The Effect of Zero-range Pseudo-potential Approximation in the Elastic Scattering Channel Using A Mass-dependent M3Y-type Interaction

The angular distributions for the elastic scattering of a deuteron from ¹²C were measured in the double model framework using a mass-dependent M3Y-type interaction. The optical potential of this study was derived using the double-folding formalism and subsequently employed in the optical model formalism to determine the reaction cross-sections of d + ¹²C at different incident energies. The calculated differential cross-sections were analyzed and compared to experimental results. A good fit of the differential cross-section to the experimental data was achieved. The suitability of the fit affirms the present formulation as a suitable tool for the study of nuclear reactions and nuclear structure. The effect of including the imaginary potential and the s-matrix elements was also analyzed.

The problem of nuclear scattering involves solving the Schrödinger equation for two interacting nuclei. This requires robust theoretical tools and models capable of reproducing fundamental nuclear matter properties such as binding energy, pressure, volume, susceptibility, and incompressibility [1-3]. Among these models, the Michigan 3-Yukawa (M3Y) interaction, derived from G-matrix elements of the harmonic oscillator potential, has been widely used and proven effective in studying nuclear structure and reactions.

In recent years, a variant of the M3Y interaction, known as the mass-dependent M3Y-type interaction, was developed using the Lowest-Order Constrained Variational (LOCV) method. This interaction incorporates mass dependence through parameters derived from nuclei with mass numbers 24, 40, and 90 [4]. Despite this mass dependence, the M3Y-type interaction was recommended for broader applications across different nuclei. However, its effect on reaction calculations remains insufficiently explored—a gap that forms the basis of the present study.

Determining the most appropriate form of nuclear potential to analyze interactions has been a longstanding focus in nuclear physics [5]. Both phenomenological and microscopic potentials have been developed and extensively used, particularly in elastic and non-elastic scattering channels [6]. Due to the presence of strong absorption effects in heavy-ion collisions, accurate modelling of the nuclear Optical Potential (OP) is essential. The real part of the OP is often derived using folding models, while the imaginary part, accounting for absorption, is typically modelled empirically [7,8].

The double-folding model combines realistic nucleon-nucleon (NN) interactions with nuclear density distributions to construct the real part of the OP [9]. This approach minimizes the ambiguities associated with phenomenological potentials [10]. Although folded potentials are inherently real, the imaginary component—representing absorption—can be included phenomenologically or through density-dependent corrections.

This study applies the double-folding model using the mass-dependent M3Y-type interaction to analyze the elastic scattering of a deuteron from ¹²C. The primary goal is to assess the impact of the zero-range pseudo-potential and the mass-dependent interactions on scattering predictions and compare the resulting cross-sections with experimental data. Both the real and imaginary parts of the optical potential are evaluated using folding techniques, and the fits to experimental angular distributions are examined to determine the model’s accuracy.

B3Y-Fetal Effective Interaction

This study adopts the isoscalar component of the central potential from the mass-dependent M3Y-type interaction, as outlined in prior work [4]. The direct and exchange components of this interaction are given by [11,12]:

$V_{00}^D\left(r\right)=7419.23\frac{ e^{-4r}}{4r}-1823.98\frac{e^{-2.5r}}{2.5s}$ (1)

Similarly, the exchange term is determined as:

$V_{00}^{Ex}\left(r\right)=4745.02\frac{e^{-4s}}{4s}-1984.144\frac{e^{-2.5r}}{2.5r}-7.8474\frac{e^{-0.7072r}}{0.7072r}$ (2)

where $V_{00}^D\left(s\right)$ , and $V_{00}^{Ex}\left(s\right)$ are the direct and exchange parts of the isoscalar central B3Y-Fetal interaction.

The bare nucleon-nucleon (NN) interactions alone are inadequate in reproducing the saturation properties of nuclear matter. To correct this, a density-dependent factor f(ρ) is introduced, ensuring that the interaction achieves nuclear matter saturation [13]. This density dependence is expressed as [14,15]:

$V_{00}^{D(Ex)}\left(\rho ,r\right)= f \left(\rho \right)V_{00}^{D(Ex)}\left(r\right)$ (3)

Concerning the density dependence, an exponential form of the density-dependent term was adopted [16,17]:

$\text{f}\left(\text{ρ}\right)=\text{ C}\left(1+\text{ α exp }\left(-\text{βρ}\right)\right)$ (4)

Here, C and β are constants determined to reproduce the correct binding energy and saturation density of symmetric nuclear matter. These constants are obtained from prior parameterizations [18].

The exchange term is analyzed using the resonating-group method in a single-channel approximation [19]. It includes both single-nucleon and core exchange effects, significant primarily when projectile and target nuclei have similar mass numbers. To localize the otherwise nonlocal exchange term, a zero-range pseudo-potential is added, yielding [13]:

${\text{V}}_{\text{pt}}\to {\text{V}}_{\text{pt}}\left(1-{\text{P}}_{\text{pt}}\right)$ (5)

Where V_pt are the direct and exchange parts of the effective interaction, P_pt is the operator that exchanges all the coordinates of the two nucleons by making the exchange part of the effective interaction local. By the zero-range pseudo-potential of equation (5), the interaction V_pt is transformed in the form [20,21],

$V_{pt}\left(1-P_{EX}\right)\to V_{pt}\left(r\right)+\widehat{J}\left(E\right)\delta \left(r\right)$ (6)

where the function $\widehat{J}\left(E\right)=J_{00}g(E)$ represents the zero-range pseudo-potential that incorporates the exchange component of the M3Y-type interaction [22-28]. The term, V_pt(r), is the direct part of the isoscalar interaction defined by Equation (1) but now expressed as the projectile-target interaction. The term g(E) represents the energy scaling factor. The magnitude of the exchange amplitude J₀₀ was evaluated to be about −361MeV [11,18]. The energy scaling factor was approximated to be $\left[1-0.005\left(\frac{E}{A}\right)\right]$ .

The double-folding model

To construct the nuclear optical potential, the double-folding model was used. The double-folding model convolves the effective NN interaction with the nuclear density distributions of the projectile and target [19]:

$V_F\left(r\right)={{\displaystyle \iint }}^{\text{}}{\rho }_p\left(r_p\right){\rho }_t\left(r_t\right)V_{pt}\left(r_{tp}\right)d{r}_{p}d{r}_t$ (7)

Where ρ_p and ρ_t are the density distributions of the projectile and target nuclei, and r_tp = r_t − r_p. The nuclear densities are modelled using a two-parameter Fermi (2pF) distribution [29-31]:

${\rho }_{n(p)}\left(r\right)={\rho }_{0n(p)}{\left[1+\exp \left(\frac{r-R_{p(n)}}{a_{p(n)}}\right)\right]}^{-1}$ (8)

Here, ρ_o is the central density, and a_p(n) are the half-density radius and the diffuseness parameter, respectively. These parameters are obtained from empirical fits to charge distributions [26].

The double-folded potential naturally represents the real part of the optical potential. To account for the imaginary part—which captures absorption into non-elastic channels—a similar folding approach is applied but with a different renormalization factor [27]:

$\text{U}\left(\text{r}\right)=\left({\text{N}}_{\text{r}}+{\text{iN}}_{\text{i}}\right){\text{V}}_{\text{F}}\left(\text{r}\right)+{\text{V}}_{\text{Coul}}\left(\text{r}\right)$ (10)

Where N_r(i) are the real and imaginary renormalization factors for the real and imaginary parts of the optical potential, and V_Coul(r) is the Coulomb potential. These renormalization constants are adjusted to achieve optimal agreement with experimental data.

Folding Model Potential of the Elastic Scattering of d + ¹²C

The double-folding model was employed to calculate the real and imaginary parts of the optical potential for the elastic scattering of a deuteron from ¹²C at incident laboratory energies of 28, 110, 120, and 170 MeV. Using these potentials, the reaction and total cross-sections were computed and compared with those obtained using both phenomenological and microscopic potentials. The results are presented in Table 1.

The results indicate good agreement between the calculated reaction cross-sections and existing literature [28-33], confirming the reliability of the double-folding approach. The optical potentials generated in this study exhibited typical Woods-Saxon shapes, and the values of the renormalization factors N_r(i) reflect strong absorption effects at higher energies. The plots of the radial dependence of the strength of the folded potentials are shown in Figures 1-4. The potentials were attractive and short-ranged over small nuclear distances.

Analysis of the Differential Cross Section of the Elastic Scattering of d + ¹²C

The plots of the angular distributions of the elastic scattering for each incident energy are shown in Figures 5-8. The theoretical differential cross-sections match well with experimental data across all angles. The large X² value at 170 MeV suggests statistical anomalies or increased sensitivity in the data fitting (Table 2).

The experimental data points were reproduced quite accurately at all energies and angular regions. However, in the data of 17^o MeV, a maximum and minimum were observed between 15^o ≤ θ_cm ≤ 25^o and 25^o ≤ θ_cm ≤ 35^o. In general, a good fit of the theoretical calculations to experimental data was obtained, and a pronounced improvement in the fit of the differential cross-sections using the present formulation was achieved as compared to those obtained using microscopic and phenomenological potentials [28,30,31]. The plots of the modulus of the S-matrix elements are also shown in Figures 9-12 as a function of total orbital angular momentum L for the different incident energies. These plots reveal regions where |S_L| < 1, indicating strong absorption and the presence of other nonelastic processes. At 28 MeV, strong absorption is observed at small impact parameters in the angular momentum range L ≈ 0 - 10. As energy increases, the contributing angular momentum values shift to higher L, indicating deeper penetration and more peripheral scattering contributions. This transition is consistent with the behaviour of the nuclear optical potential at different energies.

This study has employed the double-folding model in conjunction with the mass-dependent M3Y-type interaction to investigate the elastic scattering of [insert specific nuclei] over a range of incident laboratory energies (28–170 MeV). The analysis incorporated both the real and imaginary components of the optical potential derived through folding procedures, accounting for strong absorption effects observed in heavy-ion collisions. The calculated reaction and total cross-sections, as well as differential angular distributions, show excellent agreement with experimental data, validating the effectiveness of the M3Y-type interaction in modelling nuclear scattering phenomena. The renormalization factors obtained for the optical potentials reflect the expected energy dependence, and the folded potentials exhibit Woods-Saxon-like shapes consistent with known nuclear interaction behaviours.

Furthermore, the analysis of the elastic S-matrix elements revealed that nuclear contributions are most significant at small impact parameters and are gradually suppressed with increasing angular momentum, leaving the Coulomb interaction dominant at large distances. This confirms the realistic absorption effects modelled by the imaginary component of the potential. Overall, the results highlight the success of the mass-dependent M3Y-type interaction and the double-folding model in accurately reproducing scattering observables. It also demonstrated that the M3Y-type interaction is a robust tool for studying both elastic and potentially non-elastic nuclear reactions. These findings encourage future applications of this approach in other nuclear systems and energy regimes.

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