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arXiv:nucl-th/0202015v1 5 Feb 2002
Toward a global description of the nucleus-nucleus interaction.
L. C. Chamon1, B. V. Carlson2, L. R. Gasques1, D. Pereira1, C. De Conti2, M. A. G. Alvarez1,
M. S. Hussein1, M. A. Cˆ andido Ribeiro3, E. S. Rossi Jr.1and C. P. Silva1.
1. Departamento de F´ ısica Nuclear, Instituto de F´ ısica da Universidade de S˜ ao Paulo,
Caixa Postal 66318, 05315-970, S˜ ao Paulo, SP, Brazil.
2. Departamento de F´ ısica, Instituto Tecnol´ ogico de Aeron´ autica, Centro T´ ecnico Aeroespacial,
S˜ ao Jos´ e dos Campos, SP, Brazil.
3. Departamento de F´ ısica, Instituto de Biociˆ encias, Letras e Ciˆ encias Exatas,
Universidade Estadual Paulista, S˜ ao Jos´ e do Rio Preto, SP, Brazil.
Extensive systematizations of theoretical and experimental nuclear densities and of optical po-
tential strengths extracted from heavy-ion elastic scattering data analyses at low and intermediate
energies are presented. 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 inves-
tigated.
PACS: 24.10.Ht, 13.75.Cs, 21.10.Ft, 21.10.Gv, 21.30.-x
Keywords: Heavy-ion nuclear potential. Proton, neutron, charge, nucleon and matter density distributions. Effective
nucleon-nucleon interaction.
1. 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 DWBA or coupled-channel
formalisms. This complex and energy-dependent potential is composed of the bare and polarization potentials, the
latter containing the contribution arising from nonelastic couplings. In principle, the bare (or nuclear) 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 [1]. Apart from some structure effects, the shape of the nuclear density along
the table of stable nuclides is nearly a Fermi distribution, with diffuseness approximately constant and radius given
roughly by R = r0 A1/3, where A is the number of nucleons 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 formulae have been deduced [2–4] for the folding potential, and simple expressions have been obtained at
the surface region. An universal (system-independent) shape for the heavy-ion nuclear potential has been derived [5]
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 [5] 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 litle
rearrangement of matter and energy. Therefore, this process has been studied in a large number of experimental
investigations, and a huge body of elastic cross section data is currently available. The angular distribution for elastic
scattering provides unambiguous determination of the real part of the optical potential only in a region around a
particular distance [6] hereafter referred as the sensitivity radius (RS). At energies close to the Coulomb barrier the
sensitivity radius is situated in the surface region. In this energy region, the systematization [7,8] of experimental
results for potential strengths at the sensitivity radii has provided an universal exponential 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 [6] 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 α-particle scattering from a
variety of nuclei over a large range of energies [9–11], and later for several heavy-ion systems. However, differently from
the case for the surface region (low energy), a systematization of potential strengths at the inner distances has not
1
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been performed up to now, probably because the resulting phenomenological interactions have presented significant
dependence on the bombarding energies. Several theoretical models have been developed to account for this energy-
dependence through realistic mean field potentials. Most of them are improvements of the original double-folding
potential with the nucleon-nucleon interaction assumed to be energy- and density-dependent [6]. Another recent and
successful model [12–14] associates the energy-dependence of the heavy-ion bare potential with nonlocal quantum
effects related to the exchange of nucleons between target and projectile, resulting in a very simple expression for the
energy-dependence of the nuclear potential. Using the model of Refs. [12–14], in the present work we have realized
a systematization of potential strengths extracted from elastic scattering data analyses, considering both: low (near-
barrier) and intermediate energies. 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 number of nucleons of
the nuclei.
The paper is organized as follows. In Section 2, as a preparatory step for the potential systematization, an extensive
and systematic study of nuclear densities is presented. This study is based on charge distributions extracted from
electron scattering experiments [15,16] as well as on theoretical densities derived from the Dirac-Hartree-Bogoliubov
model [17]. In Section 3, analytical expressions for the double-folding potential are derived for the whole (surface
and inner) interaction region, and a survey of the main characteristics of this potential is presented. Section 4
contains the nonlocal model for the heavy-ion bare interaction, including several details that have not been published
before. Section 5 is devoted to the nuclear potential systematics. In Section 6, we discuss the role played by the
nucleon-nucleon interaction, and we present, in a somewhat speculative way, an alternative form for the effective
nucleon-nucleon interaction, which is consistent with our results for the heavy-ion nuclear potential. Finally, Section
7 contains a brief summary and the main conclusions.
2. Systematization of the nuclear densities
According to the double-folding model, the heavy-ion nuclear potential depends on the nuclear densities of the nuclei
in collision. Thus, a systematization of the potential requires a previous systematization of the nuclear densities. In
this work, with the aim of describing the proton, neutron, nucleon (proton+neutron), charge and matter densities, we
adopt the two-parameter Fermi (2pF) distribution, which has also been commonly used for charge densities extracted
from electron scattering experiments [15]. The shape, Eq. 1 and Fig. 1, of this distribution is particularly appealing
for the density description, due to the flatness of the inner region, that is associated with the saturation of the nuclear
medium, and to the rapid fall-off (related to the diffuseness parameter a) that brings out the notion of the radius,
R0, of the nucleus.
ρ(r) =
ρ0
1 + exp?r−R0
a
?
(1)
The ρ0, a and R0parameters are connected by the normalization condition:
4π
?∞
0
ρ(r) r2dr = X , (2)
where X could be the number of protons Z, neutrons N or nucleons A = N + Z. In our theoretical calculations, the
charge distribution (ρch) has been obtained by folding the proton distribution of the nucleus (ρp) with the intrinsic
charge distribution of the proton in free space (ρchp)
ρch(r) =
?
ρp(?r′) ρchp(? r −?r′) d?r′,(3)
where ρchp is an exponential with diffuseness achp = 0.235 fm. In an analogous way, we have defined the matter
density of the nucleus 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 distribution of the proton. For convenience,
the charge and matter 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
number of nuclei using the Dirac-Hartree-Bogoliubov (DHB) model [17]. The DHB calculations were performed using
the NL3 parameter set [18]. This set was obtained by adjusting the masses, and the charge and neutron radii of 10
nuclei in the region of the valley of stability, ranging from16O to214Pb, using the Dirac-Hartree-BCS (DH-BCS)
model. For the cases in which they have been performed, calculations using this parameter set and either the DHB [17]
or the DH-BCS [18–20] model have shown very good agreement with experimental masses and radii. In the present
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work, we have also used the results of previous systematics for charge distributions [15,16], extracted 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 the densities,
we have calculated the corresponding logarithmic derivatives (Eq. 4) at the surface region (at r ≈ R0+ 2 fm).
a ≈ −ρ(r)
dρ
dr
(4)
Fig. 2a shows the results for the experimental charge distributions: the diffuseness values spread around an average
diffuseness ¯ ac= 0.53 fm, with standard deviation 0.04 fm. Most of this dispersion arises from experimental errors.
Indeed, we have verified that different analyses (different electron scattering data set or different models for the charge
density) for a given nucleus provide diffuseness values 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 (Fig. 2b), with average value slightly smaller than the
experimental one. In this case, the observed standard deviation, 0.02 fm, is associated with the effects of the
structure of the nuclei. Despite the trend presented by the neutron and proton diffuseness (Fig. 2c), all the nucleon
distributions result in very similar diffuseness values (¯ aN= 0.48 fm), with standard deviation 0.025 fm. Due to the
folding procedure, the matter distributions present diffuseness values significantly greater (¯ aM= 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 fm and 0.56 fm, respectively. A dispersion (σa) of about 0.025 fm around these average values
is expected due to effects of the structure of the nuclei.
The root-mean-square (RMS) radius of a distribution is defined by Eq. 5:
rrms=
??r2ρ(r) d? r
?ρ(r) d? r
.(5)
We have determined the radii R0 for the 2pF distributions assuming that the corresponding RMS radii should be
equal to those of the experimental (electron scattering) and theoretical (DHB) densities. The results for R0 from
theoretical charge distributions (Fig. 3b) are very similar to those from electron experiments (Fig. 3a). This fact
indicates that the radii obtained through the theoretical DHB calculations are quite realistic. The nucleon and matter
densities give very similar radii (Fig. 3d), which are well described by the following linear fit:
R0= 1.31 A1/3− 0.84 fm.(6)
Due to effects of the structure of the nuclei, the R0values spread around this linear fit with dispersion σR0= 0.07 fm,
but the heavier the nucleus is, the smaller is the deviation. In Fig. 4 are shown the theoretical (DHB) nucleon densities
for a few nuclei, and the corresponding 2pF distributions with a = 0.50 fm and R0values obtained from Eq. 6.
3. Essential features of the folding potential
The double-folding potential has the form
VF(R) =
?
ρ1(r1) ρ2(r2) vNN(?R − ? r1+ ? r2) d? r1d? r2,(7)
where R is the distance between the centers of the nuclei, ρiare the respective nucleon distributions, and vNN(? r) is
the effective nucleon-nucleon interaction. The success of the folding model can only be judged meaningfully if the
effective nucleon-nucleon interaction employed is truly realistic. The most widely used realistic interaction is known
as M3Y [1,6], which can usually assume two versions: Reid and Paris.
For the purpose of illustrating the effects of density variations on the folding potential, we show in Fig. 5 the
results obtained for different sets of 2pF distributions. In Section 2, we have estimated the dispersions of the R0and
a parameters, σR0≈ 0.07 and σa≈ 0.025 fm, that arise from effects of the structure of the nuclei. Observe that these
standard deviations are one half of the corresponding variations considered in the example of Fig. 5, ∆R0= 0.14 fm
and ∆a = 0.05 fm. The surface region of the potential (R ≥ R1+R2) is much more sensitive to small changes of the
density parameters than the inner region. Our calculations indicate that, due to 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
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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 (Eq. 7) can easily be solved by reducing it to a product of three one-dimensional
Fourier transforms [1], but the results may only be obtained through numerical calculations. In order to provide
analytical expressions for the folding potential, we consider, as an approximation, 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:
vNN(? r) ≈ V0δ(? r) ⇒ VF(R) =2πV0
R
?∞
0
r1ρ1(r1)
??R+r1
|R−r1|
r2ρ2(r2) dr2
?
dr1. (8)
As discussed in Section 2, the heavy-ion densities involved in Eq. 8 are approximately 2pF distributions, with R0≫ a.
In the limit a → 0, the double-integral results in
VF(R ≤ R2− R1) = V0ρ01ρ02
4
3πR3
1, (9)
VF(R2− R1≤ R ≤ R1+ R2) = V0ρ01ρ02
4
3πR3
?
τ2
1 + ζτ
??3
8+τ
4+ ζτ2
16
?
,(10)
VF(R ≥ R1+ R2) = 0 , (11)
where s = R − (R1+ R2), R = 2R1R2/(R1+ R2), ζ = R/(R1+ R2), τ = s/R, R1and R2are the radii of the nuclei
(hereafter we consider R2≥ R1). We need a further approximation to obtain analytical expressions for the folding
potential in the case of finite diffuseness value.
The Fermi distribution may be represented, with precision better than 3% for any r value (see Fig. 1), by:
ρ0
1 + exp?r−R0
C(x ≤ 0) = 1 −7
a
? ≈ ρ0C
?r − R0
a
?
,(12)
8ex+3
8e2x,(13)
C(x ≥ 0) = e−x
?
1 −7
8e−x+3
8e−2x
?
.(14)
This approximation is particularly useful in obtaining analytical expressions for integrals that involve the 2pF distri-
bution. If both nuclei have the same diffuseness a, the double-integral (Eq. 8) can be solved analytically using the
approximation represented by Eq. 12, and the result expressed as a sum of a large number of terms, most of them
negligible for a ≪ R0. Rather simple expressions can be found after an elaborate algebraic manipulation:
?
R1
VF(R ≤ R2− R1+ a) ≈ V0ρ01ρ02
4
3πR3
1
1 + 9.7
?a
?2
−
?
0.875
?R3
2
R3
1
− 1
?
+
a
R1
?
2.4 +R2
2
R2
1
??
e−(R2−R1)/a
?
,
(15)
VF(R2− R1+ a ≤ R ≤ R1+ R2) ≈ V0ρ01ρ02
8η − ζτ2+
2
4
3πR3
?
1
1 + ζτ
??
τ2
?3
8+τ
4+ ζτ2
16
?
+ 2.4 η2
?
1 −5
?5
4η −1
?
eε+
?
1 +5
8η
?
e−(ε+2R1/a)
??
,(16)
VF(R ≥ R1+ R2) ≈ V0ρ01ρ02πa2R g(τ) f(s/a) ,(17)
with η = a/R, ε = s/a. The functions g and f are given by:
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g(τ) =1 + τ + τ2ζ/3 + η + (η + 1/2) e−ε
1 + ζτ
, (18)
f(s/a) = (1 + s/a) e−s/a. (19)
If the nuclei have slightly (about 10%) different diffuseness, the formulae are still valid with a ≈ (a1+ a2)/2. As an
example of the precision of the analytical expressions above, we exhibit in Fig. 6 the results of numerical calculations
(Eq. 8) and compare them with those from Eqs. 15, 16 and 17, and also with the exact expressions for a = 0, Eqs.
9, 10 and 11.
Eq. 17 presents some similarity with the proximity potential [5]:
VP = 2πΓRd Φ(s/d) , (20)
where d is the “surface width”, and Φ is an universal (system-independent) function. For a 2pF distribution, the
surface width is related to the diffuseness parameter through: d ≈ (π/√3) a [21]. The theoretical value adopted for d
is 1 fm [5], which corresponds to a diffuseness a ≈ 0.55 fm. Taking into account that the Γ value is rather system-
independent [5], systematizations of heavy-ion potential strengths extracted from elastic scattering data analyses have
been performed by using the following expression, which should be valid for surface distances,
VP(s ≫ 0)
R
= V0e−s/α. (21)
The resulting experimental α values are quite similar, α ≈ 0.62 fm [7,8], but smaller than the theoretical prediction
of the proximity potential α ≈ 0.75 fm [5]. Such systematics have included only experimental potential strengths in
the surface region, in contrast to the case of the proximity potential where V/R should be an universal function of s
also for inner distances. The proximity potential is not fully agreeing with our results for the double-folding potential
in the zero-range approach (see Fig. 7). In fact, Eq. 17 indicates that a better choice for an universal quantity at the
surface region would be
Vred(s ≥ 0) =
VF
ρ01ρ02πa2R g(τ),(22)
which results (from Eqs. 17, 19 and 22) the system-independent expression
Vred(s ≥ 0) ≈ V0(1 + s/a) e−s/a.(23)
However, it is not clear that one can find a simple form for such a universal quantity at inner distances from Eqs. 15
and 16. In Section 5, the reduced potential, Vred, is useful for addressig the potential strength systematization. Thus
we define Vredfor s ≤ 0 through the following trivial form:
Vred(s ≤ 0) = V0.
The end of this section is devoted to the study of the effect on the folding potential of a finite range for the effective
nucleon-nucleon interaction. The tri-dimensional delta function, V0δ(? r), can be represented through the limit σ → 0
applied to the finite-range Yukawa function
(24)
Yσ(r) = V0
e−r/σ
4πrσ2. (25)
Fig. 8 shows a comparison of folding potentials in the zero-range approach (Eq. 8) with the result obtained (from
Eq. 7) using an Yukawa function for the effective nucleon-nucleon interaction. The finite range is not truly significant
at small distances, and can be accurately simulated at the surface, within the zero-range approach, just by slightly
increasing the diffuseness of the nuclear densities.
4. A nonlocal description of the nucleus-nucleus interaction
Before proceeding with the systematization of the potential, we first set the stage for the model of the heavy-ion
nuclear interaction [12–14]. When dealing with nonlocal interactions, one is required to solve the following integro-
differential equation
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−¯ h2
2µ∇2Ψ(?R) + [VC(R) + Vpol(R,E) + ıWpol(R,E)]Ψ(?R) +
?
U(?R,?R′) Ψ(?R′) d?R′= EΨ(?R) . (26)
VCis the Coulomb interaction assumed to be local. Vpoland Wpolare the real and imaginary parts of the polarization
potential, and contain the contribution arising from nonelastic channel couplings. The corresponding nonlocality,
called the Feshbach nonlocality, is implicit through the energy-dependence of these terms, consistent with the disper-
sion relation [22]. U(?R,?R′) is the bare interaction and the nonlocality here, the Pauli nonlocality, is solely due to the
Pauli exclusion principle and involves the exchange of nucleons between target and projectile.
Guided by the microscopic treatment of the nucleon-nucleus scattering [23–27], the following ansatz is assumed for
the heavy-ion bare interaction [13]
U(?R,?R′) = VNL
?R + R′
2
?
1
π3/2b3e−(|?R+? R′|/b)
2
, (27)
where b is the range of the Pauli nonlocality. Introduced in this way, the nonlocality is a correction to the local model,
and in the b → 0 limit Eq. 26 reduces to the usual Schr¨ oedinger differential equation. The range of the nonlocality
can be found through b ≈ b0m0/µ [28], where b0= 0.85 fm is the nucleon-nucleus nonlocality parameter [23], m0
is the nucleon mass, and µ is the reduced mass of the nucleus-nucleus system. This type of very mild nonlocality in
the nucleon-nucleus and nucleus-nucleus interaction is to be contrasted with the very strong nonlocality found in the
pion-nucleus interaction in the ∆-region [29]. In such cases, even the concept of an optical potential becomes dubious.
In our case, however, we are on very safe ground.
The relation between the nonlocal interaction and the folding potential is obtained from [13]
VNL(R) = VF(R) .(28)
Due to the central nature of the interaction, it is convenient to write down the usual expansion in partial waves,
Ψ(?R) =
?
ıℓ(2ℓ + 1)uℓ(R)
kR
Pℓ[cos(θ)] ,(29)
U(?R,?R′) =
?2ℓ + 1
4πRR′Vℓ(R,R′) Pℓ[cos(φ)] ,(30)
Vℓ(R,R′) = VNL
?R + R′
2
?
1
bπ1/2
?
Qℓ
?2RR′
b2
?
e−?R−R′
b
?2
(−)ℓ+1Qℓ
?−2RR′
b2
?
e−?R+R′
b
?2?
,(31)
where Qℓare polynomials and φ is the angle between?R and?R′[23] . Thus, the integro-differential equation can be
recast into the following form
¯ h2
2µ
d2uℓ(R)
dR2
+
?
E − VC(R) − Vpol(R,E) − ıWpol(R,E) −ℓ(ℓ + 1)¯ h2
2µR2
?
uℓ(R) =
?∞
0
Vℓ(R,R′) uℓ(R′) dR′.(32)
When confronting theory and experiment, one usually relies on the optical model with a local potential. This brings
into light the issue of extracting from Eq. 32 a local-equivalent (LE) potential
VLE(R,E) + ıWLE(R,E) =
1
uℓ(R)
?∞
0
Vℓ(R,R′) uℓ(R′) dR′.(33)
The presence of the wave-function in Eq. 33 indicates that the LE potential is complex and also ℓ- and energy-
dependent. Despite its complex nature, the LE potential is not absorptive, ?Ψ|WLE|Ψ? = 0; this statement can
be demonstrated by considering that the nonlocal interaction is real and symmetrical, Vℓ(R,R′) = Vℓ(R′,R). For
neutron-nucleus systems, the LE potential is only weakly ℓ-dependent, and an approximate relation to describe its
energy-dependence has been obtained [23]. A generalization of this relation for the ion-ion case is given by [12,13]:
VLE(R,E) ≈ VF(R) e−γ[E−VC(R)−VLE(R,E)],(34)
with γ = µb2/2¯ h2. In order to provide an example of the precision of expression 34, in Fig. 9 the corresponding
result is compared to the exact LE potential (Eq. 33) obtained from the numerical resolution [13] of the respective
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integro-differential equations (Eq. 32). The local-equivalent potential is quite well described by Eq. 34 for any ℓ
value, except at very small distances (R ≈ 0) that are not probed by heavy-ion experiments.
Expression 34 has accounted for the energy-dependence of experimentally extracted potential strengths for several
systems in a very large energy range [12–14]. At near-barrier energies, E ≈ VC(RB) + VLE(RB), the effect of the
Pauli nonlocality is negligible and VLE(R,E) ≈ VF(R), but the higher the energy is, the greater is the effect. At
energies about 200 MeV/nucleon the local-equivalent potential is about 1 order of magnitude less intense than the
corresponding folding potential (see examples in Refs. [12,13]). In a classical physics framework, the exponent in Eq.
34 is related to the kinetic energy (Ek) and to the local relative speed between the nuclei (v) by
v2=2
µEk(R) =2
µ[E − VC(R) − VLE(R,E)] ;(35)
and Eq. 34 may be rewritten in the following form
VLE(R,E) ≈ VF(R) e−[m0b0v/(2¯ h)]2≈ VF(R) e−4v2/c2,(36)
where c is the speed of light. Therefore, in this context the effect of the Pauli nonlocality is equivalent to a velocity-
dependent nuclear interaction (Eq. 36). Another possible interpretation is that the local-equivalent potential may
be associated directly with the folding potential (Eq. 37), with an effective nucleon-nucleon interaction (Eq. 38)
dependent on the relative speed (v) between the nucleons
VLE(R,E) = VF=
?
ρ1(r1) ρ2(r2) vNN(v,?R − ? r1+ ? r2) d? r1d? r2, (37)
vNN(v,? r) = vf(? r) e−4v2/c2.(38)
5. The systematization of the nuclear potential
As already mentioned, the angular distribution for elastic scattering provides an unambiguous determination of the
real part of the optical potential in a region around the sensitivity radius (RS). For bombarding energies above (and
near) the barrier, the sensitivity radius is rather energy-independent and close to the barrier radius (RB), while at
intermediate energies much inner distances are probed. At sub-barrier energies, the RSis strongly energy-dependent,
with its variation connected to the classical turning point; this fact has allowed the determination of the potential in
a wide range of near-barrier distances, RB≤ RS≤ RB+2 fm. With the aim of avoiding ambiguities in the potential
systematization, we have selected “experimental” (extracted from elastic scattering data analyses) potential strengths
at the corresponding sensitivity radii, from works in which the RS has been determined or at least estimated. In
several articles, the authors claim that their data analyses at intermediate energies have unambiguously determined
the nuclear potential in a quite extensive region of interaction distances. In such cases, we have considered potential
strength “data” in steps of 1 fm over the whole probed region. Tables 1 and 2 provide the systems included in the
nuclear potential systematics for the sub-barrier and intermediate energies, respectively. For the energy region above
(and near) the barrier, the present systematics contains potential strengths for a large number of different heavy-ion
systems from the previous Christensen and Winther’s systematization [7]. Our systematics is not even near to being
complete, but it is rather extensive and diversified enough to account well for the very large number of data that have
been obtained in the last decades.
The experimental potential strengths represent the real part of the optical potential, which corresponds to the
addition of the bare and polarization potentials. The contribution of the polarization to the optical potential depends
on the particular features of the reaction channels involved in the collision, and is therefore quite system-dependent.
If this contribution were very significant, it would be too difficult for one to set a global description of the heavy-
ion nuclear interaction. In the present work, we neglect the real part of the polarization potential and associate the
experimental potential strengths (VExp) with the bare interaction (VLE). The success of our findings seems to support
such a hypothesis.
In analysing experimental potential results for such a wide energy range and large number of different systems,
we consider quite appropriate the use of system- and energy-independent quantities. We have removed the energy-
dependence from the experimental potential strengths through the calculation of the corresponding folding potential
strengths, VF−Exp, based on Eq. 34. The system-dependence of the potential data set has then been removed with
the use of the experimental reduced potential, Vred−Exp. For s ≥ 0 this quantity was calculated from Eq. 22, and for
inner s values we have adopted the following simple definition
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Vred−Exp= V0
VF−Exp
VF−Teo
,(39)
with VF−Teocalculated through Eq. 8. The other useful quantity is the distance between surfaces: s = RS−(R1+R2),
where RSis associated to the sensitivity radius, and the radii of the nuclei have been obtained from Eq. 6.
In Fig. 10 (top), the experimental reduced potential strengths are confronted with the theoretical prediction (Eqs.
23 and 24) for different diffuseness values. The fit to the data in the inner region (s ≤ 0) results unambiguously in the
value V0= −456 MeV fm3, and is quite insensitive to the diffuseness parameter, in agreement with the discussion
about the folding features of Section 3. The fit for s ≥ 0 is sensitive to both: V0and a, and the corresponding best fit
values are a = 0.56 fm and the same V0found for the inner region. The standard deviation of the data set around
the best fit (solid line in Fig. 10 - top) is 25%, a value somewhat greater than the dispersion (20%) expected to
arise from effects of the structure of the nuclei (as discussed in Section 3). We believe that the remaining difference
comes from two sources: uncertainties of the experimentally extracted potential strengths and the contribution of
the polarization potential that we have neglected in our analysis. We point out that the best fit diffuseness value,
a = 0.56 fm, is equal to the average diffuseness found (Section 2) for the matter distributions and greater than the
average value (a = 0.50 fm) of the nucleon distributions. This is a consistent result because we have calculated
the reduced potential strengths based on the zero-range approach (through Eqs. 8, 22, 23 and 24). As discussed
in Section 3, the effect of a finite-range for the effective nucleon-nucleon interaction can be simulated, within the
zero-range approach, by increasing the diffuseness of the (nucleon) densities of the nuclei. This subject is dealt with
more deeply in the next Section.
In order to characterize the importance of the Pauli nonlocality, in Fig. 10 (bottom) are shown the results for the
reduced potential through calculations performed without the correction (Eq. 34) due to the energy-dependence of
the LE potential, i.e. associating the experimental potential strengths directly with the folding potential. The quality
of the corresponding fit (Fig. 10 - bottom) is similar to that obtained with the nonlocality (Fig. 10 - top), but the V0
and a parameters are significantly different. In the next Section, we show that the values found without considering
the nonlocality, a = 0.61 fm and V0= −274 MeV fm3, seem to result in an unrealistic nucleon-nucleon interaction.
6. The effective nucleon-nucleon interaction
After removing the energy-dependence of the experimental potential strengths, the corresponding results are com-
patible with the double-folding potential in the zero-range approach (Eq. 8), provided that the matter densities
of the nuclei be adopted in the folding procedure instead of the nucleon densities. In this section, we study the
consistency of our results for the nuclear potential in the case that the double-folding model is treated in the more
common interpretation: the nucleon distributions and a finite-range nucleon-nucleon interaction are assumed in Eq.
7. With the purpose of keeping the comparison between experimental and theoretical results through the use of
system-independent quantities, it is necessary to change the definition of the experimental reduced potential
Vred−Exp= Vred−Teo
VF−Exp
VF−Teo
, (40)
where VF−Teois now calculated through Eq. 7. Vred−Teois still obtained from Eqs. 23 and 24, with the V0parameter
being associated to the volume integral of the effective nucleon-nucleon interaction (actually, this same procedure has
also been adopted in the zero-range case)
V0= 4π
?
vNN(r) r2dr . (41)
The effective nucleon-nucleon interaction should be based upon a realistic nucleon-nucleon force, since our goal is to
obtain a unified description of the nucleon-nucleon, nucleon-nucleus and nucleus-nucleus scattering (a discussion about
the “realism” of the interaction is found in Refs. [1,6]). For instance, a realistic interaction should match the empirical
values for the volume integral and root-mean-square radius of the nucleon-nucleon interaction, V0≈ −430 MeV fm3
and rrms≈ 1.5 fm, that were extrapolated from the main features of the optical potential for the nucleon-nucleus
scattering at Enucleon= 10 MeV [1,46–48]. The M3Y interaction has been derived [1] with basis on the G-matrix
for two nucleons bound near the Fermi surface, and certainly is representative of realistic interactions. In table 3 are
presented the volume integral and root-mean-square radius for several nucleon-nucleon interactions used in this work,
including the M3Y at 10 MeV/nucleon.
The M3Y interaction is not truly appropriate for use in the context of the nonlocal model, because it already contains
a simulation of the exchange effects included in its knock-on term. Furthermore, according to the nonlocal model the
Page 9
energy-dependence of the local-equivalent potential should be related only to the finite range of the Pauli nonlocality,
but the knock-on exchange term in the M3Y interaction is also energy-dependent. Therefore, the use of the M3Y in
the nonlocal model would imply a double counting of the energy-dependence that arises from exchange effects. In
Section 4, we have demonstrated that the LE potential is identical with the double-folding potential for energies near
the barrier, which are in a region around 10 MeV/nucleon. In this same energy range, the folding potential with the
M3Y interaction have provided a very good description of elastic scattering data for several heavy-ion systems [1].
Thus, we believe that an appropriate nucleon-nucleon interaction for the nonlocal model could be the M3Y “frozen”
at 10 MeV/nucleon [13], i.e. considering the parameters of the Reid and Paris versions as energy-independent values.
Fig. 11 (top) shows a comparison between experimental and theoretical heavy-ion reduced potentials, in which the
“frozen” M3Y-Reid was considered for the nucleon-nucleon interaction. We emphasize that no adjustable parameter
has been used in these calculations, but even so a good agreement between data and theoretical prediction has been
obtained. The “frozen” M3Y-Paris provides similar results.
With the aim of investigating how much information about the effective nucleon-nucleon interaction can be ex-
tracted from our heavy-ion potential systematics, we have considered other possible functional forms for this effective
interaction. Besides the Yukawa function (Eq. 25), we have also used the Gaussian (Eq. 42) and the exponential (Eq.
43), which reduce to the tri-dimensional delta function in the limit σ → 0,
e−r2/2σ2
(2π)3/2σ3,
Gσ(r) = V0
(42)
Eσ(r) = V0
e−r/σ
8πσ3. (43)
The fits obtained with all these functions are of similar quality and comparable with that for the M3Y interaction
(Fig. 11 - top). The resulting best fit widths (σ), volume integrals and corresponding root-mean-square radii are
found in table 3. All the V0and rrmsvalues, including those of the M3Y, are quite similar. Also the “experimentally”
extracted intensity of the nucleon-nucleon interaction in the region 1 ≤ r ≤ 3 fm seems to be rather independent of
the model assumed for this interaction (see Fig. 12).
In Section 5, we have demonstrated that the major part of the “finite-range” of the heavy-ion nuclear potential is
related only to the spatial extent of the nuclei. In fact, even considering a zero-range for the interaction vNN in Eq.
8, the shape of the heavy-ion potential could be well described just by folding the matter densities of the two nuclei.
One would ask whether the finite-range shape of the effective nucleon-nucleon interaction can be derived in a similar
way. Thus, we have considered a folding-type effective nucleon-nucleon interaction built from:
vNN(? r) ≈ vf(r) =
?
ρm(r1) ρm(r2) V0δ(?R − ? r1+ ? r2) d? r1d? r2=
2πV0
r
?∞
0
r1ρm(r1)
??r+r1
|r−r1|
r2ρm(r2) dr2
?
dr1,
(44)
where V0= −456 MeV fm3as determined by the heavy-ion potential analysis, and ρmis the matter density of the
nucleon. Based on the intrinsic charge distribution of the proton in free space, which has been determined by electron
scattering experiments, we have assumed an exponential shape for the matter density of the nucleon
ρm(r) = ρ0e−r/am. (45)
Of course, ρ0and amare connected by the normalization condition, Eq. 2. The integration of Eq. 44 results in
vf(r) = V0π a3
mρ2
0e−r/am
?
1 +
r
am
+
r2
3a2
m
?
.(46)
With this finite-range folding-type effective nucleon-nucleon interaction, a good fit of the reduced heavy-ion potential
strengths is obtained (see Fig. 11 - bottom), with realistic volume integral and root-mean-square radius (see table 3).
The folding-type interaction is quite similar to both versions of the M3Y interaction in the surface region (see Fig.
12).
The folding-type interaction in the context of the nonlocal model provides a very interesting unification between
the descriptions of the nucleus-nucleus, nucleon-nucleus and effective nucleon-nucleon interactions.
appreciated through the comparison between Eqs. 36 and 38, with the subtle detail that VF (in Eq. 36) and vf (in
Eq. 38) can both be calculated by folding the matter densities in the zero-range approach, and with the same V0
value. Therefore, the interaction between two nuclei (or nucleons) can be obtained from
This can be
Page 10
VLE(R) =
?
ρ1(r1) ρ2(r2) V0δ(?R − ? r1+ ? r2) e−4v2/c2d? r1d? r2
(47)
where V0= −456 MeV fm3, ρiare the matter densities, and v is the relative speed between the nuclei (or nucleons).
An alternative way to calculate the heavy-ion interaction is with the Eq. 37 (and 38), but in this case the nucleon
distributions must be used (in Eq. 37) instead of the matter densities. All these findings seems to be quite consistent.
However, the best fit value obtained for the diffuseness (am= 0.30 fm) of the matter density of the nucleon inside
the nucleus is considerable greater than that (achp= 0.235 fm) found for the charge distribution of the proton in
free space. This finding is consistent with the swelling of the nucleon observed in the EMC effect [49], but should be
contrasted with the opposite picture of a smaller nucleon inside the nucleus as advanced within the concept of color
transparency [50].
Finally, we mention that, if the energy-dependence of the Pauli nonlocality is not taken into account and the
experimental potential strengths are associated directly with the folding potential, our calculations indicate that the
corresponding effective nucleon-nucleon interaction should have the following unrealistic values: V0≈ −270 MeV fm3
and rrms≈ 1.9 fm.
7. Conclusion
The experimental potential strengths considered in the present systematics have been obtained at the corresponding
sensitivity radii, a region where the nuclear potential is determined from the data analyses with the smallest degree
of ambiguity. The Fermi distribution was assumed to represent the nuclear densities, with parameters consistent with
an extensive amount of theoretical (DHB calculations) and experimental (electron scattering experiments) results.
The potential data set is well described in the context of the nonlocal model, by the double-folding potential in the
zero-range as well as in the finite-range approaches. The dispersion of the potential data around the theoretical
prediction is 25%, which is compatible with the expected effects arising from the variation of the densities due to the
structure of the nuclei. If the nonlocal interaction is assumed, the heavy-ion potential data set seems to determine a
few characteristics of the effective nucleon-nucleon interaction, such as volume integral and root-mean-square radius,
in a model-independent way.
The description of the bare potential presented in this work is based only on two fundamental ideas: the folding
model and the Pauli nonlocality. We have avoided as much as possible the use of adjustable parameters, and in
the case of the “frozen” M3Y interaction no adjustable parameters were necessary to fit the experimental potential
strengths. Nowadays, the other important part of the heavy-ion interaction, the polarization potential, is commonly
treated within a phenomenological approach, with several adjustable parameters which usually are energy-dependent
and vary significantly from system to system. The association of the nonlocal bare potential presented in this work
with a more fundamental treatment of the polarization should be the next step toward a global description of the
nucleus-nucleus interaction.
This work was partially supported by Financiadora de Estudos e Projetos (FINEP), Funda¸ c˜ ao de Amparo ` a Pesquisa
do Estado de S˜ ao Paulo (FAPESP), and Conselho Nacional de Desenvolvimento Cient´ ıfico e Tecnol´ ogico (CNPq).
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Page 12
Table 1: The table presents the systems, sub-barrier bombarding energies, and corresponding references, that have
been included in the nuclear potential systematics.
System
16O +58Ni
16O +60Ni
16O +62,64Ni
16O +88Sr
16O +90Zr
16O +92Zr
16O +92Mo
16O +120Sn
16O +138Ba
16O +208Pb
18O +58Ni
18O +60Ni
ELab(MeV )
35, 35.5, 36, 36.5, 37, 38 [8,31]
35, 35.5, 36, 37, 38
34, 35, 36
43, 44, 45
46, 47, 48
45, 46, 47, 48
48, 48.5, 49
53, 54, 55
54, 55, 56, 57
74, 75, 76, 77, 78
35.1, 35.5, 37.1, 38
34.5, 35.5, 37.1, 38
Reference
[30,31]
[31]
[32]
[32]
[8,32]
[32]
[8]
[8]
[8]
[33]
[33]
Table 2: The same of table 1, but for intermediate energies.
SystemELab(MeV )
30.3
52
104
210, 318
210
350
Reference
[34]
[34]
[34]
[35,36]
[37]
[38]
p +40Ca,208Pb
d +40Ca,208Pb
4He +40Ca,208Pb
6Li +12C,28Si
6Li +40Ca,58Ni,90Zr,208Pb
7Li +12C,28Si
12C +12C
12C +208Pb
13C +208Pb
16O +16O
16O +12C,28Si,40Ca,90Zr,208Pb
40Ar +60Ni,120Sn,208Pb
300, 360, 1016, 1440, 2400 [39–41]
1440
390
250, 350, 480, 704, 1120
1504
1760
[41]
[40]
[43,44]
[42]
[45]
Table 3: The width, volume integral and root-mean-square radius for several effective nucleon-nucleon interactions
considered in this work.
Interaction σ or am(fm) V0(MeV fm3) rrms(fm)
M3Y-Reid-
M3Y-Paris-
Yukawa0.58
Gaussian 0.90
Exponential0.43
Folding-type0.30
- 408
- 447
- 439
- 448
- 443
- 456
1.62
1.60
1.42
1.56
1.49
1.47
Page 13
FIG. 1. Nucleon density for the
two-parameter Fermi distribution (2pF), with a = 0.5 fm and R0 = 4.17 fm. The small difference between the 2pF distribution
and the function ρ0 C?r−R0
56Fe nucleus represented through Dirac-Hartree-Bogoliubov calculations (DHB) and a
a
?
(Eqs. 12, 13 and 14) is hardly seen in the figure.
Page 14
FIG. 2. Equivalent diffuseness values obtained for charge distributions extracted from electron scattering experiments and
for theoretical densities obtained from Dirac-Hartree-Bogoliubov calculations.
Page 15
FIG. 3. The R0parameter obtained for charge distributions extracted from electron scattering experiments and for theoretical
densities obtained from Dirac-Hartree-Bogoliubov calculations.
Page 16
FIG. 4. Nucleon densities from Dirac-Hartree-Bogoliubov calculations (solid lines) compared with the corresponding
two-parameter Fermi distributions (dashed lines), with a = 0.50 fm and R0 obtained through Eq. 6.
Page 17
FIG. 5. Folding potential for different sets of 2pF densities that may represent the16O +58Ni system. The approximate
position of the s-wave barrier radius (RB) is indicated in the figure.
Page 18
FIG. 6. Folding potential in the zero-range approach calculated from numerical integration of Eq. 8 (solid line), for 2pF
densities that may represent the16O +58Ni system. The dashed line represents the approximate analytical expressions, Eqs.
15, 16 and 17, while the dotted line concerns the exact result for a = 0, Eqs. 9, 10 and 11. The approximate position of the
s-wave barrier radius (RB) and of the distance R = R1+ R2 (s = 0) are indicated in the figure.
Page 19
FIG. 7. Normalized folding potential VF/(V0R) in the zero-range approach (Eq.
s = R − (R1 + R2), for several sets of 2pF distributions (with a = 0.50 fm) that may represent the systems indicated in
the figure. The proximity universal function Φ is also presented in arbitrary units.
8) as a function of the distance
Page 20
FIG. 8. Double-folding potentials for 2pF distributions with different diffuseness values (a) that may represent the16O +
58Ni system. The potentials have been calculated in the zero-range approach (ZR) or with a finite-range (FR) Yukawa function
for the effective nucleon-nucleon interaction.
Page 21
FIG. 9. Double-folding (VF) and ℓ-dependent local-equivalent (VLE) potentials for the α +58Ni system at ELab= 139 MeV .
The solid line represents the approximate expression, Eq. 34, for the local-equivalent potential.
Page 22
FIG. 10. Experimental and theoretical reduced potentials in the context of the zero-range approach, with (top) or without
(bottom) considering in the calculations the energy-dependence of the local-equivalent potential (Eq. 34) that arises from the
Pauli nonlocality.
Page 23
FIG. 11. Comparison between experimental and theoretical reduced potentials in the context of the finite-range approach,
with a M3Y-Reid (top) or folding-type (bottom) effective nucleon-nucleon interaction.
Page 24
FIG. 12. The complete set of effective nucleon-nucleon interactions considered in this work.
Page 25
0123
r (fm)
456
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
−
r
a
R
C
o
o
ρ
••••
56Fe
ρ (fm
-3)
DHB
2pF
Page 26
0102030405060 708090
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
fmaC
53.0
=
a)
Z
a (fm)
Experimental Charge
0 10 20 30 405060 70 8090
fmaC
50.0
=
b)
Z
Theoretical Charge
0 20 40 6080 100120140
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Z
N
aP
aN
00083
00046
. 0
. 0
47
47
. 0
. 0
−
+
=
=
c)
a (fm)
Z or N
Proton
Neutron
020406080100120140160 180200220
fmaM
54.0
=
fmaN
48. 0
=
d)
A
Nucleon
Matter
Page 27
123456
0
1
2
3
4
5
6
7
8
84. 0
−
31 . 1
3 / 1
0
=
ARM
96. 0
−
76. 1
3/ 1
0
=
ZRC
79. 0
−
49 . 1
3 / 1
0
=
NRN
12. 1
−
81. 1
3/ 1
0
=
ZRP
a)
R0 (fm)
Z
1/3
Experimental Charge
123456
96. 0
−
76. 1
3/ 1
0
=
ZRC
b)
Z
1/3
Theoretical Charge
123456
0
1
2
3
4
5
6
7
8
c)
R0 (fm)
Z
1/3 or N
1/3
Proton
Neutron
123456
d)
A
1/3
Nucleon
Matter
Page 28
10
-2
10
-1
ρ (fm
-3)
16O
120Sn
0123456
10
-3
10
-2
10
-1
ρ (fm
-3)
r (fm)
56Fe
12345678910
r (fm)
208Pb
Page 29
0246810
10
-1
10
0
10
1
10
2
10
3
° ° ° ° °
......
R1=2.50 fm, R2=4.20 fm - a1=a2=0.55 fm
}
16O +
58Ni
RB
-VF (MeV)
R (fm)
R1=2.50 fm, R2=4.20 fm
R1=2.50 fm, R2=4.34 fm a1=a2=0.50 fm
R1=2.64 fm, R2=4.20 fm
Page 30
0246810
10
-1
10
0
10
1
10
2
S = 0
VF / (V0 ρ01 ρ02) (fm
3)
R1 = 2.46 fm
R2 = 4.23 fm
a1 = a2 = 0.50 fm
RB
16O +
58Ni
R (fm)
Page 31
-8-6-4-20246
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
Φ
VF / (V0 ?)
s (fm)
R1(fm) R2(fm) system
1.24 6.92
2.46 4.23
3.64 3.64
3.64 5.62
4He +
40Ca +
40Ca +
208Pb
58Ni
40Ca
120Sn
16O +
Page 32
024681012
10
-1
10
0
10
1
10
2
10
3
·····
RB
R1 = 2.5 fm
R2 = 4.2 fm
16O +
58Ni
FR
ZR{
- VF (MeV)
R (fm)
a = 0.50 fm
a = 0.55 fm
σ = 0.50 fm , a = 0.50 fm
Page 33
02468
0
50
100
150
200
250
300
α +
ELab= 139 MeV
58Ni
{
VLE
??
= 30
?
- V (MeV)
R (fm)
= 0
VF
Page 34
10
0
10
1
10
2
10
3
-8-6 -4-2
s (fm)
0246
10
-1
10
0
10
1
10
2
V0 = - 456 MeV fm
3
a = 0.56 fm
a = 0.50 fm
-Vred (MeV fm
3)
Sub-barrier energies
Above- and near-barrier
Intermediate energies
a = 0.61 fm
a = 0.50 fm
V0 = - 274 MeV fm
3
-Vred (MeV fm
3)
Page 35
10
0
10
1
10
2
10
3
-8-6-4-2
s (fm)
0246
10
-1
10
0
10
1
10
2
M3Y - Reid
-Vred (MeV fm
3)
Sub-barrier energies
Above- and near-barrier
Intermediate energies
Folding-type
-Vred (MeV fm
3)
Page 36
01234
10
-2
10
-1
10
0
10
1
10
2
-vNN (MeV)
r (fm)
M3Y - Reid
M3Y - Paris
Yukawa
Gaussian
Exponential
Folding-type
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