Break-up effect on ⁷Li + ²⁸Si elastic scattering angular distributions

Abstract

The ⁷Li + ²⁸Si angular distributions for elastic scattering in the laboratory energy range of 8.5–36 MeV are reanalyzed using different approaches based on both phenomenological and microscopic potentials. Special attention is paid to the α + t cluster structure for ⁷Li, which appears at a threshold energy of 2.468 MeV. The performed analysis using the different implemented potentials showed that the breakup of ⁷Li in the field of ²⁸Si plays a strong effect in forming the cross sections. Such an effect results in a significant reduction in the renormalization factors of the utilized potentials. A satisfactory description of the data is achieved using the different implemented approaches.

KEYWORDS:

• Cluster folding model
• Sao Paulo potential
• double folding
• Dynamic polarization potential

PACS number(s):

• 24.10.Ht
• 25.70.Bc
• 24.50.+g
• 24.10.Eq
• 25.60.Bx

1. Introduction

Understanding the role of the breakup channel in reactions initiated by weakly bound unstable projectiles requires a deeper understanding of the breakup effects for stable particles. As a result, many recent studies have focused on the interaction mechanism for various systems initiated by weakly bound projectiles such as ^6,7Li and ^9,10,11Be. The typical threshold anomaly (TA) phenomenon, which is well represented by tightly bound nuclei as a result of the strong couplings to inelastic channels and also to the low-lying states in both the target and projectile, is characterized by rapid change in the real and imaginary potential depth with respect to energies close to the Coulomb barrier energy (E_C), where the real part exhibits a localized peak near the E_C associated through dispersion relations, while the imaginary depth increases by increasing energy till reaching a constant plateau. Due to strong couplings to breakup channels, the TA is absent for nuclear processes initiated by weakly bound projectiles, resulting in a repulsive contribution to the real potential.

Studying the mechanism of interaction of weakly bound projectiles with various targets below, near, and above the E_C is of special interest. One of these nuclear systems, which is the topic of the current analysis, is the ⁷Li + ²⁸Si system, which was the subject of several experimental as well as theoretical studies [1–17] that aimed to probe the peculiarities of this system at energies below and comparable to the E_C. At eight near-barrier energies, Pakou et al. [1] measured the ⁷Li + ²⁸Si elastic scattering angular distributions (ADs). The data was investigated utilizing the continuum discretized coupled channel (CDCC) method, and the double folding optical model (DFOM) with the BDM3Y1 interaction was used for generating the real potential. The mechanism of interaction of both ⁶Li and ⁷Li on ²⁸Si, as well as on the heavier targets ¹³⁸Ba and ²⁰⁸Pb, was studied in Ref. [1]. It was found that both ⁶Li and ⁷Li scattered from ²⁸Si exhibit the same trend; the imaginary potential depth decreases with decreasing energy approaching the barrier, which contradicts the behaviour of ⁶Li scattered from the heavier targets (¹³⁸Ba and ²⁰⁸Pb), where the imaginary potential presents an increasing. The consistency in potential behaviour for ⁶Li and ⁷Li ions beam scattering on ²⁸Si could be attributed to a more systematic behaviour of weakly bound nuclei scattering on lighter targets. The real part of the ^6,7Li scattering potential remains nearly constant, with a very modest diminishing behaviour seen mostly in the Si and Ba data. In addition, the CDCC calculations revealed that while breakup is more critical for ⁶Li than for ⁷Li, it is still insufficient to explain the potential at the barrier.

The elastic scattering ADs for ^6,7Li + ²⁸Si in the energy interval E_lab = 11.5–26 MeV were measured by M. Sinha et al. [2]. The measured ADs were analyzed using optical potential (OP), with imaginary surface and volume potentials. Moreover, the CDCC calculations for ^6,7Li + ²⁸Si elastic scattering data were done using FRESCO code by considering only the breakup of the projectile. In Refs. [3,4], the elastic scattering ADs of ⁷Li nuclei on ²⁸Si were measured at E_lab = 20 and 36 MeV. The measured ADs were compared with optical model (OM) calculations and a good agreement was obtained. In Refs. [5,6], the ⁷Li + ²⁸Si ADs at E_lab= 45 and 48 MeV were measured and theoretically analyzed using real folded potential, which was shown to be renormalized to produce a reasonable agreement with the measured cross sections. In Refs. [7,8], the ADs for ⁷Li ions of energies 177.8 and 350 MeV scattered from ²⁸Si target have been measured. The obtained ADs revealed diffractive oscillations at forward angles followed by exponential falloff at the larger angle. Zerva et al. [9] studied the excitation functions for ⁷Li + ²⁸Si quasielastic scattering at 150° and 170° at sub-and near-barrier energies, and computed the relevant barrier distributions. The data was investigated using OM with real double-folded (DF) potential derived using the BDM3Y1 effective interaction. The barrier distributions were also investigated using coupled reaction channel (CRC) and CDCC methods to explore the effects of breakup and transfer reactions.

Different theoretical studies [10–17] investigated the ⁷Li + ²⁸Si nuclear system. In Ref. [10], the ADs for ^6,7Li elastically scattered from ¹²C, ²⁸Si, ⁴⁰Ca, ⁹⁰Zr, and ²⁰⁹Pb at E/A_p = 12.5-53 MeV/u were investigated utilizing real DF potential and an imaginary WS potential. While in Ref. [11], the same authors implemented the S1Y effective nucleon-nucleon NN interaction through the coupled channels (CC) method to fit the considered data. In Ref. [12], a phenomenological dynamical polarization potential (DPP) was introduced to compensate for the necessary reduction in potential strength reported from the analysis of ^6,7Li + ²⁸Si systems. In Ref. [14], ⁷Li + ²⁸Si elastic scattering AD at E =  350 MeV was investigated using the CDCC method, and a fair description of the data was obtained. In Ref. [15], the available ⁷Li + ²⁸Si ADs in the energy interval 7.5–32 MeV were investigated using the OM of Woods-Saxon (WS) shape utilizing SPI-GENOA code. W. Chen et al. [17] recently employed a microscopic approach to study the OP for ⁷Li + nucleus systems without any free parameters. The ⁷Li microscopic OP was used to predict the ADs for ⁷Li scattered from different targets ranging from ²⁷Al to ²⁰⁸Pb.

Finally, it is important to note that the current research expands on our earlier studies [18–25] that looked at the peculiarities and the mechanism of interaction between weakly bound projectiles and various target nuclei at energies both below and above the E_C. The article is organized as follows. The potentials employed in the computations are presented in Sec. II. The analysis and discussion of the data are presented in Section III. The summary is covered in Section IV.

2. Theoretical formalism

2.1. Optical model potential (OMP)

Within the framework of OM, the ADs for ⁷Li elastically scattered from ²⁸Si at E_lab = 8.5, 9, 10, 11, 13, 15, 16 MeV [1], 11.5, 21, and 26 MeV [2], 36 MeV [3] are reanalyzed. The utilized central potential is made up of a nuclear part of real and imaginary volume terms, each having a WS shape as well as a Coulomb part. According to previous research on nuclear processes induced by ⁷Li -projectiles, the effect of spin orbit potential (V_SO) is minor and can be ruled out. The implemented OM potential takes the form: (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) The Coulomb potential $V_C(r)$is due to a uniform sphere with a charge equal to that of the target nucleus and radius $R_C=r_C{A}_t^{1/3}$.

2.2. Double folding using CDM3Y6 interaction and Sao Paulo potential

It is preferable to construct the interaction potential utilizing microscopic methods in order to eliminate the different ambiguities that are inherited within OM potentials. Hence, the ⁷Li + ²⁸Si system is investigated from a semi-microscopic perspective by utilizing the DF models using both the CDM3Y6 interaction potential and the Sao Paulo potential (SPP). The interaction potential CDM3Y6 was folded into the ⁷Li and ²⁸Si density distributions. The density distribution of ⁷Li was taken from Ref. [26] and has the form: (2) $\rho (r)=(\xi +\gamma r^2)\exp (-\beta r^2)\text{f}{\text{m}}^{-3}$(2) while, $\xi$= 0.1387, $\gamma$= 0.0232, and $\beta$= 0.3341. The density distribution of ²⁸Si was taken from Ref. [27] and is expressed as: (3) $\begin{array}{rl}{\rho }_{\text{T}}(r)& =\frac{0.0865}{1+\exp ((r-3.087)/0.55)}\\ & +\frac{0.0873}{1+\exp ((r-3.076)/0.55)}\text{ f}{\text{m}}^{-3}\end{array}$(3) The DF potential was computed using the DFMSPH code [28]. The real DF potential is prepared by folding the ⁷Li and ²⁸Si densities with the nucleon-nucleon NN interaction potential in the CDM3Y6 form based on the M3Y-Paris potential: (4) $\begin{array}{rl}{v}_D(s)& =\left[11061.625\frac{e^{-4s}}{4s}-2537.5\frac{e^{-2.5s}}{2.5s}\right]\text{MeV},\\ v_{EX}(s)& =[-1524.25\frac{e^{-4s}}{4s}-518.75\frac{e^{-2.5s}}{2.5s}\\ & -7.8474\frac{e^{-0.7072s}}{0.7072s}]\text{ MeV,}\end{array}$(4)

The M3Y-Paris interaction is scaled by a density-dependent function F(ρ): (5) $v_{D(EX)}(E,\rho ,s)=v_{D(EX)}(s)F(\rho ),$(5) where ρ is the nuclear matter (NM) density, s is the separation between two interacting nucleons, and $v_{D(EX)}$are the direct and exchange parts of the M3Y-Paris. The $F(\rho )$has an exponential dependency [29] and is expressed as: (6) $\begin{array}{r}F(\rho )=0.2658[1+3.8033\exp (-1.41\rho )-4.0\rho ]\end{array}$(6) The implemented DF potential is the sum of direct and exchange folded potentials: (7) $V^{DF}(R)=V_D^{DF}(R)+V_{EX}^{DF}(R)$(7) Similarly, the Sao Paulo potential SPP is based on folding the projectile and target densities with NN interaction potential [30–34] and is expressed as: (8) $\begin{array}{rl}{V}_F(R)& =\int \int {\rho }_p(r_p){\rho }_t(r_t)V_0\delta (|\overrightarrow{\text{S}}|){\text{d}}^{\text{3}}{r}_p{\text{d}}^3{r}_t,\\ & s=r_t-r_p+R\end{array}$(8) where the density distributions of ⁷Li and ²⁸Si nuclei are denoted by ${\rho }_p\text{(}{r}_p\text{)}$ and ${\rho }_t\text{(}{r}_t\text{)}$, with V₀=  −456 MeV. The nuclear densities for ⁷Li and ²⁸Si are obtained from the Dirac-Hartree-Bogoliubov model [35]. Figure 1 shows the prepared DF-CDM3Y6 as well as SPP at E_lab = 8.5, 16, 26, and 36 MeV. As shown from Figure 1, the constructed potentials using both CDM3Y6 and SPP are very close to each other.

2.3. Cluster folding model

We aim to describe the ⁷Li + ²⁸Si elastic scattering ADs using both the cluster folding optical model (CFOM) as well as the cluster folding model (CFM), taking into account the considerable cluster structure of ⁷Li as α + t system. The real and imaginary components of the ⁷Li + ²⁸Si potential are created using the cluster folding (CF) procedure within the CFM framework. We define the ⁷Li + ²⁸Si potential in terms of α-particles and (triton) + ²⁸Si potentials as follows: (9) $\begin{array}{rl}{V}^{CF}(\mathbf{R})& =\int [V_{\alpha -{}^{28}\text{ Si}}\left(\mathbf{R}\mathbf{-}\frac{\text{3}}{\text{7}}\mathbf{r}\right)\\ & +V_{t{-}^{28}\text{ Si}}\left(\mathbf{R}+\frac{\text{4}}{\text{7}}\mathbf{r}\right)]|{\chi }_{\alpha -t}(\mathbf{r}){|}^{2}d\mathbf{r}\text{,}\end{array}$(9) (10) $\begin{array}{rl}{W}^{CF}(\mathbf{R})& =\int [W_{\alpha -{}^{28}\text{ Si}}\left(\mathbf{R}\mathbf{-}\frac{\text{3}}{\text{7}}\mathbf{r}\right)\\ & +W_{t{-}^{28}\text{ Si}}\left(\mathbf{R}+\frac{\text{4}}{\text{7}}\mathbf{r}\right)]|{\chi }_{\alpha -t}(\mathbf{r}){|}^{2}d\mathbf{r}\text{,}\end{array}$(10) where the relative motion of α- particle and triton in the ground state of ⁷Li is described by the intercluster wave function ${\chi }_{\alpha -t}(\mathbf{r})$. The $V_{\alpha -{}^{28}\text{Si}}$, $W_{\alpha -{}^{28}\text{ Si}}$, $V_{t-{}^{\text{28}}\text{ Si}}$, and $W_{t{-}^{28}\text{ Si}}$ are the phenomenological real and imaginary potentials for α + ²⁸Si and t + ²⁸Si channels at energies E_t ≈ 3/7E_Li and E_α ≈ 4/7E_Li taken from [36, 37]. The bound state form factor for α + t represents a 2P_3/2 state in a real WS potential identified by the parameters (R_V = 1.83 f m, a_V= 0.65 fm), and the depth is adjusted to reproduce the binding energy for the cluster (2.468 MeV). Hence, the optimal potentials for t + ²⁸Si and α + ²⁸Si at suitable energies are implemented to create the cluster folding potential (CFP) for the ⁷Li + ²⁸Si system. With a maximum energy of 36 MeV, the needed potentials are V _{t + 28Si} at E_lab= 3/7 × 36 = 15.43 MeV and V _{α + 28Si} at E_lab  = 4/7 × 36 = 20.57 MeV. The created real and imaginary CFPs are shown in Figure 2. The analysis within the CFOM is performed using the real ⁷Li + ²⁸Si CFP generated as described in Eq. 9(12) $U(R)=V_C(R)-N_{RSPP}{V}^{DF}(R)-i{N}_{ISPP}{V}^{DF}(R)$(12) , in addition to an imaginary part that has the traditional WS shape.

Figure 1. The generated DF-CDM3Y6 and SPP real potential for ⁷Li + ²⁸Si system at energies E_lab = 8.5, 16, 26 and 36 MeV.

Figure 2. The implemented CFPs in the analysis of ⁷Li + ²⁸Si system.

3. Results and discussion

3.1. OM analysis

The considered ⁷Li + ²⁸Si ADs in the energy range of 8.5–36 MeV [1–3] are investigated, utilizing pure phenomenological optical potential as a first step. The implemented potential as shown in Eq. 1(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) consists of both Coulomb and nuclear potentials. The nuclear potential has two parts: the real volume part of the WS shape and identified by three parameters; potential depth (V₀), radius parameter (r_V) and diffuseness (a_V) to simulate the effect of scattering, and an imaginary WS potential of depth (W₀), radius parameter (r_W) and diffuseness (a_W) to simulate the effect of reduction in flux. For exploring the variation of V₀ and W₀ with energy, fixed radius and diffuseness parameters for the real and imaginary parts, r_V, a_V, r_W and a_W at the values of 1.286, 0.853, 1.739, and 0.809 fm were used in the OM computations. These parameters were taken in accordance with Cook’s study [38] concerning the global potential for ^6,7Li projectiles.

As shown in Figures 3 and 4, the OM analysis agrees well with the considered data using the optimal extracted parameters presented in Table 1. The calculations are performed using the FRESCO code [39] and the SFRESCO search code. The χ² value at the various considered energies was calculated to justify the good quality of the fit.

Figure 3. Experimental ²⁸Si(⁷Li,⁷Li)²⁸Si elastic scattering ADs (solid circles) at E_lab=  8.5, 9, 10, 11, 11.5, and 13 MeV versus the OM fits (solid curves). Data is displaced by a factor of 0.5.

Figure 4. Same as Figure 3 but at E_lab=  15, 16, 21, 26, and 36 MeV.

Table 1. Optimal potential parameters for the ⁷Li + ²⁸Si system using the OM, (Real DF + Imag. DF), and (Real SPP + Imag. SPP) approaches. The OM calculations were performed using fixed geometrical parameters, r_V= 1.286 fm, a_V = 0.853, r_W= 1.739, and a_W = 0.809 fm. N_RDF, N_IDF, N_RSPP, and N_ISPP are the renormalization factors for the real and imaginary parts of the potentials constructed based on the CDM3Y6 and SPP, respectively.

E (MeV)	Implemented approach	V0 (MeV)	W0 (MeV)	NRDF	NIDF	NRSPP	NISPP	χ^2/N
8.5	OM	60.27	5.52					0.42
	Real DF + Imag. DF			0.245	0.06			0.41
	Real SPP + Imag. SPP					0.222	0.06	0.41
9	OM	51.51	12.29					0.11
	Real DF + Imag. DF			0.234	0.19			0.108
	Real SPP + Imag. SPP					0.181	0.181	0.108
10	OM	61.89	19.12					0.45
	Real DF + Imag. DF			0.309	0.288			0.45
	Real SPP + Imag. SPP					0.217	0.283	0.45
11	OM	60.31	29.59					0.07
	Real DF + Imag. DF			0.356	0.441			0.07
	Real SPP + Imag. SPP					0.216	0.430	0.07
11.5	OM	71.27	4.36					0.31
	Real DF + Imag. DF			0.291	0.159			0.4
	Real SPP + Imag. SPP					0.237	0.184	0.39
13	OM	59.23	27.7					0.4
	Real DF + Imag. DF			0.33	0.41			0.44
	Real SPP + Imag. SPP					0.209	0.423	0.44
15	OM	80.83	22.59					0.19
	Real DF + Imag. DF			0.389	0.288			0.25
	Real SPP + Imag. SPP					0.287	0.331	0.2
16	OM	88.79	28.75					0.33
	Real DF + Imag. DF			0.44	0.349			0.27
	Real SPP + Imag. SPP					0.318	0.421	0.31
21	OM	81.76	25.33					0.06
	Real DF + Imag. DF			0.379	0.304			0.09
	Real SPP + Imag. SPP					0.294	0.376	0.06
26	OM	129.61	37.26					4.0
	Real DF + Imag. DF			0.512	0.374			4.0
	Real SPP + Imag. SPP					0.466	0.554	4.2
36	OM	129.12	42.41					5.37
	Real DF + Imag. DF			0.526	0.475			11.3
	Real SPP + Imag. SPP					0.451	0.635	5.8

3.2. DF analysis using CDM3Y6 interaction and SPP

The elastic ADs for the ⁷Li + ²⁸Si system are analysed using real DF potential generated based on the interaction model CDM3Y6, and the imaginary potential was considered as a factor times the real DF part. Thus, the implemented central potential is: (11) $U(R)=V_C(R)-N_{RDF}{V}^{DF}(R)-i{N}_{IDF}{V}^{DF}(R)$(11) Hence, two adjustable parameters, N_RDF and N_IDF, namely, real and imaginary renormalization factors, respectively, were implemented to fit the data. For simplicity, we shall call this approach (Real DF + Imag. DF). The comparisons between the experimental ADs and calculations within the framework of the (Real DF + Imag. DF) approach are reasonably good, as depicted in Figures 5 and 6. The optimal N_RDF and N_IDF values obtained from the calculations using the aforementioned approach are presented in table 1. As shown in table 1, the average extracted N_RDF value is 0.365 ± 0.098 which indicates the need to reduce the real folded potential strength by ∼ 63% to reasonably fit the considered data. The observed reduction in potential strength confirms the significant effect of ⁷Li breakup in the field of ²⁸Si.

Figure 5. Experimental ²⁸Si(⁷Li,⁷Li)²⁸Si elastic scattering ADs (solid circles) at E_lab=  8.5, 9, 10, 11, 11.5, and 13 MeV versus calculations using the (Real DF + Imag. DF) approach (solid curves). Data is displaced by a factor of 0.5.

Figure 6. Same as Figure 5 but at E_lab=  15, 16, 21, 26, and 36 MeV.

Another microscopic potential based mainly on the projectile and target densities as described in Eq. 8(11) $U(R)=V_C(R)-N_{RDF}{V}^{DF}(R)-i{N}_{IDF}{V}^{DF}(R)$(11) is the microscopic SPP, which was implemented to fit the ⁷Li + ²⁸Si ADs data. The imaginary potential was considered as a factor times the real SPP. Thus, the implemented central potential is: (12) $U(R)=V_C(R)-N_{RSPP}{V}^{DF}(R)-i{N}_{ISPP}{V}^{DF}(R)$(12) The data is reproduced utilizing two varying parameters, N_RSPP and N_ISPP, namely, real and imaginary renormalization factors, respectively. For simplicity, we shall call this approach (Real SPP + Imag. SPP). The comparison between ⁷Li + ²⁸Si ADs and the theoretical calculations utilizing the (Real SPP + Imag. SPP) approach is fairly good as depicted in Figures 7 and 8 using the potential parameters listed in table 1. The results revealed that to fairly fit the experimental data, the real SPP strength had to be reduced by ∼ 72%; the average extracted N_RSPP value is 0.282 ± 0.097. The breakup effect of ⁷Li is mostly responsible for the observed decrease in potential strength.

Figure 7. Experimental ²⁸Si(⁷Li,⁷Li)²⁸Si elastic scattering ADs (solid circles) at E_lab=  8.5, 9, 10, 11, 11.5, and 13 MeV versus calculations using the (Real SPP + Imag. SPP) approach (solid curves). Data is displaced by a factor of 0.5.

Figure 8. Same as Figure 7 but at E_lab=  15, 16, 21, 26, and 36 MeV.

3.3. Data analysis using CFOM and CFM

The considered data is reanalyzed utilizing the CFOM, which employs the real CFP generated by Eq. 9(12) $U(R)=V_C(R)-N_{RSPP}{V}^{DF}(R)-i{N}_{ISPP}{V}^{DF}(R)$(12) in addition to an imaginary WS potential. The parameters of the imaginary WS potential obtained via the OM analysis are used without any adjustments in the CFOM calculations. As a result, the central potential is expressed as: (13) $\begin{array}{rl}U(R)& =V_C(R)-N_{RCF}{V}^{CF}(R)\\ & -i{W}_0{\left[1+\exp \left(\frac{r-r_W}{a_W}\right)\right]}^{-1}\end{array}$(13) Hence, one adjustable parameter N_RCF, namely, the renormalization factor for the real CFP, was implemented to describe the data. For simplicity, we shall call this approach (Real CFP + Imag. WS). The agreement between the ⁷Li + ²⁸Si ADs and the calculations utilizing the (Real CFP + Imag. WS) approach is reasonably good, as depicted in Figures 9 and 10. Table 2 shows the optimal potential parameters extracted from the CFOM analysis. The average extracted N_RCF value is 0.445 ± 0.162 which emphasizes the need to decrease the real CFP strength by ∼ 55% to reproduce the experimental data.

Figure 9. Experimental ²⁸Si(⁷Li,⁷Li)²⁸Si elastic scattering ADs (solid circles) at E_lab=  8.5, 9, 10, 11, 11.5, and 13 MeV versus calculations using the (Real CFP + Imag. WS) approach (solid curves). Data is displaced by a factor of 0.5.

Figure 10. Same as Figure 9 but at E_lab=  15, 16, 21, 26, and 36 MeV.

Table 2. Optimal potential parameters for the ⁷Li + ²⁸Si system, obtained from the analysis using both CFOM and CFM. Within the CFOM, the imaginary WS potential parameters (W₀, r_W, a_W) are kept fixed as those obtained from the OM calculations, and the data is fitted using one parameter, N_RCF. Within the CFM, the data was fitted using two parameters, N_RCF and N_ICF, which are the renormalization factors for the real and imaginary CFPs, respectively.

E (MeV)	Implemented approach	NRCF	NICF	χ^2/N
8.5	CFOM	0.311		0.41
	CFM	0.312	0.127	0.41
9	CFOM	0.274		0.11
	CFM	0.252	0.343	0.11
10	CFOM	0.339		0.45
	CFM	0.301	0.552	0.45
11	CFOM	0.342		0.07
	CFM	0.284	0.849	0.08
11.5	CFOM	0.39		0.27
	CFM	0.343	0.337	0.4
13	CFOM	0.318		0.45
	CFM	0.268	0.871	0.47
15	CFOM	0.465		0.21
	CFM	0.416	0.62	0.2
16	CFOM	0.527		0.3
	CFM	0.465	0.812	0.31
21	CFOM	0.467		0.07
	CFM	0.417	0.731	0.06
26	CFOM	0.774		4.2
	CFM	0.668	0.979	4.1
36	CFOM	0.685		7.2
	CFM	0.635	1.215	6.2

The full microscopic CFM is also tested to fit the data using both the real and imaginary CFPs that are displayed in Figure 2 and described in Eqs. (9 and 10). Within the CFM, the central potential is: (14) $U(R)=V_C(R)-N_{RCF}{V}^{CF}(R)-i{N}_{ICF}{W}^{CF}(R)$(14) this approach is denoted as the (Real CFP + Imag. CFP). The concerned data is fitted utilizing two varying parameters, N_RCF and N_ICF, which are the renormalization factors for real and imaginary CFPs, respectively. As illustrated in Figures 11 and 12, the consistency between the experimental ADs and the calculated values using the (Real CFP + Imag. CFP) approach is reasonably good. Table 2 gives the optimal potential parameters extracted from the CFM analysis. The average extracted N_RCF value is 0.396 ± 0.143, indicating that the real CFP strength must be reduced by ∼ 60% to fit the experimental data reasonably.

Figure 11. Experimental ²⁸Si(⁷Li,⁷Li)²⁸Si elastic scattering ADs (solid circles) at E_lab=  8.5, 9, 10, 11, 11.5, and 13 MeV versus calculations using the (Real CFP + Imag. CFP) approach (solid curves). Data is displaced by a factor of 0.5.

Figure 12. Same as Figure 11 but at E_lab=  15, 16, 21, 26, and 36 MeV.

As the reaction cross sections (σ_R), real (J_V) and imaginary (J_W) volume integrals are important quantities that might be utilized to observe the existence or absence of the usual TA in ⁷Li + ²⁸Si system. We are discussing the energy dependence on these quantities. The extracted J_V and J_W values from the performed analysis within the framework of the different utilized approaches showed that the so-called breakup threshold anomaly (BTA) [40] is well presented in the ⁷Li + ²⁸Si system, which agrees well with previous studies [10–17]. As shown in Figure 13, the extracted J_V and J_W values do not obey the standard dispersion relation. The extracted J_V and J_W values from the different considered approaches are close to each other and exhibit the same trend which gives an evidence for the consistency of the adopted potentials. We also note that, while the various potentials show similar fits with data, as shown in Figures 3–12 and the χ values listed in the tables are so close to each other, the presence of BTA can be further confirmed if experimental data are measured at backward angles and higher energies, where finer investigation on the considered potentials can be performed.

Figure 13. Energy dependence on the J_V (a) and J_W (b) values obtained from the different implemented approaches.

We have studied the energy dependence on the extracted σ_R values from the different implemented approaches as depicted in Figure 14, and it is found that they are nearly close to each other. This dependence shows that the σ_R values increase with increasing energy and can be approximated as: ${\sigma }_R(E)=-2597.9+449.3E-15.99E^2$.

Figure 14. Energy dependence on σ_R values for the ⁷Li + ²⁸Si system obtained from the different implemented approaches.

4. Summary

The ⁷Li + ²⁸Si elastic scattering ADs at eleven data sets in the energy range 8.5–36 MeV are studied using different approaches utilizing various nuclear potentials. The analyzed data covers the energy range below, near and above the E_C for ⁷Li + ²⁸Si system. Therefore it is interesting to explore the mechanism of interaction as well as the different peculiarities that are presented in this energy range for ⁷Li + ²⁸Si system. Five different approaches are implemented in data analysis as summarized below

1. Analysis using phenomenological OM potential of fixed radius parameters was successful in fitting the data, the obtained potential parameters showed the absence of the usual TA.

2. Within the frame work of the microscopic DF CDM3Y6 and Sao Paulo potentials, the considered data were analyzed. In both (Real DF + Imag. DF) and (Real SPP + Imag. SPP) approaches, the imaginary part was considered microscopically as a factor times the real folded potential. The analysis showed the need to reduce real potential strength by ∼ 63% and 72%, respectively.

3. Motivating by the well-observed α + t cluster structure for ⁷Li, the considered Data is investigated utilizing both CFOM and CFM. The analysis again confirmed the need to decrease the real CFP strength by ∼ 55% and 60%, respectively, to fairly reproduce the data.

4. The significant coupling effect to the breakup channel is responsible for the reported reduction in real potential strength created based on microscopic approaches such as DF and CFP.

Acknowledgement

This work was funded by the University of Jeddah, Jeddah, Saudi Arabia, under grant No. (UJ-21-DR-146). The authors, therefore, acknowledge with thanks the University of Jeddah technical and financial support.

Disclosure statement

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

Additional information

Funding

This work was funded by the University of Jeddah, Jeddah, Saudi Arabia, under grant No. (UJ-21-DR-146).