1. Introduction

Elastic scattering processes between two colliding nuclei still remain an important topic in nuclear physics studies. The scientific community has been attracted to ^4,6,8He because of its special aspects in scattering with heavy targets. The weakly bound nucleus ⁶He has a broad neutron distribution and a low 2n-binding energy, which is well known[¹-⁴]. These characteristics favor neutron transfer and breakup, and therefore should affect the angular distribution of elastic cross sections [⁵-⁹]. ⁸He has more valence neutrons but is more tightly bound, with binding energies comparable for 1n and 2n systems, whereas ⁶He favors two-neutron breakup. The elastic and reaction cross sections for collisions with heavy targets at Coulomb barrier energies are expected to reflect the differences between the two helium isotopes.

A. M. Sanchez-Bentez et al [⁵] analyzed data of elastic scattering of 6He+208Pb measured at laboratory energies of 14, 16, 18, and 22 MeV using phenomenological Woods Saxon form factors and optical model calculations. The dispersion relations that connect the real and imaginary parts of the optical potential were found to be consistent with the energy variation of the optical potential.

The goal of this research is to see how sensitive ^4,6,8He + ²⁰⁸Pb elastic scattering is to various phenomenological and semi-microscopic potentials [¹⁰-¹⁶]. Various models have been used to explain nuclear-particle interactions.

Some of these approaches include the double-folding model (DFM), the Woods-Saxon (WS) potential, and various variations of the WS potential. However, for analyzing experimental angular distributions of low-energy scattering of exotic nuclei on stable nuclei, the well-known double-folding (DF) model is the most widely used method. The phenomenological OM as well as the DF with the appropriate optimized parameter sets accurately reproduce the elastic scattering angular distributions for ^4,6,8He + ²⁰⁸Pb elastic scattering. We reanalyzed the experimental data of the ^4,6,8He + ²⁰⁸Pb reaction at 22 MeV [⁸-¹⁶] by calculating the nuclear potential using a variety of potentials, including Sao Paulo potentials, CDM3Y6 potentials with and without the rearrangement term (RT), and phenomenological WS potentials [¹⁷-²²].

The paper is organized as follows. Section 2 discusses the nuclear potentials that are used in the data analysis, while Sec. 3 discusses the results and discussion. The summary is provided in Sec. 4.

2. Theoretical methods

As a first step for probing the interaction mechanism for the considered nuclear systems, OM of the nucleus is applied. The implemented phenomenological OM potential has the following form:

The V _c (r) is the Coulomb potential due to a uniform sphere with a charge equal to that of the target nucleus and radius r _c At1/3.The nuclear potential is consisting from two parts: real volume (which simulate the scattering) and imaginary volume (simulate the reduction in flux due to absorption), both has the phenomenological Woods-Saxon (WS) shape.

According to the different parameter ambiguities both discrete and continuous associated with the OM calculations, and the fact that phenomenological representations do not include a description of the projectile or target’s structure, the real part of potential was constructed using the microscopic double folding (DF) procedure extracted from the CDM3Y6 interactions with and without rearrangement term RT as well as Sao Paulo potential (SPP) via the double convolution integral

where ρpr1, ρt(r2) are the matter densities of the projectile and the target respectively and v _NN (S) is the effective NN interaction between two nucleons where S=R→-r1→+r2→. For NN effective interactions, the widely held choosing has been based on the M3Y interactions which were designed to reproduce the G-matrix elements of Paris [¹⁸-²⁰] NN interactions. The density distribution of ^4,6,8He is deduced using the Argonne v18 two-nucleon and Urbana X three-nucleon potentials (AV18+UX) in a realistic Variational Monte Carlo (VMC) wave function [²⁶]. For the density distributions of ²⁰⁸Pb, the following fermi form is assumed [²⁷,²⁸].

R = 6.8 fm , a = 0.515 fm and root-mean-square (rms) matter radius of 5.604 fm, respectively while ρ ₀ can be determined from the normalization condition.

The DF potential consists mainly of two parts:

The direct part is

and the knock-on exchange part in the infinite-range exchange is

In this study, a modified version of the CDM3Y6 interaction is used, which includes the rearrangement term (RT) (CDM3Y6-RT). The density-dependent version (CDM3Y6) of the M3Y effective NN interaction based on the G-matrix elements of the Paris potential is used for the direct and exchange terms in Eqs. (5) and (6). [²¹], which is the full CDM3Y6 interaction form.

where, the density dependent function F is written as [²¹]

And g(E) is the additional energy dependent factor written as [²¹],

The term ΔF(ρ) is added to the folding model calculation for the modified (CDM3Y6-RT) interaction with the inclusion of the RT term, where ΔF(ρ) can be written as [²⁹],

In the SPP model, the following two equations link the real part of the local-equivalent interaction to the DF potential V _F (R) as

where ρ _p (r _p ) and ρ _T (r _T ), respectively are the nuclear matter density distributions for the two colliding nuclei. The following two equations link the real part of the local-equivalent interaction to the DF potential V _F (R) as

Where V is the nuclei’s local relative velocity and C is the speed of light. The nuclear densities were obtained from the Dirac-Hartree-Bogoliubov model [³⁰], and the Sao Paulo potential (SPP2) was calculated using the REGINA code [²⁰].

3. Results and discussion

For incident energies of E _tab = 22 MeV, elastic scattering results for the ^4,6,8He + ²⁰⁸Pb reactions were obtained. Three different theoretical approaches were used for this purpose microscopic double folding model based on Sao Paulo potential, CDM3Y6 interaction with and without RT, and phenomenological Woods Saxon (WS) model. The potentials that resulted are represented in Fig. 1. The comparisons between the experimental ^4,6,8He nuclei elastically scattered from a ²⁰⁸Pb target at E = 22 MeV and the theoretical calculations using both phenomenological OM and microscopic model are shown in the Fig. 2.

Figure 1 The generated real SPP as well as CDM3Y6 with and without RT for ^4,6,8He + ²⁰⁸Pb target at E _tab = 22 MeV. 

Figure 2 Comparison of experimental angular distributions (solid black circles) and theoretical calculations (solid red curves) for ^4,6,8He+ ²⁰⁸Pb elastic scattering using WS and real folded potential at E _tab = 22 MeV. ^{Ref. [10}-^16] provided the experimental data. Cluster folding potential for ⁶He+ ²⁰⁸Pb is performed by considering triton + triton cluster structure for ⁶He as well as the CDCC calculations. 

For the real and imaginary potentials, three parameters were used in the OM computations. Depths, radius, and diffuseness for the real (v ₀, r ₀, a ₀) and imaginary (w ₀, r _w , a _w ) volume parts of the interaction potential were freely changed until the best fit for the data was achieved by minimizing the χ ² value defined by

where N is the number of data points, σ(θ _i ) ^cal and σ(θ _i ) ^exp are the calculated and experimental differential cross sections (DCs) and Δσ(θ _i ) is the relative uncertainty in experimental data.

The parameters with the best fit are listed in the table. The theoretical fits were done with the FRESCO code [²⁷], and the optimal potential parameters were found with the SFRESCO search code. For the ^6,8He, it is obvious that the agreement between theoretical results and experimental data is almost perfect. The results of the ⁴He+ ²⁸Pb reaction, in particular, are shown to be almost excellent for the WS model, which properly represents the experimental data for both small and large angles. However, the conclusions of the SPP and CDM3Y6 models for all incidence energy exclude some angles of the experimental data.

For each reaction, we calculated volume integrals as well as chi square χ ² values. Using various versions of the nuclear model of the ^4,6,8He nucleus, we found that the phenomenological potential (WS) with 6-free parameters provides good fitting for all systems, as evidenced by the chi square values in Table I, whereas the other microscopic potentials, the obtained values for the chi square value give a closed value to each other. Real real (J _R ) and imaginary (J _I ) volume integrals are used to calculate the volume integral. The J _R and J _R volume integrals are crucial for demonstrating the evaluated potential’s strength. In this context, the Table shows the J _R and J _I values for the WS, SPP, and CDM3Y6 models. It has been found that altering the used model results in similar J _R and J _R values. The smallest values for the WS results can be seen when the χ ² values are examined. Of course, a lower chi square χ ² value corresponds to a more accurate description description of the experimental data in terms of the chosen theoretical representation. As a result, similar cross sections for various approaches can indicate similar experimental results [⁶-¹²]. As a result, it is reasonable to assume that various approaches have resulted in a successful analysis of the experimental data.

To validate our analysis, we plotted expression (12) at 22 MeV for ^4,6,8He+ ²⁰⁸Pb and, against the DF potential with finite-range exchange contribution for the densities under consideration, as shown in Fig. 1. At the center (R=0), (VSPP0/VDF0)≈1.2)(0.98(1.01)) for ⁴He, ⁶He and ⁸He respectively without RT, this means that the scattering is sensitive to the real potential for both reactions considered. This figure shows that the DF potential is about 5% deeper than the SPP potential at the center (R = 0), while they are similar for radial distances R ranging from 0.5 fm to 7 fm for ^6,8He+ ²⁰⁸Pb scattering.

A strong absorption radius R _S (closest approach) through the surface radial region can be defined as the distance between colliding particles in terms of the transmission coefficient T _1/2, which is a function of the partial wave and momentum L1/2, as well as the Sommerfield parameter η and the projectile wave number k [²⁸]. We can see that the different DF and cluster potentials agree with each other near the strong absorption radius R _S where R _S = 10.5, 10.8, 11.1 fm for the ^4,6,8He+ ²⁰⁸Pb reaction.

3.1 Cluster folding Potential for ⁶He+ ²⁰⁸Pb

Cluster folding (CF) potential obtained from the t + t probable cluster structure for 6 _He + 208 _Pb elastic scattering data is used to fit the data. Based on the suggested t + t model, the real and imaginary CF potentials for ⁶He + ²⁰⁸Pb system can be defined on the basis of t + ²⁰⁸Pb potentials as: the real and imaginary CF potentials for the ⁶He + ²⁰⁸Pb system may be defined on the basis of t + t model. Using the recommended t + t model as follows, we can write the ⁶He + ²⁰⁸Pb potential as

VCF(R)=∫‍Vt-208Pb(R-12r)+Vt-208Pb(R+12r)|χt-t(r)|2dr. (15)

The Vt-  208Pb is the phenomenological potential for the two t + ²⁰⁸Pb channels which fairly reproduce the experimental data at the appropriate energies Et≈1/2E(6He) taken from [²⁹]. The term χ _t-t(r) is the intercluster wave function for the relative motion of the two tritons in the ground state of ⁶He, and r is the relative coordinate between the centers of mass of two tritons. The bound state form factor t + t represents a 2S _1/2 state in a real Woods-Saxon potential of R = 1.15 fm, a = 0.7 fm, and the potential depth is allowed to be changed till reach the binding energy of the cluster (12.307 MeV). The constructed cluster folding potential for ⁶He + ²⁰⁸Pb is shown in Fig. 3.

Figure 3 Real cluster folding potential used in ⁶He+ ²⁰⁸Pb analysis at E = 22 MeV. 

The ⁶He + ²⁰⁸Pb experimental angular distribution at E = 22 MeV “see Fig. 2” is reanalyzed within the framework of cluster folding model using two adjustable parameters N _RCF and N _ICF , namely the renormalization factor for the real and imaginary cluster folding potentials, respectively, derived based on Eq. (14) and presented in Fig. 1. The analysis showed the necessity to reduce the real cluster folding potential strength by about 90%. Although, it is well known that, in order to fit the experimental cross sections for systems induced by weakly projectiles such as ^6,7Li and ⁶He, the potential strength should be reduced by ~ 40-60 % [³⁰-³⁵], the reported reduction (90%) in the potential strength could infer the less probability of ⁶He to be treated as a cluster of two tritons. More sophisticated approaches could be preferable to derive the interaction potential based on three body treatment (⁴He+n+n) for ⁶He. Generally, the adopted potentials either Woods-Saxon or double folded CDM3Y6 potentials with and without considering the rearrangement term fairly described the considered data for ^4,6,8He scattered from ²⁰⁸Pb at E = 22 MeV, where the extracted N _r is very close to unity.

The Continuum discretized coupled channels (CDCC) were used to investigate a three-body projectile ⁶He (n+n+ ⁴He), assuming that the two neutrons are di-neutrons and thus take the shape of a two-body projectile (⁶He+2n). Moro et al. [⁴⁰] improved this di-neutron model to account for four-body effects by using an effective di-neutron separation energy, they observed an effective binding energy of around 1.6 MeV rather than the 0.973 MeV predicted in the previous model [⁴⁰-⁴²]. To investigate the effect of couplings to the breakup channels, the (CDCC) method is used to analyze the experimental elastic scattering angular distributions for ⁶He+ ²⁰⁸Pb at E = 22 MeV. The results of the differential cross section using CDCC and other models are compared in Fig. 2. The total reaction cross section calculated by CDCC is equal 1313.4 mb, which agrees with the results of the other microscopic models.

4. Summary

The current study’s main goal is to conduct a further analysis of the interaction mechanism and peculiarities resulting from the scattering of ^4,6,8He from the heavy target ²⁰⁸Pb at energy E = 22 MeV, above the Coulomb barrier. Analysis performed within both the derived real potential and the phenomenological WS potential. The DF potentials with CDM3Y6 effective interaction is considered with and without taking into consideration the effect of rearrangement term as well as Sao Paulo calculations. For ^4,6,8He+ ²⁰⁸Pb nuclear system, the strength of the real double folded part is closed to unity to reproduce the data. All of the results are compared to the experimental data as well as to each other. On the other hand, due to the lower probability of ⁶He being treated as a cluster of two tritons, the analysis revealed the need to reduce the real cluster folding potential strength by approximately 90%. Additionally, for each theoretical approach, we provide potential parameters, cross sections, volume integrals, and chi square values. The calculated potential’s renormalization factor is maintained near unity. At all angular regions where the real part of potential plays a critical role in cross section calculations, the agreement between experimental data and calculated cross sections using folded potential is fairly good.