THE EFFECT OF DIFFUSENESS PARAMETER ON THE QUASI-ELASTIC SCATTERING OF THE 25 Mg + 90 Zr and 28 Si + ( 120 Sn, 150 Nd) SYSTEMS USING WOOD-SAXON POTENTIAL

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INTRODUCTION
The Nuclear reactions at sub-barrier energy are crucial in nature, since they are responsible for the basic behavior of stars, their development, and many aspects of element production.The nucleus potential is consist from Coulomb V C (r) and nuclear V N (r) parts [1] [2].The Woods-Saxon (WS) form is often used to represent the nuclear component, which is characterized by the deepness  , radius  , and diffuseness  parameters.It is significant in nuclear physics because it is regarded as a realistic potential [3].Experiments indicate that coupling to collective states results in a distribution of Coulomb barrier heights, which may be calculated directly from the fusion excitation function σ fus (E) or from back scattered quasi elastic events for many nuclear systems.A significant method for the investigation of barrier distributions near to the Coulomb barrier is large back-angle quasi elastic scattering [4].It may be described as the total of elastic scattering, inelastic scattering, and transfer reaction.It is very similar to the fusion process which is defined as a reaction in which two discrete nuclei combine to produce a compound system [5].Heavy-ion collisions at energies around the Coulomb barrier are strongly affected by the internal structure of colliding nuclei [6].The coupling channel model is the best instrument for simultaneously reproducing the experimental data for a variety of events, including particle transfers, fusion, elastic and inelastic scattering [7].The inter-nuclear potential is the most crucial factor in calculations involving coupled channels.Since it has an impact on the coupling strengths and the breadth of the barrier.The transfer reactions, as well as the collective vibrational and rotational movements, are coupled with the relative motion of the colliding nuclei to produce the channel coupling [8].A heavy-ion reaction to fusion has a counterpart in quasi-elastic heavy-ion scattering at reverse angles.At energies near to the Coulomb barrier, these inclusive procedures are vulnerable to channel coupling effects (due to collective inelastic excitations of the colliding nuclei).The likelihood of reflection at the Coulomb barrier causes quasi-elastic scattering, whereas transmission is connected to fusion.This fact was exploited, and barrier distributions were obtained [9].Fusion is one of the most significant near-barrier processes.The interaction between the relative velocity of two colliding nuclei and their internal structures is well known to result in a significant increase in fusion cross sections at subbarrier energy [1].Several studies on quasi elastic scattering have been studied by Khalid S. Jassim for some heavy ions systems [10][11][12].
The aim of this research study is to study quasi elastic scattering at near energies from high the coulomb potential barrier to determine the surface diffuseness parameters of the inter-nucleus potential for the systems 25 Mg+ 90 Zr and ( 28 Si, 120 Sn)+ 150 Nd single and coupled channels calculations were performed using the CQEL program [7], which includes all orders of coupling and is the most recent iteration of the computer code CCFULL.
The chi square (ᵡ 2 ) approach has been used to find the diffuseness parameters' best fitted values in comparison to the experimental data.

THEORY
The potential between two nuclei consists of two parts, the first part being the nuclear potential Vn, which may be adequately and reasonably characterized by the two parts that make up the nucleus-nucleus potential.The Woods-Saxon (WS) form provided by [12].
V N (r) = - where r is the center-of-mass separation between the target nucleus of mass number A T and the projectile nucleus of mass number A P , and  denotes the system's radius  =   +  .When they do not interact, the second part being the Coulomb potential V C between two spherical nuclei with uniform charge density distributions is given by [12]: here r is the distance between the centers of mass of the colliding nuclei and Z p and Z T are the atomic numbers of the projectile and target, respectively.The Coulomb potential is produced when the nuclei interact, and it is determined by [13].
where R c is the radius of the equivalent sphere, which corresponds to the projectile and target nuclei.
The coupling between the nuclear intrinsic motion and the relative motion of the centers of mass of the colliding nuclei, r = (r, r), which causes the collision of two nuclei.The Hamiltonian system's is provided by [4] : where V (r) is the bare potential in the absence of coupling where V(r)= V N (r)+V c (r) , H (ξ) is the Hamiltonian for the intrinsic motion, and Ccoup is the stated coupling, r stands for the center of mass distance between the colliding nuclei.
The entire wave function's Schrodinger equation is given by [14].
Generally, the internal degree of freedom has a limited spin.The coupling Hamiltonian in complexities can be written as [4] : The spherical harmonics and spherical tensors, which are constructed from the internal coordinate, are denoted by  (ȓ) and  (), respectively.That when it was taken into account in V (r), the total is taken over all values of excluding for λ = 0. (r).For a constant total angular momentum J and its z-component M, the expansion basis for the wave function in Eq. ( 5) is given by [4] : where in l denotes the orbital, I denotes internal angular momenta, and  () denotes the wave function for the internal motion that which fulfills by equation below [4] : With this basis, the total wave function ( ⇀ .)has been expansion as [15]: It is possible to write the Schrödinger equation [Eq.( 3)] as a group of coupled equations for  () [14] : The coupling matrix elements  ;ń.´.Í (), According to [4] are as follows: Where the reduced matrix elements in Eq. ( 8) is defined as follows [4]: where  ;ń.
´.Í () are separate of the index M, the index has been suppressed as seen in Eq. (11).Coupled-channels equations are the name given to the equation (10).For heavy-ion fusion reactions, these equations are usually solved using the incoming wave boundary conditions [14].
where  = 2( −  )/ℏ ,  =  = 2/ћ and the following formula defines the local wave number  : After obtaining the transmission coefficients T nlIJ, the penetrability via the Coulomb barrier is given by: The computation of quasi-elastic cross sections typically needs a high value of angular momentum in order to provide convergent results, unlike the calculation of fusion cross sections.For such a huge angular momentum, the potential pocket at (r = rabs) becomes shallow or even vanishes.Therefore, it is impossible to precisely identify the incoming flux in Eq. (13).In order to avoid employing the incoming wave boundary conditions, the quasi-elastic problem often executes the regular boundary conditions at the origin.A complex potential V N (r)= V N 0 (r)+iw(r) is required to model the fusion reaction when utilizing the usual boundary conditions.Following the acquisition of the nuclear S-matrix in Equation (Eq11), the scattering amplitude may be computed as The Coulomb phase shift is  .andgiven by the equation, As for  , which is the Coulomb scattering amplitude and is determined by [14]: here η = is the Sommerfeld parameter which is given by  =    /ћ , and utilizing Equation ( 16), the differential cross section may be evaluated may be evaluate the Rutherford cross section.
The distribution of the barrier of fusion is defined as [4] : The definition of the total scattering distribution of the barrier of scattering D tot (E) is [4]:

RESULTS AND DISCUSSION
Calculations for single-channel and coupled channels were performed using the CQEL code last version [7], which is thought to be the most recent iteration of the computer code CCFULL .This program precisely resolves the linked equations and the Schrödinger equation.The chi square methods have been used in the present work to prevent systematic mistakes, where the data with dqel /dR>1 were omitted from the fitting procedures.The chi square technique χ 2 was regarded as a normalizing factor between the theoretical calculation and the experimental data.the main potential in this work is Wood-Saxon (WS), which has both real and fictitious components.The fairly tiny internal potential was explained by the fictitious potential.A change has been made to done on the real potential's parameters to determine how best to fit the experimental data, and it was then repeated for all interactions.The value of 0.63 is considered a constant for the diffuseness parameter [15], and a higher and lower value is taken for each system, It differs from one system to another and the radius parameter is assumed to be r 0 1.2 fm.

4-1. The 24 Mg + 90 Zr reaction
In this system, the results were processed in two cases, the first case, where both the projectile and target nuclei were considered inert (SC) at different values of the diffusion parameter (0.69, 0.63, and 0.57) fm, respectively and we considered the diffusion parameter 0.63 fm is the standard value.In the second case, the projectile nucleus was 24 Mg rotates with a deformation coefficient of  = 0.374 to the state 2 + (1.368672 MeV) and this was deduced according to the ratio   ⁄ = 3.012 while the target core 90 Zr was vibrating with a deformation coefficient of  = 0.089 to the state 2 + (2.186273 MeV), where   ⁄ = 1.4 at coupled-channel (CC).We used the single phonon state of the quadrupole excitation of the projectile and target nuclei, the potential depth V 0 = 58.8MeV, and the radius parameter r 0 = 1.2 fm.From Table (1), it is shown that, according to the results of the χ2, the calculated ratio of the quasi-elastic to the Rutherford cross sections is 0.28177 at the diffuseness parameter 0.57 fm, V 0 = 58.8MeV, which was obtained from the SC data analysis where the projectile 24 Mg nucleus and target 90 Zr nucleus are inert and is represented by the hard red line in Fig.  4-2.The 28 Si+ 120 Sn system From Throughout this system, the findings were processed in two distinct ways.In the first case, we assumed that the diffusion parameter 0.63fm was the standard value and that both the projectile and target nuclei were inert (SC) at various values (0.60, 0.63, and 0.66) fm respectively.In the second instance, the target core 120 Sn was vibration with a deformation coefficient of  = 0.107 to the state 2 + (1.171265 Mev),   ⁄ = 1.8 at coupled-channel (CC), whereas the projectile nucleus was 28 Si rotation with a deformation coefficient of  = -0.478 to the state 2 + (1.77903 MeV), where   ⁄ = 2.59.The potential depth V 0 = 45.8MeV, the radius parameter r 0 =1.2 fm, and the single phonon state of the quadruple excitation of the projectile and target nuclei were employed.According to Table (2), the results of χ 2 show that the calculated ratio of the quasi-elastic to the Rutherford cross sections is 0.06072 at the diffuseness parameter of 0.60 fm, which was obtained from single channel (SC) data analysis where the projectile 28 Si nucleus and target 120 Sn nucleus are inert, and is represented by the hard red line in Fig.  shows the distribution at sub-barrier energies, using the single and coupled channels calculations, respectively.The experimental data are taken from ref [17].
4-3.The 28 Si+ 150 Nd system In this system, the change of quasi elastic scattering with the angle was studied, depending on the change in the values of the diffusion parameter, and the study was carried out in two cases, in the first case The projectile and target nuclei were each considered to be inert, single channel (SC) at different values from diffuseness parameter (0.55, 0.63, and 0.71) fm, and 0.63fm was taken to be the standard value.in the second case, we assumed that the projectile nucleus 28 Si is rotation coupled to the state 2 + (1.77903 MeV),   ⁄ = 2.59 with deformation parameter  = -0.478where the target nucleus 150 Nd was inert.The potential depth V 0 = 42.2MeV, the radius parameter r 0 =1.2 fm, and the single phonon state of the quadrupole excitation to the projectile nuclei were employed.In Table (3), According to the results of the χ 2 data, the red line in Fig. 3  shows the distribution at sub-barrier energies, using the single and coupled channels calculations, respectively.The experimental data are taken from ref [18].
It is the curve that is most nearby to the curve of the experimental data.The optimum value for the distribution D is 0.03100, shown by the hard red line in Fig. 3.b.(C) at the same diffuseness parameter value.According to the coupled-channel calculations with a rotating projectile (P) and inert target (T), it was found that the best value of is 0.00597 at the diffuseness parameter a 0 of 0.63 fm, which is represented by the hard blue line in Fig. 3

.b. (B).
It is the curve closest to the curve of the experimental data, the best value of the distribution D is 0.00880 at the diffuseness parameter of 0.55 fm denoted by the red-colored curve.

CONCLUSIONS
We conclude from this study the following: 1-The standard value of the diffuseness parameter a 0 is not the only one that shows the best match between the theoretical calculations and the practical values, but it is possible to take lower and higher values than the standard value of the diffuseness parameter by 9, and thus we were able to obtain the best match within this range of values.

Figure 1 .
Figure 1.a.The ratio of the quasi-elastic scattering to the Rutherford cross sections for 25 Mg+ 90 Zr system at sub-barrier energies.Banal A and B using the single channel and coupled-channels calculations, respectively.Figure 1.b.(C, D) shows the distribution at sub-barrier energies, using the single and coupled channels calculations, respectively.The experimental data are taken from Ref[16].

Figure 1 .
Figure 1.a.The ratio of the quasi-elastic scattering to the Rutherford cross sections for 25 Mg+ 90 Zr system at sub-barrier energies.Banal A and B using the single channel and coupled-channels calculations, respectively.Figure 1.b.(C, D) shows the distribution at sub-barrier energies, using the single and coupled channels calculations, respectively.The experimental data are taken from Ref [16].It is the curve closest to the curve of the experimental data; the best value of the distribution D is 0.33499 at the same value of a 0 which is represented by the black colored.According to the coupled-channel calculation with a rotating projectile (P) and a vibrating target (T), the best value of is 0.01071 at the diffuseness parameter a 0 = 0.57 fm is represented by the hard red line in Fig. 1.b (B).From the Fig. 1.b.(C) the best value of the distribution D is 0.015 at the diffuseness parameter of 0.69 fm, which is represented by the black colored.

Figure 2
Figure 2.a.The ratio of the quasi-elastic scattering to the Rutherford cross sections for 28 Si+ 120 Sn system at sub-barrier energies.Banal A and B using the single channel and coupled-channels calculations, respectively.Figure 2.b. (C, D)shows the distribution at sub-barrier energies, using the single and coupled channels calculations, respectively.The experimental data are taken from ref[17].

Figure 2 .
Figure 2.a.The ratio of the quasi-elastic scattering to the Rutherford cross sections for 28 Si+ 120 Sn system at sub-barrier energies.Banal A and B using the single channel and coupled-channels calculations, respectively.Figure 2.b. (C, D)shows the distribution at sub-barrier energies, using the single and coupled channels calculations, respectively.The experimental data are taken from ref[17].

Figure 3 .
Figure 3.a.The ratio of the quasi-elastic scattering to the Rutherford cross sections for 28 Si+ 150 Nd system at sub-barrier energies.Banal A and B using the single channel and coupled-channels calculations, respectively.Figure 3.b.(C,D) shows the distribution at sub-barrier energies, using the single and coupled channels calculations, respectively.The experimental data are taken from ref[18].

Figure 3 .
Figure 3.a.The ratio of the quasi-elastic scattering to the Rutherford cross sections for 28 Si+ 150 Nd system at sub-barrier energies.Banal A and B using the single channel and coupled-channels calculations, respectively.Figure 3.b.(C,D) shows the distribution at sub-barrier energies, using the single and coupled channels calculations, respectively.The experimental data are taken from ref[18].

Table 1 .
The values of the WS potential's parameters and χ 2 fitting between experimental and theoretical data for the 24 Mg + 90 Zr reaction.

Table 2 .
The values of the WS potential's parameters and χ 2 fitting between experimental and theoretical data for the 28 Si+ 120 Sn reaction

Table 3 .
The values of the WS potential's parameters and χ 2 fitting between experimental and theoretical data for the 28 Si+ 150 Nd reaction