2.1 The analysis using optical model

As a first step, the experimental elastic scattering ⁷Li +⁵⁸Ni ADs at energies of 13 and 13.5 MeV ^[1], 14.22, 16.25, 18.28, and 20.31 MeV ^[2], 19 MeV ^[3], 34 MeV ^[4], and 42 MeV ^[5] are reanalyzed using the phenomenological OM with a central potential which has the following form: ear systems, OM of the nucleus is applied. The implemented phenomenological OM potential has the following form:

The first term VC(r) is the Coulomb potential due to a uniform sphere with a charge equal to that of the target nucleus and radius rC  At1/3.

The second and third terms are the real and imaginary parts of nuclear potential which describe the scattering and absorption processes, respectively. Both parts are of volume shape and expressed in the conventional WS form. The spin orbit potential (VSO) for ⁷Li has been excluded as it has a little effect as well as for the sake of simplicity.

2.2 The analysis using Sao Paulo potential

It is more preferable to construct the interaction potential using more microscopic methods in order to get rid of the different ambiguities inherited with the OM potential. The SPP is similar to the usual DF potential as it is based on folding the projectile and target densities with nucleon-nucleon interaction potential (VNN)^[20-24] expressed as.

where the density distributions of ⁷Li and ⁵⁸Ni nuclei are denoted by ρp(r1) and ρt(r2), respectively. The nuclear densities for ⁷Li and ⁵⁸Ni were taken from the tabulated values created by REGINA code and were prepared based on the Dirac-Hartree-Bogoliubov (DHB) model ^[25]. The prepared SPP at Elab=14.22,,18.28, 20.31, 34, and 42 MeV are shown in Fig. 1.

Figure 1 The generated real SPP at energies Elab=14.22,18.28, 20.31, 34, and 42 MeV. 

2.3 The analysis using cluster folding model

Due to the well-developed α+t cluster structure in ⁷Li nucleus that appears at Ex=2.468 MeV, various break-up effects are observed in different system induced by this weakly bound projectile. Better understanding the dissociation nature of ⁷Li would clarify the dissociation of loosely unstable radioactive isotopes. The resulted break-up of ⁷Li into a triton (valence particle) orbiting an α-particle (core) can have a significant effect even on elastic scattering data. In order to check this effect, it is desirable to derive the ⁷Li + ⁵⁸Ni nuclear interaction potential based on a microscopic method that considers the ⁷Li cluster nature, such as the fully microscopic cluster folding model (CFM). In the CFM, the real and imaginary cluster folding potentials (CFPs) for the ⁷Li +⁵⁸Ni system are generated based on the t+58Ni and α+58Ni potentials as follows:

The suitable needed potentials are: Vt-58  Ni at Elab=18 MeV (Et≈3/7ELi) and V  α-  58  Ni at Elab=24 MeV (Eα≈4/7ELi). As the ⁷Li +⁵⁸Ni AD data at 42 MeV is the greatest regarded energy, the t + ⁵⁸Ni at Elab=17 MeV ^[26] and α+⁵⁸Ni at Elab=24.1 MeV ^[27] are the most appropriate data for determining the CFPs for ⁷Li +⁵⁸Ni based on prior investigations regarding t and α scattered from ⁵⁸Ni target. In the ground state of ⁷Li, the χα-t(r) is the intercluster wave function that characterizes the relative motion of α and t. The α-t bound state form factor represents a 2P_3/2 state in a real WS potential with a radius of 1.83 fm, a diffuseness of 0.65 fm, and a depth that can change until the cluster’s binding energy (2.468 MeV) is reached. The created CFPs are shown in Fig. 2, these potentials well agree with the previously created CFPs by Nishioka et al., ^[13].

Figure 2 The implemented real and imaginary CFPs in the ⁷Li +⁵⁸Ni CFM analysis. 

3.1 Analysis of ⁷Li +⁵⁸Ni data using OM potential

The elastic scattering ADs for ⁷Li +⁵⁸Ni system at energies ranging from 13 - 42 MeV ^[1-5] are reanalyzed phenomenologically using OM of central potential presented in Eq. (1). The implemented OM potential consists of a Coulomb part of radius 1.4 x (58)^1/3 in addition to nuclear part. The calculations were performed using fixed geometry parameters to observe how the real and imaginary potential depths vary with bombarding energy. In accordance with Cook study ^[28] concerning the systematic global optical potential for ⁷Li projectile, the following parameters rV, rW, aV, and aW were fixed to 1.286, 1.739, 0.853, and 0.809 fm, respectively. Consequently, two adjustable parameters V0 (real potential depth) and W0 (imaginary potential depth) were used to fit the data. The experimental data for ⁷Li +⁵⁸Ni ADs at all the considered energies below and above the Ec is fairly reproduced utilizing the OM approach as presented in Figs. 3 and 4 using the extracted parameters displayed in Table I.

Figure 3 Comparison between ⁵⁸Ni(⁷Li, ⁷Li)⁵⁸Ni elastic scattering ADs (circles) and OM fits (curves) at Elab=13,13.5, 14.22, 16.25, and 18.28 MeV. 

Figure 4 Same as Fig. 3 but at Elab=19,20.31, 34 and 42 MeV. The dashed line refers to OM fit using another potential set which is applied to wash up the observed oscillations at E = 19 MeV. 

In order to check the applicability of the dispersion relation on real (JV) and imaginary (JW) volume integrals, these quantities were calculated as their values are displayed in Table I. It is clearly shown that, the extracted (JV) and (JW) values don not obey the usual dispersion relation (localized peak followed by a continuous decrease in JV values, and direct increase followed by nearly constant energy dependence for JW values). The performed OM analysis gives an evidence for the absence of usual threshold anomaly (TA) which is presented in many systems induced by tightly bounded projectiles ^[29-32]. This absence of TA is observed in some systems induced by weakly projectiles ^[33,34], the so called break-up threshold anomaly (BTA) phenomenon.

The quality of fitting and hence the optimal potential parameters were obtained by minimizing the χ 2 value which defines the deviation between experimental data and calculations, and defined as follow:

The σ(θi)exp and σ(θi)cal are the experimental and calculated (DCs), Δσ(θi) is the relative uncertainty in experimental data. FRESCO code ^[35] upgraded with χ 2 minimization SFRESCO code were implemented to fit the data and to get the optimal potential parameters. Although the OM calculations using real and imaginary WS potentials were successful in reproducing the considered data, the fitting at E = 19 MeV showed some oscillations which are mainly linked to the used potential parameters. In order to wash up these oscillations, we had to reduce the applied constraints by allowing the real and imaginary depth (V0 and W0) and diffuseness (aV and aW) to freely change till the best fit is reached, and the radius parameters were still fixed (rV=1.286 fm and rW=1.739 fm). The used parameters to fit the data at E = 19 MeV are: V0=99.15 MeV, aV=0.908 fm, W0=28.6 MeV and aW=0.502 fm, and the obtained fit utilizing such parameters is represented by the dashed line in Fig. 4.

3.2 Analysis of ⁷Li +⁵⁸Ni data using SPP

Despite the well-known success of the OM to reproduce the experimental results for a variety of nuclear systems, it is preferable to construct the interaction potential on more microscopic methods such as SPP. This fact arises from the different parameter ambiguities both discrete and continuous associated with OM calculations. In order to overcome these ambiguities and to take the internal structure of the interacting nuclei into consideration, the new version SPP2 ^[36] was applied to investigate the concerned ⁷Li +⁵⁸Ni ADs data. The real part of SPP was created using (Eq. (2) and the imaginary part was taken of WS shape with potential parameters fixed to their optimal values obtained from OM analysis. The implemented potential has the following shape:

where NRSPP is the renormalization factor for the real part of the adopted SPP, and its optimal extracted values are displayed in Table II. As shown in Figs. 5 and 6, reasonable agreement was obtained between the experimental ⁷Li +⁵⁸Ni ADs and the theoretical calculations utilizing (real SPP + Imag. WS) approach.

Figure 5 Comparison between ⁵⁸Ni(⁷Li,⁷Li)⁵⁸Ni ADs (circles) and calculations using (real SPP + Imag. WS) approach (curves) at Elab=13,13.5, 14.22 16.25 and 18.28 MeV. 

Figure 6 Same as Fig. 5 but at Elab=19,20.31, 34 and 42 MeV. 

Two important facts are presented in the performed analysis. First, an observed reduction in the NRSPP value was found to be necessary for fitting well the data at the different considered energies with an average value 0.64 ± 0.18. These results show that, in order to reproduce the ⁷Li + ⁵⁸Ni ADs, the strength of the real SPP should be reduced by ~36%. This observed reduction is basically due to the break-up effect of ⁷Li. Second, the extracted (JV) and (JW) values from calculations using (real SPP + Imag. WS) approach do present the BTA which agree with the previously reported findings ^[33,34,37]. ⁵⁸Ni density distributions which were obtained from DHB model ^[25] as was done in SPP calculations. In other words, both SPP and DF-CDM3Y6 calculations implemented the same densities for ⁷Li and ⁵⁸Ni, so the main difference here was only in the utilized interaction potential. The DF potentials at the different considered energies were generated using DFMSPH code ^[38]. The real DF potential is prepared by folding the ⁷Li and ⁵⁸Ni densities with the (VNN).

The VNN has CDM3Y6 form based on the M3Y-Paris potential and consists of two parts, namely, direct vD(s) and exchange vEX(s) parts.

and the knock-on exchange part in the infinite-range exchange is

The M3Y-Paris interaction is scaled by a density-dependent function F(ρ):

where ρ is the nuclear matter (NM) density, s is the distance between the two interacting nucleons. The F(ρ) function was taken in the following form ^[39]:

So, the nuclear potential has the following form:

The data in this case is fitted by two free parameters NRDF and NIDF, namely, the renormalization factors for the real and imaginary parts of DF potential, respectively.

In addition to the previously described DF calculations using CDM3Y6 interaction, a modified version of CDM3Y6 is employed, which is denoted by CDM3Y6-RT and includes the effect the rearrangement term (RT). In the CDM3Y6-RT interaction, the term ΔF(ρ) is added and expressed as ^[40],

The experimental ⁷Li +⁵⁸Ni ADs are in a reasonable agreement with the performed calculations using DF-CDM3Y6 potential as shown in Figs. 7 and 8, the extracted optimal NRDF and NIDF alues are listed in Table III. The investigation revealed that the real DF potential strength had to be reduced by ~63%, the average extracted NRDF value is 0.37 ± 0.17 Then, we have applied the CDM3Y6-RT approach by considering the impact of including the rearrangement term shown in Eq. (13). Nearly the same quality of fitting was produced as shown in Figs. 7 and 8. The analysis using CDM3Y6-RT confirmed the need to reduce the real DF strength by ~62%, where the average extracted NRDF value is 0.38 ± 0.17 This observed reduction in potential strength in basically due to the break-up effect observed in the weakly bound ⁷Li nuclei. The slightly higher NRDF values extracted from the analyses using CDM3Y6-RT approach in comparison with those extracted from CDM3Y6 approach showed that the inclusion of RT has not a significant effect.

Figure 7 Comparison between ⁵⁸Ni(⁷Li,⁷Li)⁵⁸Ni ADs (circles) and calculations using DF Real + DF Imag. (CDM3Y6) approach (solid curves) and CDM3Y6-RT approach (dashed curves) fits at Elab=13,13.5, 14.22, 16.25 and 18.28 MeV. 

Figure 8 Same as Fig. 7 but at Elab=19,20.31, 34, and 42 MeV. 

3.3 Analysis of ⁷Li +⁵⁸Ni data using CFP

Due to the highly clusterization probability of ⁷Li and its break-up into t+α at low excitation energy ~2.468, it is interesting to explore the CFP based on the microscopic CFM and to test its applicability to describe the ⁷Li +⁵⁸Ni ADs utilizing the following central potential:

In terms of the so called (CFP Real + CFP Imag.) approach, the considered data was reproduced using two varying parameters NRCF and NICF, namely, the renormalization factors for the real and imaginary CFPs, respectively, generated as defined in Eqs. (3) and (4). Good description for the data was achieved using CFM at the different considered energies and in the whole angular range as shown in Figs. 9 and 10 except at energies 16.25 and 18.28 MeV which showed a slight deviation at angles >120^o. The extracted NRCF value in the energy range 13-42 MeV is close to each other with an average value 0.51± 0.2 as shown in Table IV. These results confirm the need to reduce the real cluster folding potential strength by ~49% in order to well describe the ⁷Li + ⁵⁸Ni ADs. This behavior is similar to our findings from the analyses using (real SPP + Imag. WS), DF-CDM3Y6 and DF-CD M3Y6 RT approaches. In general, the observed reduction in potential strength is one of the features for systems induced by the weakly bounded ⁷Li projectile ^[34].

Figure 9 Comparison between ⁵⁸Ni(⁷Li,⁷Li)⁵⁸Ni elastic scattering ADs (circles) and OM fits (curves) at Elab=13,13.5, 14.22, 16.25, and 18.28 MeV. 

Figure 10 Same as Fig. 9 but at Elab=19,20.31, 34, and 42 MeV. 

3.4 Analysis of ⁷Li +⁵⁸Ni data using CDCC method

The DF calculations using both CDM3Y6 and CDM3Y6-RT interactions, SPP, and CFP confirmed the need to reduce the real potential strength by ~63%, 62%, 36 %, and 49 %, respectively. This observed reduction is mainly due to the effect of ⁷Li break-up in the field of ⁵⁸Ni target. This effect can be simulated by applying the more sophisticated CDCC method using FRESCO code. The couplings to the unbound α+t resonant and non-resonant continuum states play a significant role as they produce a repulsive real dynamical polarization potential (DPP) ^[41,42]. The cluster folding (CF) procedures described in Eqs. (3) and (4) are used to calculate the coupling and diagonal potentials. With respect to the momentum of the α+t relative motion, the α+t continuum above the break-up threshold (2.468 MeV) was discretized into a series of momentum bins, with k restricted to 0.0≤k≤0.75 fm^-1 and width Δk=0.25 fm^{-1 [43]}. The t+58Ni, α+58Ni, as well as the α+t binding potentials (for 3/2^- and 1/2^- states) are the same as those used in the performed CFM calculations. As, the most significant contributions come from L = 3resonances ^[34,43-45], two resonant states (7/2^- and 5/2^-) with widths of 0.2 and 2.0, respectively, as well as one non-resonant (1/2^-, Ex=0.4776 MeV) are included inthe CDCC calculations. As shown in Figs. 11 and 12, good agreement between the experimental ⁷Li +⁵⁸Ni ADs and the CDCC calculations was achieved except at the lower energies: 13, 13.5 and 14.22 MeV which exhibited some deviations especially at larger angles > 100^o, but the over whole agreement is still acceptable as no adjustable parameters were used in the CDCC calculations.

Figure 11 Experimental ⁵⁸Ni(⁷Li,⁷Li)⁵⁸Ni ADs (circles) versus CDCC calculations (curves) at Elab=13,13.5, 14.22, 16.25, and 18.28 MeV. 

Figure 12 Same as Fig. 11 but at Elab=19,20.31, 34, and 42 MeV. 

Figure 13 depicts the variation of reaction cross section values obtained for the ⁷Li +⁵⁸Ni system by utilizing the OM, SPP, CDM3Y6, CDM3Y6-RT, and CFM potentials as well as those obtained from CDCC calculations with energy at the various examined bombarding energies. The following logarithmic formula can be used to express this behavior:

Figure 13 Energy versus the extracted σR values for ⁷Li +⁵⁸Ni system using the different implemented approaches: OM, SPP, CDM3Y6, CDM3Y6-RT, CFM, and CDCC. 

As shown in Fig. 13, there is an observed drop in the value of reaction cross section at 19 MeV. Such drop is presented in the different implemented approaches. The possible explanation of such drop in reaction cross sectionat E = 19 MeV may be due to the presence of resonance effect. In order to experimentally justify the presence of this resonance, a study for the excitation function is required. It worth to mention, although the extracted σR from CDCC method is quite reasonable (1080 mb), as it agrees well with the neighboring values, but the general agreement between the data and the CDCC calculations at 19 MeV is quite worse than those obtained from the other implemented potentials. Consequently, similar value for σR could be obtained from the other implemented approaches if we sacrificed the quality of fitting by reducing the applied constraints on the imaginary part of potential.

The performed analysis within the scope of the various approaches revealed that the BTA phenomenon is well represented in ⁷Li +⁵⁸Ni system, as the retrieved JV values shown in Tables I, II, III and IV do not obey the conventional dispersion relation. This is seen in Fig. 14, which depicts the energy dependence on JV values. Although the retrieved JV values from the different approaches are significantly differ due to the different methodologies followed in preparing the implemented nuclear potentials, but the general trend for the variation of JV values with energy is quite similar.

Figure 14 Energy dependence on the JV values obtained from the different implemented approaches.