2.1 Phenomenological OM potential

The data on elastic scattering were firstly analyzed from the phenomenological point of view within the framework of the standard optical model of the nucleus, where the influence of inelastic channels is taken into account by introducing a phenomenological imaginary absorptive part in the interaction potential between the two colliding nuclei. In this model the elastic scattering is described by a complex interaction potential with a radial dependence in the form of WS. For both the real and imaginary parts of the potential, the WS shape is taken in addition to the Coulomb potential of a uniformly charged sphere. So, the utilized interaction potential can be written as:

with radius Ri=ri(AT1/3),i = V, W, C

Parameters of optical potential (OP) were selected to achieve the best agreement between theoretical and experimental angular distributions. The χ 2 value, which is the measure for the deviation of theoretical calculations from experimental measurements, is defined by:

where N is the number of data points. The quantities σ(θi)cal and σ(θi)exp are the calculated and experimental differential cross sections, while the quantity Δσ(θi) is the relative uncertainty in experimental data. The theoretical calculations as well as searching for the optimal potential parameters were performed using FRESCO and SFRESCO search code ^[33].

2.2 Sao Paulo potential

According to the different parameter ambiguities both discrete and continuous associated with the OM calculations, and the fact that phenomenological representations do not include a description of the projectile or target’s structure, the real part of potential was constructed using the microscopic double folding (DF) procedure extracted from the Sao Paulo potential (SPP) via the double convolution integral as described in Refs. ^[34,35,36].

where ρPrP and ρTrT, are the nuclear matter density distributions of ⁶Li and ²⁰⁸Pb nuclei, respectively, with V₀ = — 456 MeV.

In this model, the following two equations link the real part of the local-equivalent interaction to the DF potential V _F (R) as

where V is the local relative velocity between the two nuclei and C is the speed of light. The new version of the Sao Paulo potential (SPP2) was calculated using the REGINA code ^[37] with nuclear densities obtained from the Dirac-Hartree-Bogoliubov model ^[38].

2.3 Cluster folding potential

The ⁶Li + ²⁰⁸Pb elastic scattering data are analyzed from microscopic point of view using the cluster folding model (CFM). The importance of various break-up mechanisms in nuclear systems induced by ⁶Li is of special interest due to its very low binding energy and the dissociation into α (core) + d (valence). Based on the cluster nature of ⁶Li, we describe the ⁶Li + ²⁰⁸Pb elastic scattering angular distributions using CFM where, both the real and imaginary parts of potential are created based on cluster folding. The real and imaginary cluster folding parts of ⁶Li + ²⁰⁸Pb potential can be defined based on α + ²⁰⁸Pb and d + ²⁰⁸Pb potentials as:

where (Vα-208Pb,Vd-208Pb) and (Wα-208Pb, Wd-208Pb) are the real and imaginary parts of potentials for α + ²⁰⁸Pb and d + ²⁰⁸Pb channels which fairly reproduce the experimental data at the appropriate energies Ed  ≈1/3E _Li and Eα≈2/3E _Li taken from Refs. ^[39,40]. χαd(r) is the intercluster wave function for the relative motion of α and d in the ground state of ⁶Li, and r is the relative coordinate between the centres of mass of α and d. The α -d bound state form factor represents a 2S state in a real WS potential with V ₀=79.0 MeV, R=1.83 fm, a=0.7 fm ^[6] plus Coulomb potential. The main parameters required to prepare the cluster folding potential for ⁶Li + ²⁰⁸Pb are the optimal potentials for d + ²⁰⁸Pb and α + ²⁰⁸Pb a t appropriate energies. The highest energy under consideration is 210 MeV, so the required potentials are Vd-208  Pb at E _lab = 1/3 ( 201 = 70 MeV and Vα-208Pb at E _lab = 2/3 ( 201 = 120 MeV. By searching through the previous experimental studies for d + ²⁰⁸Pb and α + ²⁰⁸Pb nuclear systems, the most suitable potentials which could be used to generate the cluster folding potential for ⁶Li + ²⁰⁸Pb are: d + ²⁰⁸Pb at E _lab = 80 MeV ^[39] and α + ²⁰⁸Pb at E _lab = 139 MeV ^[40]. These potentials are used to generate the real and imaginary CF potentials expressed in Eqs. (6) and (7) as shown in Fig. 1.

Figure 1 Real and imaginary cluster folding potentials used in the current study.  

3.1 ⁶Li + ²⁰⁸Pb data analysis using phenomenological OM potential

Within the framework of OM, the available angular distributions for ⁶Li elastically scattered from ²⁰⁸Pb in the energy range 25 - 210 MeV ^{[1,2,3,4,5,6,7,8]} have been reanalyzed. The potential parameters considered by C. Fulmer et al. ^[5] for both the real and imaginary parts of potential are taken as starting parameters. The utilized central potential consists of Coulomb part as well as nuclear part of real and imaginary volume terms each of WS shape. In accordance with previous studies concerning nuclear processes induced by ⁶Li -projectile, the influence of spin orbit potential (V _so ) for ⁶Li is little and its effect can be excluded. Data are fitted using four varying parameters, namely, depth and diffuseness for the real (V ₀ and a _V ) and imaginary (W ₀ and a _W ) parts of the potential. Where, the radius parameters for the real (r _V ) and imaginary (r _W ) parts was fixed at 1.3 fm and 1.7 fm, respectively. The agreement between the experimental ⁶Li + ²⁰⁸Pb elastic scattering angular distributions and the theoretical calculations using OM is fairly good at the different considered energies as shown in Figs. 2-4. The extracted optimal OM parameters are listed in Table I, real (J _V ) and imaginary (J _W ) volume integrals as well as total reaction cross section σR values are also presented.

Figure 2 Comparison between ²⁰⁸Pb(⁶Li, ⁶Li) ²⁰⁸Pb elastic scattering angular distributions (solid circles) and OM (solid curves) fits at E _lab = 25, 29, 30, 31, and 33 MeV. The data are displaced by 0.5 for the sake of clarity.  

Figure 3 Same as Fig. 2 but at E _lab = 35, 36, 42, 43, 46 and 48 MeV. The data are displaced by 0.2 for the sake of clarity.  

Figure 4 Same as Fig. 2 but at E _lab = 7307, 88, 99, 156 and 210 MeV. The data are displaced by 0.2 for the sake of clarity.  

As shown from Figs. 2-4, ⁶Li + ²⁰⁸Pb data show unmistakable Coulomb rainbow phenomenon which results in the so called Fresenl peak. It is pronounced that the position of this peak is shifted toward smaller angles with increasing the bombarding energy. At lowest energy 25 MeV which is very close to Ec, this peak is not presented and it starts to be clearly appear at E > 33 MeV which is slightly above Ec.

3.2 ⁶Li + ²⁰⁸Pb data analysis using Sao Paulo potential

The ⁶Li + ²⁰⁸Pb elastic scattering angular distributions are then analyzed semi-microscopically using SPP. Two approaches were used: in the first approach, the real part of the potential was derived using SPP and the imaginary part was taken as a factor times the real SPP. In other words, the calculations were performed using two free parameters, namely, N_RSPP (renormaliztion factor for the real part created based on SPP) and N_{I SPP} (renormaliztion factor for the imaginary part). The total potential in this case has the following form:

As shown in Figs. 5-7, the agreement between the experimental ⁶Li + ²⁰⁸Pb angular distributions at energies 25 - 210 MeV and the calculations using real and imaginary SPP is fairly good. The optimal extracted potential parameters using this approach are listed in Table II, (J _v), (J_w), and σR values are also presented.

Figure 5 Comparison between ²⁰⁸Pb(⁶Li, ⁶Li) ²⁰⁸Pb elastic scattering angular distributions (solid circles) and SPP (solid curves) fits at E_lab= 25, 29, 30, 31, and 33 MeV. The data are displaced by 0.5 for the sake of clarity.  

Figure 6 Same as Fig. 5 but at E_lab = 35, 36, 42, 43, 46 and 48 MeV. The data are displaced by 0.2 for the sake of clarity.  

Figure 7. Same as Fig. 5 but at E_lab = 73.7, 88, 99, 156 and 210 MeV. The data are displaced by 0.2 for the sake of clarity.  

The extracted N_RSPP value at the different considered energies is close to each other with an average value 0.507 ± 0.05 except at energies 25, 43, and 46 MeV. At E = 25 MeV which is less than to E _C , the N _RSPP value is 0.927. At energies 43 and 46 MeV, an anomaly is observed where the N _RSPP value is 0.172, and 0.104 respectively. These results clearly show that, in order to reproduce the ⁶Li + ²⁰⁸Pb data, the strength of the real SPP should be reduced by about 49 %.

This reduction is essential to reproduce the experimental data for nuclear system induced by weakly bound nuclei such as ⁶Li, ⁷Li, and ⁹Be.

The energy dependence on the obtained values for both the real J_V and imaginary J_W volume integrals from SPP calculations is illustrated in Fig. 8, which is based on the obtained values listed in Table II. The behavior deduced was fitted using the following equation:

Figure 8 Energy dependence on volume integral for ⁶Li + ²⁰⁸Pb nuclear system.  

With a = 269.09(49.1), b = -1186.6(21964.57), c = 303905.12(-489825.38) for real and imaginary volume integral respectively.

In the second approach, the real part of the potential was derived using SPP exactly as in the first approach in addition to an imaginary volume part in WS form to simulate the reduction in flux due to absorption, the radius parameter for the imaginary volume term was fixed at 1.7 fm similar to OM calculations. So, the calculations were performed using three parameters, namely, N_RSPP (renormaliztion factor for the real part created based on SPP), depth (W₀) and diffuseness (a _W) of the imaginary volume part. The total potential in this case has the following form:

As shown in Figs. 9-11, the agreement between the experimental ⁶Li + ²⁰⁸Pb angular distributions at energies 25 - 210 MeV and the calculations using real SPP and an imaginary WS potential is fairly good. The optimal extracted potential parameters using this approach are listed in Table III, (J_V), (J_W), and σR values are also presented. The average extracted N_RSPP value at the different considered energies is 0.472 ± 0.144 except at energy 25 MeV. At E = 25 MeV, the N_RSPP value is 0.993 which is close to the extracted value from the first approach. The observed anomaly at energies 43 and 46 MeV from the first approach is not exists in the results of the second approach. These results again emphasize that the strength of the real SPP should be reduced by about 52 % in order to reproduce the ⁶Li + ²⁰⁸Pb data.

Figure 9 Comparison between ²⁰⁸Pb(⁶Li, ⁶Li) ²⁰⁸Pb elastic scattering angular distributions (solid circles) and SPP for the real part +WS imaginary part (solid curves) fits at E _lab = 25, 29, 30, 31, and 33 MeV. The data are displaced by 0.5 for the sake of clarity.  

Figure 10 Same as Fig. 9 but at E_lab = 35, 36, 42, 43, 46 and 48 MeV. The data are displaced by 0.2 for the sake of clarity.  

Figure 11 Same as Fig. 9 but at E_lab = 73.7, 88, 99, 156 and 210 MeV. The data are displaced by 0.2 for the sake of clarity.  

3.3 ⁶Li + ²⁰⁸Pb data analysis using CF potential

Motivating by the well-known d + α cluster structure for ⁶Li, we tried to reproduce the available experimental data for ⁶Li + ²⁰⁸Pb elastic scattering angular distributions using the fully microscopic cluster folding (CF) model. Within the framework of this model, the real and imaginary parts of potential ware constructed based on CF potential (Eqs. 6 and 7). The total potential in this case has the following form:

The comparisons between the experimental ⁶Li + ²⁰⁸Pb elastic scattering angular distributions in the energy range 25 - 210 MeV [¹, ², ³, ⁴, ⁵, ⁶, ⁷, ⁸] and the theoretical calculations within the framework of the CF model are shown in Figs. 12-14, with potential parameters listed in Table IV. The data are fitted using two parameters - NR CF and NICF - renormalization factor for the real and imaginary CF potentials. To obtain good fitting with the experimental data, the strength of the real cluster folding potential should be reduced by ∼ 62 %. In general, the observed reduction in the strength of the real part created based on either CF or DF is one of the signatures for ⁶Li+X nuclear systems. The extracted average NRCF is 0.378 ± 0.139 from our CFM calculations. At energies 43 and 46 MeV, the data require more reduction in the strength of the real CF potential since the extracted NRCF = 0.1 at these two aforementioned energies. Such anomaly was also observed in the calculations using SPP first approach.

Figure 12 Comparison between ²⁰⁸Pb(⁶Li, ⁶Li) ²⁰⁸Pb elastic scattering angular distributions (solid circles) and CFP (solid curves) fits at E_lab = 25, 29, 30, 31, and 33 MeV. The data are displaced by 0.5 for the sake of clarity.  

Figure 13 Same as Fig. 12 but at E_lab = 35, 36, 42, 43, 46 and 48 MeV. The data are displaced by 0.2 for the sake of clarity.  

Figure 14. Same as Fig. 12 but at E_lab = 73.7, 88, 99, 156 and 210 MeV. The data are displaced by 0.2 for the sake of clarity.  

3.4 ⁶Li + ²⁰⁸Pb data analysis using CFM plus a dynamical polarization potential

It is clearly shown from the current study and also previous studies concerning ⁶Li + ²⁰⁸Pb nuclear system that, the real real part of the potential constructed on microscopic procedures needs a renormalization by about 40-60 % in order to reproduce the experimental data. This reduction was assumed to be due to the break-up effect observed in the loosely bound ⁶Li nucleus. The analysis of ⁶Li + ²⁰⁸Pb elastic scattering data using real part of potential constructed on DF and CF procedures showed the same trend, in order to reproduce the data, NRCF should be reduced by ∼ 62 %. This well-known reduction in the real DF and CF potentials’ strength to reproduce the experimental data could be compensated by the introducing an additional dynamical polarization potential (DPP) of surface repulsive shape. This DPP simulates the polarization effects caused by the break-up of the weakly bounded projectiles. The data are then reanalyzed using the following total potential:

where the nuclear potential consists of three parts: a) real part derived from CF procedure without any renormalization (NRCF = 1), b) an imaginary CF potential with renormaliztion factor NICF fixed to the same values obtained from CFM analysis, and c) a dynamical polarization potential, ΔVpol(R) term which takes into account the dynamic contributions from all the allowed inelastic channels due to coupling. The DPP is taken as a surface potential with a repulsive real part and is characterized by three parameters (Vpol, rpol, apol) and their corresponding values at the different concerned energies are listed in Table V. As shown in Figs. 15-17, the agreement between ⁶Li + ²⁰⁸Pb experimental data and theoretical calculations constructed utilizing the non-renormalized real CFP plus DPP is fairly good except at E = 46 MeV, where calculations showed some deviation especially at large angles.

Figure 15 Comparison between ²⁰⁸Pb(⁶Li, ⁶Li) ²⁰⁸Pb elastic scattering angular distributions (solid circles) and CFP fits with non-renormalized real cluster folding potential (NRCF = 1) plus a DPP term (solid curves) at E_lab = 25, 29, 30, 31, and 33 MeV. The data are displaced by 0.5 for the sake of clarity.  

Figure 16 Same as Fig. 15 but at E_lab = 35, 36, 42, 43, 46 and 48 MeV. The data are displaced by 0.2 for the sake of clarity.  

Figure 17 Same as Fig. 15 but at E_lab = 73.7, 88, 99, 156 and 210 MeV. The data are displaced by 0.2 for the sake of clarity.  

The energy dependence on total reaction cross section σR values extracted from OM, SPP Real + SPP Imag. (first approach), SPP Real + WS Imag. (second approach), and CFM calculations at the different concerned energies for ⁶Li + ²⁰⁸Pb nuclear system is shown in Fig. 18. The extracted values for σR from the different implemented models are very close to each other especially at energies < 80 and with a slight deviation at higher energies.

Figure 18 Energy dependence on reaction cross-section for 6 ⁶Li + ²⁰⁸Pb nuclear system.