PHYSICAL REVIEW C 72, 014603 (2005)
Gene al beha io o he e ec i e nucleon-nucleon in e ac ion as a unc ion o he ela i e eloci y
L. C. Chamon
Ins i u o de F´
ısica da Uni e sidade de S˜
ao Paulo, Caixa Pos al 66318, 05315-970 S˜
ao Paulo, SP, B azil
M. A. G. Al a ez
Depa amen o de Fisica A omica Molecula y Nuclea (FAMN), Apa ado 1065, Uni e sidade de Se illa, E-41080, Se illa, Spain
(Recei ed 23 Ma ch 2005; published 7 July 2005)
We ha e s udied olume in eg als o he cen al pa o op ical po en ials ex ac ed om da a analyses o a
a ie y o ligh and hea y sys ems. The da a-ex ac ed in eg als p esen a qui e simple beha io as a unc ion o
he ela i e eloci y be ween a ge and p ojec ile. This beha io is sys em independen and, he e o e, i e lec s
a ea u e o he e ec i e nucleon-nucleon in e ac ion i sel . The o e all esul s a e in good ag eemen wi h he
p edic ions o he S˜
ao Paulo po en ial, which is a model o he nuclea in e ac ion ha so a has been employed
mos ly in analyses o hea y-ion eac ions.
DOI: 10.1103/PhysRe C.72.014603 PACS numbe (s): 24.10.H , 21.30.Fe, 25.10.+s
I. INTRODUCTION
Elas ic sca e ing o nucleus-nucleus sys ems has been an
impo an subjec o s udy o se e al decades. Wi hin he
con ex o he op ical model, such s udies usually ha e been
pe o med wi h he aim o de e mining he co esponding
mean- ield in e ac ion. Indeed, he op ical model has p o ided
good da a i s o angula dis ibu ions o nume ous di e en
sys ems in a wide ene gy ange. The esul ing op ical po en ial
pa ame e s should p o ide physical in o ma ion on he nuclea
in e ac ion. Howe e , in se e al cases he da a-ex ac ed
po en ials a e a he ambiguous in he sense ha di e en
se s o pa ame e s esul in qui e simila da a i s. This is
clea ly obse ed a nea -ba ie ene gies whe e po en ials ha
a e simila in he su ace egion gi e equi alen i s o he da a.
A hese low ene gies, he Coulomb epulsion and s ong ab-
so p ion p e en he p ojec ile om pene a ing he a ge and
he e o e i p obes only he su ace egion. Wi h he ad en o
highe ene gy beams, much less ambigui y has been obse ed
in he ex ac ion o he op ical po en ial a sho dis ances om
da a analyses a in e media e ene gies. Such da a we e i s
ob ained o α-pa icle sca e ing and la e o se e al hea y-
ion sys ems [1]. The esul ing phenomenological po en ials
p esen signi ican dependence on he bomba ding ene gies,
and heo e ical models ha e been de eloped o accoun o his
ene gy dependence h ough ealis ic mean- ield in e ac ions.
Among hem, he S˜
ao Paulo (SP) po en ial associa es his
beha io wi h nonlocal quan um e ec s ela ed o he exchange
o nucleons be ween a ge and p ojec ile [2–4]. The SP
po en ial has success ully been used in he da a desc ip ion
o a la ge numbe o hea y-ion sys ems [3,5–13]. In he
p esen pape , we show ha he ene gy dependence o olume
in eg als ob ained o po en ials ex ac ed om da a analyses
is in ag eemen wi h he p edic ions o he SP po en ial [4], no
only o hea y-ion sys ems bu also o qui e ligh ones, such
as nucleon-nucleus and he αnucleus.
II. S ˜
AO PAULO POTENTIAL
Pauli nonlocali y a ises om quan um exchange e ec s
and has been s udied in he con ex o neu on-nucleus [14],
αnucleus [15], and hea y-ion collisions [2–13]. The nonlocal
in e ac ion has been assumed in he desc ip ion o he elas ic
sca e ing p ocess h ough an in eg o-di e en ial equa ion
[3,4,14]. I is possible o de ine a local-equi alen po en ial
ha , wi hin he usual amewo k o he Sch ¨
odinge di e en ial
equa ion, ep oduces he esul s o he in eg o-di e en ial
app oach [3,4,14]. Fo hea y-ion sys ems, we p oposed a
model o he eal pa o he local-equi alen in e ac ion, he
SP po en ial, which is de ined h ough wo equi alen e sions
[4]. The i s e sion is a con en ional olding o he nucleon
dis ibu ions o he nuclei wi h an e ec i e nucleon-nucleon
in e ac ion
VN(R)=ρ1( 1)ρ2( 2)unn(
R− 1+ 2)d 1d 2.(1)
The SP po en ial consis s in using Eq. (1) in he con ex o he
olding- ype nucleon-nucleon in e ac ion
unn( )=ρm( 1)ρm( 2)V0δ(
R− 1+ 2)e−4 2/c2d 1d 2,
(2)
whe e ρmis he ma e densi y o he nucleon, V0=
−456 MeV m3, is he local ela i e eloci y be ween he
nucleons (o nuclei), and cis he speed o ligh . Because o
he del a unc ion, he olding in Eq. (2) is called he ze o-
ange app oach. Based on he esul s o he cha ge den-
si y o he p o on in ee space, ob ained om elec on
sca e ing expe imen s, we assumed an exponen ial shape,
ρm( )=ρ0e− /am, o he ma e densi y o he nucleon. This
assump ion esul s in
unn( )=V0
64πa3
m
e− /am1+
am
+ 2
3a2
me−4 2/c2.(3)
A di useness alue o am=0.30 m was ound [4] o he
ma e densi y o he nucleon inside he nucleus.
The o he e sion o he SP po en ial is ob ained h ough
he ma e densi y ρMo he nucleus, which is de ined as he
olding o he nucleon dis ibu ion o he nucleus wi h he
0556-2813/2005/72(1)/014603(5)/$23.00 014603-1 ©2005 The Ame ican Physical Socie y
L. C. CHAMON AND M. A. G. ALVAREZ PHYSICAL REVIEW C 72, 014603 (2005)
ma e densi y o he nucleon:
ρM( )=ρ(
)ρm( −
)d
.(4)
Thus, we dis inguish he ma e and nucleon densi ies o he
nucleus by aking in o accoun he ini e size o he nucleon.
By inse ing Eqs. (2) and (4) in Eq. (1), he SP po en ial can
be ecas as
VN(R)=ρM1( 1)ρM2( 2)V0δ(
R− 1+ 2)e−4 2/c2d 1d 2.
(5)
The simila o m o Eqs. (5) and (2) implies a qui e in e es ing
uni ica ion o he nucleus-nucleus and e ec i e nucleon-
nucleon in e ac ions: Bo h a e calcula ed by olding he
co esponding ma e dis ibu ions in he ze o- ange app oach
wi h he same V0 alue. Fu he mo e, he same dependence on
he ela i e eloci y is p esen in bo h kinds o in e ac ion. The
model also includes sys ema ics o he densi ies [4]. Wi hin
his con ex , he SP po en ial does no con ain any adjus able
pa ame e and he e o e i is a o ally pa ame e ee model o
he eal pa o he nuclea in e ac ion.
An essen ial cha ac e is ic o he SP po en ial is he
dependence on he local ela i e eloci y (R). In a classical
physics amewo k, he eloci y is ela ed o he kine ic ene gy
EK(R)by
2=2EK
µ=2
µ[E−VC(R)−VN(R)],(6)
whe e µis he educed mass o he sys em, Eis he ene gy
in he cen e -o -mass ame and VCis he Coulomb po en ial.
Al hough Eq. (6) connec s ene gy and eloci y, i one akes
in o accoun he o m o exp essions (2) and (5) i seems
mo e app op ia e o conside he nuclea in e ac ion as eloci y
dependen ins ead o ene gy dependen .
A s anda d physical quan i y in op ical model analyses is
he olume in eg al pe nucleon o he eal pa o he nuclea
in e ac ion:
JR=4π
A1A2∞
0
VN(R)R2dR. (7)
Wi hin he SP model, a simple exp ession o JRis ob ained i ,
as an app oxima ion, we eplace in Eq. (5) he local eloci y
by he asymp o ic one, ∞= (R→∞), ha is,
VN(R)≈V0e−4 2
∞/c2ρM1( 1)ρM2( 2)δ(
R− 1+ 2)d 1d 2.
(8)
In his case and aking in o accoun he no maliza ion o he
densi ies, one ob ains
JR=V0e−4 2
∞/c2.(9)
One should obse e ha Eq. (9) is deduced om Eqs. (7) and
(8) wi hou any hypo hesis on he shape o he densi ies. Thus,
acco ding o he SP model, he olume in eg al pe nucleon
should be app oxima ely a sys em-independen quan i y wi h
a simple dependence on he ela i e eloci y be ween a ge
and p ojec ile.
III. VOLUME INTEGRAL SYSTEMATICS
Fo hea y-ion sys ems, usually he op ical po en ial in-
ol ed in da a analyses p esen s only cen al eal and imag-
ina y pa s. Fo ligh e sys ems, o he kind o in e ac ions,
such as spin-o bi , a e o en included in he calcula ions. In he
p esen wo k, we conside only he olume in eg al pe nucleon
o he cen al eal pa o he op ical po en ial. Conside ing he
ambigui ies p esen in he de e mina ion o op ical po en ial
pa ame e s om da a analyses, one should expec simila
ambigui ies in he co esponding olume in eg als. Fo ligh
sys ems, in spi e o ambigui ies ound o he da a-ex ac ed
pa ame e s, he olume in eg al o he in e ac ion is a a he
well-de ined quan i y, ee o pa ame e co ela ions (see, e.g.,
[16]). Fo hea y sys ems, he deg ee o ambigui y is g ea e
a low ene gies, nea he ba ie . Thus, we selec ed po en ials
ex ac ed om da a a highe ene gies and mos ly om anal-
yses in ol ing ealis ic po en ials based on he olding model.
E en so, some deg ee o ambigui y s ill emains. Fo ins ance,
o 16O+16Oa ELab =480 MeV i is possible o ob ain
good elas ic sca e ing da a i s wi hin abou 40 MeV m3
unce ain y, ha is, wi h JR anging om 242 o 279 MeV m3
[17,18].
Figu e 1(a) shows he da a-ex ac ed olume in eg als
[17–23] o a ew hea y-ion sys ems as a unc ion o he
asymp o ic eloci y ha was calcula ed wi hin he scena io
o he heo y o ela i i y:
EK=m0c2
1− 2/c2−m0c2,(10)
whe e m0and EKa e he es -mass and asymp o ic kine ic
ene gy o he p ojec ile, espec i ely. The s aigh line in
Fig. 1(a) ep esen s he SP po en ial p edic ion [Eq. (9)]. The
sys em-independen slope o he “da a” is clea ly compa ible
wi h he eloci y dependence o he heo e ical model. How-
e e , he alue V0=−456 MeV m3 om he SP po en ial
seems o be oo la ge o ep esen he da a. This disc epancy
is, in ac , no e y signi ican , because he SP po en ial has
been success ully used in he elas ic sca e ing da a desc ip ion
o se e al hea y-ion sys ems, also including a ew angula
dis ibu ions ha co espond o olume in eg al da a p esen ed
in Fig. 1(a). Thus, we belie e ha such a disc epancy is due
o he app oxima ion assumed in he deduc ion o Eq. (9), o
using he asymp o ic eloci y, and also due o he ambigui y
al eady men ioned in he de e mina ion o olume in eg als
om elas ic sca e ing da a analyses. To illus a e his poin
wi h an example, Fig. 2 p esen s an elas ic sca e ing angula
dis ibu ion o he 12C+12Csys ema ELab =1016 MeV.
The solid line in he igu e was ob ained om op ical model
calcula ions, whe e he SP po en ial and a Woods-Saxon
po en ial (wi h h ee adjus able pa ame e s) we e assumed o
he eal and imagina y pa s o he in e ac ion, espec i ely.
As one can see in Fig. 2, he ag eemen be ween da a
and heo e ical esul s is excellen . In his case, he exac
olume in eg al o he SP po en ial, ob ained om Eq. (7),
is JR=227 MeV m3, whe eas exp ession (9) esul s in a
sligh ly g ea e alue JR=241 MeV m3. Simila i s o he
same angula dis ibu ion ha e also been ob ained wi hin o he
models o he nuclea in e ac ion, wi h olume in eg als ha
014603-2
GENERAL BEHAVIOR OF THE EFFECTIVE NUCLEON- . . . PHYSICAL REVIEW C 72, 014603 (2005)
102
103
ELab= 1016 MeV
16O+
16O
16O+
28Si
(a)
12C+
12C
16O+
12C
-J
R(MeV m3)
(c)
n+X
p+X
p+α
0.0 0.1 0.2 0.3 0.4 0.5 0.6
101
102
(b)
α+X
d+X
-J
R(MeV m3)
2/c2
0.1 0.2 0.3 0.4 0.5 0.6 0.7
(d)
M3Y
LAX PP
PN
Reid
Pa is
2
/c2
FIG. 1. Volume in eg als pe nucleon as a
unc ion o he asymp o ic ela i e eloci y o
(a) hea y-ion sys ems, (b) α, d +nucleus,
(c) nucleon +nucleus, and (d) e ec i e nucleon-
nucleon in e ac ions. The solid lines in he igu e
ep esen Eq. (9). The a ow in (a) poin s o he
da e ha ep esen s ea lie esul s [17,20] ob-
ained om analyses o he angula dis ibu ion
p esen ed in Fig. 2.
a y om 175 o 189 MeV m3[17,20] [see a ow in Fig. 1(a)].
The ange 175 ⩽JR⩽227 MeV m3o alues compa ible wi h
his angula dis ibu ion shows, again, ha a dispe sion o
JR≈50 MeV m3is expec ed o JR alues ex ac ed om
elas ic sca e ing da a analyses.
Figu e 1(b) shows olume in eg als o α-nucleus and
deu e on-nucleus sys ems. The da a co espond o a e age
alues o e a la ge numbe o di e en a ge nuclei [25].
Volume in eg als o nucleon-nucleus sys ems a e p esen ed
in Fig. 1(c). Again he da a co espond o a e age alues o e
se e al nuclei [26]. Also in Fig. 1(c), we p esen da a o he
e y ligh sys em p o on +α[27].
As shown in Figs. 1(a)–1(c), Eq. (9) is in easonable
ag eemen wi h he da a o all hea y and ligh sys ems s udied
in he p esen wo k. The beha io o he olume in eg al
161284020
10-3
10-2
10-1
100
101
ELab =1016MeV
12C+12C
σel./σMo
θc.m. (deg ee)
FIG. 2. Elas ic sca e ing angula dis ibu ion o he 12C+12C
sys em a ELab =1016 MeV ( om [24]). The solid line ep esen s
heo e ical esul s om op ical model calcula ions wi h he pa ame e -
ee SP po en ial assumed o he eal pa o he in e ac ion.
as a unc ion o he eloci y is clea ly sys em independen ,
and he e o e i p obably e lec s a ea u e o he e ec i e
nucleon-nucleon in e ac ion i sel . Ob iously, he olding ype
nucleon-nucleon in e ac ion [Eq. (2) o (3)] obeys Eq. (9). We
ha e checked whe he o he models o he nucleon-nucleon
in e ac ion a e also in ag eemen wi h ha equa ion. Two
di e en e sions, Pa is and Reid, o he M3Y in e ac ion
ha e been used o desc ibe low-ene gy (5–20 MeV/nucleon)
elas ic sca e ing da a [28]. Ano he model used in he analyses
o elas ic sca e ing is he LAX in e ac ion [29], which is he
op ical limi o he Glaube high-ene gy app oxima ion. The
LAX in e ac ion is essen ially a ze o- ange double- olding
po en ial used o bo h he eal and imagina y pa s o he
op ical po en ial. The eal pa o he LAX in e ac ion is w i en
as
V(R)=−α(E)¯h
2σ(EN)ρ1( )ρ2(
R− )d , (11)
whe e σis he ene gy-dependen o al nucleon-nucleon c oss
sec ion. The pa ame e αis de ined by Re nn =αIm nn,
whe e nn is he nucleon-nucleon ampli ude. The olume
in eg al pe nucleon co esponding o Eq. (11) is gi en by
JR=−
¯h
2α(E)σ(E).(12)
Equa ion (11) has been de i ed om mul iple-sca e ing
heo ies and should be alid only a high ene gies. In Fig. 1(d)
we p esen olume in eg als o he M3Y and LAX e ec i e
nucleon-nucleon in e ac ions. Fo he LAX in e ac ion, we
ha e ob ained alues o σand αin able 1 o Re . [29], o bo h
p o on-p o on (pp) and p o on-neu on (pn) in e ac ions. One
can see in Fig. 1(d) ha he M3Y and LAX in e ac ions p esen
a simila beha io in compa ison wi h ha o he olding ype
nucleon-nucleon in e ac ion.
In Fig. 3 (bo om), we p esen he same olume in eg al
da a se o Fig. 1, bu on a linea scale o show ha he
nuclea in e ac ion becomes epulsi e o ⩾0.7c, whe eas
014603-3
L. C. CHAMON AND M. A. G. ALVAREZ PHYSICAL REVIEW C 72, 014603 (2005)
0.0 0.2 0.4 0.6 0.8 1.0
-400
-200
0
200
400
-200
0
200
400
600
X+X
α,d + X
p+α
nucleon + X
nucleon-nucleon
-J
R(MeV m3)
/c
a e age alues
|V
0|e
-4 2
/c2
-J
R(MeV m3)
FIG. 3. (Bo om) Volume in eg als pe nucleon as a unc ion o
he asymp o ic ela i e eloci y. (Top) A e age alues o he olume
in eg als o e 0.1 /c bins. The solid lines in he igu e ep esen
Eq. (9).
he p edic ion o he SP po en ial anishes. Howe e , o
⩽0.7c he da a ollow he beha io o Eq. (9). In his egion,
he s anda d de ia ion o he da a ela i e o he heo e ical
p edic ions is JR≈70 MeV m3. This alue is close o he
a o emen ioned da a dispe sion (40–50 MeV m3) expec ed
as a esul o he ambigui y in he de e mina ion o JR om
elas ic sca e ing da a-analyses. The e o e, wi hin he inhe en
unce ain y o da a-ex ac ed JR, he comple e da a se ( o
⩽0.7c) is in ag eemen wi h he heo e ical p edic ions. To
ob ain smalle unce ain y, in Fig. 3 ( op) we p esen a e age
alues o e 0.1 /c bins o he comple e olume in eg al da a
se . The ag eemen o hese a e age alues wi h he p edic ion
o he SP po en ial is ema kable up o abou 0.7c. Indeed,
o ⩽0.7c he dispe sion o he a e age da a a ound he
heo e ical p edic ion is only JR≈11 MeV m3, which
co esponds o abou 2.4% o he V0 alue. We poin ou
ha such e y good p ecision was ob ained wi hou using any
adjus able pa ame e in he calcula ions.
In se e al wo ks (see, e.g., [25–27]), he olume in eg als
ha e been pa ame ized as a unc ion o he kine ic ene gy pe
nucleon (EN) wi h he ollowing exp ession:
JR=J0+βln EN.(13)
This ype o ene gy dependence is in acco d wi h Passa o e’s
[30–32] applica ion o Feshbach’s dispe sion ela ion [33].
Figu e 4 p esen s he a e age olume in eg al alues as a
unc ion o he kine ic ene gy pe nucleon. The solid line in
100101102103104
-400
-200
0
200
400
600
-V0e-4 2/c2
-[J0+βln(EN)]
-J
R(MeV m3)
EN(MeV/nucleon)
FIG. 4. A e age alues o he olume in eg als as a unc ion o
he ene gy pe nucleon. The solid and dashed lines ep esen Eqs. (9)
and (13), espec i ely.
Fig. 4 ep esen s Eq. (9); he dashed line co esponds o a i o
he da a in he egion EN>20 MeV/nucleon using Eq. (13).
We ound J0=−812 MeV m3and β=130 MeV m3, which
should be compa ed, o example, wi h J0=−872 ±44 MeV
m3and β=136 ±7MeV m
3ob ained o p o on-nucleus
sys ems [26]. Figu e 4 shows ha Eqs. (9) and (13) p o ide
e y simila JR alues o 30 ⩽EN⩽300 MeV/nucleon, which
co esponds o 0.25 ⩽ /c ⩽0.65. Fo low ene gies Eq. (9) is
much be e han Eq. (13) whe eas he in e se is obse ed a
e y high eloci ies.
IV. CONCLUSION
We ha e s udied olume in eg als o he cen al eal
pa o op ical po en ials ex ac ed om elas ic sca e ing
da a analyses o a la ge a ie y o sys ems. The beha io
o he olume in eg als pe nucleon as a unc ion o he
ela i e eloci y be ween a ge and p ojec ile is clea ly sys em
independen and he e o e i e lec s a cha ac e is ic o he
e ec i e nucleon-nucleon in e ac ion i sel . The dependence
o hese in eg als on he ela i e eloci y is compa ible wi h
he p edic ion o he SP po en ial up o ≈0.7c. So a , he SP
po en ial had been es ed only up o E=200 MeV/nucleon
[3,5], which ep esen s ≈0.57c, and i has been employed
mos ly in he s udy o hea y-ion eac ions. The p esen
indings indica e ha he model p obably could be success ully
applied also in elas ic sca e ing da a analyses o much ligh e
sys ems.
ACKNOWLEDGMENTS
This wo k was pa ially suppo ed by Fundac¸˜
ao de Ampa o
`
a Pesquisa do Es ado de S˜
ao Paulo (FAPESP), Conselho Na-
cional de Desen ol imen o Cien ´
ı ico e Tecnol´
ogico (CNPq),
and he Spanish MEC.
014603-4
GENERAL BEHAVIOR OF THE EFFECTIVE NUCLEON- . . . PHYSICAL REVIEW C 72, 014603 (2005)
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