# Toward a global description of the nucleus-nucleus interaction

**ABSTRACT** Extensive systematization of theoretical and experimental nuclear densities and of optical potential strengths exctracted 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 indicate 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 poissibility of extracting information about the nucleon-nucleon interaction from the heavy-ion potential is investigated. Comment: 12 pages,12 figures

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- E. Williams, D. J. Hinde, M. Dasgupta, R. du Rietz, I. P. Carter, M. Evers, D. H. Luong, S. D. McNeil, D. C. Rafferty, K. Ramachandran, A. Wakhle[Show abstract] [Hide abstract]

**ABSTRACT:**Background: Quasifission, a fission-like reaction outcome in which no compound nucleus forms, is an important competitor to fusion in reactions leading to superheavy elements. The precise mechanisms driving the competition between quasifission and fusion are not well understood.Purpose: To understand the influence reaction parameters have on quasifission probabilities, an investigation into the evolution of quasifission signatures as a function of entrance channel parameters is required.Methods: Using the Australian National University's (ANU) CUBE detector for two-body fission studies, measurements were made for a wide range of reactions forming isotopes of curium. Important quasifission signatures—namely, mass-ratio spectra, mass-angle distributions, and angular anisotropies—were extracted.Results: Evidence of quasifission was observed in all reactions, even for those using the lightest projectile (12C+232Th). But the observables showing evidence of quasifission were not the same for all reactions. In all cases, mass distributions provided some evidence of the possible presence of quasifission but were not sufficient in most cases to clearly identify reactions for which quasifission was important. For reactions using light projectiles (12C, 28,30Si, 32S), experimental angular anisotropies provided the clearest signature of quasifission. For reactions using heavier projectiles (48Ti, 64Ni), the presence of mass-angle correlations in the mass-angle distributions provided strong evidence of quasifission and also provided information about quasifission timescales.Conclusions: The observable offering the clearest signature of quasifission differs depending on the reaction timescale.Physical Review C 09/2013; · 3.72 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We have analyzed the backward angle quasi-elastic excitation function and the barrier distribution for the 7Li + 144Sm system by considering not only the direct breakup of the projectile, the inelastic excitations of the target and the one-neutron transfer channel, but also the sequential breakup of 7Li as a two-step process: the stripping of one neutron followed by the breakup of 6Li. The agreement of the theoretical calculations with the experimental barrier distribution is good, even without any fit procedure. This result confirms some recent experimental evidences showing the importance of this two-step process to 7Li breakup.Journal of Physics G Nuclear and Particle Physics 12/2013; 40(12):5105-. · 5.33 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The 12C + 12C fusion reaction is investigated in a multichannel folding model, using the density-dependent DDM3Y nucleon-nucleon interaction. The 12C(01+,2+,02+,3-) states are included, and their densities are taken from a microscopic cluster calculation. Absorption to fusion channels is simulated by a short-range imaginary potential, and the model does not contain any fitting parameter. We compute elastic and fusion cross sections simultaneously. The role of 12C + 12C inelastic channels, and in particular of the 12C(01+) + 12C(02+) channel involving the Hoyle state, is important even at low energies. In the Gamow region, the energy range relevant in astrophysics, inelastic channels increase the S factor by a factor of three.Physics Letters B 06/2013; 723(4):355-359. · 4.57 Impact Factor

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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

Page 2

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

Page 3

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

Page 4

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:

Page 5

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

Page 6

−¯ 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

Page 7

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

Page 8

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

Gaussian0.90

Exponential 0.43

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.

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