A STUDY THE NUCLEAR POTENTIAL USING QUASI-ELASTIC SCATTERING CALCULATION FOR THE 9 , 10 , 11 Be + 208 P b REACTIONS

Specific systematic studies on the nuclear potential parameter for the heavy-ion reactions which includes the systems have been achieved by using large-angle quasi elastic scattering at deep sub-barrier energies close to the Coulomb barrier height.The single-channel (SC) and coupled-channels calculations have been carried out to elicit the nuclear potential. The chi-square method χ 2 has been used find the best value of the nuclear potential in comparison with the experimental data. The best values of the nuclear potential were found from the calculations of the coupled channels for an inert projectile and a vibrating target for systems: 9 Be + 208 Pb , 10 Be + 208 Pb , 11 Be + 208 Pb , which are equal to 45 MeV, 65 MeV, 53 MeV, respectively.


INTRODUCTION
Knowing the nucleus-nucleus interaction potential is the key ingredient in nuclear reaction analysis [1] and it has played an important role in describing nucleus-nucleus collisions.It has been well recognized that heavyion collisions at energies about the Coulomb barrier are strongly influenced by the internal structure of colliding nuclei [2].The couplings of the relative motion to the substantial degrees of freedom (such as collective inelastic excitation of the colliding nuclei and/or transfer processes) result in a single potential barrier being changed by many distributed barriers.The nucleus-nucleus potential is the cause of the interaction energy of colliding nuclei [5] it has been used to appreciate the cross sections of different nuclear reactions.too,indeformed nucleus interaction the nucleus-nucleus potential depends on the orientation angle of the deformed nucleus prorated to the beam direction.We can define the nucleus-nucleus potential as the sum of the nuclear potential V N (r) which is less defined and the Coulomb potential V C (r) which is well-known.By the specific description of the Coulomb or Rutherford scattering.The barrier height of the nucleus-nucleus reaction depends on the ratio between the nuclear and Coulomb potentials, which work at teeny distances between the surfaces of reactant nuclei [6] Ṡo,the nucleus-nucleus potential includes Coulomb and nuclear parts, so that long-range disharmony Coulomb potential acts between the protons in nuclei while the nuclear interaction between nucleons, the nuclear fraction is commonly expressed by the Woods-Saxon (W S) form, which is characterized by the deepness V o , radius r o and diffuseness a parameter [2].The truth is that the W S form of a simple exponential had been exploited to study the surface-characteristic of nuclear potential.Quasi-elastic scattering can be defined as the sum of elastic scattering, inelastic scattering, and transfer reaction, it is very well equivalent to the fusion reaction, which is defined as a reaction where two discrete nuclei integrate to form a compound system [7].Fusion and quasi-elastic scattering are both considered extensive operations that work in tandem.As a result,these interactions share the same potential and information about the mechanism of interaction, and both are sensitive to channel coupling impacts (due to collective in elastic excitements of colliding nuclei) at energies near the Coulomb barrier [8].
Experimentally, the measurement of quasi-elastic scattering is easier than that of fusion interaction, particularly at deep sub-barrier energies.As well as note that the scattering operation is sensitive fundamentally to the surface area of the nuclear potential, whilst the fusion reaction is also comparatively sensitive to the internal fraction [9].The experimental measurement process for large-angle quasi-elastic scattering cross sections is more efficient and straight forward than the measurement process for fusion cross sections.At deep sub-barrier energies, the perversion of the rate of the quasi-elastic to the Rutherford cross sections from unity provides a clear way to set the account of the surface diffuseness parameter in the nucleus-nucleus potential [10].
As a result, diffuseness parameter can be defined as a landing of the nuclear potential and thus has a direct impact on the barrier width and coupling strong points,which to first order rely on the derivative of the potential.A coupling channel model is an ideal tool for simultaneously reproducing experimental data for EEJP.3(2023) several processes such as elastic and inelastic scattering,particle transfers, and fusion with in a unified framework [11].
The inter-nuclear potential is the most important component in coupled-channels calculations, with nuclear potential influencing barrier width and coupling strengths.The channel coupling is caused by the interaction of the internal degrees of freedom, which include transfer reactions and collective vibrational and rotational motions, with the relative motion of colliding nuclei [12].
The effect of coupling channels can be ignored in nucleus-nucleus collisions at deep sub-barrier energies near the Coulomb barrier because reflection probability is nearly unity at such energies; however, this analysis would be acceptable for spherical nuclei collisions.The use of coupling channel accounts does not play a significant role in determining the best value for the diffuseness parameters at deep sub-barrier energies, but their primary purpose is to achieve the effects of some calculation inputs on the resulting diffuseness parameters.The excitation states of colliding nuclei are critical for performing coupled-channel calculations.[13] K. Washiyama et al. used large-angle quasi-elastic scattering at energies much less than the Coulomb barrier to investigate the surface characteristics of nucleus-nucleus potential in heavy-ion reactions.As a result, a singlechannel potential model was appropriate for describing these energies [2,14].
The goal of this study is to obtain the nuclear potential parameters for the systems 9,10,11 Be + 208 P b by using large-angle quasi-elastic scattering at deep sub-barrier energies close to the Coulomb barrier height,and the single-channels and coupled-channels calculations were performed using the CQEL program, which includes all orders of coupling and is considered the most recent version of computer code CCFULL [15].The chi-square χ 2 method was used to obtain the best-fitting values of the nuclear potential in comparison to the experimental data [9,16].

THEORY
The nucleus-nucleus potential is divided into two parts nuclear part V N , which can be well and reasonably described by the Woods-Saxon (W S)form given by [17]: where R 0 is a radius parameter of the system, V o , a and r o represent the potential depth,surface diffuseness parameter,and radius parameter,respectively, whilst r refers to the center-of-mass distance between the target nucleus of mass number A T and the projectile nucleus of mass number A P [18].From another side,Coulomb part V C between two spherical nuclei with regular charge density distributions and when they do not interfere is given by [18]: where Z p and Z T represent the atomic number of the projectile and target,respectively, r the distance between the centers of mass of the colliding nuclei.When the nuclei interfere, then the Coulomb potential is given by [19]: where R c is the radius of the ball equivalent to the of the target and the projectile.The collision between two nuclei through the presence of coupling between the relative motion of the center of mass of the colliding nuclei r − → = (r, r) and the nuclear intrinsic motion ξ.The Hamiltonian for the system is given by [1]: where r refers to the center of mass distance between the colliding nuclei, the reduced mass of the system while V (r) is the naked potential in the absence of the coupling where V (r) = V N (r) + V c (r),H 0 (ξ), represents the Hamiltonian for the intrinsic motion V coup is the mentioned coupling.The Schrodinger equation for the total wave function would be given by [1]: The internal degree of freedom ξ principally has a limited spin.We can write the coupling Hamiltonian in complications as [1]: Y λµ (r)refers to the spherical harmonics and T λµ (ξ)refers to the spherical tensors,which are built from the internal coordinate.The sum is taken over all values of excluding for λ=0 since it is originally considered inV (r) The expansion basis for the wave function in equation ( 5) for a fixed total angular momentum J and its z-component M is defined as [20]: where l refers to the orbital, Irepresent the internal angular momenta and represents the wave function for the internal motion which fulfills [16]: The total wave function ψ(r.ξ) has been expanded with this basis as [1]: The Schrödinger equation [equation ( 2)] can then be written as a group of coupled equations for u J n l I (r) [1].
Terms of the coupling matrix elements are given by:[1] The reduced matrix elements in equation ( 11) is defined by:[14] As can be observed in the equation,the coefficient has been suppressed since it is independent of the coefficient M .Coupled-channels equations are the name given to the equation.These equations are frequently solved using the incoming wave boundary conditions for heavy-ion fusion interactions [14].
The definition of the local wave number is [9]: The penetrability across the Coulomb barrier is calculated using the transmission coefficients and is given by: EEJP.3(2023) is the wave number for the entrance channel.The fusion cross section for the unpolarized target is given by: Equation ( 17) therefore reads When the initial intrinsic spin=0,the initial angular momentum =J,with the coefficients and are suppressed in the penetrability [9]: σ l is the Coulomb phase shift which is given by [9]: While c is the Coulomb scattering amplitude which is given by [9]: η is the Summerfield parameter,Equation ( 19) may be used to evaluate the differential cross-section, which is given by [9]: Equation ( 21) may be used to evaluate the Rutherford cross section [9]: 3. PROCEDURE The CQEL software,which is regarded as the most recent version of the computer code CCFULL,was used to do the computations for single-channel and coupled-channels.The Schrödinger equation and the linked equations are precisely solved by this program [15].To prevent systematic mistakes in the current study, the chi-square approach was used as a normalizing factor between the theoretical computation and the experimental results.The nuclear potential, which has both real and fictitious components, was calculated using a W S form [21].The research was done on the true potential parameters to find the one that suited the experimental data the best, allowing it to be replicated for all interactions [21].
The Woods-Saxon (W S) The radius parameter r 0 is taken to be 1.2f m, while the values of potential depth V 0 depended on the diffuseness parameter are taken to be [(45, 50, 60)M eV, (49, 55, 65)M eV and (43, 53, 59)M eV for the 9 Be+ 208 P b, 10 Be+ 208 P b and 11 Be+ 208 P b systems, respectively.The radius of the target was taken as R T = r T A 1/3 such that r T = 1.16 f m while for the projectile R p = r p A 1 /3 so r p = 1.22f m.The beam energies at the center of the reaction target were 88M eV , 127M eV and 140 M eV for 9 Be, 10 Be, and 11 Be, respectively [21].The experimental data for the quasi-elastic cross sections at deep sub-barrier energy for all systems were taken from the references [21].To verify that the calculations are suitably compatible with the available experimental data, we analyze and display the calculated ratio of the quasi-elastic to the Rutherford cross sections as functions of the center of mass energies.

RESULT AND DISCUSSION
In the 9 Be+ 208 P b system, the nuclear potential parameter has been discussed in four states, in the first state we considered the projectile 9 Be as well as target 208 P b as inert nuclei (SC).As for the three cases,we assumed the target nucleus 20 P b is vibrational coupling with deformation parameter β 0 = 0.055 to the state 2 + (4.085M eV ) and the projectile 9 Be was inert.We used single-quadruple phonon excitation for the projectile and target nuclei that were vibrationally excited.The values of the nuclear potential parameters (V 0 ) have been obtained from SC and CC analysis, as well as other parameters of W S potential(radius r 0 and diffuseness a 0 ) and the values of χ 2 fitting between experimental and theoretical data for the 9 Be+ 208 P b reaction were shown in Table (1).
By looking at the outcomes in Table (1), we find that the better suitable value nuclear potential parameter which has obtained from CC analysis (where the projectile 9 Be was inert and target 208 P b nuclei was vibrational coupling) is 45M eV with χ 2 =0.00242, this result considered very near for standard value and represented by the solid line in Fig. (1)(B), while the dashed line represents the single-channel accounts with the nuclear potential parameter, was drawn for the comparison, that our calculation outputs bred by all computational models are near one another and in consensus with the experimental outputs.The dσqe l dσ R at the best fitted nuclear potential parameter is 45M eV , with χ 2 =0.00242 using a coupled channel calculation at deep sub-barrier energies.In this reaction, we assumed that projectile 9 Be is inert whilst the target 208 P b is vibrational coupling to the state 2 + .The Figure (C) represents a comparison between the best value of the single channel and the coupling channels, which was found using the chi-square χ 2 code.We concluded that CC and phonon excitation influences augment the calculated cross-sections at energies near the barrier district.It is observed that the influence of vibrational states for the spherical nuclei states for the deformed nuclei, is the effective couplings leading to big fusion cross-sections around the barrier regions.
Table 1.The parameters of the W S potential a 0 ,r 0 ,andV 0 , as well as the values of χ 2 fitting between experimental and theoretical data for various types of reactions when the excited nuclei are in a vibrational excitation state with a single-quadruple phonon.In the 10 Be+ 208 P b system, the nuclear potential parameter has been discussed in four states,in the first state we considered the projectile 10 Be as well as 208 P b as inert nuclei,while in the three cases,we assumed the target nucleus 208 P b is vibrational coupling with deformation parameter β 0 =0.055 to the state 2 + (4.085M eV ) and the projectile 10 Be was inert.We used single-quadruple phonon excitation for the projectile and target nuclei that were vibrationally excited.The values of the nuclear potential parameters (V 0 ) have been obtained from SC and CC analysis, as well as other parameters of W S potential (radius r 0 and diffuseness a 0 ) and the values of χ 2 fitting between experimental and theoretical data for the 10 Be+ 208 P b reaction were shown in Table (2).
By observing the results in Table (2),We find that the nuclear potential parameter's more appropriate value,as determined via CC analysis (where the projectile 10 Be was inert and target 208 P b nuclei was vibrational coupling) is 65M eV with χ 2 =0.00550,these results are perceived to be near the conventional value,and represented by the dash dot line in Fig. (2)(B), the Figure (C) represents a comparison between the best value of the single channel and the coupling channels, which was found using the chi-square χ 2 code.We have shown that in heavy ion fusion reactions, higher order couplings to nuclear surface vibrations play an important role.
In the 11 Be+ 208 P b system, the nuclear potential parameter has been discussed in four states,in the first state we considered the projectile 11 Be as well as 208 P b as inert nuclei, while in the As for the three cases, we assumed the target nucleus 208 P b is vibrational coupling with deformation parameter β 0 =0.055 to the state 2 + (4.085M eV ) and the projectile 11 Be was inert.We used single-quadruple phonon excitation for the projectile and target nuclei were vibrationally excited.The values of the nuclear potential parameters (V 0 ) have been obtained from SC and CC analysis, as well as other parameters of W S potential (radius r 0 and diffuseness a 0 ) and the values of χ 2 fitting between experimental and theoretical data for the 11 Be+ 208 P b reaction were shown in Table (3).
By observing the results in Table (3), we find that the nuclear potential parameter's more appropriate value, as determined via CC analysis (where the projectile 11 Be was inert and target 208 P b nuclei was vibrational EEJP.3(2023)Table 2.The parameters of the W S potential a 0 ,r 0 ,andV 0 , as well as the values of χ 2 fitting between experimental and theoretical data for various types of reactions when the excited nuclei are in a vibrational excitation state with a single-quadruple phonon.

System
Case r 0 (f m) a 0 (f m) V )(B), the figure (C) represents a comparison between the best value of the single channel and the coupling channels, which was found using the chi-square χ 2 code.We have shown that in heavy ion fusion reactions, higher order couplings to nuclear surface vibrations play an important role.5. CONCLUSION We found, through micro methodology analyses of the data, that the method of large angle quasi-elastic scattering at deep sub-barrier energies near to the Coulomb barrier height is the perfect instrument for examining the surface property of Inter nucleus potential for the spherical systems discussed in this article.Single-channel EEJP.3(2023) analyses fit experimental data gives nuclear potential parameters for the systems 9 Be+ 208 P b, 10 Be+ 208 P b and 11 Be+ 208 P b respectively, does not differ substantially from the nuclear potential parameter's best fitted value, which was obtained using CC analysis (with an inert projectile and a vibrating target) and is exactly in line with the standard value.All coupling channel accounts produced values that were quite close to the nuclear potential parameter's standard value.

Figure 1 .
Figure 1.Comparison of accounts for single-channel and kinds of linked channels using experimental data[21]Referred to as points with error bars for the system.In the upper panel (a) the hard and dashed lines represent the results of SC analysis at V0 = 45M eV ,V0 = 50M eV and V0 = 60M eV respectively,while the hard, dashed and dot-dashed lines in the lower panel (b) represent the results of CC analysis at V0 = 45M eV (represents the better suitable value of the nuclear potential parameter)V0 = 50M eV and V0 = 60M eV respectively (C) comparison between the best value of the single channel and the coupling channels.

Figure 2 .
Figure 2. Comparison of accounts for single-channel and kinds of linked channels using experimental data[21]Referred to as points with error bars for the system.In the upper panel (a) the hard and dashed lines represent the results of SC analysis at V0=49M eV ,V0=55M eV and V0=65M eV respectively,while the hard,dashed and dot-dashed lines in the lower panel (b) represent the results of CC analysis at V0=49M eV ,V0=55M eV and V0=65M eV (represents the better suitable value of the nuclear potential parameter) respectively (C) comparison between the best value of the single channel and the coupling channels.

Figure 3 .
Figure 3.Comparison of accounts for single-channel and kinds of linked channels using experimental data [21].Referred to as points with error bars for the system.In the upper panel (a) the hard and dashed lines represent the results of SC analysis at V0 = 43M eV ,V0 = 53M eV and V0 = 59M eV respectively, while the hard, dashed and dot-dashed lines in the lower panel (b) represent the results of CC analysis at V0 = 43M eV ,V0 = 53M eV (represents the better suitable value of the nuclear potential parameter) and V0 = 59M eV respectively (C) comparison between the best value of the single channel and the coupling channels.
=0.00639, these results are perceived to be near the conventional value, and represented by the solid line in Fig.(

Table 3 .
The parameters of the W S potential a 0 ,r 0 ,andV 0 ,as well as the values of χ 2 fitting between experimental and theoretical data for various types of reactions when the excited nuclei are in a vibrational excitation state with a single-quadruple phonon.