Investigation of the Nuclear Structure of Some Ni and Zn Isotopes with Skyrme-Hartree-Fock Interaction

: The inelastic C2 form factors and the charge density distribution (CDD) for 58,60,62 Ni and 64,66,68 Zn nuclei has been investigated by employing the Skyrme-Hartree-Fock method with (Sk35-Skzs*) parametrization. The inelastic C2 form factor is calculated by using the shape of Tassie and Bohr-Mottelson models with appropriate proton and neutron effective charges to account for the core-polarization effects contribution. The comparison of the predicted theoretical values was conducted with the available measured data for C2 and CDD form factors and showed very good agreement.


Introduction:
In order to know the nuclear forces and the laws that govern the interactions between elementary particles, the only possible method is that of scattering a diversity of particles by a diversity of targets. There will be three types of scattering experiments, the first one is that the incident particle has no atomic force, but interacts with the nucleus via only electromagnetic interaction, such as speedy electron scattering. The data on charge and current densities can be given 1 . Secondly, the scattering of neutral electrical particles, for example neutrons that only contribute to nuclear fields. Lastly, the scattering of particles charged by electricity, oftentimes experiences strong nuclear contact such as proton and alpha particle. The scattering of nuclear electrons provides the most detailed knowledge about the nuclear size and distribution of charge and offers valuable information about electromagnetic currents within nuclei. Electron scattering can be a good test for calculation because it is sensitive to spatial load dependency and current densities 2,3 . To explain nuclei using the nuclear shell model, one must understand the effective one-and two-body interactions. Several techniques for calculating the effective interaction for nuclei in the fp-shells have been proposed in the last 30 years 4 . Kuo and Brown proposed the first comprehensive technique in the 1960s, using the Hamada-Johnston potential 5 . In model space calculations, Radhi et al [6][7][8][9] showed that it is advisable to include CP effect in form-factors for nuclei in the beginning of p-shell and the end of sd-shell. F. A. Majeed et al [10][11][12] demonstrated that form factor calculations based on large-basis of the shell model to study the nuclei in the p, sd and fp shells considering the contribution of high-energy configurations beyond the p, sd and fp shell space model space denoted as the core polarization effects. The proton and neutron matter density distribution and their related charge radii along with the ground state form factors have been investigated by Hameed BS 13 for some selected potassium isotopes, their conducted study showed that taking into consideration the core polarization effects along with effective g-factors made remarkable enhancement in their calculations which leads to good agreement with the measured data.
The recent developments with new modern effective interactions designed for fp-shell region are still not universal and need more tuning. The nuclei at the upper limit of the fp-shell region attract attention in recent years due to their importance in understanding the supernova explosions, therefore this study aimed to investigate the charge density distribution (CDD) and the inelastic C2 form factors for 58,60,62 Ni and 64,66,68 Zn nuclei by employing the Skyrme-Hartree-Fock method with (Sk35-Skzs*) parametrization and comparing he predicted theoretical calculations with the available measured data.

Theoretical Background Skyrme Effective Interaction
Skyrme presented an effective interaction for HF calculations of nuclei 14,15 . In describing the ground conditions properties of nuclei with approximately ten adjustable parameters, it is the more convenient force used to fit experimental nuclei data such as binding energies and charged radii 16 . The Skyrme forces are zero-range interactions that are composed of a momentumdependent two-body term and a zero-range threebody term. A density-dependent two-body term can be replaced for the three-body term in the HF calculations. As a result, as an extended Skyrme force 17 , the Skyrme forces are unified in a single form. The interaction between two bodies in Skyrme is written as follows 14 : are free parameters that describe the strengths of various interaction factors that are fitted to nuclear structure data. ̂ is the Pauli matrices, ̂= ( +̂•̂) is operator of spin-exchange, and the Dirac delta function (⃗⃗ − ⃗⃗ ). In the typical Skyrme-Hartree-Fock (SHF) model, the total energy density of a nucleus is expressed as 14 = is the kinetic energy of the nucleons (proton and neutron) can be calculated using the following formula 15 is the energy functional of the Skyrme force and given by 15 , is the local densities for protons and neutrons (depending on the value of q), is the total density, is the kinetic energy densities for protons and neutrons and is the spin-orbit current density. They are given by these relations 15 ( The nucleon densities ( ) and ( ) may be calculated from a single particle wave function defined by the HF calculation. In the (SHF) theory using the Skyrme forces, the most general product wave functions ( ) consist of independently moving single particles. In this method the neutron and proton densities are given in where represents the state, is the probability of the state, and the angular momentum of the state. The Coulomb energy part is the third part of the total energy equation. The exchange component contributes a modest amount of energy to the Coulomb energy. This contribution is owing to the fact that the Coulomb interaction is infinite range. The Coulomb energy is calculated as follows 15,17 : where is the radial part of the harmonics oscillator wavefunction and is given as is the associated Laquerre polynomials and b is the harmonic oscillator size parameter.
where = ∑̂ is the total momentum operator, A the nucleon number and m the average nucleon mass.

Tassie and Bohr-Mottelson
The Tassie model (TM) used in NushellX@MSU 18 for core polarization is a more elastomeric and modified modeling that allows for a non-uniform mass and charge density distribution. In the TM model, the CP charge density is determined by the nucleus ground state charge density. The two-body charge density for all occupied shells, including the core, has been used to obtain the ground state charge density. The Tassie shape core polarization transition density, which is based on the collective modes of the nuclei is given by 19 where is a constant of the proportionality and 0 is the charge density distribution for the ground state two-body, which is given 20 , where the first term gives zero contribution, the second and the third term can be combined together as 19 ,

Results and Discussion:
There are many theoretical tools available to describe the properties of fp-shell nuclei that are available. Such calculations vary with the completion of the space of the shell model, the mass dependent of the involved parameters and the twomatrix elements is appropriate in terms of experimental results, in the present work 58,60,62 Ni and 64,66,68 Zn. The jun45 interaction has been used to give the 1f 5/2 2p 1/2 1g 9/2 -shell model wave functions. The core is taken as 56 Ni for all the nuclei under study with different valence neutrons for Ni isotopes and two valence protons with different values of valence neutrons for Zn isotopes. The Bohr-Mottelson and Tassie models were employed to perform the present calculations. The curve with red dots displays the results of the calculations Tassie in any of the following diagrams (see Fig.1). The Bohr-Mottelson calculations are shown by the solid blue curve.
The parameters used with Sk35-Skzs* used to perform the calculations are listed in Table1.  Fig.   1. The effective interaction jun45 is utilized as effective interaction for the calculations of the shell model and the (Sk35-Skzs*) residual interaction to calculate the matrix element for the interactions from initial to final states. The first maxima are very well reproduced by both Tassie Model and Bohr-Mottelson calculations. The second maxima did not agree in comparison with the calculations. The experimental data for this state is taken from Ref. 21 .
The effective proton and neutron charges used in these calculations are 0.5 and 1.5, respectively. Figure 2. shows the CDD obtained with the investigated Skyrme parameterization Sk35-Skzs* used in this work as well as the experimental charge distributions for 58 Ni nucleus. Theoretically the calculated CDD is quite consistent with the theoretical calculations with the Skyrme parameterization.  The C2 longitudinal form factor for the state ( J = 2 1 + 2) of 60 Ni nucleus has been calculated by Tassie model and Bohr-Mottelson models. The calculations according to standard shell model the core is taken at 56 Ni with four neutrons as valence particles outside the core, as shown in Fig.  3. The core polarization is considered by using the effective proton and neutron charges. Figure 4 shows the CDD obtained with the investigated Skyrme parameterization Sk35-Skzs*used in this work as well as the experimental charge distributions for 60 Ni nucleus. The comparison of the predicted theoretical values was compared with the available measured data for CDD and C2 form factors and showed good agreement.  MeV. The C2 longitudinal form factor of 62 Ni nucleus in 1f 5/2 2p 1/2 1g 9/2 -shell is presented in Fig. 5. The jun45 is utilized as an effective residual interaction for the calculations of the shell model. The third maxima are very well reproduced by both Tassie and Bohr-Mottelson calculations. The experimental data for this state are taken from Ref. 23 . Figure 6 shows the CDD obtained with the investigated Skyrme parameterization Sk35-Skzs* used in this work as well as the experimental charge distributions for 62 Ni nucleus .The comparison of the predicted theoretical values was compared with the available measured data for CDD and C2 form factors and showed good agreement.

Conclusion:
The present study detailed theoretical investigation to study the form factors and charge density distribution (CDD) of inelastic electron scattering for some selected fp-shell nuclei. The theoretical calculations were employed the Skyrme-Hartree-Fock method with (Sk35-Skzs*) parametrization. The comparison of the predicted theoretical values was compared with the available measured data for CDD and C2 form factors and showed good agreement.