Simulation-Based Determination of Angular Differential Cross Sections of (n,el) Scattering via Evaluated Interaction Potentials

Abstract

In this study, the nucleon density distributions of tungsten isotopes were calculated using the Skyrme interaction within Hartree–Fock approximation model based on the Woods–Saxon potential. Then, using these densities, the interaction potentials were calculated for the elastic scattering of neutrons from tungsten isotopes through the Single Folding method. The obtained potentials were used as input parameters in the TALYS 1.96 nuclear reaction simulation code to evaluate the angular differential cross sections of the ^{182,183,184,186}W(n,el) scattering. The evaluated angular differential cross sections were graphically compared with theoretical values found in the literature as well as experimental data available in EXFOR nuclear data library.

1. Introduction

One of the main challenges in nuclear physics and related fields is the high cost associated with experimental studies. The first requirement to overcome this difficulty is that the experiments to be conducted must be highly efficient. Consequently, it is essential to plan, model, and simulate experiments in advance to optimize their efficiency and reduce costs. Therefore, in research and development studies in nuclear physics, nuclear reactions, and related fields, modeling of nuclear systems and simulation of experiments are necessary to ensure rapid progress, reduce costs, and increase safety.

Nuclear modeling refers to the process of using theoretical calculations to understand the structure and reactions of nuclei, ranging from light-mass to heavy and neutron-rich nuclei. It involves microscopic calculations based on interactions between nucleons to make predictions that can be compared with experimental data.

In recent years, significant advancements have been made in the field of neutron elastic scattering, both experimentally and theoretically. In particular, dispersive optical model provide more accurate parameterizations in energy regions where traditional optical model potentials (OMP) prove insufficient, offering a more consistent explanation of the relationship between nuclear structure and scattering data [1,2]. Additionally, ab initio-based calculations—especially for light nuclei—have demonstrated that nucleon–nucleon (NN) interactions directly influence elastic scattering cross sections [3].

In parallel with these developments, high-precision experimental studies have also increased. For instance, facilities such as Oak Ridge National Laboratory (Oak Ridge, TN, USA) and n_TOF-CERN (Geneva, Switzerland) have produced neutron scattering data with high angular resolution in the energy range of 1–20 MeV [4,5]. Such data are critically important for validating theoretical models. Additionally, studies involving deformed nuclei—particularly heavy targets like tungsten—have highlighted the sensitivity of elastic scattering patterns to deformation parameters [6,7].

For example, Watanabe et al. [8] demonstrated that nuclear deformation parameters are 15–35% larger compared to traditional models by using high-energy heavy-ion scattering data. This finding underscores the importance of microscopic coupled-channel calculations for more accurately modeling the effects of deformation on elastic scattering cross sections. Similarly, Gkatis et al. [9] made critical contributions to the validation of theoretical models by obtaining differential cross sections from high-precision neutron scattering experiments on the ⁵⁴Fe nucleus, employing time-of-flight techniques and advanced detector systems.

On the theoretical side, Yang et al. [10] employed deep learning methods to predict the neutron density distribution and neutron skin thickness of the ⁴⁸Ca nucleus, notably improving model accuracy for large-angle scattering. Additionally, Catacora-Rios and colleagues [11] applied Bayesian approaches to reduce uncertainties in elastic scattering and single-nucleon transfer reaction cross sections, achieving significant improvements particularly for the ⁴⁸Ca and ²⁰⁸Pb nuclei.

The Skyrme–Hartree–Fock Woods–Saxon (SHF-WS)-based density distributions and single-folding potential approach used in this study are consistent with these modern methods and demonstrate high agreement with experimental data in neutron elastic scattering simulations involving tungsten isotopes. However, for more precise modeling of deformation effects and reduction of theoretical uncertainties, the integration of advanced microscopic models and statistical analyses will constitute a crucial direction for future research.

Validating simulation results of nuclear systems against existing experimental data enhances the credibility and reliability of the simulation software. This is why experimentally obtained nuclear data are needed for the development and validation of such modeling. Such data are available from the IAEA Experimental Nuclear Reaction Data library (EXFOR (Vienna, Austria)) [12]. In cases where experimental data cannot be obtained, validated simulated data, as mentioned above, are used. Accordingly, simulation-generated databases such as ENDF (Vienna, Austria) [13] and TENDL (Paris-Saclay, France) [14] provide valuable reference points for experimental studies. These simulated databases are based on well-accepted nuclear reaction calculation software (TALYS 1.96 (Paris-Saclay, France) [15], EMPIRE (Upton, MA, USA) [16] (available online: https://www-nds.iaea.org/empire/, accessed on 30 May 2025), etc.). The data presented in these computed nuclear databases are used by researchers all over the world to simulate various systems and phenomena—including nuclear reactors, radiation detectors, and particle accelerators—through deterministic modeling codes.

In addition, simulation of experiments and related studies are essential to understand current experimental research and to further develop existing theories. Verified simulation programs offer a great convenience in performing such calculations. In this way, making the necessary calculations before conducting the intended experiments allows us to be better prepared for the conditions that may arise during the experiments, as well as for the results that might be obtained.

In short, simulation programs are used:

• To design experimental systems tailored to specific research objectives,
• To compare experimental results with simulations to determine error margins and assess the accuracy of the analysis,
• To test the validity and reliability of existing theoretical models.

Each particle in nuclear matter, at single-particle energy levels, moves in a mean field formed by the surrounding particles. The nucleons that make up atomic nuclei interact with each other through the nuclear force. These nuclear forces are short-ranged, extremely strong, and attractive. Nuclear forces are characterized as short-ranged, highly attractive, and strong, counterbalancing the repulsive components of the mean field.

There are two primary factors that support the existence of effective, short-range, and hence saturable nuclear forces:

• Binding energy per nucleon;
• Nuclear density.

Hartree–Fock (HF) calculations using phenomenological nuclear forces commonly employ the force developed by Skyrme [17,18], and these are known as SHF calculations [19]. In nuclear physics, many phenomenological forces are used, whose parameters are tuned according to experimental results and perform successfully; however, most are designed for specific purposes only. With the introduction of the Skyrme forces, HF calculations began to be widely applied in nuclear physics. Since then, these calculations have been used in a variety of contexts, such as determining nuclear deformation properties, describing superheavy nuclei, modeling vibrational states, and analyzing heavy-ion collisions. Notably, these applications are especially effective in describing the ground-state properties of spherical nuclei.

The writing of the HF equations is simplified by the analytically solvable Skyrme interaction. In its most general form, the SHF method, which incorporates the shell model, describes a system in which a nucleon moves independently within an average central potential formed by other nucleons. However, due to the quantitative nature of the formalism, such calculations—especially those requiring higher-order correlations—can become increasingly complex. Another empirical method is to fit the two-body matrix elements derived from a realistic NN interaction to experimental data [20,21]. In nuclear physics, especially for finite many-body systems, a general energy density functional is often phenomenologically adjusted to match experimental results. In this way, many-body correlations—i.e., in-medium effects—are effectively incorporated through an effective potential [22]. Many relativistic and non-relativistic density functional theories (DFT) have been developed and are still successfully applied to describe a wide range of nuclear phenomena, from the heaviest to the lightest nuclei, from exotic states to the valley of β-stability, and from excited to ground states. Commonly used energy functionals include the relativistic energy functionals [23,24,25], finite-range Gogny/BCP(Barcelona–Catania–Paris)/M3Y(Michigan 3 Yukawa) interactions [26,27,28], and the Skyrme energy functional [22,29,30]. Interestingly, atomic nuclei exhibit single-particle motion well described by the mean-field approximation. The M3Y interaction was chosen due to its well-established performance in folding model analyses and its compatibility with experimental elastic scattering data across a wide energy range. In this study, the NN interaction potential used in the folding model calculations is based on the M3Y effective interaction. The M3Y interaction was developed in the late 1970s by Bertsch, Love, and collaborators, using G-matrix elements derived from the Reid-Elliott NN potential [28,31]. This interaction represents the direct component of the effective NN force in the nuclear medium and models it as a sum of three Yukawa functions, which gives the model its name.

In its original form, the M3Y interaction is energy-independent. However, to simulate the effects of the nuclear medium, density dependence can be introduced phenomenologically, leading to the widely used DDM3Y (density-dependent M3Y) version in the literature [32,33,34]. With this modification, the M3Y interaction becomes highly suitable for deriving both NN and nucleus–nucleus (nn) potentials within the framework of the folding model.

The M3Y interaction has been particularly successful in reproducing experimental data in folding model analyses of elastic scattering at intermediate energies [35]. Furthermore, its analytical form allows for efficient numerical implementation, making it a practical and reliable choice in constructing microscopic OMP [36].

The SHF method can be used to determine essential nuclear properties such as binding energy, nucleon density, electromagnetic moments, and mass and charge radii of certain heavy, radioactive nuclei [19,29].

Essentially, a nucleon incident on a nucleus can either scatter elastically or induce various nuclear reactions. To model this phenomenon, the many-body nature of the interaction is approximated by a single-body (effective) potential. Bethe’s [37] study of neutron reactions using a purely real potential failed to reproduce physical observables consistent with experimental data. The inability of such a real potential to account for inelastic processes was addressed by Bohr’s compound nucleus model [38]. However, the Bohr model was also found to be inadequate at higher incident energies. The OMP was introduced as a solution, defined as a complex potential with both real and imaginary components—an analogy to the scattering of light in an optical medium. This concept was first implemented in a semi-classical framework by Fernbach et al. [39]. The use of a complex potential to describe proton scattering was initially applied by Ostrofsky et al. [40]. The first formal expression of the OMP in the literature used a square well potential [41]. Later, Le Levier and Saxon [42] performed a complete quantum mechanical calculation using a complex potential to describe the scattering of 17 MeV protons by aluminum, successfully reproducing the then imprecise experimental data. OMP remains a widely used model in nuclear physics today. For example, Koning and Delaroche [43] calculated global OMP parameters for spherical nuclei with mass numbers in the range 24 $\le$ A $\le$ 209 and for incident energies between 0.001 $\le$ E $\le$ 200 MeV. Arnould and Goriely [44] employed the optical model (OM) in studies related to nuclear astrophysics, while Herrmann [45] applied it in inelastic scattering calculations.

In the folding model, the nn potential can be obtained by integrating the effective NN interaction over the matter distributions of the two nuclei, as detailed by Satchler and Love [46,47]. In the specific case of scattering involving a single nucleon and a target nucleus, the real part of the OMP for nucleon–nucleus (Nn) scattering can be derived by integrating the effective NN interaction with the density distribution of the target nucleus [46].

Greenlees et al. [48] used the folding model potential to successfully explain nn elastic scattering data. In this approach, the Nn potential is calculated as a single-fold integral of the target’s density distribution with a suitable effective NN interaction. Accordingly, the Single Folding (SF) model is established using the standard single-fold formulation of the OMP [46].

In this study, the elastic scattering of neutrons from tungsten isotopes was investigated through a simulation-based approach comprising three main stages:

The nucleon density distributions for the selected tungsten isotopes (¹⁸²W, ¹⁸³W, ¹⁸⁴W, and ¹⁸⁶W) were first calculated using the Skyrme–Hartree–Fock (SHF) method with a Woods–Saxon (WS) potential framework. Based on the obtained density distributions, Nn interaction potentials were derived using the SF method and compared with values reported in the literature. These interaction potentials were then implemented as input into the TALYS 1.96 nuclear reaction simulation code to compute the angular differential cross sections for neutron elastic scattering.

To evaluate the accuracy and validity of the modeling approach, the simulation results were compared graphically with both theoretical predictions from the literature and experimental data. The theoretical potential set used in the cross-section calculations was taken from recent studies available via the NRV database (Dubna, Russia) (available online: http://nrv2.jinr.ru, accessed on 30 May 2025) and generated using the Fresco nuclear reaction code (Guildford, UK) (available online: https://www.fresco.org.uk, accessed on 30 May 2025) [49,50,51]. The experimental cross-section data were obtained from the EXFOR database [12].

2. Materials and Methods

The nucleon density distributions of tungsten isotopes were calculated using the SHF-WS model, which is based on the WS potential [52]. The nucleon distribution describes how protons and neutrons are spatially arranged within an atomic nucleus. This arrangement is determined by the strong nuclear force that binds the nucleons together. Typically, a parametric form is assumed, and its parameters are fitted to experimental data to determine the distribution. For the nuclear potential ($V_N$), emphasizing the short-range nature of the nuclear force, it is described as

$$V_N(\overrightarrow{r})=-V_0\rho (\overrightarrow{r}).$$

(1)

Here, $V_0$ is expressed as the interaction potential, and $\rho$ is the nucleon density. Equation (1) shows that the number density of nucleons in the nucleus and the nuclear part of the average potential at which nucleons interact in the nucleus have similar radial dependence. Hence, the nuclear potential for nucleons in the nucleus is usually assumed to be

$$V_N\left(\overrightarrow{r}\right)=-{\displaystyle \frac{V_0}{1+exp\left(\frac{\left|\overrightarrow{r}\right|-R}{\alpha }\right)}} ,$$

(2)

where $V_N(\overrightarrow{r})$ is the nuclear potential at a radial position $\overrightarrow{r}$, and $\left|\overrightarrow{r}\right|$ denotes the radial distance from the center of the nucleus (in fm); $V_0$ is the depth of the potential (in MeV), representing the strength of the attractive nuclear force; $R$ is the radius parameter (in fm), typically defined as $R=r_0{A}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}$, where $A$ is the mass number; and $\alpha$ is the surface diffuseness parameter (in fm), which determines how rapidly the potential transitions from its maximum value to zero at the nuclear surface.

This form of the potential accounts for the finite size and diffuse edge of the nucleus, providing a more realistic description compared to a simple square well. This function is called the WS potential [52]. It should be noted that if a spherical core is assumed, one can obtain Equation (3) for the WS-type functional form, which is a widely adopted choice [53]:

$$\rho \left(\overrightarrow{r}\right)={\displaystyle \frac{{\rho }_0}{1+exp\left(\frac{\left|\overrightarrow{r}\right|-R}{\alpha }\right)}} ,$$

(3)

where ${\rho }_0$ is the central density. In this case, the density depends only on the magnitude of the spatial coordinate, $\left|\overrightarrow{r}\right|$. This reflects the assumption of spherical symmetry in the system. While calculating the density with the SHF-WS method, nucleon displacement was made according to the shell model.

The neutron elastic scattering energies at which nucleon density distributions were evaluated using the SHF-WS model for the tungsten isotopes ^{182,183,184,186}W are as follows: for ¹⁸²W, 1.5, 2.5, 3.4, 4.87, 6, and 14 MeV; for ¹⁸³W, 3.4 and 14 MeV; for ¹⁸⁴W, 1.5, 2.5, 3.4, 4.84, 6, and 14 MeV; and for ¹⁸⁶W, 1.5, 2.5, 3.4, and 14 MeV. These were used to obtain the interaction potential between the projectile and the target in scattering reactions. In these calculations, the projectile–target interaction is modeled by integrating the NN interaction over the density distributions of both the projectile and the target nucleus. The single-fold integral referred to here is the SF Equation. These calculations, using nucleon density data, were performed by writing a code in a mathematical software.

Transforming the problem into momentum space via Fourier transformation simplifies the calculations by reducing the complexity inherent in coordinate space. The interaction potentials ($V_0$) derived from the calculations were fitted to a Gaussian form to obtain theoretical potential parameters. The theoretical potential sets used for the scattering and cross-section analyses were taken from the NRV website and obtained with the help of the Fresco nuclear reaction code [49,50,51]. The interaction potential parameters calculated using the densities obtained for the SHF-WS model are compared with the theoretical data and tabulated.

2.1. Skyrme–Hartree–Fock Model

The HF method [54] provides an approximate solution to determine the ground state energy and wave function of a quantum many-body system. In the HF approximation, the harmonic oscillator or WS [33] wave functions are proposed. Based on these wave functions, nucleon densities ($\rho (\overrightarrow{r})$) are generated. Then, iterations are performed between the density ($\rho (\overrightarrow{r})$) and the energy potential ($U(\overrightarrow{r})$) for the wave function ($\phi (\overrightarrow{r})$).

In the HF approximation, employing a density-dependent effective Skyrme-type NN interaction [17,18] facilitates analytical tractability and computational efficiency. Appropriate Skyrme parameter sets can be found by comparing with experimental results.

The Skyrme interaction used in the calculation of nuclear properties is defined in its simplest form as follows:

$${\overrightarrow{V}}_{Skyrme}={\sum }_{i<j}{\overrightarrow{V}}_{i,j}+{\sum }_{i<j<k}{\overrightarrow{V}}_{i,j,k}.$$

(4)

Here, the first term refers to the two-body interaction, and the second term refers to the three-body interaction.

Writing the Skyrme energy density in terms of densities allows the Skyrme interaction to be obtained by directly deriving the HF equations, without calculating the two- and three-body matrix elements. Here, the energy density $H_{Skyrme}({\rho }_q,{\tau }_q,{\overrightarrow{J}}_q)$ for the Skyrme interaction is an algebraic function of the nucleon densities (${\rho }_q$), kinetic energy (${\tau }_q$), and spin–orbit (${\overrightarrow{J}}_q$) densities.

$$E_{Skyrme}=\int H_{Skyrme}({\rho }_q,{\tau }_q,{\overrightarrow{J}}_q)d^{3}r$$

(5)

The form of the Skyrme interaction simplifies the HF equations, making them solvable with standard numerical techniques. The SHF model, based on the shell model framework [55,56], in its most general form, describes a system in which a nucleon moves independently within an average central potential formed by other nucleons.

To calculate the nucleon densities of nuclei with the Skyrme-interactive HF method, the computer program Hartree–Fock [57], written in Fortran, is used. These evaluated nucleon densities were used in the SF method to calculate the interaction potentials of neutron and tungsten isotopes.

When using the Hartree–Fock code, protons and neutrons for each nucleus are placed into energy levels in accordance with the WS potential shell model. These assignments are processed separately and incorporated into the subroutines of the code. The subroutines perform calculations for 11 widely used Skyrme parameter sets—namely, Ska [58], GS6 [59], SKM* [60], SGII [61], SLy4, SLy5, SLy6, SLy7, SLy10 [62], SLy8, and SLy9 [63]. These are among the most frequently employed sets out of over 240 SHF parameterizations.

Although a spherical WS potential is used in this study, deformation effects are implicitly accounted for through the parametrization of nucleon densities obtained via the HF method. In this approach, the occupation of orbitals in shell-model reflects the aspects of nuclear deformation, particularly through the density distribution function. Though the WS potential is spherical, the occupation of higher-ℓ orbitals in the HF calculation reflects quadrupole deformation, which modifies the radial shape and spatial asymmetry of the resulting density distribution.

The parameters of the Skyrme interaction are obtained by fitting the HF results to experimental data. In this study, the SLy4 Skyrme parametrization was employed due to its proven success in accurately describing the ground-state properties of heavy nuclei, including neutron-rich isotopes such as tungsten.

Table 1. Skyrme parameter sets used in this study and their corresponding values for effective interaction parameters.

	$t_0$ $(\mathbf{M}\mathbf{e}\mathbf{V}{.\mathbf{f}\mathbf{m}}^3)$	$t_1$ $(\mathbf{M}\mathbf{e}\mathbf{V}{.\mathbf{f}\mathbf{m}}^5)$	$t_2$ $(\mathbf{M}\mathbf{e}\mathbf{V}{.\mathbf{f}\mathbf{m}}^5)$	$t_3$ $(\mathbf{M}\mathbf{e}\mathbf{V}{.\mathbf{f}\mathbf{m}}^{3\mathsf{\alpha }})$	$t_4$ $(\mathbf{M}\mathbf{e}\mathbf{V}{.\mathbf{f}\mathbf{m}}^5)$	$x_0$	$x_1$	$x_2$	$x_3$	$\alpha$
Ska	−1602.78	570.88	−67.70	8000	125	−0.02	0	0	−0.286	1/3
GS6	−1012	209	−76.3	10,619	105	0.139	0	0	1	1
SKM*	−2645	410	−135.0	15,595.0	130	0.09	0	0	0	1/6
SGII	−2645	340	−41.9	15,595	105	0.09	−0.0588	1.425	0.06044	1/6
SLy4	−2488.91	486.82	−546.39	13,777	123	0.834	−0.344	−1	1.354	1/6
SLy5	−2483.45	484.23	−556.69	13,757	125	0.776	−0.317	−1	1.263	1/6
SLy6	−2479.50	462.18	−448.61	13,673	122	0.825	−0.465	−1	1.355	1/6
SLy7	−2480.8	461.29	−433.93	13,669	125	0.848	−0.492	−1	1.393	1/6
SLy8	−2481.41	480.78	−538.34	13,731	125	0.8024	−0.3424	−1	1.3061	1/6
SLy9	−2511.13	510.6	−429.8	13,716	110	0.7998	−0.6213	−1	1.3727	1/6
SLy10	−2506.77	430.98	−304.95	13,826.41	105	1.0398	−0.6745	−1	1.6833	1/6

presents several commonly used Skyrme parameter sets from the literature. These sets define the strengths of the effective NN interactions via adjustable parameters: $t_0$, $t_1$, $t_2$, $t_3$, $t_4$ are coefficients representing different components of the effective interaction, including central, momentum-dependent, and density-dependent terms, while $x_0$, $x_1$, $x_2$, $x_3$ are dimensionless parameters that modify the strength of each corresponding t-term, allowing for isospin dependence in the interaction, and α is the exponent of the density-dependent term ($t_3$), controlling the nonlinearity of the density dependence.

Together, these parameters allow the Skyrme interaction to simulate a wide range of nuclear phenomena by effectively capturing the bulk and shell properties of nuclei.

2.2. 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 Satchler and Love, is as follows [46] with an arrangement of neglection of coulomb interaction term:

$$V_{NN}\left(s\right)=\left[7999 {\displaystyle \frac{\mathrm{e}\mathrm{x}\mathrm{p}\left(-4s\right)}{4s}}-2134{\displaystyle \frac{\mathrm{e}\mathrm{x}\mathrm{p}\left(-2.5s\right)}{2.5s}} \right]$$

(6)

Phenomenological models generally neglect detailed internal structures of the projectile and target cores. Elastic scattering data can be evaluated using HF-based models, which substitute the phenomenological real potentials with microscopically derived ones. The Double Folding (DF) Potential is used in the nn interaction [28,46].

$$V_F(\overrightarrow{R})=N\iint {\rho }_1({\overrightarrow{r}}_1){\rho }_2({\overrightarrow{r}}_2)V_{NN}(\overrightarrow{s})d{r}_{1}d{r}_2,$$

(7)

where $V_F(\overrightarrow{R})$ is the folded potential between the projectile and the target nucleus, depending on the relative distance $R$ between their centers. The vector $(\overrightarrow{s}=\overrightarrow{R}+{\overrightarrow{r}}_2-{\overrightarrow{r}}_1)$ 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 ($\overrightarrow{s}$) is the distance between the pair of interacting nucleons, while N is the renormalization factor. $V_{NN}\left(\overrightarrow{s}\right)$ 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:

$$V_{NN}^{\prime }=V_{NN}+J_{00 }(E)\delta (s)$$

(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 $\rho (\overrightarrow{r})$, while the effective NN interaction is represented by $V_{NN}$ [28,31,46,64]. Accurate specification of the NN interaction plays a vital role in folding model calculations. The SF potential for Nn elastic scattering is given by

$$V_{SF}(\overrightarrow{R})=\int \rho (\overrightarrow{r})V_{NN}\left(|\overrightarrow{R}-\overrightarrow{r}|\right)dr.$$

(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.

$$V_{SF}(\overrightarrow{R})=\int {\rho }_2({\overrightarrow{r}}_2)V_{NN}\left(|\overrightarrow{R}-{\overrightarrow{r}}_2|\right)d{r}_2,$$

(10)

where ${\rho }_2({\overrightarrow{r}}_2)$ is the target nucleon density and $V_{NN}\left(|\overrightarrow{R}-{\overrightarrow{r}}_2|\right)$ is the effective NN interaction (see Equation (6)).

The latter can itself be expressed by folding over the projectile’s internal coordinate,

$$V_{NN}\left(\left|\overrightarrow{R}-{\overrightarrow{r}}_2\right|\right)=\int d{r}_2{\rho }_2\left({\overrightarrow{r}}_2\right)V(\overrightarrow{s}).$$

(11)

The vector $\overrightarrow{s}=\overrightarrow{R}+{\overrightarrow{r}}_2$ represents the relative distance between the incident nucleon and a target nucleon. The specific form of the effective NN interaction $V_{NN}$ employed in this study is given in Equation (6).

The interaction potentials, evaluated through the SF method as functions of incident neutron energy, were used as input parameters in the TALYS 1.96 [15] nuclear reaction code to compute angular differential cross sections for elastic neutron scattering.

It is widely accepted that the interaction potential is density-dependent, as the folding integral (Equation (7)) inherently relies on the density profiles of both interacting nuclei. An experimental NN potential includes effects that depend only on the density of a single nucleus. When employing the phenomenological potential in the SF integral (Equation (10)), the density contribution of the second nucleus is generally omitted. However, the $V_{NN}$ potential (Equation (11)), though not explicitly, still reflects density-dependent effects that must not be ignored [65,66].

The HF method incorporating an effective Skyrme-type NN interaction is utilized in this study. For this purpose, codes which use HF approximation based on various wave functions are widely used. This approach enables the calculation of ground-state properties—such as binding energy per nucleon, charge density distributions, and root-mean-square (rms) charge radii—particularly for closed-shell nuclei.

In this study, the ground-state properties of the ^{182,183,184,186}W isotopes were calculated using the SHF method implemented via a Fortran-based computational code [67]. The Hartree–Fock code basically uses the WS wave function as the single-particle trial function. In this study, the ground state properties of the ^{182,183,184,186}W isotopes were analyzed using this code.

2.3. TALYS 1.96

TALYS 1.96 [15] is a nuclear reaction simulation code developed for the analysis and prediction of nuclear processes, written in Fortran and designed to operate on Linux-based systems. Its primary purpose is to simulate nuclear reactions induced by various projectiles—including neutrons, protons, deuterons, tritons, ³He, α-particles, and photons—over an energy range of 1 eV to 200 MeV for target nuclei with mass numbers $\ge 12$. TALYS 1.96 integrates multiple nuclear reaction models into a unified code structure to ensure broad applicability and accurate predictions. This allows for the evaluation of nuclear reactions from the resonance range to medium energies.

In the absence of experimental data, TALYS 1.96 [15] can generate theoretical nuclear data for all open reaction channels over user-defined energy and angular intervals, either in default mode or by adjusting model parameters to better align with available measurements. TALYS 1.96 output files can include diverse data such as total and partial cross sections, angular distributions for elastic and inelastic channels, discrete level populations, isomeric and ground-state cross sections, (n,xn), (n,xp), and other reaction products, as well as energy and double-differential spectra.

An example input file used in this study for TALYS 1.96 simulations is shown in Figure 3, which illustrates the structure and parameters employed.

2.4. The Numerical Calculations

• The nucleon densities, $\rho$, are calculated using the SHF-WS model.
• The interaction potential, $V(\overrightarrow{r})$, is constructed via the folding integral.
• The Schrödinger equation is solved numerically.
• The scattering amplitude, $f(\theta )$, and the differential cross section, $d\sigma /d\mathsf{\Omega }$, are computed.
• Results are compared with experimental data.

In the calculation of the differential cross section for neutron elastic scattering, the fundamental physical quantity is the interaction potential between the target nucleus and the neutron. This potential is determined microscopically using the folding model and subsequently converted into the differential cross section through scattering theory within the framework of quantum mechanics.

Satchler and Love [46] explained how the folding-model-based OM can be used in solving the Schrödinger equation to obtain the differential cross section. Within this framework, the scattering problem is typically solved through the following steps.

2.4.1. Generation of the Folding Potential

To determine the interaction potential microscopically, the folded potential $V_F$ is obtained using the nucleon density of the target nucleus, ${\rho }_1({\overrightarrow{r}}_1)$, and the NN interaction potential, $V_{NN}(\overrightarrow{s})$. This process is defined by the integral Equation (7).

2.4.2. Incorporation of the OMP into the Schrödinger Equation

The folded potential $V_F$ is used within the Schrödinger equation to obtain the wave function $\phi (\overrightarrow{r})$:

$$\left[{\displaystyle \frac{{\mathbf{\hslash }}^2}{2\mu }}{\nabla }^2+V\left(\overrightarrow{r}\right)\right]\phi \left(\overrightarrow{r}\right)=E\phi (\overrightarrow{r})$$

(12)

Here, $\mu$ is the reduced mass of the system and $E$ represents the energy eigenvalue.

2.4.3. Calculation of the Cross Section via Partial Wave Expansion

Using the asymptotic solution of the wave function, the scattering amplitude $f(\theta )$ calculated, and this amplitude is converted into the differential cross section as:

$${\displaystyle \frac{d\sigma }{d\mathsf{\Omega }}}={\left|f(\theta )\right|}^{2.}$$

(13)

2.4.4. Use of TALYS 1.96 Software

The TALYS 1.96 software, when provided with folded potentials (in this study obtained via the SHF-WS + SF approach) as input, solves the Schrödinger equation and automatically generates differential cross sections by utilizing angular distribution functions expressed through Legendre expansions.

Satchler [47] provided a detailed mathematical formulation of the OM theory and scattering functions. Furthermore, the study by Greenlees et al. [48] experimentally demonstrated how OM parameters relate to nuclear matter radii and scattering data.

2.4.5. Neutron Densities and Folded Potential for W Isotopes

In this study, neutron densities for tungsten isotopes were calculated using the SHF-WS theoretical method. These density distributions represent the spatial arrangement of nucleons within the nucleus and form the fundamental input for the folding potential.

The interaction potential between the tungsten nucleus and the neutron is obtained via the SF integral using these densities and the NN interaction (M3Y-Reid). This potential is Equation (9). The NN interaction potential used is Equation (11). The folded potential is then adapted as an OMP using a WS form. The imaginary part is taken in WS form:

$$U\left(\overrightarrow{r}\right)=V\left(\overrightarrow{r}\right)+iW(\overrightarrow{r}).$$

(14)

For this potential, the parameters $r_v$ and $a_v$ were employed, and the value $V_0$ was obtained.

2.4.6. Calculation and Comparison of Results

The Schrödinger equation is solved numerically using the potential described above. The TALYS 1.96 software was employed for the numerical solution. The scattering amplitude $f(\theta )$ obtained from this solution is then used to calculate the differential cross section $d\sigma /d\mathsf{\Omega }$ as a function of angle. These calculations are performed for various scattering angles, and the resulting data are compared with experimental measurements.

3. Results

Table 2. Calculated theoretical $r_v$ and $a_v$ parameters.

$W$	$r_v(\mathbf{f}\mathbf{m})$	$a_v(\mathbf{f}\mathbf{m})$
182W	1.2324	0.6507
183W	1.2325	0.6506
184W	1.2326	0.6504
186W	1.2329	0.6501

presents a comparison between the interaction potential parameters calculated at various neutron energies using SHF-WS densities and the theoretical values reported by Koning and Delaroche [43]. A neutron incident energy of 14 MeV was selected as a reference point for cross-section calculations across all ^{182,183,184,186}W isotopes. Additional neutron energies were selected in accordance with the available experimental cross-section data for these isotopes.

The spherical neutron parameters’ energy and mass number for nuclei in the range 0.001 $\le$ E $\le$ 200 MeV and 24 $\le$ A $\le$ 209 are given by [43]:

$$r_v=1.3039-0.4054A^{-1/3},$$

(15)

$$a_v=0.6778-1.487\times {10}^{-4}\mathrm{A}.$$

(16)

In the OM, the real volume potential is typically described using a WS form, which includes two important geometrical parameters:

$r_v$: This is the geometric radius parameter of the real central (volume) potential.

$a_v$: This is the diffuseness parameter, controlling the surface smoothness of the real central potential. It defines how smoothly the potential falls off at the nuclear surface. A typical value is around 0.65 fm, though it may vary slightly depending on the specific global or local fit.

These parameters are crucial in determining the shape and strength of the nuclear potential felt by the incoming nucleon (neutron or proton). They are adjusted as part of the fitting process to match experimental scattering and reaction data.

The angular differential cross-section curves, computed using TALYS 1.96 with the $r_v$ and $a_v$ parameters from Table 2 and the interaction potentials $V_0$ listed in

Table 3. Interaction potentials $V_0$ evaluated for incident neutrons at various energies, compared to theoretical values.

Reaction	Energy (MeV)	Experimental $V_0$ (MeV)	Koning and Delaroche $V_0$ (MeV)	SHF-WS $V_0$ (MeV)
182W	1.5	49.5 [68]	−48.234	−47.7239
	2.5	49.0 [68]	−47.893	−47.5772
	3.4	50.1 [6]	−47.588	−47.4452
	4.87	50.3 [69]	−47.092	−47.2297
	6	50.3 [69]	−46.713	−47.0640
	14	NA	−44.090	−45.8927
183W	3.4	49.0 [6]	−47.488	−47.3332
	14	NA	−43.998	−45.7878
184W	1.5	49.5 [68]	−48.032	−47.4708
	2.5	49.0 [68]	−47.693	−47.3254
	3.4	50.1 [6]	−47.389	−47.1947
	4.84	50.3 [69]	−46.895	−46.9812
	6	50.3 [69]	−46.518	−46.8171
	14	NA	−43.907	−45.6567
186W	1.5	49.5 [68]	−47.834	−47.1878
	2.5	49.0 [68]	−47.496	−47.0439
	3.4	50.1 [6]	−47.194	−46.9144
	14	NA	−43.727	−45.391

, are compared with both theoretical predictions and experimental data. The values calculated with the NRV-based FRESCO code, using theoretical parameters from the literature [32,33], alongside the experimental results obtained from the EXFOR database, are presented in Figure 4, Figure 5, Figure 6 and Figure 7 for each tungsten isotope. Neutron incident energies used in the analysis of ^{182,183,184,186}W isotopes were selected based on the corresponding experimental data available in the EXFOR database [12]. In addition, 14 MeV commonly used in neutron scattering experiments was included as a standard benchmark energy. The projectile energies used as input parameters for neutrons are also provided in Table 3.

4. Discussion

Based on nucleon densities obtained via the SF method, interaction potentials were calculated for each tungsten isotope at the corresponding energies reported in EXFOR [12]. It is observed that the evaluated values are in good agreement with the theoretical values, as the differences are around 1–2%. The calculated interaction potentials exhibit strong agreement with theoretical values, with deviations generally remaining within the 1–2% range.

As illustrated in Figure 4, Figure 5, Figure 6 and Figure 7, the angular differential cross-section curves obtained using the SHF-WS + SF + TALYS 1.96 model generally show good agreement with experimental data, particularly in the forward scattering region. Notably, experimental features such as the diffraction-like structures observed around 50^o in the ¹⁸²W(n,el) and ¹⁸⁴W(n,el) reactions are also reproduced in the SHF-WS simulations, supporting its applicability as an alternative microscopic model. Although some discrepancies remain at larger angles—possibly due to limitations in accounting for spin–orbit interactions—the overall shape and energy dependence of the distributions are captured reasonably well. This suggests that the evaluated model employed in this study offers a comparable level of consistency with experimental data, making it a viable alternative to the widely used Koning and Delaroche global OM available on the NRV platform [49,50,51]. At larger scattering angles, although a deviation from theoretical models is observed, the overall shape of the distribution remains qualitatively consistent. We only fail to capture the peak decrease that occurs around 150° of the scattering angle.

Furthermore, the number of oscillatory features (i.e., minima and maxima) in angular differential cross sections increases with higher incident neutron energies. The reason for this is that as the neutron incident energy increases, the interaction also increases, and therefore the scattering centers change. This is attributed to changes in the effective scattering centers, which broaden the angular distribution. In the graphs, we can see that there is less interaction because there are fewer scattering points at low energies. As the energy of the incident neutron increases, the wavelength decreases ($\lambda =h/\sqrt{2mE}$) and the neutron particle starts to scatter back from the nucleons instead of the entire nucleus. Our model effectively captures the energy-dependent enhancements in backward-angle scattering by incorporating detailed Nn interactions through the SF approach, which significantly influences the NN interaction potential. This aspect will be explored in greater depth in future studies.

Most tungsten (W) isotopes, especially the heavier ones (¹⁸²W, ¹⁸³W, ¹⁸⁴W, and ¹⁸⁶W), exhibit non-spherical, i.e., deformed nuclear structures. This deformation can be observed as a deviation from spherical symmetry and is typically quantified by the quadrupole deformation parameter β₂. Due to the asymmetry in their internal structure, deformed nuclei exhibit different scattering behaviors in interactions with incoming particles. This effect becomes particularly pronounced in angular differential cross sections. While spherical nuclei display symmetric scattering patterns, deformed nuclei break this symmetry, resulting in more complex angular distributions.

Nuclear deformation also influences the nucleon density distribution, thereby directly affecting the accuracy of input data used in SF potential calculations, such as the densities obtained from the SHF-WS model. Consequently, accounting for deformation plays a critical role in achieving agreement with experimental data, especially at backward scattering angles.

The HF based models and the folding potential approach employed in this study implicitly incorporate these deformation effects, which contributes to the good agreement observed between the calculated angular differential cross sections and experimental measurements.

Numbered lists can be added as follows:

• ¹⁸²W(n,el) Scattering Reaction (Figure 4)
  • For low-energy neutrons between 1.5 and 3.4 MeV: The observed scattering cross sections are relatively flat, with few minima and maxima. This can be explained by the long neutron wavelength interacting with the entire nucleus, resulting in limited angular resolution. The SHF-WS + SF + TALYS 1.96 model successfully reproduces this flat pattern.
  • At 4.87 and 6 MeV: A pronounced minimum around ~50° is observed in the experimental data, which is also clearly seen in the SHF-WS + SF calculations. This indicates that the model can capture fine angular structures due to the interaction potentials derived from the density distributions.
  • At 14 MeV: The scattering structure becomes more complex with more peaks and valleys at higher energy. This arises from the dominance of nucleon-level interactions due to the shorter wavelength. The simulation model reproduces this pattern quite well.
• ¹⁸³W(n,el) Reaction (Figure 5)
  • Comparison between 3.4 MeV and 14 MeV: At lower energy, wider angular scattering is dominant, whereas at 14 MeV, distinct angular modulations and more peak/trough structures emerge. The model performs successfully in capturing both the general trend and the proximity to experimental data at both energies.
• ¹⁸⁴W(n,el) Reaction (Figure 6)
  • At 4.84 and 6 MeV energy levels: Especially at 6 MeV, a distinct oscillation is observed in the experimental data, which the SHF-WS + SF approach largely reproduces. This result, showing higher sensitivity than the Koning and Delaroche model, suggests a theoretical framework better aligned with the experimental diffraction patterns.
• ¹⁸⁶W(n,el) Reaction (Figure 7)
  • Generally flatter scattering curves with fewer structural variations: The more stable nature of this isotope may contribute to fewer structural features in the scattering spectrum. However, angular differentiations are still observed at 14 MeV, and the model successfully predicts these trends.

4.1. Comparison of the Global OM by Koning and Delaroche [43] and the Microscopic Approach Proposed in This Study Is Presented

In this study, a method for determining the existing parameters to re-evaluate experimental data is presented.

In nuclear physics calculations, parameters are generally derived from the analysis of experimental data for a large number of stable target nuclei. The SHF method has been in use since the 1970s and serves as a fundamental technique for determining nucleon density distributions. The SF method, developed primarily by Satchler and Love since the 1970s, provides a microscopic approach to obtaining the NN interaction potential.

These methods offer high theoretical accuracy but are typically limited to local applications (i.e., specific isotopes or energy ranges).

The work by Koning and Delaroche introduced a parametric global model based on an extensive dataset. It enables broad applicability, software integration, and comprehensive optimization with experimental data. However, it does not incorporate microscopic physics (such as SHF or Folding). This reflects a different methodological choice rather than a deficiency (

Table 4. Comparison of global OM by Koning and Delaroche [43] and microscopic approach proposed in this study.

Feature	Koning and Delaroche [43]	The Microscopic Approach Proposed in This Study (SHF-WS)
Method	Parametric global OMP	Microscopic: SHF + SF
Physical foundation	Systematic fit based on experimental data	Skyrme NN interaction + potential derived from density
Scope of application	Fast and broad (wide mass number and energy range)	Specific nuclei (182W–186W), detailed resolution
Computational approach	Fitted parametric functions	Numerical integration from densities (folding)
Level of innovation	Innovation lies in scope and parametrization, not method	Methodological detail and microscopic foundation

).

While studies such as that of Koning and Delaroche employ global OM parameters, the present work microscopically computes nucleon densities using SHF-WS to calculate interaction potentials, and compares these potentials with actual interaction surfaces. This enables the use of more physically grounded potentials that can be directly correlated with experimental data.

4.2. The Distinction and Contribution of This Study in the Literature

• Original Methodological Approach and Application Scope
  This study advances beyond traditional methods in determining angular differential cross sections for (n,el) elastic scattering reactions by integrating nucleon densities computed using the Skyrme-interaction-based HF model with a WS potential (SHF-WS), the SF model, and implementing them within the TALYS 1.96 nuclear reaction simulation code. This tripartite approach—SHF-WS + SF + TALYS 1.96—has not been widely adopted in the existing literature.
• Use of Density-Based Realistic Potentials in Calculations:
  Whereas studies such as Koning and Delaroche utilize global OM parameters, this work generates interaction potentials by microscopically calculating nucleon densities through SHF-WS and compares these with the realistic interaction surfaces. This enables the use of more physically grounded potentials that are directly relatable to experimental data.
• Comprehensive Surface Analysis Across Energies
  The study evaluates angular differential cross sections for the isotopes ^{182,183,184,186}W across various energy ranges (1.5–14 MeV), providing separate comparisons for each energy point. This broader energy interval and multi-isotope analysis go beyond many previous studies, which were often limited to a narrow set of energies and isotopes.
• High Agreement of Simulation Results with Experimental Data
  When compared with experimental results from the EXFOR database, the study produced highly consistent outcomes, particularly at forward scattering angles and in intermediate energy ranges. This consistency demonstrates the accuracy and validity of the employed model and indicates that it yields more reliable results than traditional models such as that of Koning and Delaroche [43].
• Filling a Gap in the Literature
  The contributions of this study stand out in the following ways:
  • To date, an integrated approach combining nucleon densities calculated using the SHF-WS model for tungsten isotopes, folded into potentials via the SF method, and subsequently used in (n,el) scattering simulations with TALYS 1.96, has not been applied in the literature.
  • This method goes beyond theoretical modeling by enabling direct comparison with experimental data, thereby enhancing the reliability of the simulations.
  • In this regard, the study serves as an interdisciplinary example that bridges theoretical nuclear physics with applied simulation technologies.

5. Conclusions

The SHF-WS-based approach, in combination with the SF model and the TALYS 1.96 simulation framework, yields angular differential cross sections that align closely with experimental observations. Although the global Koning and Delaroche model offers slightly better average agreement across all energies, the SHF-WS model matches or exceeds it in specific cases. These results suggest that the SHF-WS method constitutes a promising and physically grounded alternative, particularly for detailed studies of nuclear structure and scattering in heavy deformed nuclei. By incorporating nucleon densities derived from microscopic principles and realistic folding potentials, this model can serve as a complementary tool alongside traditional OM approaches. Future work may focus on further improving backward-angle accuracy by including spin–orbit interactions and coupling effects. For a more detailed structural analysis, the observed deviations in terms of shell model effects, spin-parity configurations, and collective motions in specific isotopes are discussed below.

Additionally, this model employs the $a_v$ and $r_v$ parameters developed in the 2003 study by Koning and Delaroche. The rationale behind these choices can be explained as follows:

From the shell model perspective, the energy levels, spin-parity configurations, and pairing properties of these isotopes differ. These differences directly influence the scattering cross sections and resonance behavior. The OM represents these variations in an averaged manner, which can sometimes lead to deviations in the model predictions.

Although the global nucleon OMP developed by Koning and Delaroche in [43] is applicable over a wide range of energies and mass numbers, it exhibits discrepancies with experimental data for certain target nuclei and energy regions. One significant reason for these deviations is that the model does not explicitly account for the microscopic structure of the nucleus, particularly its shell structure.

A major part of the discrepancies in Koning and Delaroche’s global OM arises from its inconsistency with the shell model. While the shell model posits that nucleons occupy discrete energy levels that directly influence scattering behavior, the OM represents this interaction as an average potential. This approximation leads to pronounced deviations in the model, especially in cases involving:

• Nuclei near shell closures,
• Resonance structures,
• Spin-parity dependent transitions.

As explained by the shell model, each isotope possesses a unique energy level structure, spin-parity distribution, and collective behaviors, whereas the global OMP represents these features in an averaged, parametric manner. This leads to an inability to fully capture:

• Spin effects in single nuclei such as ¹⁸³W,
• Collective vibrations in open-shell nuclei like ^182,184W,
• Resonance damping in more rigid nuclei such as ¹⁸⁶W.

• The goal of the global model is broad applicability across wide mass (A) and energy (E) ranges.
• Since the shell structure differs for each isotope, a separate parameterization would be required for each.
• Such a detailed approach would compromise the “global” nature of the model.

Therefore:

• The model attempts to represent shell effects by incorporating variations within its parameters.
• However, this approach is insufficient to explain abrupt structural changes around, for example, magic numbers.

Reasons for deviations (due to differences between OMP and shell model):

• Example of ¹⁸³W (single neutron)
  • The ¹⁸³W isotope contains a single neutron, so the spin–orbit splitting affects the angular distribution of scattering.
  • Shell model: The placement of this neutron in a specific orbital (e.g., 2f7/2) creates resonances during scattering.
  • OMP: The spin–orbit potential is fixed, and individual orbitals are not considered deviations occur at large angles.
• ^182,184W (open shell)
  • These isotopes may exhibit collective behaviors from the shell model perspective (e.g., surface vibrations).
  • Shell model: These collective modes cause oscillations or resonance structures in the angular distribution.
  • OMP: Does not include these effects → fine structures in angular distributions cannot be explained, only average behavior is captured.
• ¹⁸⁶W (near closed shell)
  • This isotope has a “stiffer” potential, with less absorption and smoother scattering.
  • Shell model: Surface collective effects are weak, and transitions between levels are limited.
  • OMP: The average parameterization generally fits better here, but shell closure effects at low energies can still make a difference.

When calculating densities using the SHF-WS method, nucleon configurations are assigned according to the shell model.

In the SHF-WS approach, placing protons and neutrons into energy levels based on the shell model allows for more precise calculations. Additionally, the density distribution having a WS form is also effective (Equation (3)).

The parameters $r_v$ (the radius parameter of the real potential) and $a_v$ (the diffuseness parameter) used in the OMP generally yield results that are in good agreement with experimental scattering data. This outcome stems both from the intrinsic structure of the OMP and the physical significance of these parameters, and it can also be understood in relation to the shell model.

Why are $r_v$ and $a_v$ consistent with experimental data?

• ¹⁸⁶W (near closed shell)
  • This isotope has a “stiffer” potential, with less absorption and smoother scattering.
  • Shell model: Surface collective effects are weak, and transitions between levels are limited.
  • OMP: The average parameterization generally fits better here, but shell closure effects at low energies can still make a difference.
• They are geometrical parameters
  • These parameters directly characterize the global structure of the nucleus: the total volume and surface thickness.
  • Especially in heavy nuclei (e.g., ¹⁸²W to ¹⁸⁶W), the nuclear volume and surface properties are relatively independent of shell structure, thus they can be accurately represented by an average effect.
• Regions where the OMP achieves the most accurate fit
  • The most significant contribution in scattering data is related to the overall size of the nucleus, which directly correlates with $r_v$.
  • The surface potential diffuseness affects the spread of the potential and the total cross section; $a_v$ effectively controls this.
  • Therefore, the general shape of the angular distribution can be well fitted using these two parameters.

Shell model:

• Assumes nucleons occupy discrete energy levels in specific orbitals. This level structure mainly affects the internal nuclear structure and transition phenomena (resonances, spin splitting, etc.).

However:

• $r_v$ and $a_v$ represent an average geometrical distribution of nucleon placement.
• Shell effects only cause minor deviations in this average density distribution.
• In other words, shell model details primarily influence reaction dynamics (resonances, fine structures, spin effects), rather than the overall potential shape.

In conclusion:

• In other words, shell model details primarily influence reaction dynamics (resonances, fine structures, spin effects), rather than the overall potential shape.
• Geometrical parameters like $r_v$ and $a_v$ are relatively independent of the shell model.
• Therefore, the absence of shell model details in the OMP does not prevent these parameters from aligning well with experimental data.

The strong consistency observed between the angular differential scattering cross sections simulated using the TALYS 1.96 nuclear reaction code and the EXFOR experimental data—based on nucleon densities calculated via the SHF-WS model and the corresponding SF interaction potentials—provides encouraging evidence for the accuracy and applicability of the theoretical model as a complementary approach.

In particular, the distinct structures observed at approximately 50° in the 4.87 MeV and 6 MeV energy regions of the ¹⁸²W(n,el) and ¹⁸⁴W(n,el) reactions are not only clearly present in the experimental data but are also accurately reproduced using the SHF-WS + SF + TALYS 1.96 approach. These features, which are not well captured by traditional OMP (e.g., Koning and Delaroche), indicate that this simulation-based methodology can potentially resolve finer details in certain cases where conventional models show limitations.

This success contributes a new dimension to our understanding of nuclear structure in several respects:

Since the SHF model is based on microscopic principles, it describes the distribution of nucleons within the nucleus according to fundamental interactions, rather than relying on parameterized fits to experimental data. As a result, structural properties of neutron-rich nuclei can be represented more accurately. This enables reliable predictions of nucleon density distributions, especially for exotic or unstable isotopes, where experimental data are limited or inaccessible.

The SF approach derives the Nn interaction potential by integrating the NN interaction over the target nucleus density, offering a more physically grounded method compared to classical OMP parameterizations. Through this technique, the diffraction-like structure of the angular distribution—reflecting changes in the scattering centers with energy—is naturally reproduced.

The graphical analysis reveals several new insights:

• Proliferation of energy-dependent diffraction structures
  The increasing number of peaks and valleys observed at higher energies can be attributed to the shortening of the wavelength. This suggests that scattering occurs not over the entire nucleus but rather on individual nucleons, thereby enhancing the visibility of nuclear shell effects.
• Discrepancies in backward scattering
  Although the model generally preserves the overall shape of the distribution, it fails to fully reproduce certain peak structures around 150°. This indicates that there is still room for improvement in the model, particularly in better accounting for spin–orbit effects and many-body correlations.

Detailed modeling of neutron elastic scattering reactions enables more precise results in practical applications such as reactor physics and radiation shielding design. Moreover, this modeling approach allows for the generation of high-accuracy nuclear reaction data for isotopes lacking experimental measurements, thereby contributing to nuclear data libraries such as ENDF and TENDL.

This study demonstrates that, beyond the conventional OM, constructing interaction potentials based on microscopically derived nuclear densities can achieve better agreement with experimental data and provide deeper insights into nuclear structure. In this respect, it represents a significant step toward theoretical data generation, particularly for exotic or experimentally inaccessible systems.

Based on the graphical analysis, the following general conclusions can be drawn:

Advantages of the model:

• The angular resolution is particularly high at medium to high energies.
• The positions and numbers of the diffraction-like structures observed experimentally are largely well reproduced.

Limitations and areas for improvement:

• Some minima and maxima at backward scattering angles have been overlooked.
• This may be due to the model’s current exclusion of additional effects such as spin–orbit forces or channel coupling.

Regression Analysis

Regression analysis is a statistical method used to determine the strength and nature of the relationship between a dependent variable (e.g., experimental cross-section data) and one or more independent variables (e.g., theoretical model predictions). In this study, the coefficient of determination (R²) was used to evaluate how well the angular differential cross sections predicted by the SHF-WS and Koning and Delaroche models correlate with the experimental data (

Table 5. R² values obtained from regression analysis between theoretical angular differential cross sections and experimental data, for both SHF-WS and Koning and Delaroche models.

Isotope—Energy (MeV)	R2 Koning and Delaroche [43]	R2 The Microscopic Approach Proposed in This Study (SHF-WS)
182W—1.5	0.9768	0.5479
182W—2.5	0.4525	0.4586
182W—3.4	0.9935	0.9874
182W—4.87	0.9951	0.9967
182W—6.0	0.9969	0.9992
183W—3.4	0.9895	0.9839
184W—1.5	0.9798	0.9940
184W—2.5	0.2950	0.2935
184W—3.4	0.9934	0.9978
184W—4.84	0.9964	0.9991
184W—6.0	0.9980	0.9982
186W—1.5	0.9848	0.9958
186W—2.5	0.1502	0.1540
186W—3.4	0.9918	0.9992

Average R² (Koning and Delaroche): 0.8424; average R² (SHF-WS Model): 0.8147.

). An R² value close to 1 indicates a strong agreement between the model and experiment, whereas a value closer to 0 indicates weaker correlation. This approach provides a quantitative measure of model performance across different neutron energies and isotopes, allowing for an objective comparison between the global OM and the microscopic model proposed in this work.

To further quantify the agreement between the theoretical models and the experimental angular differential cross sections, a regression analysis was performed using the coefficient of determination (R²). The average R² values across multiple energy levels and isotopes indicate that the global Koning and Delaroche model achieves a slightly higher overall correlation (R² ≈ 0.8424) compared to the SHF-WS model (R² ≈ 0.8147). However, this difference is relatively minor. In fact, the SHF-WS model demonstrates comparable or even better performance in specific isotopic and energy combinations, such as for ¹⁸⁴W and ¹⁸⁶W at 3.4, 4.84, and 6 MeV. These findings suggest that the SHF-WS model is not only a consistent alternative but also capable of capturing the nuclear deformation effects more effectively in some cases. Therefore, despite its slightly lower average R², the SHF-WS model offers a physically grounded and reliable microscopic framework, complementing the global OM approach with a more structure-sensitive methodology.