1. INTRODUCTION

In the last decades, considerable efforts have been made to analyze nuclear reactions
induced by weakly-bound nuclei (^{Canto, Gomes,
Donangelo, & Hussein, 2006}; ^{Canto,
Gomes, Donangelo, Lubian, & Hussein, 2015}; ^{Gomes, Lubian, Padron, & Anjos, 2005}; ^{Kolata & Aguilera, 2009}). There are different important
dynamic and static properties of weakly-bound nuclei that can strongly affect the
fusion mechanism of such systems (^{Bush, AlKhalili,
Tostevin, & Johnson, 1996}; ^{Lubian et
al., 2007}; ^{Gomes, Lubian, & Canto,
2009}; ^{Shrivastava et al., 2013};
^{Lukyanov et al., 2017}). Compared to the
tightly-bound nuclei, the low separation energy of weakly-bound nuclei increases the
probability of breakup, which is a typical dynamic feature in the collision of these
systems. The static property of weakly-bound nuclei also makes the study of such
systems different than that of the tightly-bound systems. A longer tail density
distribution of a weakly-bound projectile can affect the bare potential and
therefore the reaction mechanism. Considering the difference in the predictions of
density distributions of a weakly-bound nucleus, it is of great interest to probe
the sensitivity of the respective theoretical calculations to the variations of the
density distribution. For this purpose, the 6Li nucleus is selected to investigate
the importance of the density distribution in the analysis of weakly-bound
nuclei.

The ^{6}Li nucleus is one of the weakly-bound projectiles which has
attracted much attention in the last years (^{Kukulin,
Krasnopol´sky, Voronchev, & Sazonov, 1984}; ^{Otomar, Gomes, Lubian, Canto, & Hussein, 2013}). Considering
the exotic structure of ^{6}Li, i.e., the extended density distribution and
a cluster nature (α + d) with a break-up threshold energy of 1.48 MeV, the nuclear
reactions induced by this stable, weakly-bound projectile have been the subject of
many experimental and theoretical studies (^{Beck,
Keeley, & Diaz-Torres, 2007}; ^{Shrivastava et al., 2009}; ^{Luong et al.,
2013}; ^{Aguilera, Martinez-Quiroz et al.,
2017}; ^{Camacho et al., 2010}; ^{So, Udagawa, Kim, Hong, & Kim, 2007}). The
present work is devoted to a comparative study done by using different density
distributions for ^{6}Li in the optical-model (OM) analyses of
^{6}Li + ^{58}Ni. In these calculations, we probe the effect of
variations of the diffuseness parameter of ^{6}Li on the predictions of the
recently measured fusion cross sections for the ^{6}Li + ^{58}Ni
system (^{Aguilera, Martinez-Quiroz et al.,
2017}). Toward this goal, we calculate three bare potentials for
^{6}Li + ^{58}Ni by the use of different two-parameter Fermi (2pF)
density distributions, which include different diffuseness parameters, for
^{6}Li. Employing these bare potentials in the OM, we derive the
appropriate polarization potentials and calculate the fusion cross sections for the
selected system. The obtained results are used to discuss the sensitivity of the OM
analysis to the density distributions of a weakly-bound projectile.

In the next section, we describe the optical model potentials used, discussing in
particular the bare potentials employed in the analysis of ^{6}Li +
^{58}Ni. The results and discussion are presented in Section 3 and
finally the conclusions are drawn in Section 4.

2. THEORETICAL OUTLINE

To perform the OM analysis for the nuclear reactions, the total potential can be described in the following form

projectile and target nuclei and V_{Coul} is the Coulomb potential with
*r*
_{
C
} = 1.2 fm. The strong absorption after ions pass through the Coulomb barrier
is described by adding a short-range imaginary potential, (*i*W_{
int
} ), to the real part of the interaction. This imaginary potential has a volume
Woods-Saxon shape with the depth, radius and diffuseness of *W*
_{0} = 50 MeV, *r*
_{0} = 1.0 fm, *a*
_{0} = 0.2 fm, respectively. In the OM, the effect of breakup on the fusion
analysis has been taken into account, in a phenomenological way, by including the
dynamic polarization potentials, i.e., *U*
_{
D
} (E) and *U*
_{
F
} (E), which account for couplings of the elastic channel to direct-reaction
and fusion channels, respectively (^{So, Udagawa, Kim,
Hong, & Kim, 2007}; ^{Kim, So, Hong,
& Udagawa, 2002b}). The polarization potentials are composed of real
and imaginary parts as

for which, the real and imaginary parts should satisfy the dispersion relation
expressed by (^{Mahaux, Ngo, & Satchler,
1986}; ^{Nagarajan, Mahaux, & Satchler,
1985}; ^{Satchler & Love,
1979}).

In this equation, V (*E*
_{
S
} ) shows the value of V (*E*) at the reference energy
*E* = *E*
_{
S
} and *P* stands for the principal value.

The real and imaginary parts of the direct-reaction polarization potential,
*U*
_{
D
} , are included in the OM analysis by using the energy-dependent form as

where

Here, *r*
_{
D
} and *a*
_{
D
} are the reduced radius and diffuseness, respectively. These parameters were
obtained by using the experimental elastic scattering angular distributions of
^{6}Li + ^{58}Ni (^{Aguilera et
al., 2009}) and the method employed in Ref. (^{Aguilera, Kolata et al., 2017}). The values found in this way are
*r*
_{
D
} =1.61 fm and *a*
_{
D
} =0.68 fm. In addition to U_{D}, the energy-dependent form of the
fusion polarization potential is also included in the OM with the following form

where _{F} = 1.4 fm and a_{F} = 0.43 fm (^{So et al., 2007}; ^{Kim et al.,
2002b}; ^{Kim, So, Hong, & Udagawa,
2002a}).

Using the total potential in the optical-model analysis, the fusion cross sections
can be obtained from the following expression (^{Udagawa & Tamura, 1984})

where *χ*
^{(+)} represents the distorted wave function that can satisfy the
Schrodinger equation with *U*
_{
TOT
} .

The density distribution plays a major role in the calculations of the bare
potential. Therefore, considering the extended density distribution of the
weakly-bound nucleus, the OM analyses of ^{6}Li + ^{58}Ni are
carried out through the use of bare potentials obtained from different density
distributions of ^{6}Li.

*2.1 BARE POTENTIALS USED IN THE OM CALCULATIONS*

In order to calculate the bare potential, use was made of the parameter-free Sao
Paulo potential (SPP) which is given by ^{Chamon
et al. (2002)}.

Here *v* defines the relative velocity of the colliding ions,
*c* is the speed of light, and *V*
_{
fold
} is obtained from

In this equation, *V*
_{0} = -456 MeV fm^{3} and *ρ*(*r*
_{1}) and *ρ*(*r*
_{2}) represent the matter densities of the colliding nuclei described
by the two-parameter Fermi density distribution as

The diffuseness and radius are *a* = 0.56 fm and
*R*
_{0} = 1.31*A*
^{
1/3
} - 0.84 fm extracted from a systematic study of densities (^{Chamon et al., 2002}), and
*ρ*
_{0} is determined from normalization to the nucleus mass. In addition
to the diffuseness mentioned, to examine the effect of density variation of
^{6}Li on the OM analysis of ^{6}Li + ^{58}Ni, we
selected two other diffuseness values for the density distribution of
^{6}Li as well. Considering the extended density distribution of
^{6}Li and according to the studies done on its density distribution
(^{Antonov et al., 2005}; ^{Dobrovolsky et al., 2002}), we also used the
2pF density distributions with diffuseness values of 0.59 and 0.63 fm for this
nucleus. Using each diffuseness mentioned, i.e., *a* = 0.56,
0.59, and 0.63 fm, for the ^{6}Li density, we calculated the respective
bare potential of the selected system. Each *V*
_{
bare
} obtained was employed in the OM and the respective appropriate
polarization potentials were estimated for the system by using the FRESCO code
and its search version, SFRESCO (^{Thompson,
1988}).

3. RESULTS AND DISCUSSIONS

The OM analyses and calculations of the polarization potentials for
^{6}Li + ^{58}Ni were done in two steps. First, we used SFRESCO
to find the real and imaginary parts of the direct-reaction polarization
potential by considering *U*
_{
F
} =0. Based on each of the bare nuclear potentials obtained from different
density distributions of ^{6}Li, we carried out a *χ*
^{2} fitting to the elastic scattering data of ^{6}Li +
^{58}Ni (^{Aguilera et al.,
2009}) and extracted the strength parameters of *U*
_{
D
} , i.e., *V*
_{
D0
} and *W*
_{
D0
} . The values calculated by minimizing the *χ*
^{2}
*/N* in each of the analyses are shown in Fig. 1 (a-c). The dashed curves in this figure demonstrate
the results of the dispersion relation, Eq. (3), when the linear approximation
shown by the solid line is used for *W*
_{
D0
} (*E*). As it is seen, the depths found based on all three
bare potentials can well satisfy the dispersion relation. From Fig. 1 it is also clear that the use of
different bare potentials in the OM analyses leads to different sets of values
for the real part of *U*
_{
D
} . More specifically, the OM analysis made by the bare potential derived
from the more extended density distribution (*a*=0.63 fm) results
in more negative values for the real part of *U*
_{
D
} . The physical interpretation is that the more diffuse

3. RESULTS AND DISCUSSIONS

The OM analyses and calculations of the polarization potentials for 6Li + 58Ni were
done in two steps. First, we used SFRESCO to find the real and imaginary parts of
the direct-reaction polarization potential by considering UF =0. Based on each of
the bare nuclear potentials obtained from different density distributions of 6Li, we
carried out a χ2 fitting to the elastic scattering data of 6Li + 58Ni ^{(Aguilera et al., 2009)} and extracted the
strength parameters of UD, i.e., VD0 and WD0. The values calculated by minimizing
the χ2/N in each of the analyses are shown in Fig. 1
(a-c). The dashed curves in this figure demonstrate the results of the
dispersion relation, Eq. (3), when the linear approximation shown by the solid line
is used for WD0(E). As it is seen, the depths found based on all three bare
potentials can well satisfy the dispersion relation. From Fig. 1 it is also clear that the use of different bare
potentials in the OM analyses leads to different sets of values for the real part of
UD. More specifically, the OM analysis made by the bare potential derived from the
more extended density distribution (a=0.63 fm) results in more negative values for
the real part of UD. The physical interpretation is that the more diffuse densities
actually extend the influence of the attractive nuclear forces to longer distances,
which tends to be compensated by more repulsive polarization potentials. The bare
potentials and direct reaction polarization potentials obtained here are used in the
second step χ2 analysis to calculate the fusion polarization potential for the 6Li +
58Ni system.

Three bare potentials and their corresponding *U*
_{
D
} were independently used in the optical model and for each of them the real
and imaginary depths of fusion polarization potential, *V*
_{
F0
} and *W*
_{
F0
} , were extracted from the simultaneous *χ*
^{2} fitting to the fusion and elastic scattering data of ^{6}Li +
^{58}Ni (^{Aguilera, Martinez-Quiroz et
al., 2017}, ^{Aguilera et al., 2009}).
In these calculations, we kept *U*
_{
D
} found in Fig. 1 (a-c) fixed, and treated
the real and imaginary depths of *U*
_{
F
} as adjustable parameters. The results showed that, independent of the
*V*
_{
bare
} + *U*
_{
D
} used in the OM analyses, identical values for *V*
_{
F0
} and *W*
_{
F0
} are obtained. The values extracted are shown in Fig. 2. The result of Eq. (3) using the linear approximation for
*W*
_{
F0
} is demonstrated by the dashed curve in this figure. As it is seen, the
*V*
_{
F0
} and *W*
_{
F0
} values obtained for *U*
_{
F
} can also satisfy the dispersion relation.

The resulting values of *U*
_{
D
} and *U*
_{
F
} , together with the respective bare potentials, were used to calculate the
fusion cross sections of the ^{6}Li + ^{58}Ni system in the OM. The
theoretical values obtained are compared with the experimental data in Fig. 3. As it is observed, three different OM
calculations done by using the bare potentials derived from different density
distributions and their corresponding polarization potentials, result in similar
predictions for the fusion cross sections of the weakly-bound system. As it is
clear, the cross sections found are in agreement with the experimental data (^{Aguilera, Martinez-Quiroz et al., 2017}) at
higher energies. One can also see that the theoretical results cannot reproduce the
experimental values at lowest energies. Similar to other ^{6}Li-induced
fusion reactions, such as ^{6}Li + ^{64}Zn (^{Di Pietro et al., 2013}), the disagreement found between the
theoretical values and the experimental ones may represent the possibility of
contributions of direct reaction channels in the fusion measurements of
^{6}Li + ^{58}Ni. This possibility has been discussed in Ref. (^{Aguilera, Martinez-Quiroz et al., 2017}).

4. CONCLUSIONS

In this study, we examined the sensitivity of the OM analysis of elastic scattering
and fusion data for ^{6}Li + ^{58}Ni to the density distribution of
^{6}Li. Different bare potentials for the selected system were
calculated by the use of three 2pF density distributions with different values of
the diffuseness parameter for ^{6}Li. Using the bare potentials obtained in
the *χ*
^{2} analyses of the data resulted in different strengths for the direct
polarization potential, but identical predictions for the fusion polarization
potential, and similar values for the fusion cross sections of ^{6}Li +
^{58}Ni. Considering the results obtained, one can conclude that the
density variations of the weakly-bound projectile in the OM analysis of
^{6}Li + ^{58}Ni can affect the description of the direct-reaction
polarization potential and make it more repulsive for more extended density
distributions, however, it cannot significantly influence the accuracy of the final
fusion results.