Mic oscopic desc ip ion o Coulomb and nuclea exci a ion o mul iphonon s a es
in 40Ca¿40Ca collisions
M. V. And e
´s
Depa amen o de Fı
´sica A o
´mica, Molecula y Nuclea , Uni e sidad de Se illa, Apdo 1065, E-41080 Se illa, Spain
F. Ca a a and E. G. Lanza
Dipa imen o di Fisica Uni e si a
´di Ca ania and INFN, Sezione di Ca ania, I-95129 Ca ania, I aly
Ph. Chomaz
GANIL, Boı
ˆ e Pos ale 5027, F-14021 Caen Cedex, F ance
M. Fallo and J. A. Sca paci
Ins i u de Physique Nucle
´ai e, IN2P3-CNRS, F-91406 O say Cedex, F ance
共Recei ed 2 Janua y 2001; published 14 Decembe 2001兲
We calcula e he inelas ic sca e ing c oss sec ions o popula e one- and wo-phonon s a es in hea y ion
collisions wi h bo h Coulomb and nuclea exci a ions. S a ing om a mic oscopic app oach based on andom-
phase app oxima ion, we go beyond i in o de o ea anha monici ies and nonlinea e ms in he exci ing
ield. These anha monici ies and nonlinea i ies a e shown o ha e impo an e ec s on he c oss sec ions bo h
in he low ene gy pa o he spec um and in he ene gy egion o he double gian quad upole esonance. By
p ope ly in oducing an op ical po en ial, he inelas ic c oss sec ion is calcula ed semiclassically by in eg a ing
he exci a ion p obabili y o e all impac pa ame e s. A sa is ac o y ag eemen wi h he expe imen al esul s is
ob ained.
DOI: 10.1103/PhysRe C.65.014608 PACS numbe 共s兲: 25.70.De, 21.60.E , 21.60.Jz, 24.30.Cz
I. INTRODUCTION
All heo e ical app oaches used o calcula e he c oss sec-
ion o he mul iple exci a ion o gian esonances 共GR兲in
hea y ion collisions a e based on a semiclassical desc ip ion
o he p ocess 关1兴whe e he exci a ion o one eac ion pa -
ne is assumed o be due o he ac ion o he mean ield o he
o he and is ea ed quan um mechanically while he ela i e
mo ion is de e mined classically.
Fo each eigens a e
␣
o he in e nal Hamil onian o one
nucleus, one can calcula e i s exci a ion p obabili y P
␣
(b)by
pe u ba ion heo y o by sol ing a sys em o coupled equa-
ions. This is done by in eg a ing he equa ions o mo ion
along he classical ela i e mo ion ajec o y co esponding
o he impac pa ame e b. The o al exci a ion c oss sec ion
␴
␣
is hen e alua ed by in eg a ing he p obabili y o e all
he impac pa ame e s s a ing om a minimum one, bmin .In
Coulomb exci a ion s udies, he alue o he la e is chosen
acco ding o a sys ema ics 关2兴 ollowing some p esc ip ion
based on he condi ion ha he con ibu ions om he
nuclea ield should be elimina ed. E en so, howe e , some
ambigui ies a e p esen since he calcula ed c oss sec ions
can a y app eciably o small a ia ions o bmin . Mo eo e ,
when he bomba ding ene gy is no e y high and he wo
nuclei a e no e y hea y, he nuclea exci a ion is he domi-
nan p ocess. In his si ua ion one canno apply ha p oce-
du e because, in p inciple, one should add mo e in e nal a-
jec o y o he de e mina ion o
␴
␣
. On he o he hand, he
ajec o ies co esponding o small impac pa ame e s would
no con ibu e oo much o he inelas ic c oss sec ion i he
abso p ion due o all o he channels is aken in o accoun .
This can be done by in oducing an op ical po en ial as was
al eady done in a quali a i e way in Re . 关3兴.
In his pape we p esen calcula ions o he exci a ion
c oss sec ion o one- and wo-phonon s a es in he
40Ca⫹40Ca eac ion a 50 MeV/m o which expe imen al
esul s exis 关4兴. The calcula ions a e done wi hin he ex-
ended andom-phase app oxima ion 共RPA兲model desc ibed
in ou p e ious wo ks 关5–7兴whe e we ha e in oduced an-
ha monici ies in he in e nal Hamil onian and nonlinea
e ms in he ex e nal ield. This model has been success ul in
he desc ip ion o he exci a ion o he double gian eso-
nances 共DGR兲, educing he disc epancy be ween he mea-
su ed c oss sec ion and he s anda d heo e ical es ima e.
He e he model is ex ended by in oducing an op ical po en-
ial in o de o a oid he unce ain y on he in eg a ion o e
he impac pa ame e . Since he op ical po en ial akes in o
accoun he abso p ion due o all channels, we ha e in o-
duced a p ocedu e in o de o a oid double coun ing he
e ec s o he channels explici ly included in ou calcula ions.
In he ollowing sec ion we will ecall b ie ly ou model and
ex ensi ely desc ibe i s imp o emen s. In Sec. III we p esen
ou esul s and he quan i a i e compa ison wi h he expe i-
men al indings. We hen d aw ou conclusions and discuss
some pe spec i es.
II. THE MODEL
The bes mic oscopic heo y o desc ibe collec i e exci a-
ions in nuclei is he RPA whose Hamil onian can be w i en
as
PHYSICAL REVIEW C, VOLUME 65, 014608
0556-2813/2001/65共1兲/014608共7兲/$20.00 ©2001 The Ame ican Physical Socie y65 014608-1
HRPA⫽兺
␯
E
␯
Q
␯
†Q
␯
,共2.1兲
whe e he phonon c ea ion ope a o is
Q
␯
†⫽兺
p,h共Xph
␯
Bph
†⫺Yph
␯
Bph兲.共2.2兲
The bosonic ope a o s Ba e he lowes -o de e ms o he
bosonic expansion o he e mionic ope a o s 关8兴
ap
†ah→Bph
†⫹共1⫺&兲兺
p⬘h⬘
Bp⬘h⬘
†Bp⬘h
†Bph⬘⫹¯.共2.3兲
He e, he index p(h) labels he pa icle 共hole兲s a es wi h
espec o he Ha ee-Fock 共HF兲g ound s a e. The o he
e ms a e he i s one co ec o he Pauli p inciple.
In he ha monic RPA Hamil onian 共2.1兲only he Vph,p⬘h⬘
and Vpp⬘,hh⬘ e ms o he esidual in e ac ion a e aken in o
accoun . I we conside also he o he e ms Vpp⬘,p⬙,p⵮,
Vhh⬘,h⬙h⵮,Vpp⬘,p⬙h, and Vph,h⬘,h⬙and in oduce he map-
pings 关8兴
ap
†ap⬘→共ap
†ap⬘兲B⫽兺
hBph
†Bp⬘h,
共2.4兲
ahah⬘
†→共ahah⬘
†兲B⫽兺
pBph
†Bph⬘.
We end up wi h a Hamil onian con aining cubic, qua ic, e c.
e ms in he phonon c ea ion and annihila ion ope a o s. In
he space spanned by one- and wo-phonon s a es he bosonic
Hamil onian is
H⫽兺
␯
E
␯
Q
␯
†Q
␯
⫹
冋
兺
␯
1
␯
2
␯
3
V
␯
1
␯
2
␯
3
21 Q
␯
1
†Q
␯
2
†Q
␯
3
†
⫹兺
␯
1
␯
2
␯
3
␯
4
V
␯
1
␯
2
␯
3
␯
4
22 Q
␯
1
†Q
␯
2
†Q
␯
3Q
␯
4
册
⫹H.c., 共2.5兲
whe e V21 (V22) a e he ma ix elemen s connec ing one-
pho on s a es wi h wo-phonon s a es 共 wo-pho on s a es wi h
wo-phonon s a es兲. The eigens a es o he Hamil onian 共2.5兲
a e
兩
⌽
␣
典
⫽兺
␯
c
␯
␣
兩
␯
典
⫹兺
␯
1
␯
2
d
␯
1
␯
2
␣
兩
␯
1
␯
2
典
共2.6兲
and he co esponding eigen alues do no o m a ha monic
spec um.
In he semiclassical models o g azing ion-ion collisions
he exci a ion o one o he wo nuclei is due o he mean
ield o he o he . Since he mean ield is a one-body ope a-
o , he exci a ion ope a o has he ollowing o m:
W共 兲⫽兺
␣␤
具
␣
兩
UB„R共 兲…
兩
␤
典
a
␣
†a
␤
,共2.7兲
whe e UBis he mean ield o he o he nucleus. The ime
dependence comes in h ough he ela i e dis ance Rbe-
ween he wo nuclei. In he s anda d app oach W( ) is linea
in he phonon ope a o s because only he ph e ms o Eq.
共2.7兲a e conside ed and he lowes -o de boson expansion is
aken. I we include also he pp and hh e ms, hei mapping
关Eq. 共2.4兲兴 leads o a quad a ic o m in Q
W⫽W00⫹兺
␯
W
␯
10Q
␯
†⫹H.c.⫹兺
␯␯
⬘
W
␯␯
⬘
11 Q
␯
†Q
␯
⬘
⫹兺
␯␯
⬘
W
␯␯
⬘
20 Q
␯
†Q
␯
⬘
†⫹H.c. 共2.8兲
The i s e m in Eq. 共2.8兲 ep esen s he in e ac ion o he
wo colliding nuclei in hei g ound s a e; in he p esen case
i has also an imagina y pa ha desc ibes he abso p ion
due o he nonelas ic channels. The W10 pa connec s s a es
di e ing by one phonon, he W11 e m couples exci ed s a es
wi h he same numbe o phonons, while W20 allows ansi-
ions om he g ound s a e o wo-phonon s a es. All he
o m ac o s Wa e calcula ed by double olding he Coulomb
and nuclea nucleon-nucleon in e ac ions wi h he Ha ee-
Fock g ound-s a e densi y o he p ojec ile and wi h he
g ound s a e densi y o he ansi ion densi ies o he consid-
e ed exci ed s a es o he a ge .
In he space o he g ound s a e and he
兩
⌽
␣
典
s a es we
can cas he Sch o
¨dinge equa ion in o a se o linea di e -
en ial coupled equa ions o he ime-dependen ampli ude
p obabili ies A
␣
( ). Then he c oss sec ion is calcula ed non-
pe u ba i ely as desc ibed in Re . 关6兴whe e we in eg a ed
he p obabili y o exci ing he s a e
兩
⌽
␣
典
s a ing om a
minimum impac pa ame e . In he calcula ion p esen ed he e
we in eg a e o e all impac pa ame e s since we ha e in o-
duced in W00 he op ical po en ial, which, in an e ec i e
way, akes ca e o he mos inne ajec o ies.
The imagina y pa Wim o he op ical po en ial is usually
de e mined by i ing he expe imen al elas ic c oss sec ion.
This po en ial desc ibes he abso p ion due o all nonelas ic
channels. The e o e, i canno be inse ed di ec ly in W00
关Eq. 共2.7兲兴 since he abso p ion due o he inelas ic channels
explici ly included in he coupled equa ions would be
coun ed wice.
Le us i s discuss how o sol e his p oblem when no
anha monici ies a e p esen and he e o e, he s a es
兩
⌽
␣
典
a e
pu e mul iphonon s a es. In such a case one can sol e he
Sch o
¨dinge equa ion in a semiclassical app oach by in e-
g a ing i along each classical ela i e mo ion ajec o y. The
s a e o he sys em 兩⌿典 is a cohe en s a e and he p obabili y
o exci e m
␯
imes a phonon
␯
is 关9兴
P
␯
,m
␯
0⫽共N
␯
兲m
␯
m
␯
!Pg.s.
0,共2.9兲
whe e N
␯
is he a e age numbe o
␯
phonons in 兩⌿典. In he
abo e equa ion, as well as in he ollowing discussion, he
dependence on he impac pa ame e bis unde s ood. The
M. V. ANDRE
´Se al. PHYSICAL REVIEW C 65 014608
014608-2
supe sc ip ‘‘0’’ e e s o he ac ha only he abso p ion due
o he mul iple exci a ion o phonons is aken in o accoun . In
such a case
Pg.s.
0⫽e⫺N,共2.10兲
whe e N⫽⌺
␯
N
␯
. We s ess ha he same su i al p obabil-
i y o he g ound s a e appea s as a ac o in all he p obabili-
ies in Eq. 共2.9兲.
The su i al p obabili y associa ed wi h he imagina y
op ical po en ial Wim is calcula ed as
Pg.s.
W⫽exp
再
2
បc
冕
⫺⬁
⫹⬁Wim共 兲d
冎
,共2.11兲
whe e he in eg al is again done along a classical ajec o y.
The depopula ion o he g ound s a e due only o he ne-
glec ed channels can be, in p inciple, calcula ed as in Eq.
共2.11兲bu wi h an auxilia y imagina y po en ial W
¯
, which
does no con ain he abso p ion due o he adop ed ones.
Then
Pg.s.
W⫽Pg.s.
0⫻Pg.s.
W
¯
共2.12兲
and
P
␯
,m
␯
⫽P
␯
,m
␯
0⫻Pg.s.
W
¯
.共2.13兲
When anha monici ies a e aken in o accoun , he s a e o
he sys em is no mo e a cohe en s a e. The p obabili y o
exci e he s a e
兩
⌽
␣
典
is equal o
P
␣
0⫽
兩
A
␣
兩
2,共2.14兲
whe e A
␣
is solu ion o he coupled equa ions o mo ion
wi hou any imagina y po en ial. The e o e, P
␣
0con ains only
he abso p ion due o all he adop ed channels. The emain-
ing pa , due o all he o he channels, can be in oduced by
w i ing, in analogy wi h Eq. 共2.13兲,
P
␣
⫽P
␣
0⫻Pg.s.
W
¯
.共2.15兲
This equa ion can be o mally de i ed by assuming ha he
abso p ion due o he excluded channels is he same in all he
adop ed ones. This is ce ainly an app oxima ion, howe e ,
we would like o emphasize ha many impo an inelas ic
channels a e explici ly aken in o accoun in he coupled
equa ions and ha we sol e he la e exac ly. The e o e, he
co esponding abso p ion is calcula ed co ec ly, including
he Q- alue e ec s. The unknown auxilia y imagina y po en-
ial W
¯
can be elimina ed by inse ing Eq. 共2.12兲in Eq. 共2.15兲,
P
␣
⫽P
␣
0⫻Pg.s.
W
Pg.s.
0,共2.16兲
which is he exp ession we ha e used in o de o calcula e
he inelas ic c oss sec ion. We would like o s ess ha he
pa o he nuclea abso p ion ha co esponds o noninelas-
ic channels is o en aken in o accoun as a sha p cu o
ansmission coe icien . So he in oduc ion o he imagina y
po en ial can be seen as an impo an imp o emen .
III. RESULTS AND DISCUSSION
The abo e desc ibed model has been applied o he eac-
ion 40Ca on 40Ca a E/nucleon⫽50MeV. The one-phonon
basis has been ob ained wi h a sel -consis en HF⫹RPA cal-
cula ion wi h Sky me in e ac ion SGII 关10兴. Only he mos
collec i e one-phonon s a es, exhaus ing a leas 5% o he
ele an ene gy-weigh ed sum ule 共EWSR兲, a e aken in o
accoun . They a e lis ed in Table I. We hen ha e conside ed
all possible wo-phonon s a es ha can be cons uc ed ou o
hem, wi h all possible alues o he o al angula momen um
L, and in his space we ha e diagonalized he Hamil onian
共2.5兲 o ge he s a es 共2.6兲. In Table II we ha e epo ed
some p ope ies o he quad upole s a es, each one labeled
wi h he name o i s main componen and whose unpe u bed
ene gy is gi en in he second column. In he hi d column
he e a e he ene gy shi s due o he anha monici ies. Thei
o e laps wi h he single and double isoscala gian quad u-
pole esonance 共ISGQR兲s a es a e shown in he las wo
columns. Simila ables o he DGR s a es a e epo ed in
关6兴.
The elemen a y nuclea o m ac o s W o pu e one- and
wo-phonon con igu a ions 关Eq. 共2.8兲兴 we e calcula ed by
double olding he M3Y nucleon-nucleon in e ac ion 关11兴
wi h he RPA ansi ion densi ies. The ansi ion ma ix ele-
men s be ween mixed s a es
兩
⌽
␣
典
we e compu ed by mixing
he elemen a y o m ac o s acco ding o he uni a y ans-
o ma ion 共2.6兲. The same p ocedu e was used wi h he Cou-
lomb in e ac ion o calcula e he Coulomb o m ac o s. The
ela i e mo ion ajec o ies we e de e mined by sol ing he
classical equa ion o mo ion in he p esence o bo h he Cou-
lomb ield and he eal pa o he nuclea po en ial.
The eal pa o he op ical po en ial was ob ained by
double olding he M3Y nucleon-nucleon po en ial wi h he
Ha ee-Fock densi ies o he wo nuclei while i s imagina y
pa was chosen wi h he same geome y and mul iplied by a
scale ac o whose alue 共0.627兲was de e mined by a i o
he expe imen al elas ic c oss sec ion o he collision 40Ca
TABLE I. RPA one-phonon basis o he nucleus 40Ca. We ha e
wo gian monopole esonance 共GMR兲s a es as well as wo dipole
共GDR兲and wo quad upole: isoscala 共ISGQR兲and iso ec o
共IVGQR兲. Fo he oc upole esonance we ha e he low ene gy
共LEOR兲and he high ene gy one 共HEOR兲. Fo each s a e i s spin,
pa i y, isospin, ene gy, and pe cen age o he EWSR a e epo ed.
Phonons J
␲
TE共MeV兲EWSR 共%兲
GMR10⫹0 18.25 30
GMR20⫹0 22.47 54
GDR11⫺1 17.78 56
GDR21⫺1 22.03 10
ISGQR 2⫹0 16.91 85
IVGQR 2⫹1 29.59 26
3⫺3⫺0 4.94 14
LEOR 3⫺0 9.71 5
HEOR 3⫺0 31.33 25
MICROSCOPIC DESCRIPTION OF COULOMB AND . . . PHYSICAL REVIEW C 65 014608
014608-3
on 40Ca a E/nucleon⫽50MeV o Re . 关12兴.
In hese calcula ions bo h he nuclea and Coulomb exci-
a ions we e included. Ac ually, he Coulomb exci a ion
alone does no p oduce a sizable c oss sec ion because he
colliding nuclei a e no e y hea y, bu when i is conside ed
oge he wi h he nuclea exci a ion i p oduces an in e e -
ence e ec , which can be impo an . This is due o he ac
ha on one hand we ha e a coupled channel e ec and, on
he o he hand, some wo-phonon s a es a e exci ed only
when bo h ields a e ac ing. This was clea ly demons a ed in
ou p e ious wo k 关7兴.
Since ou calcula ions a e based on a disc e e RPA we ge
a disc e e exci a ion spec um and a c oss sec ion
␴
␣
co e-
sponding o each s a e
兩
⌽
␣
典
. The ene gy di e en ial c oss
sec ions p esen ed in Fig. 1 a e ob ained by summing up all
he con ibu ions coming om he s a es
兩
⌽
␣
典
a e a
smoo hing o each indi idual line by a Lo en zian wi h a 3
MeV wid h. The dashed line e e s o a calcula ion whe e he
in e nal Hamil onian is ha monic and he ex e nal ield is
linea . The solid line co esponds o a calcula ion whe e he
anha monici y and nonlinea i y we e in oduced, which p o-
duce a sizable inc ease wi h espec o he s anda d case. In
he igu e we can clea ly dis inguish h ee ene gy egions.
The c oss sec ions gi en in Tables III–V a e ob ained by
summing up he
␴
␣
’s o he disc e e s a es
兩
⌽
␣
典
lying in
each egion. As al eady obse ed in Re s. 关6,7兴, he inc ease
a low ene gies is due bo h o he anha monici ies and non-
linea i ies. In pa icula , he anha monici ies a e impo an
because he low-lying wo-phonon s a es can be exci ed by
he W10 pa o he ex e nal ield h ough hei la ge one-
phonon componen . A high ene gies he main con ibu ion
comes om he nonlinea i ies because hei p esence in-
c eases he numbe o exci a ion ou es. This is seen be e in
Table IV whe e he exci a ion c oss sec ion in he double
gian quad upole esonance ene gy egion is epo ed. Fo
each mul ipola i y we ha e summed he exci a ion c oss sec-
ion in he ene gy egion be ween 28 and 38 MeV, and his is
done o ou di e en cases as shown in he able. The L
⫽3 con ibu ion is due o he HEOR a 31.33 MeV, while he
L⫽0, 2, and 4 con ibu ions a e domina ed by he double
exci a ion o he double ISGQR. As we can see in Table V,
he nonlinea e ms a e also esponsible o he inc ease o
he c oss sec ion in he ISGQR egion, especially o he L
⫽2 s a e whose main componen is he ISGQR. This is a
a iance wi h he ela i is ic Coulomb exci a ion s udied in
Re . 关6兴because he Coulomb in e ac ion e y selec i ely
popula es dipole ansi ions and he e o e canno exci e he
mos impo an wo-phonon componen s o he ISGQR,
which a e buil wi h monopole and quad upole phonons 共see
Table II兲.
The ob ained a io be ween he c oss sec ion in he gian
TABLE II. Cha ac e is ics o he
兩
⌽
␣
典
quad upole 2⫹s a es whose majo componen s a e in he i s
column. In he second column we show he ene gies o he majo componen s in he ha monic app oach. The
shi in he ene gy p oduced by he anha monici ies is indica ed by ⌬E共in KeV兲. We can compa e hese
alues wi h he diagonal ma ix elemen s o he esidual in e ac ion, ⌬E0共in KeV兲. In he las columns we
epo he ampli ude wi h which he single and double ISGQR componen s appea in he mixed s a es.
Quad upole S a es E0共MeV兲⌬E(⌬E0)cISGQR cISGQR-ISGQR
ISGQR 16.910 ⫺402 0 0.985 ⫺0.014
IVGQR 29.594 ⫺506 0 ⫺0.005 0.017
GMR1丢ISGQR 35.155 87 ⫺11 ⫺0.073 ⫺0.028
GMR1丢IVGQR 47.845 ⫺42 ⫺187 ⫺0.000 0.002
GMR2丢ISGQR 39.378 246 ⫺31 ⫺0.108 ⫺0.014
GMR2丢IVGQR 52.067 190 ⫺178 ⫺0.002 0.003
GDR1丢GDR135.560 ⫺464 ⫺505 0.034 0.087
GDR1丢GDR239.814 ⫺436 ⫺439 0.009 0.006
GDR1丢3⫺22.722 ⫺31 ⫺35 0.029 ⫺0.000
GDR1丢LEOR 27.486 ⫺444 ⫺442 ⫺0.013 ⫺0.007
GDR1丢HEOR 49.110 ⫺278 ⫺288 ⫺0.005 0.006
GDR2丢GDR244.058 ⫺435 ⫺436 0.004 0.002
GDR2丢3⫺26.976 ⫺6 7 0.003 0.001
GDR2丢LEOR 31.740 ⫺307 ⫺309 0.600 ⫺0.007
GDR2丢HEOR 53.364 ⫺212 ⫺217 0.000 0.000
ISGQR 丢ISGQR 33.819 0 4 ⫺0.020 0.995
ISGQR 丢IVGQR 46.508 39 40 0.002 0.002
IVGQR 丢IVGQR 59.198 ⫺247 ⫺250 ⫺0.007 ⫺0.004
3⫺丢3⫺9.884 750 776 ⫺0.045 ⫺0.005
3⫺丢LEOR 14.648 ⫺267 ⫺241 0.086 0.001
3⫺丢HEOR 36.272 ⫺104 ⫺120 0.025 ⫺0.003
LEOR 丢LEOR 19.413 ⫺271 ⫺269 ⫺0.021 ⫺0.000
LEOR 丢HEOR 41.037 ⫺192 ⫺197 ⫺0.005 0.002
HEOR 丢HEOR 62.660 ⫺212 ⫺215 ⫺0.006 ⫺0.001
M. V. ANDRE
´Se al. PHYSICAL REVIEW C 65 014608
014608-4
esonance egion and ha in he wo-phonon egion a ies
om 3.7 in he anha monic and nonlinea case o 4.6 in he
ha monic and linea calcula ion. I we only conside he
c oss sec ion up o he single and double isoscala gian
quad upole esonance, hese a ios inc ease o 6.5 and 9.6,
espec i ely. Those alues a e smalle han he ones epo ed
in Re . 关3兴 o he c oss sec ions a he g azing angle. This
di e ence can be aced back o he p esen a ailabili y o
he expe imen al elas ic c oss sec ion needed o ix he
imagina y pa o he op ical po en ial and o he ac ha he
heo e ical app oach has been imp o ed in se e al aspec s,
especially in he calcula ion o he o m ac o s.
Ou calcula ion can be compa ed wi h he expe imen al
da a o Re . 关4兴whe e he eac ion 40Ca⫹40Ca a 50 MeV/
nucleon has been s udied. Le us esume he impo an e-
sul s o Re . 关4兴and he mos c i ical poin s. We discuss i s
he inclusi e spec um and la e on we will analyze he one
ob ained in coincidence wi h backwa d emi ed pa icles. The
inelas ic spec um was ex ac ed o ejec iles sca e ed be-
ween 3.4° and 10° in he cen e o mass ame. The GR
con ibu ion was ob ained om he inclusi e inelas ic spec-
um by decon olu ion o he angula dis ibu ions in o in-
elas ic exci a ions and a noninelas ic backg ound. Fo he
inelas ic exci a ion, a dis o ed-wa e Bo n app oxima ion
共DWBA兲p edic ion was used. As o he backg ound, i s
angula dis ibu ion was assumed o be simila o he one o
he ene gy egion loca ed immedia ely abo e he GRs. This
p ocedu e ga e 113 mb/s be ween 12 and 22 MeV o he
inelas ic exci a ion co esponding o 40% o he quad upole
EWSR. Howe e , i should be no iced ha he es ima e o
he noninelas ic backg ound unde lying he GR is no unam-
biguous. Indeed, i inelas ic exci a ion is s ill p esen in he
egion abo e he esonance as expec ed om Fig. 1, he as-
sumed backg ound is o e es ima ed. In his case he ex-
ac ed alue should be unde s ood as a minimum. The maxi-
mum inelas ic con ibu ion compa ible wi h he measu ed
angula dis ibu ion is 223 mb/s . This co esponds o he
o he ex eme when no noninelas ic backg ound is consid-
e ed. The e o e he GR c oss sec ion ex ac ed om he in-
clusi e spec um is be ween 113 and 223 mb/s depending
upon he backg ound hypo hesis. The associa ed EWSR
would hus ange be ween 40% and 80% i he whole c oss
sec ion is assumed o be coming om quad upole s a es.
In o de o ge he o al c oss sec ion one has o ex apo-
la e he measu ed di e en ial c oss sec ion beyond he solid
angle co e ed by he ejec ile de ec o . This was done by
assuming ha he DWBA angula dis ibu ion used o i he
measu ed angula dis ibu ion in Re . 关4兴was also alid in
FIG. 1. Inelas ic c oss sec ion o he sys em 40Ca⫹40Ca a 50
MeV/nucleon as a unc ion o he exci a ion ene gy. Bo h cu es a e
he esul o a smoo hing p ocedu e wi h a Lo en zian o wid h ⌫
⫽3 MeV. The shadowed a eas a e he ene gy egions o e which
we ha e summed he c oss sec ions epo ed in he ables.
TABLE III. Coulomb plus nuclea exci a ion c oss sec ion o
40Ca⫹40Ca a 50 MeV/nucleon. Each mul ipola i y con ibu ion is
shown o se e al anha monic and nonlinea combina ions. The al-
ues o L⫽1 and 5 a e e y small and hey a e no shown. The
c oss sec ions 共in milliba ns兲a e summed o e he ene gy egion
(0⭐E⭐12 MeV).
Phonons Ha m.
and lin. Ha m.
and nonlin. Anh.
and lin. Anh.
and nonlin.
L⫽0 0.1 0.3 1.3 2.3
L⫽2 0.2 0.4 0.2 0.1
L⫽3 14.2 16.9 14.3 16.8
L⫽4 0.2 0.3 0.2 0.3
L⫽6 0.5 0.7 0.4 0.7
To al 15.2 18.6 16.4 20.2
TABLE IV. Same as Table III bu o he double ISGQR egion.
The c oss sec ions 共in milliba ns兲a e summed o e he ene gy e-
gion (28⭐E⭐38MeV). The alues in pa en heses co espond o
he double ISGQR s a e.
Phonons Ha m.
and lin. Ha m.
and nonlin. Anh.
and lin. Anh.
and nonlin.
L⫽0 0.2 共0.15兲0.3 共0.26兲0.2 共0.14兲0.3 共0.21兲
L⫽2 0.6 共0.33兲1.0 共0.51兲0.6 共0.33兲1.1 共0.53兲
L⫽3 2.2 2.5 2.3 2.5
L⫽4 1.0 共0.90兲1.9 共1.83兲0.9 共0.85兲1.8 共1.73兲
L⫽6 0.2 0.2 0.2 0.2
To al 4.2 共1.38兲5.9 共2.60兲4.2 共1.32兲5.9 共2.47兲
TABLE V. Same as Table III bu o he ISGQR egion. The
c oss sec ions 共in milliba ns兲a e summed o e he ene gy egion
(14⭐E⭐20 MeV). In his egion he e a e no s a es wi h L⫽3 and
5. The alues in pa en heses co espond o he ISGQR s a e.
Phonons Ha m.
and lin. Ha m.
and nonlin. Anh.
and lin. Anh.
and nonlin.
L⫽0 2.7 2.8 2.2 2.2
L⫽1 3.0 2.9 3.6 3.3
L⫽2 13.3共13.2兲16.1共16.0兲13.8共13.6兲16.0共16.0兲
L⫽4 0.1 0.1 0.1 0.1
L⫽6 0.2 0.3 0.2 0.3
To al 19.3 22.2 19.9 21.9
MICROSCOPIC DESCRIPTION OF COULOMB AND . . . PHYSICAL REVIEW C 65 014608
014608-5

he egion in which no da a a e a ailable. The a io be ween
he in eg als o he DWBA c oss sec ion o e he ull angula
ange and ha o e he angles co e ed by he de ec o is
3.16. Taking in o accoun he ac ion o he solid angle co -
e ed by he spec ome e one ge s a o al compensa ing ac o
o 6.67⫻10⫺2. Such a ac o ans o ms he double di e en-
ial c oss sec ion in o he ene gy di e en ial one. The esul -
ing o al c oss sec ion is hen 7.5 and 15 mb, espec i ely.
These alues ha e o be compa ed wi h he heo e ical in-
elas ic c oss sec ion, which, in he anha monic and nonlinea
case, adds up o 22 mb in he GR egion. Taking in o accoun
he unce ain ies o he analysis o he expe imen al da a and
he ac ha ou heo e ical esul s a e ob ained wi hou ad-
jus ing any pa ame e , he compa ison can be conside ed sa -
is ac o y. In o de o d aw quan i a i e conclusions one
should elabo a e on di e en issues bo h om he expe i-
men al and heo e ical sides. A ecen expe imen on he
same eac ion 关13兴using an imp o ed appa a us is expec ed
o elimina e mos o he expe imen al unce ain ies. These
new da a will allow a mo e eliable de e mina ion o some
pa ame e s en e ing in he heo e ical calcula ion, mainly in
he op ical po en ial.
Coincidences wi h backwa d emi ed pa icles p o ide an
unambiguous signal o he inelas ic exci a ions and could, in
p inciple, be used o a oid he noninelas ic backg ound p ob-
lems. This was he idea o Re . 关4兴, bu some o he sou ces o
unce ain ies appea . The coincidence a e wi h backwa d
emi ed p o ons was con e ed in o a di e en ial c oss sec-
ion, co ec ing o he ene gy dependence o he p o on mul-
iplici y. A ha ime i was al eady s essed ha his co ec-
ion ac o can be subjec o many unce ain ies. Fi s o all,
his p o on mul iplici y unc ion was calcula ed wi h a s a is-
ical decay code, which does no include any di ec decay
componen . Fu he mo e, due o he absence o ou -o -plane
de ec o s, he azimu hal angula dis ibu ion was no mea-
su ed and was assumed o be uni o m. This p ocedu e gi es
a c oss sec ion o he GR ex ac ed om he coincidence
da a 共339 mb/s 兲la ge han he one ob ained om he inclu-
si e inelas ic spec um 共be ween 113 and 223 mb/s 兲. This
shows ha he hypo heses used a e no co ec . The use o a
4
␲
de ec o in a ecen expe imen 关13兴should sol e hese
ambigui ies since i will p o ide he angula dis ibu ion o
he emi ed p o ons and he e will be no need o ely on a
s a is ical code o in e hei mul iplici y. Tha was no he
case in he expe imen o Re . 关4兴. The e o e, only he a io
was deduced om he coincidence da a. Two alues o his
a io we e epo ed by assuming wo backg ounds o he
wo-phonon egion, while he GR peak was conside ed wi h
no backg ound sub ac ion in he coincidence spec um. The
alues o he second phonon c oss sec ion we e, a e sub-
ac ion o he wo backg ounds, 30 and 17 mb/s , espec-
i ely, o an ene gy unning om 28 o 40 MeV, while he
GR c oss sec ion was 339 mb/s in he coincidence spec um
in he ange 12–22 MeV, leading o he a ios 11 and 20
quo ed in Re . 关4兴.1Such alues a e he a ios be ween he
single GR c oss sec ion and only a small ac ion o he DGR
c oss sec ion. We wan o s ess he e ha he co ec p oce-
du e should be no o sub ac any backg ound in he wo-
phonon egion. Indeed, on one hand, coincidence wi h back-
wa d emi ed pa icles a oids any con ibu ion om
noninelas ic backg ound in he expe imen al da a. On he
o he hand, in ou heo e ical calcula ion, no only double
GQR has been included bu many con ibu ions om di e -
en inelas ic exci a ions ha e been aken in o accoun . These
wo ema ks plead in a o o a di ec compa ison o he one-
and he wo-phonon egions wi h no backg ound sub ac ion.
In o de o ha e a mo e di ec compa ison we p esen , in
Fig. 2, he expe imen al coincidence inelas ic spec um o
Re . 关4兴关Fig. 16共b兲兴 wi h no backg ound sub ac ion. The
igh scale is he double di e en ial c oss sec ion while he
le scale is he ene gy di e en ial c oss sec ion ob ained
wi h he abo e men ioned ac o o 6.67⫻10⫺2. In he igu e
we p esen he heo e ical esul s smoo hed by a Lo en zian
o 5 MeV wid h, a he han he 3 MeV used in Fig. 1. F om
he igu e we see ha wi h his alue he shape o he expe i-
men al peak in he GR egion is well ep oduced. I should
be no iced ha some con ibu ion o he expe imen al c oss
sec ion is p esen jus below 14 MeV. Howe e , due o he
p oximi y o he p o on emission h eshold, he co ec ion o
he mul iplici y is mo e delica e in ha ene gy egion. Dis-
ega ding hese wo poin s, he o e all ag eemen be ween
heo y and expe imen is a he sa is ac o y. A ough es ima e
o he one-phonon and wo-phonon c oss sec ions can be
ob ained by in eg a ing bo h cu es in he ene gy anges
shown in Fig. 2 as shadowed a eas. By doing ha one would
ge an expe imen al and heo e ical a io o 2.4 and 2.3, e-
spec i ely. We wan o s ess ha he expe imen al a io
1One digi -in e sion e o was spo ed in he ex o Figs. 16共c兲
and 16共d兲o Re . 关4兴共e a um o be published兲.
FIG. 2. The do s ep esen he expe imen al coincidence inelas-
ic spec um o Re . 关4兴关Fig. 16共b兲兴 wi h no backg ound sub ac ion
共 igh scale兲. The solid line is he esul o a smoo hing p ocedu e
wi h a Lo en zian wi h a wid h ⌫⫽5 MeV o he heo e ical inelas-
ic c oss sec ion o he anha monic and nonlinea case 共le scale兲.
The shadowed a eas a e he ene gy egions o e which we ha e
in eg a ed he ene gy di e en ial c oss sec ions. The esul ing al-
ues in milliba ns a e he numbe s epo ed in he wo a eas. Those
abo e he cu es e e o he heo e ical esul s, while he ones
below e e o he expe imen al da a. In he inse we epo he
a ios be ween he single GR c oss sec ion and he DGR ones o
he wo cases.
M. V. ANDRE
´Se al. PHYSICAL REVIEW C 65 014608
014608-6
quo ed abo e is di e en om he one deduced in Re . 关4兴
because he la e is he a io be ween he ull peak o he
single GR and he DGR wi h backg ound sub ac ion, while
he o me one is ob ained wi hou backg ound sub ac ion in
bo h single GR and DGR. Fu he mo e, he i s wo expe i-
men al poin s in Fig. 2 we e no included as explained be-
o e. Finally, we would like o commen on he dependence
o he heo e ical a ios upon he smoo hing wid h. This a io
is dec easing wi h he inc easing wid h, due o he ac ha
while he in eg al o he single GR is dec easing, ha o e
he egion o he DGR emains almos unchanged. This is
ela ed o he ac ha in he single GR ene gy egion he
peaks o he single ⌽
␣
s a es a e qui e sepa a e while he
densi y o s a es in he DGR egion is e y high. In any case
he dependence on he wid h is no e y s ong; by a ying ⌫
om 3 o 6 MeV he a io changes om 2.75 o 2.20. These
alues canno be di ec ly compa ed wi h he alues epo ed
in Tables III–V because he la e ha e been ob ained jus by
summing he c oss sec ions associa ed wi h each disc e e
s a e.
IV. CONCLUSIONS
We ha e calcula ed he inelas ic sca e ing c oss sec ions
o one- and wo-phonon s a es o he 40Ca⫹40Ca collision a
E/nucleon⫽50MeV. Se e al e ec s ha e been e idenced.
In pa icula , we ha e analyzed he ole played by anha mo-
nici ies in he exci a ion spec um and nonlinea i ies in he
ope a o desc ibing he mu ual in e ac ion o he collision
pa ne s. The anha monici ies a e pa icula ly impo an a
ela i ely low ene gy whe e he exci a ion comes h ough he
one-phonon componen o he mixed s a es. The nonlinea i-
ies gi e hei main con ibu ion a high ene gy, in pa icula ,
in he egion o he double quad upole gian esonance.
Namely, in he in e al be ween 28 and 38 MeV, hey gi e an
inc ease o abou 40% wi h espec o a ha monic and linea
calcula ion. This inc ease is due o he exci a ion o o he
wo-phonon s a es ha a e popula ed because o he p esence
o he anha monici ies and nonlinea i ies. Wi h all he p e i-
ously discussed ca ea s, he compa ison o he smoo hed he-
o e ical esul wi h he expe imen al coincidence inelas ic
spec um o Re . 关4兴is sa is ac o y. The inclusion o h ee-
phonon s a es in he calcula ion will inc ease he inelas ic
c oss sec ion a highe exci a ion ene gies. A he same ime,
a ac ion o he popula ion o he wo-phonon s a es will
mo e o highe ene gies. In Re . 关14兴i has been shown,
wi hin a simple model, ha he spec um calcula ed by di-
agonalizing he Hamil onian ob ained by a boson expansion
unca ed a he qua ic o de in a space including up o h ee
phonons is in easonable ag eemen wi h he exac one. A
simila calcula ion is easible also in a ealis ic case. This,
oge he wi h he esul s shown he e, encou ages us o p o-
ceed in he di ec ion o calcula ing he h ee-phonon exci a-
ion c oss sec ion o he sys em 40Ca⫹40Ca a E/nucleon
⫽50MeV o which expe imen s ha e al eady been done
关13兴.
ACKNOWLEDGMENTS
This wo k has been pa ially suppo ed by he Spanish
DGICyT unde Con ac No. PB98-1111, by he Spanish-
I alian ag eemen be ween he CICyT and he INFN, and by
he Spanish-F ench ag eemen be ween he CICyT and he
IN2P3.
关1兴H. Emling, P og. Pa . Nucl. Phys. 33, 729 共1994兲; Ph.
Chomaz and N. F asca ia, Phys. Rep. 252, 275 共1995兲; T. Au-
mann, P. F. Bo ignon, and H. Emling, Annu. Re . Nucl. Pa .
Sci. 48, 351 共1998兲.
关2兴C. Benesh, B. Cook, and J. Va y, Phys. Re . C 40, 1198
共1989兲.
关3兴F. Ca a a, Ph. Chomaz, and A. Vi u i, Nucl. Phys. A471, 661
共1987兲.
关4兴J. A. Sca paci e al., Phys. Re . C 56, 3187 共1997兲; Phys. Re .
Le . 71, 3766 共1993兲.
关5兴C. Volpe, F. Ca a a, Ph. Chomaz, M. V. And e
´s, and E. G.
Lanza, Nucl. Phys. A589, 521 共1995兲;A599, 347c 共1996兲.
关6兴E. G. Lanza, M. V. And e
´s, F. Ca a a, Ph. Chomaz, and C.
Volpe, Nucl. Phys. A613, 445 共1997兲;A654, 792c 共1999兲.
关7兴E. G. Lanza, M. V. And e
´s, F. Ca a a, Ph. Chomaz, and C.
Volpe, Nucl. Phys. A636, 452 共1998兲.
关8兴M. Hage-Hassan and M. Lambe , Nucl. Phys. A188, 545
共1972兲.
关9兴K. Alde and A. Win he , Elec omagne ic Exci a ion 共No h-
Holland, Ams e dam, 1975兲.
关10兴N. V. Giai and H. Sagawa, Phys. Le . 106B, 379 共1981兲.
关11兴G. R. Sa chle and W. G. Lo e, Phys. Rep. 55, 183 共1979兲.
关12兴J. A. Sca paci, Ph.D. hesis, Uni e si e
´d’O say, F ance, 1990.
关13兴N. F asca ia, Nucl. Phys. A687, 154 共2001兲;共p i a e commu-
nica ion兲.
关14兴C. Volpe, Ph. Chomaz, M. V. And e
´s, F. Ca a a, and E. G.
Lanza, Nucl. Phys. A647, 246 共1999兲.
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