Reanalysis of ⁶Li+⁹⁰Zr angular distributions using different nuclear potentials

Abstract

The experimental angular distributions for ⁶Li elastically scattered from a ⁹⁰Zr target in the energy range 14.89–240 MeV are theoretically reanalyzed using phenomenological and semi-microscopic potentials. The phenomenon of the break-up threshold anomaly (BTA) is well represented in the various used potentials. In analyzing nuclear systems induced by weakly bound projectiles such as ⁶Li, the well-known reduction in the strength of the real part of potential generated based on double folding (DF) and cluster folding (CF) models is also observed. The Sao Paulo potential (SPP) is used to perform double folding calculations. In order to fit the data fairly, a reduction of ∼ 42% and 45% in the strength of the microscopic real part of potential constructed using SPP and CFM, respectively, is found to be required. By including an additional dynamical polarization potential, the data are also reproduced without renormalizing the real cluster folding potential.

KEYWORDS:

• Elastic scattering
• Sao Paulo potential
• phenomenological potential
• cluster folding
• dynamical polarization potential

1. Introduction

Study of nuclear processes involving weakly bound projectiles is a hot topic of research as it could play a key role in understanding the mechanism of interactions induced by radioactive unstable beams. Weakly bound projectiles have pronounced cluster structure such as the α+d observed at threshold energy E = 1.47 MeV in ⁶Li, and α+t at E = 2.467 MeV in ⁷Li. This high clusterization probability causes strong coupling to the break-up channel, resulting in a significant reduction in the microscopic real potential strength. This observed reduction in real potential strength could be simulated by the inclusion of a dynamical polarization potential (DPP) [1–3], which was taken in numerous studies as a real repulsive surface potential. The inclusion of such DPP has led to a successful reproducing for the experimental data for X + nucleus systems (X is a weakly bound projectile such as ^6,7Li and ^9,10Be) without renormalizing the potential [4–8]. Another interesting feature for systems involving weakly projectiles is the absence of the usual threshold anomaly (TA) normally observed in systems involving tightly bound projectiles which exhibited a localized peak in the real part with a sharp decrease in the imaginary part with decreasing the projectile’s energy towards the Coulomb barrier energy (E_C.B.). This absence is mainly due to the strong coupling to the break-up channel, which has a significant effect, and it is commonly known as the break-up threshold anomaly (BTA) [9]. Therefore it is sometimes difficult to apply the dispersion relation to system of high break-up nature.

In our previous studies [4–8], the mechanism of interaction for the weakly bound ⁶Li and ^9,10Be projectiles scattered from different targets were investigated. The current study complements such aforementioned studies. In the current study, the ⁶Li+⁹⁰Zr system at various energies ranging from near E_C.B up to relatively high energies is subjected to further investigation. In connection with this goal, we summarize briefly the most important experimental and theoretical studies concerning this system.

Fortunately, extensive angular distribution (AD) measurements and theoretical studies for the ⁶Li+⁹⁰Zr system [10–24] are available to assist us in analyzing this system over a wide range of energies. In Ref. [10], the ⁶Li+⁹⁰Zr elastic scattering ADs in the energy range of 11–30 MeV were performed. The data was analyzed using optical model (OM) and double folding (DF) potentials. Continuum discretized coupled channel (CDCC) method was adopted and a good fit with the data was achieved. The performed analyses predict the break-up coupling effects. The DPPs extracted from the CDCC calculations for ⁶Li+⁹⁰Zr were compared to those for ⁶Li +²⁰⁸Pb and ⁶Li +²⁸Si systems, and similar energy dependence was obtained. In Ref. [11], the elastic and inelastic ⁶Li+⁹⁰Zr ADs at E_lab = 34 MeV were measured, leading to the 2⁺ (2.18 MeV) and 3^- (2.75 MeV) ⁹⁰Zr excited states. Data was analyzed using OM and the distorted wave Born approximation (DWBA) method, and the deformation parameters for 2⁺ and 3^- excited states were extracted. In Ref. [12], elastic scattering and transfer (proton pick up) ADs for ⁸⁹Y and ⁹⁰Zr induced by ⁶Li projectiles at E_lab = 60 MeV were measured. Data were analyzed using both OM and finite range DWBA. The extracted spectroscopic factor for the configuration ⁷Be→⁶Li + p were in a good agreement with theoretically reported values. In Ref. [13], ADs for 6Li ions of energy 70 MeV is elastically and inelastically scattered from ^90,92,94,96Zr were measured leading to the 2⁺ and 3^- excited states for the different considered Zirconium isotopes. DF potentials were adopted in data analysis which leads to good description for the 2⁺ state but underestimated the cross-sections for the 3^- state. In Ref. [14–17], the measured ADs for 73.7, 99, 156, and 210 MeV ⁶Li ion beams, respectively, scattered from different targets, among them ⁹⁰Zr were analyzed using OM Woods-Saxon (WS) potential, and the extracted potential parameters exhibited both discrete and continuous ambiguities. In Ref. [15], in addition to the performed OM calculations for the ⁶Li+⁹⁰Zr system, the single folding ⁶Li potential gave a good description of the data only after renormalization of the real potential by a factor of ∼ 0.5. In Ref. [18], elastic and inelastic scattering of 240-MeV 6 Li particles from ⁵⁸Ni and ⁹⁰Zr were measured. Data was analyzed using the DF approach with the density-dependent M3Y NN interaction; the obtained renormalization factors (N_r) for the two considered targets were ∼ 0.87.

In Ref [20], ADs for ⁶Li at energy of 40 MeV /nucleon elastically scattered from different targets were analyzed within the framework of the optical double folding model based upon the density-independent M3Y effective interaction. In Ref. [21], a large body of experimental data for ⁶Li elastically scattered from ²⁴Mg to ²⁰⁹Bi targets at energies below 250 MeV was analyzed systematically in order to obtain the ⁶Li global phenomenological OM potential. The extracted OM potential was found to be applicable to other targets outside of the considered mass range. In Ref. [22], relativistic corrections to the reaction kinematic parameters were made for some nuclear systems, among them ⁶Li+⁹⁰Zr elastic scattering ADs at energies between 20 and 100 MeV/nucleon. The main finding of the study is that the effects of such corrections are important when describing the ADs for heavy ion scattering at energies as low as around 40 MeV/nucleon. In Ref. [23], the break-up coupling effects on the formation of the nuclear rainbow were investigated. To that end, data from the refractive elastic scattering of ⁶Li from various targets, including ⁹⁰Zr at 35 MeV/nucleon, were reanalyzed using the (CDCC) method to account for the coupling effects that coming from the break-up of ⁶Li into α and d. The calculations were performed by switching on and then by switching off the break-up coupling for comparison. The main finding of this study was that the break-up coupling can boost the nuclear rainbow, although it also damps the elastic scattering cross-sections.

2. Data analysis methodology

2.1 Data analysis within the framework of OM

Optical model (OM) is a powerful tool for understanding and describing the interaction between two colliding nuclei by reducing the solution of Schrödinger equation from n-body problem – n, is the total number of nucleons existed in both projectile and target nuclei – into one body system of mass equal the reduced mass (μ) and is moving in a potential well created by all other nucleons. So, in order to investigate the mechanism of interaction for a specified nuclear system, it is preferable to start with OM analysis as a first step, and hence the analysis could be developed further by adopting more microscopic approaches which use different forms for the density distributions of the interacting nuclei and also the effective nucleon-nucleon interaction potential (V_NN). For these reasons, the available ADs for ⁶Li elastically scattered from a ⁹⁰Zr target in the energy range of 14.89–240 MeV [10–18] are reanalyzed first from the phenomenological point of view using OM. The utilized central potential consists of both the Coulomb part $V_C(r)$ of radius$r_C{A}_t^{1/3}$and nuclear part. The nuclear part has a real volume part which describes the scattering, and imaginary volume term to simulate the absorption. Both of these have a Woods-Saxon (WS) shape. The parameters considered by A. Nadasen et al. [17] are taken as starting parameters. In was found that the effect of spin orbit potential (V_SO) on the ⁶Li-prjectile is small and it can be excluded. The utilized OM potential has the following shape: (1) $\begin{array}{rl}U(r)& =V_C(r)-V{\left[1+\exp \left(\frac{r-R_V}{a_V}\right)\right]}^{-1}\\ & -i{W}_0{\left[1+\exp \left(\frac{r-R_W}{a_W}\right)\right]}^{-1}\end{array}$(1) $\begin{array}{rl}{R}_i& =r_i(A_T^{1/3}),i=C,V,W\end{array}$

The $V_C(r)$ is the Coulomb potential between two charged spheres representing the projectile and target nuclei.

2.2. Data analysis using real Sao Paulo potential (SPP)

Due to the parameter ambiguities associated with OM calculations, and the fact that the phenomenological representations do not include a description of the internal structure of both the projectile and target nuclei, it is appreciable to derive the interaction potential from microscopic approaches such as double folding (DF). From this perspective, the real part of nuclear potential is constructed by the double folding procedures extracted from the Sao Paulo potential (SPP) NN interaction [25–28]. (2) $\begin{array}{r}{V}^F(R)=\int {\rho }_p(r_P){\rho }_T(r_T){\text{V}}_0\delta \mid \overrightarrow{S}\mid d^3{r}_P{d}^3{r}_T,\overrightarrow{S}=\overrightarrow{R}-\overrightarrow{r_P}+\overrightarrow{r_T}\end{array}$(2) where ${\rho }_{\text{P}}({\text{r}}_{\text{P}})$ and ${\rho }_{\text{T}}({\text{r}}_{\text{T}})$ are the nuclear matter density distributions of ⁶Li and ⁹⁰Zr nuclei, respectively, with ${\text{V}}_0$ =  −456 MeV. The new Sao Paulo potential (SPP2) has already been implemented with nuclear densities obtained from the Dirac-Hartree-Bogoliubov model [28].

The following two equations link the real part of the local-equivalent interaction to the DF potential $V_F(R)$ as (3) $V_N(R,E)=V_F(R)e^{(-4v^2/C^2)}$(3) and (4) $V^2(R,E)=\frac{2}{\mu }[E_{c.m.}-V_C(R)-V_N(R,E)]$(4) where v is the local relative velocity between the two nuclei and C is the speed of light. The effects of the Pauli non-locality can be numerically obtained by solving Equations (3) and (4) in an iterative process. The effects of the Pauli non-locality cause the SPP to be velocity dependent. Nuclear densities for ⁶Li and ⁹⁰Zr, which were obtained from the Dirac-Hartree-Bogoliubov model [29], are employed to calculate the nuclear potentials. The obtained SPP at E_lab = 20.91, 34, 60, 73.7, 99, 156, 210 and 240 MeV as expressed in Equation (2) is shown in Figure 1.

Figure 1. The generated real SPP at energies E_lab = 20.91, 34, 60, 73.7, 99, 156, 210 and 240 MeV.

2.3. Data analysis within the framework of CFOM

Due to the well-known α + d cluster nature of ⁶Li and its very low binding energy (⁶Li nucleus is bound by 1.47 MeV). It is appreciable to construct the ⁶Li+⁹⁰Zr potential based on the ⁶Li cluster structure. This weakly bound nature is responsible for the various break-up effects in systems induced by ⁶Li-b projectiles. Consequently, it is necessary to examine the applicability of cluster folding (CF) model based on previous works [30], where the ⁶Li cluster model was well established in reproducing the ⁶Li+⁹⁰Zr elastic scattering ADs. In connection with this aim, the ⁶Li+⁹⁰Zr data are reproduced utilizing the cluster folding optical model (CFOM) where, real part of the potential is defined based on the α + ⁹⁰Zr and d + ⁹⁰Zr potentials: (5) $\begin{array}{rl}{V}^{CF}(\mathbf{R})& =\int \left[V_{\alpha -{}^{90}\text{ Zr}}\left(\mathbf{R}\mathbf{-}\frac{\text{1}}{\text{3}}\mathbf{r}\right)+V_{d{-}^{90}\text{ Zr}}\left(\mathbf{R}+\frac{\text{2}}{\text{3}}\mathbf{r}\right)\right]\\ & \times |{\chi }_{\alpha d}(\mathbf{r}){|}^{2}d\mathbf{r}\text{,}\end{array}$(5) where ($V_{\alpha -{}^{90}\text{ Zr}}$and $V_{d{-}^{90}\text{ Zr}}$) is the real potential for α+⁹⁰Zr and d+⁹⁰Zr channels, which reasonably fit the data at suitable energies E_d ≈ 1/3E_Li and E_α ≈ 2/3E_Li [31, 32], ${\chi }_{\alpha d}(\mathbf{r})$ is the intercluster wave function that describes the relative motion of α and d in the ground state of ⁶Li, and $\mathbf{r}$ is the relative coordinate between the centres of mass of d and α. The α-d bound state form factor represents a 2S state in a real WS potential as was taken in Ref. [4]. The parameters needed to prepare the cluster folding potential (CFP) for ⁶Li + ⁹⁰Zr are the optimal potentials for d + ⁹⁰Zr and α + ⁹⁰Zr at suitable energies. The highest energy under consideration is 240 MeV. Therefore, the required potentials are $V_{d{-}^{90}\text{ Zr}}$at E_lab= 1/3 × 240 = 80 MeV and $V_{\alpha -{}^{90}\text{ Zr}}$at E_lab = 2/3 × 240 = 160 MeV. Based on the previous experimental studies for d + ⁹⁰Zr and α + ⁹⁰Zr systems, the most suitable available data, which could be implemented to generate the CFP for ⁶Li + ⁹⁰Zr are: d + ⁹⁰Zr at E_lab = 56 MeV [31] and α + ⁹⁰Zr at E_lab = 141.7 MeV [32], these data reasonably agree with the second highest energy under consideration (210 MeV). On one side, the AD for d + ⁹⁰Zr at E_lab = 56 MeV is fitted using real DF potential in addition to an imaginary surface and spin orbit terms “see Equation (6)”. Two forms for the real DF potential “SPP and CDM3Y6” were tested and both of them give equally good fitting as shown in Figure 2. (6) $\begin{array}{rl}U(r)& =V_C(r)-N_R{V}^{DF}(r)-i{W}_D\frac{d}{dr}\\ & \times {\left[1+\exp \left(\frac{r-R_D}{a_D}\right)\right]}^{-1}-\frac{2V_{so}}{r}{\left(\frac{\hslash }{m_{\pi }{C}^2}\right)}^2\\ & \times \frac{d}{dr}{\left[1+\exp \left(\frac{r-r_{so}{A}^{1/3}}{a_{so}}\right)\right]}^{-1}\end{array}$(6) With N_R is the renormalization factor for the real DF potential. The imaginary surface potential is characterized by three parameters, namely, potential depth (W_D), radius parameters (r_D) and diffuseness (a_D). The spin orbit potential is also characterized by three parameters, namely, potential depth (V_SO), radius parameters (r_SO) and diffuseness (a_SO).

On the other side, the AD for α + ⁹⁰Zr at E_lab = 141.7 MeV is fitted with a real DF potential of CDM3Y6 interaction [33] in addition to an imaginary volume part of the conventional WS shape “ see Equation (7)”. (7) $U(r)=V_C(r)-N_R{V}^{DF}(r)-i{W}_0{\left[1+\exp \left(\frac{r-R_W}{a_W}\right)\right]}^{-1}$(7)

The ADs for d + ⁹⁰Zr at E_lab = 56 MeV and α + ⁹⁰Zr at E_lab = 141.7 MeV and theoretical calculations are in a good agreement as shown in Figures 2 and 3, respectively using the potential parameters listed in Table 1. The potentials obtained from Table 1 are used to generate the real CF potential expressed in Equation (5) as shown in Figure 4.

Figure 2. The d + ⁹⁰Zr elastic scattering AD at E_lab = 56 MeV and the theoretical fit using real (SPP and CDM3Y6) plus an imaginary WS term.

Figure 3. Comparison between α + ⁹⁰Zr elastic scattering AD at E_lab = 141.7 MeV and calculations using real DF-CDM3Y6 potential plus an imaginary WS term.

Figure 4. Real CFP used in the current study to fit ⁶Li+⁹⁰Zr ADs at the various considered energies.

Table 1. Parameters of the d + ⁹⁰Zr and α + ⁹⁰Zr potentials used as inputs to the ⁶Li + ⁹⁰Zr CF potential. Radii R_W,D,SO  = r_W,D,SO x 90^1/3 fm.

Nuclear system	Model	NR	W0 (MeV)	rW (fm)	aW (fm)	WD (MeV)	rD (fm)	aD (fm)	VSO (MeV)	rSO (fm)	aSO (fm)
d + ^90Zr	SPP	1.02				16.78	1.304	0.684	2.63	0.499	0.45
d + ^90Zr	CDM3Y6	0.733				16.78	1.304	0.684	2.63	0.499	0.45
α + ^90Zr	CDM3Y6	1.09	20.28	1.533	0.748

Note: The real part was generated using SPP for d + ⁹⁰Zr, and using DF-CDM3Y6 for α + ⁹⁰Zr with renormalization factors N_R.

3. Results and discussions

3.1. Analysis of ⁶Li+⁹⁰Zr system using OM

Initially, the experimental ADs for ⁶Li, which are elastically scattered from ⁹⁰Zr at energies of 14.89, 16.9, 18.9, 20.91, 24.92 and 29.93 MeV [10], 34 MeV [11], 60 MeV [12], 70 MeV [13], 73.7 MeV [14], 99 MeV [15], 156 MeV [16], 210 MeV [17] and 240 MeV [18] are reanalyzed within the framework of OM using the potential presented in Equation (1). The performed OM analysis employed real and imaginary volume terms for each WS shape. The radii parameters for both real and imaginary parts are fixed to r_V= 1.182 fm and r_W= 1.627 as those in Ref. [17], with four free changing parameters – potential depth and diffuseness for the two parts – till the best agreement between data and calculations is reached through minimizing the value χ². The theoretical calculations performed in the current study using the different considered potentials were done utilizing FRESCO code [34] and upgraded with χ² minimization SFRESCO search code for searching the optimal potential parameters. As shown in Figures 5–7, the adopted OM is successfully reproduced the experimental data in the whole angular range using the optimal extracted parameters listed in Table 2. Table 2 also presents the reaction cross-section (σ_R), real (J_V) and imaginary (J_W) volume integrals at all energies.

Figure 5. Comparison between ⁹⁰Zr(⁶Li,⁶Li) ⁹⁰Zr elastic scattering ADs (black solid circles) at E_lab=  14.89, 16.9, 18.9, 20.81 and 24.92 MeV and OM calculations (red solid curves). The data are displaced by 0.5 for the sake of clarity.

Figure 6. Same as Figure 5 but at E_lab=  29.93, 34, 60, 70 and 73.7 MeV. The data are displaced by 0.1 for the sake of clarity.

Figure 7. Same as Figure 5 but at E_lab=  99, 156, 210 and 240 MeV. The data are displaced by 0.01 for the sake of clarity.

3.2 Analysis of ⁶Li+⁹⁰Zr system using SPP

In order to eliminate the parameter ambiguities which could be associated with OM calculations, analysis using more microscopic approaches is highly recommended. In the following approach, the analysis adopted a real folded potential based on the SPP with density distributions obtained from the Dirac-Hartree-Bogoliubov model [29] as expressed in Equation (2), in addition to a WS imaginary potential of fixed r_W=  1.627 fm as in the previously performed OM calculation. Using this approach, namely, (SPP Real + WS Imag.), the considered data are fitted using three parameters – N_RSPP, W₀, and a_w (renormalization factor for real SPP, depth, and diffuseness for the imaginary WS potential). The implemented potential has the following form: (8) $\begin{array}{rl}U(r)& =V_C(r)-N_{RSPP}{V}^{DF}(r)\\ & -i{W}_0{\left[1+\exp \left(\frac{r-R_W}{a_W}\right)\right]}^{-1}\end{array}$(8)

It is worth to mentioning that, although W₀, and a_w were free parameters, their extracted values using this approach were close to their values extracted from OM calculations. As shown in Figures 8–10, the adopted (SPP Real + WS Imag.) is successful in fitting the data in the whole angular range using the optimal extracted parameters listed in Table 2, and the quality of fitting is as good as for OM calculations. It is found that the real folded SPP strength should be reduced by ∼ 42% in order to obtain a good description of the data. The average extracted N_RSPP is 0.58 ± 0.13. The main cause of such an observed reduction in N_RSPP value is the significant coupling to break-up channel. The potential behaviour shows the characteristics of the BTA as it exhibits a reduction in the real part strength near the E_C.B..

Figure 8. Comparison between ⁹⁰Zr(⁶Li,⁶Li) ⁹⁰Zr elastic scattering ADs (black solid circles) at E_lab=  14.89, 16.9, 18.9, 20.81 and 24.92 MeV and SPP calculations (blue solid curves). The data are displaced by 0.5 for the sake of clarity.

Figure 9. Same as Figure 8 but at E_lab=  29.93, 34, 60, 70 and 73.7 MeV. The data are displaced by 0.1 for the sake of clarity.

Figure 10. Same as Figure 8 but at E_lab=  99, 156, 210 and 240 MeV. The data are displaced by 0.01 for the sake of clarity.

3.3. Analysis of ⁶Li+⁹⁰Zr system using CFP

As reported in the previously performed calculations using (SPP Real + WS Imag.) approach, the real folded SPP strength should be reduced by ∼ 42% in order to obtain a good description of the data. So, it is necessary to reanalysis the considered data using another microscopic approach which takes into consideration the appreciable a + d cluster structure for ⁶Li to check if such a characteristic is still present. In the following approach, the analysis adopted a real microscopic potential based on the CFOM as expressed in Equation (3), while an imaginary WS potential of fixed r_W=  1.627 fm as in the previously performed OM calculation, was included to simulate the absorption. Using this approach, namely, (CFP Real + WS Imag.), the considered data is fitted using three parameters: N_RCF (renormalization factor for real CFP), W₀, and a_w. The implemented potential has the following form: (9) $\begin{array}{rl}U(r)& =V_C(r)-N_{RCF}{V}^{CF}(r)-i{W}_0\\ & \times {\left[1+\exp \left(\frac{r-R_W}{a_W}\right)\right]}^{-1}\end{array}$(9)

As shown in Figures 11–13, the adopted (CFP Real + WS Imag.) approach is successful in fitting the data in the whole angular range using the optimal extracted parameters listed in Table 3 with nearly the same quality of fitting except at energies 210 and 240 MeV which underestimate the data especially at θ_c.m. > 30°. It is found that the real CFP strength should be reduced by ∼ 45% in order to obtain a good description of the data. The average extracted N_RCF is 0.55 ± 0.14. The main cause of such an observed reduction in N_RCF value is the break-up effects. The BTA phenomenon is also presented in the calculation using the (CFP Real + WS Imag.) approach and consequently does not obey the usual dispersion relation.

Figure 11. Comparison between ⁹⁰Zr(⁶Li,⁶Li) ⁹⁰Zr elastic scattering ADs (black solid circles) at E_lab=  14.89, 16.9, 18.9, 20.81 and 24.92 MeV and CFP calculations (black solid curves). The data are displaced by 0.5 for the sake of clarity.

Figure 12. Same as Figure 11 but at E_lab=  29.93, 34, 60, 70 and 73.7 MeV. The data are displaced by 0.1 for the sake of clarity.

Figure 13. Same as Figure 11 but at E_lab=  99, 156, 210 and 240 MeV. The data are displaced by 0.01 for the sake of clarity.

3.4. Analysis of ⁶Li+⁹⁰Zr system using CFP with the inclusion of dynamic polarization potential

The observed reduction in real DF and CF potential strength as shown in the calculations performed using both (SSP Real + WS Imag.) and (CFP Real + WS Imag.) approaches is basically due to the significant coupling effect to the break-up channel. This effect is possible to be compensated using two techniques. The first technique is to perform the full microscopic CDCC method where the continuum is discretized into momentum bins of a certain width, and consequently, coupling to the continuum states of certain angular momentum is involved. The CDCC method is powerful, as the data can be fitted nearly without any renormalization, as was adopted in the work of Sakuragi et al. [1–3]. The second technique which is used in the present work is the inclusion of a DPP of real repulsive surface potential. As shown in the aforementioned section, the ⁶Li+⁹⁰Zr ADs were reproduced fairly using the (CFP Real + WS Imag.) approach, if the real CF potential strength is reduced by ∼ 45%. Now, and with adopting the DPP technique, the data are fitted using non-renormalized real CF potential “N_RCF is fixed to unity” and using the same imaginary potential parameters extracted from (CFP Real + WS Imag.) approach in addition to a real repulsive surface DPP potential taken as a factor times the derivative of the previously created real cluster folding potential. In other words, the data was fitted using one free parameter (N_DPP), which is the renormalization factor for the repulsive surface DPP. As shown in Figures 14–16, the agreement between the experimental ⁶Li+⁹⁰Zr ADs and the theoretical calculations using the (non-renormalized CFP Real + WS Imag. + DPP) approach is fairly good using the potential parameters listed in Table 5.

Figure 14. Comparison between ⁹⁰Zr(⁶Li,⁶Li) ⁹⁰Zr elastic scattering ADs (black solid circles) at E_lab=  14.89, 16.9, 18.9, 20.81 and 24.92 MeV and CFP calculations in addition to DPP (green solid curves). The data are displaced by 0.5 for the sake of clarity.

Figure 15. Same as Figure 14 but at E_lab=  29.93, 34, 60, 70 and 73.7 MeV. The data are displaced by 0.1 for the sake of clarity.

Figure 16. Same as Figure 14 but at E_lab=  99, 156, 210 and 240 MeV. The data are displaced by 0.01 for the sake of clarity.

In order to observe the presence or absence of the usual TA in ⁶Li+⁹⁰Zr system, the energy dependence on both real and imaginary volume integrals extracted from the different considered approaches is plotted as shown in Figure 17. The plotted data emphasizes the absence of the usual TA, and that the BTA is well presented. This dependence doesn’t obey the usual dispersion relation. The energy dependence on σ_R values extracted from the different adopted approaches is plotted as shown in Figure 18. The extracted σ_R values are found to be in a good agreement with the values reported in Refs. [10, 14, 15, 18]. The σ_R value increases with increasing energy and can be expressed by the following polynomial function ${\sigma }_R(E)=-637.5+70.1E-0.453E^2$ as shown in Figure 18.

Figure 17. the energy dependence on (a) real and (b) imaginary volume integrals

Figure 18. the energy dependence on σ_R values for ⁶Li+⁹⁰Zr system extracted from the different adopted theoretical approaches.

4. Summary

The weakly bound stable nucleus ⁶Li, exhibit a predominant α+d cluster structure at threshold energy 1.47 MeV, and hence it could be a useful tool to understand the break-up nature of the weakly bound unstable radioactive beams. In connection with this importance, experimental ⁶Li+⁹⁰Zr elastic scattering ADs at fourteen energy sets ranging from 14.89 MeV and up to 240 MeV are investigated using different potentials constructed based on phenomenological, semi microscopic, and microscopic models. Firstly, OM calculations using real and an imaginary volume terms each has a WS shape of fixed radius parameter (r_V= 1.182 fm and r_W= 1.627 fm), were successful in reproducing the concerned data.

Then, the semi microscopic calculation based on a real potential part constructed using the SPP was performed in order to eliminate the ambiguities inherited in the phenomenological OM approach. The real part of the potential was derived using SPP and an imaginary volume part has a WS shape with fixed r_W= 1.627 fm, the approach namely (SPP Real + WS Imag.). Calculations using the aforementioned approach showed the necessity to reduce the strength of the real DF potential by ∼ 42% in order to reproduce the experimental data.

Finally, the considered data is analyzed using a real part derived based on CFM due to the well-known d + α cluster structure for ⁶Li. Reasonable fitting could be obtained if the strength of the real CF potential is reduced by ∼ 45%. The observed reduction in the real CF and DF potential strength is mainly due to the ⁶Li break-up effect. The CFM + DPP calculation is performed with a non-renormalized real CF potential (N_RCF  = 1.0) in addition to a repulsive real surface potential which takes into account the coupling to the d + α break-up channel. From the current and our previous studies [4–8], the break-up effect is strongly presented in systems involving weakly projectiles and it is independent of the considered target and the projectile’s energy. By comparing the extracted χ² values listed in Tables 2–5, the performed analysis for ⁶Li+⁹⁰Zr system using the different adopted potentials nearly give equally good fitting except at E_lab = 210 and 240 MeV. The calculations using OM and (SPP Real + WS Imag.) approaches are in a better agreement with experimental data at E_lab = 210 and 240 MeV in comparison with those using (CFP Real + WS Imag.) approach. The main reason of such observed deviation is probably due to the d + ⁹⁰Zr and α + ⁹⁰Zr potentials at energies E_lab = 56 and 141.7 MeV, respectively, which are the most appropriate energies found in the literature for preparing the CFP for ⁶Li + ⁹⁰Zr system. While, the actually required d + ⁹⁰Zr and α + ⁹⁰Zr potentials to fairly create the CFP for ⁶Li + ⁹⁰Zr system at E_lab  = 240 MeV are those at: 80 MeV for d + ⁹⁰Zr channel and 160 MeV for α + ⁹⁰Zr channel.

Table 2. Extracted OM potential parameters for the ⁶Li+⁹⁰Zr nuclear system with fixed r_V=  1.182 fm and r_W=  1.627 fm.

E (MeV)	V0 (MeV)	aV (fm)	W0 (MeV)	aW (fm)	χ^2/N	${\mathit{\sigma }}_{\mathit{R}}$ (mb)	JV (MeV.fm^3)	JW (MeV.fm^3)
14.89	223.47	0.871	15.97	0.87	0.37	44.00	326.41	54.77
16.9	109.2	0.871	21.62	0.87	0.43	201.3	159.50	74.14
18.9	111.92	0.871	17.56	0.87	0.05	403.9	163.47	60.22
20.91	108.73	0.898	25.67	0.79	0.15	644.3	160.92	86.13
24.92	104.77	0.883	21.41	0.843	0.47	1079	153.92	72.87
29.93	112.51	0.902	23.98	0.792	0.29	1387	166.85	80.50
34	107.38	0.95	36.31	0.712	0.64	1576	163.11	119.45
60	127.14	0.921	33.03	0.703	0.43	2107	190.33	108.43
70	209.44	0.849	25.08	0.846	0.32	2384	302.70	85.43
73.7	179.68	0.826	22.49	0.858	1.0	2365	256.86	76.86
99	173.57	0.846	25.75	0.828	0.42	2475	250.50	87.28
156	189.45	0.872	26.67	0.93	2.7	2767	267.85	93.07
210	184.32	0.91	30.94	0.881	5.3	2748	274.42	106.44
240	165.8	0.934	29.63	0.806	2.1	2574	249.83	99.84

Note: The values σ_R, J_V and J_W are presented.

Table 3. Optimal potential parameters extracted from (SPP + WS Imag.) approach for the ⁶Li+⁹⁰Zr nuclear system.

E (MeV)	NRSPP	W0 (MeV)	aW (fm)	χ^2/N	${\mathit{\sigma }}_{\mathit{R}}$ (mb)	JV (MeV.fm^3)	JW (MeV.fm^3)
14.89	0.717	17.33	0.879	0.38	49.35	297.61	59.58
16.9	0.41	21.62	0.87	0.39	201.8	169.80	74.14
18.9	0.42	17.56	0.87	0.2	408.7	173.56	60.22
20.91	0.437	25.18	0.79	0.2	639.7	180.19	84.48
24.92	0.413	23.44	0.824	0.47	1079	169.54	79.36
29.93	0.563	29.36	0.725	0.32	1356	229.58	96.89
34	0.595	34.86	0.723	0.74	1596	240.61	114.98
60	0.62	33.05	0.703	0.46	2120	244.92	108.49
70	0.647	24.0	0.882	0.34	2435	255.59	82.58
73.7	0.522	23.0	0.853	0.8	2349	203.15	78.50
99	0.541	25.27	0.832	0.58	2474	204.85	85.75
156	0.701	26.67	0.95	2.6	2806	249.68	93.63
210	0.787	30.65	0.946	5.8	2871	264.77	107.45
240	0.757	29.23	0.889	1.75	2716	246.74	100.78

Note: The values σ_R, J_V and J_W are presented.

Table 4. Optimal parameters extracted from the ⁶Li+⁹⁰Zr analysis using (CFP Real + WS Imag.) approach.

E (MeV)	NRCF	W0 (MeV)	aW (fm)	χ^2/N	${\mathit{\sigma }}_{\mathit{R}}$ (mb)	JV (MeV.fm^3)	JW (MeV.fm^3)
14.89	0.763	15.98	0.87	0.37	44.09	236.45	54.80
16.9	0.355	21.52	0.874	0.41	203.6	110.01	73.88
18.9	0.407	17.66	0.85	0.09	387.2	126.13	60.22
20.91	0.414	25.66	0.79	0.17	643.3	128.30	86.09
24.92	0.367	21.38	0.84	0.52	1073	113.73	72.71
29.93	0.43	23.89	0.791	0.4	1383	133.26	80.18
34	0.549	34.79	0.721	0.69	1587	170.14	114.70
60	0.552	32.65	0.701	0.44	2104	171.06	107.27
70	0.64	26.89	0.794	0.34	2315	198.34	90.32
73.7	0.505	24.67	0.788	1.4	2275	156.50	82.73
99	0.537	26.57	0.748	0.57	2340	166.42	88.19
156	0.694	25.29	0.916	3.6	2714	215.07	87.89
210	0.75	29.08	0.897	10.1	2748	232.43	100.50
240	0.708	27.27	0.891	4.8	2687	219.41	94.08

Note: The values σ_R, J_V and J_W are presented.

Table 5. Optimal potential parameters for ⁶Li+⁹⁰Zr extracted from (CFP Real + WS Imag. + DPP) calculations using non-renormalized real cluster folding potential (N_RCF  = 1), an imaginary WS potential of the same parameters extracted from (CFP Real + WS Imag.) approach in addition to DPP of repulsive surface shape taken as a factor times real CFP.

E (MeV)	W0 (MeV)	aW (fm)	NDPP	χ^2/N	${\mathit{\sigma }}_{\mathit{R}}$ (mb)	JV (MeV.fm^3)	JW (MeV.fm^3)
14.89	15.98	0.87	0.124	0.5	44.96	271.47	54.80
16.9	21.52	0.874	0.551	0.44	203.8	139.15	73.88
18.9	17.66	0.85	0.503	0.07	388.6	154.02	60.22
20.91	25.66	0.79	0.491	0.15	646.1	157.74	86.09
24.92	21.38	0.84	0.527	0.49	1078	146.58	72.71
29.93	23.89	0.791	0.468	0.31	1390	164.87	80.18
34	34.79	0.721	0.385	0.68	1589	190.59	114.70
60	32.65	0.701	0.379	0.45	2108	192.45	107.27
70	26.89	0.794	0.286	0.51	2321	221.27	90.32
73.7	24.67	0.788	0.405	2.5	2283	184.39	82.73
99	26.57	0.748	0.399	3.8	2345	186.25	88.19
156	25.29	0.916	0.336	4.6	2710	205.77	87.89
210	29.08	0.897	0.321	9.9	2742	210.42	100.50
240	27.27	0.891	0.439	8.04	2677	173.85	94.08

Note: The values σ_R, J_V and J_W are presented.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

A.A. Ibraheem extend his appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups programme under [grant number R.G.P.1/1/42].