Toward a global description of the nucleus-nucleus interaction
L. C. Chamon,
1
B. V. Carlson,
2
L. R. Gasques,
1
D. Pereira,
1
C. De Conti,
2
M. A. G. Alvarez,
1
M. S. Hussein,
1
M. A. Ca
ˆ
ndido Ribeiro,
3
E. S. Rossi, Jr.,
1
and C. P. Silva
1
1
Departamento de Fı
´
sica Nuclear, Instituto de Fı
´
sica da Universidade de Sa
˜
o Paulo,
Caixa Postal 66318, 05315-970 Sa
˜
o Paulo, SP, Brazil
2
Departamento de Fı
´
sica, Instituto Tecnolo
´
gico de Aerona
´
utica, Centro Te
´
cnico Aeroespacial, Sa
˜
o Jose
´
dos Campos, SP, Brazil
3
Departamento de Fı
´
sica, Instituto de Biocie
ˆ
ncias, Letras e Cie
ˆ
ncias Exatas, Universidade Estadual Paulista,
Sa
˜
o Jose
´
do Rio Preto, SP, Brazil
共Received 10 September 2001; published 17 July 2002兲
Extensive systematizations of theoretical and experimental nuclear densities and of optical potential
strengths extracted from heavy-ion elastic scattering data analyses at low and intermediate energies are pre-
sented. The energy dependence of the nuclear potential is accounted for within a model based on the nonlocal
nature of the interaction. The systematics indicates that the heavy-ion nuclear potential can be described in a
simple global way through a double-folding shape, which basically depends only on the density of nucleons of
the partners in the collision. The possibility of extracting information about the nucleon-nucleon interaction
from the heavy-ion potential is investigated.
DOI: 10.1103/PhysRevC.66.014610 PACS number共s兲: 24.10.Ht, 13.75.Cs, 21.10.Ft, 21.10.Gv
I. INTRODUCTION
The optical potential plays a central role in the description
of heavy-ion collisions, since it is widely used in studies of
the elastic scattering process as well as in more complicated
reactions through the distorted-wave Born approximation
共DWBA兲 or coupled-channel formalisms. This complex and
energy-dependent potential is composed of the bare and po-
larization potentials, the latter containing the contribution
arising from nonelastic couplings. In principle, the bare 共or
nuclear兲 potential between two heavy ions can be associated
with the fundamental nucleon-nucleon interaction folded into
a product of the nucleon densities of the nuclei 关1兴. Apart
from some structure effects, the shape of the nuclear density
along the table of stable nuclides is nearly a Fermi distribu-
tion, with diffuseness approximately constant and radius
given roughly by R⫽ r
0
A
1/3
, where A is the number of nucle-
ons of the nucleus. Therefore, one could expect a simple
dependence of the heavy-ion nuclear potential on the number
of nucleons of the partners in the collision. In fact, analytical
formulas have been deduced 关2–4兴 for the folding potential,
and simple expressions have been obtained at the surface
region. A universal 共system-independent兲 shape for the
heavy-ion nuclear potential has been derived 关5兴 also in the
framework of the liquid-drop model, from the proximity
theorem which relates the force between two nuclei to the
interaction between flat surfaces made of semi-infinite
nuclear matter. The theorem leads 关5兴 to an expression for
the potential in the form of a product of a geometrical factor
by a function of the separation between the surfaces of the
nuclei.
The elastic scattering is the simplest process that occurs in
a heavy-ion collision because it involves very little rear-
rangement of matter and energy. Therefore, this process has
been studied in a large number of experimental investiga-
tions, and a huge body of elastic cross section data is cur-
rently available. The angular distribution for elastic scatter-
ing provides unambiguous determination of the real part of
the optical potential only in a region around a particular dis-
tance 关6兴 hereafter referred as the sensitivity radius (R
S
). At
energies close to the Coulomb barrier the sensitivity radius is
situated in the surface region. In this energy region, the sys-
tematization 关7,8兴 of experimental results for potential
strengths at the sensitivity radii has provided a universal ex-
ponential shape for the heavy-ion nuclear potential at the
surface, as theoretically expected, but with a diffuseness
value smaller than that originally proposed in the proximity
potential.
In a recent review article 关6兴 the phenomenon of rainbow
scattering was discussed, and it was emphasized that the real
part of the optical potential can be unambiguously extracted
also at very short distances from heavy-ion elastic scattering
data at intermediate energies. Such a kind of data has been
first obtained for
␣
-particle scattering from a variety of nu-
clei over a large range of energies 关9–11兴 and later for sev-
eral heavy-ion systems. However, differently from the case
for the surface region 共low energy兲, a systematization of po-
tential strengths at the inner distances has not been per-
formed up to now, probably because the resulting phenom-
enological interactions have presented significant
dependence on the bombarding energies. Several theoretical
models have been developed to account for this energy de-
pendence through realistic mean field potentials. Most of
them are improvements of the original double-folding poten-
tial with the nucleon-nucleon interaction assumed to be en-
ergy and density dependent 关6兴. Another recent and success-
ful model 关12–14兴 associates the energy dependence of the
heavy-ion bare potential with nonlocal quantum effects re-
lated to the exchange of nucleons between target and projec-
tile, resulting in a very simple expression for the energy de-
pendence of the nuclear potential. Using the model of Refs.
关12–14兴, in the present work we have realized a systemati-
zation of potential strengths extracted from elastic scattering
data analyses, considering both low 共near-barrier兲 and inter-
mediate energies. The systematics indicates that the heavy-
ion nuclear potential can be described in a simple global way
PHYSICAL REVIEW C 66, 014610 共2002兲
0556-2813/2002/66共1兲/014610共13兲/$20.00 ©2002 The American Physical Society66 014610-1
through a double-folding shape, which basically depends
only on the number of nucleons of the nuclei.
The paper is organized as follows. In Sec. II, as a prepa-
ratory step for the systematization of the potential, an exten-
sive and systematic study of nuclear densities is presented.
This study is based on charge distributions extracted from
electron scattering experiments 关15,16兴 as well as on theoret-
ical densities derived from the Dirac-Hartree-Bogoliubov
model 关17兴. In Sec. III, analytical expressions for the double-
folding potential are derived for the whole 共surface and in-
ner兲 interaction region, and a survey of the main character-
istics of this potential is presented. Section IV contains the
nonlocal model for the heavy-ion bare interaction, including
several details that have not been published before. Section
V is devoted to the nuclear potential systematics. In Sec. VI,
we discuss the role played by the nucleon-nucleon interac-
tion, and we present, in a somewhat speculative way, an
alternative form for the effective nucleon-nucleon interac-
tion, which is consistent with our results for the heavy-ion
nuclear potential. Finally, Sec. VII contains a brief summary
and the main conclusions.
II. SYSTEMATIZATION OF THE NUCLEAR DENSITIES
According to the double-folding model, the heavy-ion
nuclear potential depends on the nuclear densities of the nu-
clei in collision. Thus, a systematization of the potential re-
quires a previous systematization of the nuclear densities. In
this work, with the aim of describing the proton, neutron,
nucleon 共proton⫹neutron兲, charge, and matter densities, we
adopt the two-parameter Fermi 共2pF兲 distribution, which has
also been commonly used for charge densities extracted from
electron scattering experiments 关15兴. The shape, Eq. 共1兲 and
Fig. 1, of this distribution is particularly appealing for the
density description, due to the flatness of the inner region,
which is associated with the saturation of the nuclear me-
dium, and to the rapid falloff 共related to the diffuseness pa-
rameter a) that brings out the notion of the radius R
0
of the
nucleus:
共
r
兲
⫽
0
1⫹exp
冉
r⫺R
0
a
冊
. 共1兲
The
0
, a, and R
0
parameters are connected by the normal-
ization condition
4
冕
0
⬁
共
r
兲
r
2
dr⫽ X, 共2兲
where X could be the number of protons Z, neutrons N,or
nucleons A⫽ N⫹ Z. In our theoretical calculations, the
charge distribution (
ch
) has been obtained by folding the
proton distribution of the nucleus (
p
) with the intrinsic
charge distribution of the proton in free space (
chp
):
ch
共
r
兲
⫽
冕
p
共
r
ជ
⬘
兲
chp
共
r
ជ
⫺ r
ជ
⬘
兲
dr
ជ
⬘
, 共3兲
where
chp
is an exponential with diffuseness a
chp
⫽ 0.235 fm. In an analogous way, we have defined the mat-
ter density by folding the nucleon distribution of the nucleus
with the intrinsic matter distribution of the nucleon, which is
assumed to have the same shape of the intrinsic charge dis-
tribution of the proton. For convenience, the charge and mat-
ter distributions are normalized to the number of protons and
nucleons, respectively.
In order to systematize the heavy-ion nuclear densities,
we have calculated theoretical distributions for a large num-
ber of nuclei using the Dirac-Hartree-Bogoliubov 共DHB兲
model 关17兴. The DHB calculations were performed using the
NL3 parameter set 关18兴. This set was obtained by adjusting
the masses and the charge and neutron radii of ten nuclei in
the region of the valley of stability, ranging from
16
Oto
214
Pb, using the Dirac-Hartree-BCS 共DH-BCS兲 model. For
the cases in which they have been performed, calculations
using this parameter set and either the DHB 关17兴 or the DH-
BCS 关18–20兴 model have shown very good agreement with
experimental masses and radii. The quality of the description
of nuclear masses and charge radii, calculated in various mi-
croscopic approaches, has been presented in a recent paper
关21兴. In this work, the difference between experimental rms
charge radii of stable nuclei with the corresponding theoret-
ical predictions has been found to be around 0.05 fm for all
models, including the DH-BCS model with the NL3 param-
eter set. This precision is quite satisfactory taking into ac-
count our purpose of systematizing the optical potential
strengths. In the present paper, we have also used the results
of previous systematics for charge distributions 关15,16兴, ex-
tracted from electron scattering experiments, as a further
check of our DHB results. All the theoretical and most of the
‘‘experimental’’ densities are not exact Fermi distributions.
Thus, with the aim of studying the equivalent diffuseness of
FIG. 1. Nucleon density for the
56
Fe nucleus represented
through Dirac-Hartree-Bogoliubov calculations 共DHB兲 andatwo-
parameter Fermi distribution 共2pF兲, with a⫽ 0.5 fm and R
0
⫽ 4.17 fm. The small difference between the 2pF distribution and
the function
0
C„(r⫺ R
0
)/a… 关Eqs. 共12兲, 共13兲, and 共14兲兴 is hardly
seen in the figure.
L. C. CHAMON et al. PHYSICAL REVIEW C 66, 014610 共2002兲
014610-2
the densities, we have calculated the corresponding logarith-
mic derivatives 关Eq. 共4兲兴 at the surface region 共at r⬇R
0
⫹ 2 fm):
a⬇⫺
共
r
兲
d
dr
. 共4兲
Figure 2共a兲 shows the results for the experimental charge
distributions: the diffuseness values spread around an aver-
age diffuseness a
¯
c
⫽ 0.53 fm, with standard deviation
0.04 fm. Most of this dispersion arises from experimental
errors. Indeed, we have verified that different analyses 共dif-
ferent electron scattering data set or different models for the
charge density兲 for a given nucleus provide diffuseness val-
ues that differ from each other by about 0.03 fm. Therefore,
the experimental charge distributions are compatible, within
the experimental precision, with a constant diffuseness value.
The theoretical charge distributions present similar behavior
关Fig. 2共b兲兴, with an average value slightly smaller than the
experimental one. In this case, the observed standard devia-
tion, 0.02 fm, is associated with the effects of the structure
of the nuclei. Despite the trend presented by the neutron and
proton diffuseness 关Fig. 2共c兲兴, all the nucleon distributions
result in very similar diffuseness values (a
¯
N
⫽ 0.48 fm),
with standard deviation 0.025 fm. As a result of the folding
procedure, the matter distributions present diffuseness values
significantly greater (a
¯
M
⫽ 0.54 fm) than those for the
nucleon distributions. Taking into account that the theoretical
calculations have slightly underestimated the experimental
charge diffuseness, we consider that more realistic average
values for the nucleon and matter density diffuseness are
0.50 and 0.56 fm, respectively. A dispersion (
a
) of about
0.025 fm around these average values is expected due to
effects of the structure of the nuclei.
The rms radius of a distribution is defined by Eq. 共5兲:
r
rms
⫽
冑
冕
r
2
共
r
兲
dr
ជ
冕
共
r
兲
dr
ជ
. 共5兲
We have determined the radii R
0
for the 2pF distributions
assuming that the corresponding rms radii should be equal to
those of the experimental 共electron scattering兲 and theoretical
共DHB兲 densities. The results for R
0
from theoretical charge
distributions 关Fig. 3共b兲兴 are very similar to those from elec-
tron experiments 关Fig. 3共a兲兴. This fact indicates that the radii
obtained through the theoretical DHB calculations are quite
realistic. The nucleon and matter densities give very similar
radii 关Fig. 3共d兲兴, which are well described by the following
linear fit:
R
0
⫽ 1.31A
1/3
⫺ 0.84 fm. 共6兲
As a result of effects of the structure of the nuclei, the R
0
values spread around this linear fit with dispersion
R
0
⫽ 0.07 fm, but the heavier the nucleus is, the smaller is the
deviation. In Fig. 4 are shown the theoretical 共DHB兲 nucleon
FIG. 2. Equivalent diffuseness values obtained for charge distributions extracted from electron scattering experiments and for theoretical
densities obtained from Dirac-Hartree-Bogoliubov calculations.
TOWARD A GLOBAL DESCRIPTION OF THE NUCLEUS- . . . PHYSICAL REVIEW C 66, 014610 共2002兲
014610-3
densities for a few nuclei and the corresponding 2pF distri-
butions with a⫽ 0.50 fm and R
0
values obtained from Eq.
共6兲.
III. ESSENTIAL FEATURES OF THE FOLDING
POTENTIAL
The double-folding potential has the form
V
F
共
R
兲
⫽
冕
1
共
r
1
兲
2
共
r
2
兲
v
NN
共
R
ជ
⫺ r
ជ
1
⫹ r
ជ
2
兲
dr
ជ
1
dr
ជ
2
, 共7兲
where R is the distance between the centers of the nuclei,
i
are the respective nucleon distributions, and
v
NN
(r
ជ
) is the
effective nucleon-nucleon interaction. The success of the
folding model can only be judged meaningfully if the effec-
tive nucleon-nucleon interaction employed is truly realistic.
The most widely used realistic interaction is known as M3Y
关1,6兴, which can usually assume two versions: Reid and
Paris.
For the purpose of illustrating the effects of density varia-
tions on the folding potential, we show in Fig. 5 the results
FIG. 3. The R
0
parameter obtained for charge distributions extracted from electron scattering experiments and for theoretical densities
obtained from Dirac-Hartree-Bogoliubov calculations.
FIG. 4. Nucleon densities from Dirac-
Hartree-Bogoliubov calculations 共solid lines兲
compared with the corresponding two-parameter
Fermi distributions 共dashed lines兲, with a
⫽ 0.50 fm and R
0
obtained through Eq. 共6兲.
L. C. CHAMON et al. PHYSICAL REVIEW C 66, 014610 共2002兲
014610-4
obtained for different sets of 2pF distributions. In Sec. II, we
have estimated the dispersions of the R
0
and a parameters,
R
0
⬇0.07 fm and
a
⬇0.025 fm, that arise from effects of
the structure of the nuclei. Observe that these standard de-
viations are one-half of the corresponding variations consid-
ered in the example of Fig. 5, ⌬R
0
⫽ 0.14 fm and ⌬a
⫽ 0.05 fm. The surface region of the potential (R⭓R
1
⫹ R
2
) is much more sensitive to small changes of the density
parameters than the inner region. Our calculations indicate
that, as a result of such structure effects, the strength of the
nuclear potential in the region near the barrier radius may
vary by about 20%, and the major part of this variation is
connected to the standard deviation of the parameter a.
Therefore, concerning the nuclear potential, the effects of the
structure of the nuclei are mostly present at the surface and
mainly related to the diffuseness parameter.
The six-dimensional integral 关Eq. 共7兲兴 can easily be
solved by reducing it to a product of three one-dimensional
Fourier transforms 关1兴, but the results may only be obtained
through numerical calculations. In order to provide analytical
expressions for the folding potential, we consider, as an ap-
proximation, that the range of the effective nucleon-nucleon
interaction is negligible in comparison with the diffuseness
of the nuclear densities. In this zero-range approach, the
double-folding potential can be obtained from
v
NN
共
r
ជ
兲
⬇V
0
␦
共
r
ជ
兲
⇒ V
F
共
R
兲
⫽
2
V
0
R
冕
0
⬁
r
1
1
共
r
1
兲
冋
冕
兩
R⫺ r
1
兩
R⫹ r
1
r
2
2
共
r
2
兲
dr
2
册
dr
1
.
共8兲
As discussed in Sec. II, the heavy-ion densities involved in
Eq. 共8兲 are approximately 2pF distributions, with R
0
Ⰷ a.In
the limit a→0, the double integral results in
V
F
共
R⭐R
2
⫺ R
1
兲
⫽ V
0
01
02
4
3
R
1
3
, 共9兲
V
F
共
R
2
⫺ R
1
⭐R⭐R
1
⫹ R
2
兲
⫽ V
0
01
02
4
3
R
3
冉
2
1⫹
冊
冋
3
8
⫹
4
⫹
2
16
册
,
共10兲
V
F
共
R⭓R
1
⫹ R
2
兲
⫽ 0, 共11兲
where s⫽ R⫺ (R
1
⫹ R
2
), R⫽ 2R
1
R
2
/(R
1
⫹ R
2
),
⫽ R/(R
1
⫹ R
2
),
⫽ s/R, and R
1
and R
2
are the radii of the nuclei
共hereafter we consider R
2
⭓R
1
). We need a further approxi-
mation to obtain analytical expressions for the folding poten-
tial in the case of finite diffuseness value.
The Fermi distribution may be represented, with precision
better than 3% for any r value 共see Fig. 1兲,by
0
1⫹exp
冉
r⫺R
0
a
冊
⬇
0
C
冉
r⫺R
0
a
冊
, 共12兲
C
共
x⭐0
兲
⫽ 1⫺
7
8
e
x
⫹
3
8
e
2x
, 共13兲
C
共
x⭓0
兲
⫽ e
⫺ x
冉
1⫺
7
8
e
⫺ x
⫹
3
8
e
⫺ 2x
冊
. 共14兲
This approximation is particularly useful in obtaining ana-
lytical expressions for integrals that involve the 2pF distri-
bution. If both nuclei have the same diffuseness a, the double
integral 关Eq. 共8兲兴 can be solved analytically using the ap-
proximation represented by Eq. 共12兲, and the result ex-
pressed as a sum of a large number of terms, most of them
negligible for aⰆ R
0
. Rather simple expressions can be
found after an elaborate algebraic manipulation:
V
F
共
R⭐R
2
⫺ R
1
⫹ a
兲
⬇V
0
01
02
4
3
R
1
3
再
1⫹9.7
冉
a
R
1
冊
2
⫺
冋
0.875
冉
R
2
3
R
1
3
⫺ 1
冊
⫹
a
R
1
冉
2.4⫹
R
2
2
R
1
2
冊
册
e
⫺ (R
2
⫺ R
1
)/a
冎
, 共15兲
V
F
共
R
2
⫺ R
1
⫹ a⭐R⭐R
1
⫹ R
2
兲
⬇V
0
01
02
4
3
R
3
冉
1
1⫹
冊
再
2
冋
3
8
⫹
4
⫹
2
16
册
⫹ 2.4
2
冋
1⫺
5
8
⫺
2
⫹
冉
5
4
⫺
1
2
冊
e
⫹
冉
1⫹
5
8
冊
e
⫺ (⫹ 2R
1
/a)
册
冎
, 共16兲
V
F
共
R⭓R
1
⫹ R
2
兲
⬇V
0
01
02
a
2
Rg
共
兲
f
共
s/a
兲
, 共17兲
FIG. 5. Folding potential for different sets of 2pF densities that
may represent the
16
O⫹
58
Ni system. The approximate position of
the s-wave barrier radius (R
B
) is indicated in the figure.
TOWARD A GLOBAL DESCRIPTION OF THE NUCLEUS- . . . PHYSICAL REVIEW C 66, 014610 共2002兲
014610-5
