1. Introduction

The nuclear shell model proved to be very successful tool to investigate the nuclear structure: by choosing an adequate residual effective interaction, the shell model able to account for various observables accurately and systematically. The nuclear structure study progressed by developing the nuclear shell model¹. Although the shell model is basically simple, it explains many nuclear properties such as spin, magnetic moment, and nuclear spectra. The shell model is composed of two fundamental kinds of models which are related to the basis of the shell model: the models of the mean-field and they configuration mixing models². The sd-shell is an interesting region for shell model calculations which can be investigated by elastic magnetic electron scattering, where the nucleus is considered as an inert ¹⁶ O and the full 1d5/2 , 2s1/2 , 1d3/2 space is used for the valance nucleons. Excitation to the higher shell are ignored in the model. Calculations based on this model may not be able to reproduce the experimental observations or to agree with the experimental form factors. Effective charges are adopted in many previous studies in which the effective g-factors were implied. The (q)-dependence of form factors on the momentum transfer resulting from configuration mixing is very different for a different major shell, and cannot in general be considered as a q independent scaling³. The shell model electron scattering form factors needs to be modified by including higher configurations, called core polarization effect. These effects are considered as a supplement to the usual shell-model treatments, which gives more practical efforts to account for the Coulomb excitations collectivity. A model based on micorscopic approach has been used to account for CP effects between states of single particle with LS closed shell⁴. Radhi et al.⁵-⁸ have argued previously that the CP effects are essential to be taken into consideration for nuclei lies in the p -shell and sd-shell to improve the calculations of the form factors. The single quadrupole transitions Coulomb form factors for electron scattering in the p-shell ¹⁰ B nucleus have been investigated by F. A. Majeed⁹, in which 2ℏω excitations were considered by prompting necleons from the core orbits to higher orbits to account for CP effects. The high energy configuration effect were considered by means of core polarization effect have been investigated by Majeedet al.¹⁰ on the C2 and C4 form factors of some selected nuclei lies in the fp-shell region. The core polarization were calculated by employing harmonic oscillator (HO) and Skyrme-Hartree Fock (Skx) as residual interactions. Majeed et al.¹¹ investigated the effect of the configuration of high energy for the positive and negative parity states form factors for longitudinal and transverse cases.

The goal of the current study is to investigate the nuclear structure of ²⁰ Ne, ²² Ne and ²⁴ Mg nuclei. In particular, energy levels, inelastic electron scattering form factors and transition probabilities using shell model codes CP and NushellX@MSU for windows¹². The study of C2 and C4 for this nuclei including configurations of high-energies by utilizing the first-order perturbation theory to account for the CP effects. The zeroth contribution for the single-particle wave functions are used and the effect of CP, is taken into consideration by perturbation theory of first order with the residual interaction which is the modified surface delta interaction (MSDI) ¹³. The potential of harmonic-oscillator (HO) with the size parameter b is taken to account for the root-mean-square (rms) charge radii of the studied nuclei.

The effect of CP on the form factors is derived from a microscopic theory that allows the basis function of the shell and the configurations with higher-energy to be combined as perturbations of first order. The electron scattering operator T​Λ , written in terms of the reduced matrix elements which consists of the residual interaction Vres as the sum of the product of the density matrix of one-body (DMOB) elements χΓfΓiΛ (α, β) times he matrix element of single particle, and is given by,

where ∣Oi⟩ and ∣Of⟩ are the model space states. The quantum numbers are denoted by Greek symbols in space and isospace coordinates, i.e. Oi≡JiTi , Of≡JfTf and Λ=JT .

The matrix of the model space (ms) consists of the sum of the product of the matrix element of the density matrix of one-body (DMOB) χOfOiΛ (α,β) times the matrix element of the single-particle as follows ,

with the single-particle states α and β are to account for the model space where the isospin is included.

The matrix element of the configurations with higher energy (CP) is similarly written as

where Q represent the projection operator, which projects onto the model space. The MSDI is adopted as the residual interaction Vres and Ei and Ef are the energies of the initial and final states, respectively.

The MSDI effective interaction that was adopted for the calculation of the CP effects is a very adequate choice due to its adjustable parameters that allows us to reliably consider the CP effects with respect to the model space. The MSDI can be written as

where ⟨j2-(1/2)j1(1/2)|J0⟩ , ⟨j2(1/2)j1(1/2)|J1⟩ are Clebsch-Gordan coefficients¹⁵, T is the nuclear isospin, B and C are parameters obtained from the fitting to the experimental data for various mass region. The parameters AT with T taken as 0 or 1, B and C are approximated as¹³

where A is the mass number.

The CP terms are written as

where α1 and α2 indices which runs over particles states and e is the energy for single-particle states. The CP terms are determined from the intermediate states up to the 2p1f-shells. The matrix element of the single-particle is reduced in both spin and isospin is expressed in terms of the matrix element of the single reduced in spin only¹³.

with

where tz=-1/2 for neutrons and 1/2 for protons. The matrix element of the single-particle Coulomb operator is expressed as^[14]

where jJ(qr) is the Bessel function in spherical coordinates and Rnl(r) is the radial wavefunction for the single particle.

The form factors for electron scattering involves the momentum transfer q and angular momentum J, between the initial and final nuclear shell model states of spin Ji,f and isospin Ti,f ³ is ³

where Tz is the final isospin states projected along z-axis and is evaluated by the relation Tz=(Z-N)/2 . The form factor for the finite size of the nucleon (f.s) is Ff.s(q)=exp(-0.43q2/4) and Fc.m(q)=exp(q2b2/4A) represent the translational invariance lack in the shell model. A is the mass number and b is the size parameter for harmonic oscillator.

The strength of the electric transition is

where k=Ex/ℏ c .

The Tassie model (TM) used for the core polarization in NushellX@MSU is a modelling of more elasticity and modification that allows a non-uniform mass and charge density distribution. The CP charge density in TM model depends on the ground state charge density of the nucleus. The ground state charge density is expressed in terms of the two-body charge density for all occupied shells including the core. Based on the collective modes of the nuclei, the Tassie shape core polarization transition density is given by¹⁶.

where N is a proportionality constant and ρ0 is the ground state two- body charge density distribution, which is given

where

i and j are all the required quantum numbers, i.e. i≡ni,li,ji,mi,ti,mti and j≡nj,lj,jj,mj,tj,mtj where the functions f(rij) are the two body short range correlation (SRC). In this work, a simple model form of short range correlation has been adopted, i.e.

where rc is the radius of a suitable hard-core and β is a correlation parameter. The Coulomb form factor for this model becomes:-

The radial integral

can be written as:-

where the first term gives zero contribution, the second and the third term can be combined together as

from the recursion of the spherical Bessel function:

Therefore, the form factor of Eq. ( number ) takes the form:

The proportionality constant N can be determined from the form factor evaluated at q=k , i.e. substituting q=k in the equation above we obtained

2. Results and Discussion

3. Conclusion

The nuclear structure of ²⁰Ne, ²²Ne and ²⁴Mg nuclei have been studied by employing the shell model with usda and usdb effective interactions designed for the sd-shell region. The core polarization effects have been considered by a microscopic theory that allows the excitation to 4ℏω , without any adjustable parameters that were used previously when the core polarization effects is taken by the concept of the effective proton and neutron effective charges. The level excitation spectra, transition probabilities and inelastic electron scattering form factors have been addressed in the present study. The shell model prediction have a shortfall in describing the form factors and the CP effect must be taken into consideration to be able to reproduce the longitudinal C2 and C4 form factors. The Tassie model with proton and neutron effective charges is able to reproduce the C2 and C4 form factors for all the studied states of the nuclei under study.