## 2 Materials and Methods
### Folding Potential  
The choice of NN interaction is crucial in the analysis of the folding model. In M3Y [14], effective NN interactions are used to derive these potentials. The M3Y effective potential is derived by adapting G-matrix elements based on the Reid-Elliott NN interaction. The parameterized version of the M3Y interaction, introduced by [PERSON] and [PERSON], is as follows [46] with an arrangement of neglection of coulomb interaction term:  
\[V_{NN}(s)=\left[7999\;\frac{\exp(-4s)}{4s}-2134\frac{\exp(-2.5s)}{2.5s}\;\right] \tag{6}\]  
Phenomenological models generally neglect detailed internal structures of the projectile and target cores. Elastic scattering data can be evaluated using HF-based models,  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & \(\mathbf{t_{\theta}}\) & \(\mathbf{t_{1}}\) & \(\mathbf{t_{2}}\) & \(\mathbf{t_{3}}\) & \(\mathbf{t_{4}}\) & & & & & & \\  & (**MeV.fm\({}^{3}\)**)  
which substitute the phenomenological real potentials with microscopically derived ones. The Double Folding (DF) Potential is used in the nn interaction [28; 46]. \[V_{F}(\stackrel{{\rightarrow}}{{R}})=N\iint\rho_{1}(\stackrel{{ \rightarrow}}{{r}}_{1})\rho_{2}(\stackrel{{\rightarrow}}{{r}}_{2})V _{NN}(\stackrel{{\rightarrow}}{{s}})dr_{1}dr_{2}, \tag{7}\]  
where \(V_{F}(\stackrel{{\rightarrow}}{{R}})\) is the folded potential between the projectile and the target nucleus, depending on the relative distance \(R\) between their centers. The vector \((\stackrel{{\rightarrow}}{{s}}=\stackrel{{ \rightarrow}}{{R}}+\stackrel{{\rightarrow}}{{r}}_{2}- \stackrel{{\rightarrow}}{{r}}_{1})\) is the distance between interacting nucleons in the two nuclei, and \(\rho_{1}\) and \(\rho_{2}\) are their respective nucleon density distributions. The relative vector \((\stackrel{{\rightarrow}}{{s}})\) is the distance between the pair of interacting nucleons, while N is the renormalization factor. \(V_{NN}(\stackrel{{\rightarrow}}{{s}})\) gives the effective NN interaction. The schematic representation of the DF is shown in Figure 1.  
In this scenario, a single nucleon exchange between two ions requires some corrections, primarily due to antisymmetrization. During nn scattering, the displacement of the two interacting nucleons is called the \"knock-on exchange\" [31; 64]. The knock-on exchange effect is incorporated into the DF integral through the following formalism:  
\[V^{\prime}_{NN}=V_{NN}+J_{00}\;(E)\delta(s) \tag{8}\]  
The schematic representation of the DF is shown in Figure 1.  
Performing only one of the integrals in Equation (7) yields the NN interaction potential, referred to as the SF potential. The schematic representation of the SF is shown in Figure 2. In this study, Nn interaction potentials are calculated using SHF-WS-based nucleon densities and energy-dependent NN effective interactions within the SF framework. Single Folding Potential  
In this context, the nucleon density of the target nucleus is denoted by \(\rho(\stackrel{{\rightarrow}}{{r}})\), while the effective NN interaction is represented by \(V_{NN}\)[28; 31; 46; 64]. Accurate specification of  
Figure 1: Schematic diagram of nn interaction in DF potential calculations. A: nucleon number, n: nucleus, \(\rho_{1}\),\(\rho_{2}\): nucleon density. Figure 2: Diagram of Nn interaction based on SF potential. A: nucleon number, N: nucleon, n: nucleus, \(\rho_{2}\): nucleon density. the NN interaction plays a vital role in folding model calculations. The SF potential for NN elastic scattering is given by  
\[V_{SF}(\overrightarrow{R})=\int\rho(\overrightarrow{r})V_{NN}\bigg{(}\bigg{|} \overrightarrow{R}-\overrightarrow{r}\bigg{|}\bigg{)}dr. \tag{9}\]  
In the case of neutron-nucleus (i.e., Nn) scattering, the incoming neutron is treated as a point-like particle with no internal structure. Therefore, the folding model reduces to a single-folding expression. Accordingly, if only one of the integrals in Equation (7) is performed, the result--as shown in the following equation--reduces to a single nn potential.