p
e
wh' as ed line is h -o -mass ene ie
omgies as indica -
asned using V old +

M. V. ANDRES, F.CATARA, PH. CHOMAZ, AND E.G. LANZA 39
10
10
E10
1
10 c.m. =" 0
10 0I
10 I
20 50 40 50
FIG. 7. Same as Fig. 6bu a E, =120 MeV.
In all he conside ed cases, he ag eemen wi h he ex-
pe imen al da a is a he good. This is an indica ion ha
he op ical po en ial is well es ima ed in ou model and
shows he ele ance o he collec i e modes. We s ess
ha we do no ha e any adjus able pa ame e . Besides,
in gene al, o he deg ees o eedom, such as nucleon
ans e and noncollec i e pa icle-hole exci a ion,
neglec ed in he p esen calcula ion, a e expec ed o play
some ole.
V. CONCLUSION
We ha e calcula ed adynamical pola iza ion po en ial
as due o he coupling o he collec i e ib a ional s a es,
desc ibed wi hin he RPA. The dynamics is in oduced
by in eg a ing he equa ions o mo ion o he ele an
deg ees o eedom along classical ajec o ies go e ned
by he eal pa o he op ical po en ial. Ene gy and
angula -momen um loss a e aken in o accoun con-
sis en ly. The op ical po en ial is hen cons uc ed by
adding he pola iza ion po en ial o he olding one.
We ha e pe o med calcula ions o elas ic di e en ial
c oss sec ion o he Ca+ Ca sys em a se e al ene -
gies. Ou esul s a e in good ag eemen wi h he expe i-
men al da a, showing ha he op ical po en ial is well es-
ima ed. The inclusion o he eal pa o he pola iza ion
po en ial imp o es his ag eemen . A ene gies nea he
ba ie , his beha io can be ela ed o he indings o
Re . 10 whe e a eno maliza ion o he olding po en ial
was shown o be necessa y in o de o i he elas ic
sca e ing c oss sec ion. Ade ailed analysis o he con i-
bu ions coming om he di e en collec i e modes shows
ha a low ene gies he pola iza ion po en ial a ises
essen ially om he low-lying 3s a e. Con e sely, a
high ene gy, namely a E/A =44 MeV, he GQR plays a
e y impo an ole in he de e mina ion o he op ical
po en ial and has as ong e ec on he elas ic di e en ial
c oss sec ion a ound he g azing angle.
The Di ision de Physique Theo ique is "Labo a oi e
Associe au Cen e Na ional de la Reche che
Scien i ique. "
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