Global and consis en analysis o he hea y-ion elas ic sca e ing and usion p ocesses
L. R. Gasques,1L. C. Chamon,1D. Pe ei a,1M. A. G. Al a ez,1E. S. Rossi, J .,1C. P. Sil a,1and B. V. Ca lson2
1Labo a ó io Pelle on, Ins i u o de Física da Uni e sidade de São Paulo, 05315-970 São Paulo, São Paulo, B azil
2Depa amen o de Física, Ins i u o Tecnológico de Ae onáu ica, Cen o Técnico Ae oespacial, São José dos Campos, São Paulo, B azil
(Recei ed 17 Oc obe 2003; published 10 Ma ch 2004)
We ha e de eloped a model o he nuclea in e ac ion which is based on he e ec s o he Pauli nonlocali y.
In ea lie wo ks, we ha e success ully used his in e ac ion o desc ibe he elas ic sca e ing o se e al sys ems
in a e y wide ene gy ange. In he p esen wo k, we ha e checked he alidi y o he same in e ac ion in he
desc ip ion o abou 2500 usion c oss sec ion da a o 165 di e en sys ems. By in oducing only one ene gy-
and sys em-independen e ec i e pa ame e , he nonlocal model desc ibes he global beha io o he usion
p ocess wi h good p ecision.
DOI: 10.1103/PhysRe C.69.034603 PACS numbe (s): 24.10.H , 25.60.Pj, 25.70.Jj
The hea y-ion usion p ocess has been ex ensi ely s ud-
ied o e he las decades [1]. I is well known ha usion
c oss sec ions o hea y-ion sys ems ha e shown la ge en-
hancemen s a sub-ba ie ene gies in compa ison wi h heo-
e ical p edic ions om he ba ie pene a ion model [2].
These enhancemen s can be desc ibed by in oducing e ec-
i e ba ie pa ame e s, which ha e been s udied in a global
way o a la ge numbe o sys ems [2–4]. On he o he hand,
he enhancemen s ha e been explained o se e al pa icula
sys ems by conside ing he in e nal s uc u e o he pa ici-
pa ing nuclei h ough couped-channel calcula ions (e.g.,
Re s. [5,6]). A ew models ha e also been p esen ed o de-
sc ibe he elas ic sca e ing p ocess and he ene gy and sys-
em dependences o he co esponding op ical po en ial [7].
Howe e , he consis ency be ween he models o he usion
and elas ic sca e ing p ocesses has only been e i ied in ce -
ain pa icula cases (e.g., Re . [8]). We ha e de eloped a
model o he nuclea in e ac ion which has been success ul
in desc ibing he elas ic sca e ing o se e al sys ems in a
wide ene gy ange [9–15]. The model is based on he e ec s
o he Pauli nonlocali y and is o ally pa ame e ee. In his
wo k, we use his in e ac ion in he desc ip ion o abou 2500
usion c oss sec ion da a [16–53] o 165 di e en sys ems
om sub-ba ie o high ene gies.
Wi hin he nonlocal model, he ba e in e ac ion VNis con-
nec ed wi h he olding po en ial VF h ough [54]
VN共R,E兲=VF共R兲e−4 2/c2,共1兲
whe e cis he speed o ligh and is he local ela i e e-
loci y be ween he wo nuclei,
2共R,E兲=2
␮
关E−VC共R兲−VN共R,E兲兴.共2兲
The olding po en ial 关Eq. 共3兲兴 can be ob ained in wo di e -
en ways 关54兴:共i兲using he nucleon dis ibu ions o he nu-
clei and an app op ia e o m o he nucleon-nucleon in e -
ac ion, and 共ii兲using he ma e dis ibu ions o he nuclei
wi h a ze o- ange app oach o 共
ជ
兲. We dis inguish he ma -
e densi y om he nucleon one by aking in o accoun he
ini e size o he nucleon. Bo h al e na i es a e equi alen in
desc ibing he hea y-ion nuclea po en ial 关54兴, and we ha e
adop ed he ze o- ange app oach o desc ibe he usion p o-
cess:
VF共R兲=
冕
␳
1共 1兲
␳
2共 2兲 共R
ជ
−
ជ
1+
ជ
2兲d
ជ
1d
ជ
2.共3兲
Wi h he aim o p o iding a global desc ip ion o he nuclea
in e ac ion, we p oposed 关54兴an ex ensi e sys ema iza ion o
nuclea densi ies, based on expe imen al cha ge dis ibu ions
and heo e ical densi ies calcula ed h ough he Di ac-
Ha ee-Bogoliubo model. In ha wo k, we adop ed he
wo-pa ame e Fe mi 共2pF兲dis ibu ion o desc ibe he
nuclea densi ies. The adii o he 2pF dis ibu ions a e well
desc ibed by
R0= 1.31A1/3 − 0.84 m, 共4兲
whe e Ais he numbe o nucleons o he nucleus. The ma -
e densi ies p esen an a e age di useness alue a
=0.56 m. Owing o speci ic nuclea s uc u e e ec s
共single pa icle and/o collec i e兲, he pa ame e s R0and a
show small a ia ions a ound he co esponding a e age
alues h oughou he pe iodic able. This sys ema iza ion
o he nuclea dis ibu ions is essen ial o ob ain a
pa ame e - ee in e ac ion, since he olding po en ial de-
pends on he densi ies o he pa ne s in he collision. In
he p esen wo k, we use he nonlocal model in he con-
ex o his sys ema ics, i.e., assuming he a e age di use-
ness alue and Eq. 共4兲 o he adii o he dis ibu ions.
The e o e, he in e ac ion does no con ain any ee pa-
ame e and is qui e app op ia e o connec expe imen al
esul s o di e en sys ems in a ealis ic manne . In he
p esen wo k, we ha e also calcula ed he Coulomb po en-
ial h ough a olding p ocedu e using he ealis ic cha ge
densi ies o Re . 关54兴.
In he con ex o he ba ie pene a ion model (BPM), he
e ec i e po en ial is a sum o he Coulomb, nuclea , and
cen i ugal pa s:
Ve 共R,E兲=VC共R兲+VN共R,E兲+ᐉ共ᐉ+1兲ប2
2
␮
R2.共5兲
The usion c oss sec ion is associa ed wi h he ansmi ed
lux h ough
PHYSICAL REVIEW C 69, 034603 (2004)
0556-2813/2004/69(3)/034603(5)/$22.50 ©2004 The Ame ican Physical Socie y69 034603-1
␴
BPM共E兲=
␲
k2兺共2ᐉ+1兲Tᐉ.共6兲
In ou calcula ions, he sum in Eq. 共6兲is pe o med up o a
maximum ᐉwa e, which is he g ea es ᐉ alue ha esul s a
pocke 共and a ba ie 兲in he co esponding e ec i e po en-
ial. Fo ᐉwa es wi h e ec i e ba ie heigh s below he
cen e o mass ene gy, we ha e app oxima ed he e ec i e
po en ial by a pa abola wi h cu a u e ប
␻
ᐉ. In such cases, he
ansmission coe icien s can be ob ained h ough he Hill-
Wheele o mula 关55兴:
Tᐉ=
再
1 + exp
冋
2
␲
共VBᐉ−E兲
ប
␻
ᐉ
册
冎
−1,共7兲
ប
␻
ᐉ=冏ប2
␮
d2Ve
dR2冏RBᐉ
1/2 ,共8兲
whe e VBᐉand RBᐉa e he ba ie heigh and he co espond-
ing adius, espec i ely. On he o he hand, o ᐉwa es wi h
e ec i e ba ie s abo e he cen e o mass ene gy, ins ead o
he Hill-Wheele o mula we ha e used he mo e app op ia e
WKB me hod:
Tᐉ=关1 + exp共Sᐉ兲兴−1,共9兲
Sᐉ=
冕
R1
R2冑8
␮
ប2关Ve 共R,E兲−E兴dR,共10兲
whe e R1and R2a e he classical u ning poin s. A low
ene gies, he WKB me hod esul s in alues o he ansmis-
sion coe icien s qui e di e en om hose o he Hill-
Wheele o mula. In his case, we ha e de ined he ba ie
cu a u e connec ing exp essions 共7兲and 共9兲as
ប
␻
ᐉ=2
␲
共VBᐉ−E兲
Sᐉ
.共11兲
Wi hin he con ex o pa abolic ansmission coe icien s,
and conside ing RBᐉ⬇RB0and ប
␻
ᐉ⬇ប
␻
0, Wong has dem-
ons a ed 关56兴 ha
␴
Wong共E兲=RB0
2ប
␻
0
2Eln
再
1 + exp
冋
2
␲
共E−VBᐉ兲
ប
␻
0
册
冎
.共12兲
Wi h he aim o ob aining sys em-independen quan i ies, we
ha e de ined he ollowing educed c oss sec ion and ene gy:
␴
ed =2E
RB0
2ប
␻
0
␴
us,共13兲
E ed =E−VB0
ប
␻
0.共14兲
By using hese adimensional quan i ies, Eq. 共12兲can be e-
cas in he ollowing sys em-independen o m:
␴
ed
Wong =ln关1 + exp共2
␲
E ed兲兴.共15兲
Howe e , we ha e e i ied ha , in some cases, he esul s
ob ained om he Wong exp ession 关Eq. 共12兲兴 p esen sig-
ni ican di e ences in compa ison wi h hose om he ull
BPM calcula ions 关Eq. 共6兲兴. Thus, in o de o compa e ex-
pe imen al esul s o e y di e en sys ems, we ha e de-
ined he expe imen al educed usion c oss sec ion h ough
he ollowing i ial equa ion:
␴
ed
exp =
␴
us
␴
BPM
␴
ed
Wong.共16兲
The educed usion c oss sec ion da a, acco ding o Eq.
(16), a e p esen ed in Fig. 1, as a unc ion o he educed
ene gy [Eq. (14)]. The da a se includes 165 qui e di e en
sys ems anging in educed mass om abou 4 o 40 amu. A
e y good ag eemen be ween da a and heo e ical p edic-
ions is ob ained o ene gies abo e he s-wa e ba ie
共E ed艌0兲 o he whole se o sys ems, while sub-ba ie da a
p esen la ge enhancemen s in compa ison wi h he BPM cal-
cula ions. The sub-ba ie de ia ion is negligible o
␮
艋8
and inc eases as a unc ion o he educed mass o he sys em
in a ela i ely smoo h manne . A simila beha io o he sub-
ba ie usion da a has al eady been e y well es ablished [2].
An inspec ion o he slopes o he da a in Fig. 1 indica es ha
he enhancemen s a e mo e closely connec ed wi h he ba -
ie cu a u e han wi h he ba ie heigh i sel . Taking in o
accoun all hese conside a ions, we p opose a simple model
o desc ibe he enhancemen s by in oducing e ec i e ba ie
cu a u es. This e ec i e pa ame e is assumed o be a
simple linea unc ion o he educed mass o he sys em, as
gi en in Eq. (17). The i o he da a esul s in ␭=0.1/amu:
ប
␻
ᐉe =
再
ប
␻
ᐉ o
␮
艋8 amu
ប
␻
ᐉ关1+␭共
␮
−8兲兴 o
␮
艌8 amu. 共17兲
Using Eq. 共17兲wi h ␭=0.1, he ba ie pene a ion model
desc ibes wi h good p ecision he global beha io o he
2500 expe imen al usion c oss sec ion da a in he whole
ene gy ange 共see Fig. 2兲. In ac , i is possible o desc ibe
FIG. 1. The educed usion c oss sec ion as a unc ion o he
educed ene gy, using ba ie pa ame e s ob ained wi h he nonlocal
model. The igu e shows he esul s o di e en anges o he e-
duced mass o he sys em. The solid lines ep esen he educed
Wong equa ion.
L. R. GASQUES e al. PHYSICAL REVIEW C 69, 034603 (2004)
034603-2
he abo e-ba ie da a wi hin 20% p ecision 共s anda d de-
ia ion兲and he sub-ba ie da a wi hin an a e age ac o
o abou 3. We conside his dispe sion a he small be-
cause ou analysis has been pe o med o a la ge numbe
o di e en sys ems and o e a e y wide ene gy ange
共 om abou 18 MeV below he ba ie o 120 MeV abo e
i 兲wi h only one sys em- and ene gy-independen ee pa-
ame e ␭. Wi h he aim o including he whole da a se ,
Figs. 1 and 2 ha e been made wi h educed scales. Figu es
3–5 p esen he esul s in usual scales o a ew pa icula
sys ems, in o de o p o ide u he examples o he qual-
i y o ou p edic ions.
We belie e ha he e ec i e cu a u e simula es, in some
way, he a e age e ec o a la ge numbe o coupled chan-
nels ha con ibu e o he usion p ocess. The e ec o he
couplings depends on he educed mass o he sys em be-
cause hea ie sys ems p esen a g ea e numbe o eac ion
channels and also la ge coupling ampli udes (which a e con-
nec ed wi h he size o he nuclei). The dispe sion obse ed
in Fig. 2 is p obably due [57] o pa icula s ong couplings
ha we e no included in ou model, and is also connec ed
wi h a ia ions o he densi ies, a ising om nuclea s uc-
u e e ec s, which ha e an in luence on he nuclea po en ial
[54]. Fo example, he g ea iso opic dependence (see Fig.
6— op)o he sub-ba ie usion c oss sec ion o he
16O+144,148,150,152,154Sm sys ems is dec eased by using he
e ec i e cu a u e (Fig. 6—bo om). E en so, he di e ence
is no o ally elimina ed p obably due o s uc u e e ec s no
included in ou model.
The e ec o he nonlocali y is no e y signi ican a
nea -ba ie ene gies (low eloci ies), whe e Eq. (1)indica es
ha VN共R,E兲⬇VF共R兲. The e ec becomes g ea e as he en-
e gy inc eases, such ha a ene gies o abou
200 MeV/nucleon VN共R,E兲is abou one o de o magni ude
less in ense han he co esponding olding po en ial [9,10].
FIG. 2. The same as Fig. 1, conside ing he co ec ion [Eq. (17)
wi h ␭=0.1/amu] o he ba ie cu a u es.
FIG. 3. The usion c oss sec ion o he 12C+12C, 12C+16O,
16O+208Pb, and 58Ni+64Ni sys ems. The lines ep esen ull ba ie
pene a ion model calcula ions wi h (solid lines)o wi hou (dashed
lines)including he e ec o he e ec i e cu a u e.
FIG. 4. The same o Fig. 3 o he 16O+144,148,150,154Sm sys-
ems.
FIG. 5. The usion c oss sec ion o he 12C+27Al, 32S sys ems.
The lines ep esen ull ba ie pene a ion model calcula ions wi h
(do ed lines)o wi hou (solid lines) he e ec o he Pauli nonlo-
cali y.
GLOBAL AND CONSISTENT ANALYSIS OF THE HEAVY-ION…PHYSICAL REVIEW C 69, 034603 (2004)
034603-3
Howe e , he e a e ew a ailable usion da a a high ene -
gies. Thus, o mos o he cases conside ed in his wo k, he
heo e ical esul s o he BPM c oss sec ions ob ained using
Eq. (1)di e by less han 5% in compa ison om hose
ob ained conside ing VN共R,E兲=VF共R兲. E en so, we ha e
ound wo cases in he p esen da a se in which he ene gy is
high enough o emphasize such a di e ence (see Fig. 5).
He e he dec ease o he c oss sec ion o highe ene gies is
connec ed wi h he compe i ion be ween cen i ugal epul-
sion and nuclea a ac ion. The ene gy dependence o he
nuclea in e ac ion ha a ises om nonlocali y plays an im-
po an ole by p o iding u he educ ion o he c oss sec-
ion. On he o he hand, he nonlocali y has been undamen-
al in ou desc ip ion o he hea y-ion elas ic sca e ing
p ocess, in which he ene gy dependence o he po en ial has
been success ully aken in o accoun by Eq. (1). The e o e,
we conside he majo eason o using he nonlocal in e ac-
ion o be he goal o ob aining a uni ied desc ip ion o bo h
he elas ic sca e ing and usion p ocesses, wi hin a consis-
en model om he sub-ba ie egion o in e media e ene -
gies.
In summa y, ou pa ame e - ee nonlocal model o he
nuclea in e ac ion has been success ul in desc ibing he
hea y-ion elas ic sca e ing p ocess om sub-ba ie ene gies
up o 200 MeV/nucleon. This in e ac ion has also been suc-
cess ully es ed in some cases o inelas ic sca e ing and
ans e a sub-Coulomb and in e media e ene gies
[11,12,15]. In he p esen wo k, we ha e also ob ained good
p edic ions o usion c oss sec ions, using he nonlocal in-
e ac ion in he con ex o he ba ie pene a ion model, o a
e y la ge numbe o di e en sys ems and o e a wide en-
e gy ange ( om he sub-ba ie egion o ene gies abou
15 MeV/nucleon, whe e he e a e s ill a ew exis ing expe i-
men al da a). We emphasize ha he heo e ical calcula ions
ha e no ee pa ame e s, excep o he sys em- and ene gy-
independen one connec ed wi h he e ec i e ba ie cu a-
u e. We also emphasize ha ou model p o ides ema kable
p edic ions o he ligh hea y-ion sys ems o e nine o de s
o magni ude (see Fig. 1 o
␮
艋8). In his ange o educed
mass he e is no need o include any co ec ion in he ba ie
cu a u es. The e o e, he pa ame e - ee nonlocal model
seems o be a good basis o s udying he usion p ocess in
impo an cases o as ophysics, which in ol e mainly ligh
sys ems. The model also p o ides a good desc ip ion, again
wi hou using any ee pa ame e , o he whole se o sys-
ems in ene gies abo e he ba ie (see Fig. 1 o E ed艌0).
Clea ly, he e is oom o imp o emen o he model [57] o
hea ie sys ems in lowe ene gies, by conside ing wo
poin s: (i) he de elopmen o a mo e ealis ic model o he
e ec i e ba ie pa ame e s, based on esul s ob ained om
coupled-channel calcula ions; and (ii)inclusion o he ex-
pe imen al esul s ob ained o usion ba ie dis ibu ions in
he analysis.
This wo k was pa ially suppo ed by Financiado a de
Es udos e P oje os (FINEP), Fundação de Ampa o à Pesquisa
do Es ado de São Paulo (FAPESP), and Conselho Nacional
de Desen ol imen o Cien í ico e Tecnológico (CNPq).
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