ELECTROEXCITATION FORM FACTORS AND DEFORMATION OF 20,22 NE ISOTOPES BASED ON THE SHELL MODEL AND HARTREE-FOCK PLUS BCS CALCULATIONS

Nuclear structure of 20,22 Ne isotopes has been studied via the shell model with Skyrme-Hartree-Fock calculations. In particular, the transitions to the low-lying positive and negative parity excited states have been investigated within three shell model spaces; sd for positive parity states, spsdpf large-basis (no-core), and zbme model spaces for negative parity states. Excitation energies, reduced transition probabilities, and elastic and inelastic form factors were estimated and compared to the available experimental data. Skyrme interaction was used to generate a one-body potential in the Hartree-Fock calculations for each selected excited state, which is then used to calculate the single-particle matrix elements. Skyrme interaction was used to calculate the radial wave functions of the single-particle matrix elements, from which a one-body potential in Hartree-Fock theory with SLy4 parametrization can be generated. Furthermore, we have explored the interplays among neutron and proton density profiles in two dimensions, along with the deformations of 20,22 Ne using Hartree-Fock plus BCS calculations.


I. INTRODUCTION
For a microscopic description of the nucleus, different nuclear models have been utilized [1].The most efficient one is the Shell Model (SM) [2], based on the idea of an independent nucleon freely orbiting in a spherically symmetrical core potential generated by all the other nucleons within the nucleus.In actual SM calculations, nuclear states are linear combinations of states rather than pure states [3].Always, SM computations are performed in a configuration space with a limited number of single-particle states outside of an inert core, which is typically a doubly-magical nucleus.As a result of this truncation, the residual interactions must be regarded as effective interactions, and choosing the proper N-N interaction is not simple [4,5].The ground states of nuclei are created when nucleons fill shells to the Fermi level.Fermi's level is the same for protons and neutrons in stable nuclei, but the Coulomb repulsion between protons [6] explains why the line of stability and the N = Z line on the chart of the nuclides do not correspond for heavier nuclei.The laws of quantum mechanics govern the location and characteristics of the nucleus' discrete energy levels, just like they do for the atom.The positions of excited states vary from nucleus to nucleus.Excitation energy (Ex) is influenced by each nucleus's internal structure.Quantum numbers denote each excited state's angular momentum, parity, and isospin, in addition to its electromagnetic and strong properties.Positive-parity spectra can be generated by considering only the 1d 5/2 , 2s 1/2 , and 1d 3/2 orbits in the sd-shell configuration space.Any realistic negative parity calculation must account for both 1p and 2p-1f active shells.In the absence of such theoretical work, approaches to comprehending the structure of negative-parity states typically rely on more generalized descriptions [7].In a system of identical Fermions, the Pauli exclusion principle dictates that the properties of a nucleus with a given number of protons and neutrons are defined by the filling of the lowest energy single-particle levels (the nucleons in this case).The Pauli Exclusion Principle states that a particular set of quantum numbers can only be occupied by a single proton or neutron.The average nuclear potential is determined by the shape of the nuclear density distribution and the attractive short-range nucleon-nucleon interaction [8].

II. THEORETICAL FRAMEWORK (a) Shell model calculations
The reduced matrix elements of the electron scattering ( ) X λ operator between the final f and initial i states can be expressed as the sum of the one-body density matrix (OBDM) times the reduced single-particle matrix elements [9]; The OBDM of multipolarity ( λ ) can be expressed in term of the second quantization notation as; , , where the single-partical state (k), and t z = 1/2 and -1/2 for proton and neutron, respectively and i and f contain all the quantum numbers needed to separate the states.M1 operator is used to define the nuclear magnetic dipole moment as [10]

OBDM(
where the nuclear magneton  = ħ = 0.1051 efm.While, in terms of the E2 operator, the electric quadrupole moment is defined as 2 16 ˆ(E2) 0 5 where the initial and final nuclear states |J> contain all the quantum numbers necessary to differentiate the nuclear states.The reduced of transition probability given as [11].
where k= E X /ħc, B(Eλ) is in unit of e 2 fm 2λ and B(Mλ) is in the unit of μ N 2 fm 2λ-2 .With the realization that the energy functional could be written as a zero-range expansion, the Skyrme interaction was developed for nuclear structure computations, resulting in a straightforward derivation of the Hartree-Fock (HF) equations in which exchange terms have the same mathematical structure as direct terms.This approach decreases significantly the number of single-particle state integrations necessary to solve the equations.The Skyrme energy (E Sky ) reflects the strong force in the particle-hole channel in coordinate space and consists of central, spin-orbit, and tensor contributions [6].Central potential is represented by Skyrme potential.It is a one-body potential as a mean-field potential.It is designed to approach the realistic nucleon-nucleon forces by providing the average field owing to all of the nucleons constituting the nucleus.VSky equals the combination of two and three bodily components.[12] as: (2) (3) The two-body part is given by: ( where ( ) The  and  operators represent the relative wave vectors of two nucleons acting to the right and left, respectively (i.e., complex conjugate wave functions with coordinate  ), They possess the shape; Electron scattering form factor between final and initial nuclear shell model states, including angular momentum λ and momentum transfer q, is given by [13].
where J f and J i are the total angular momentum λ of final and initial state, F cm (q) is the correction (center-of-mass) and F fs (q) is the finite size of the nucleon, with χ involving the transverse (T) and longitudinal (C) form factors.The nuclear

EEJP. 2 (2023)
Omar A. Alswaidawi, Ali A. Alzubadi structure enters into the electron scattering through the longitudinal F C form factor and the transverse F T form factors (F E and F M ) are the electric and magnetic transverse form factor, respectively).The total longitudinal and transverse form factors for electron scattering are given by: These form factors are functions of the momentum transfer q only. ) ) ) The related to the electromagnetic transition operators as ( , , ) where ( ) ˆr ρ , ( ) Ĵ r , and ( ) ˆr μ , are the nuclear charge, the magnetization current density operators is ( ) function of order λ , Y λμ is the spherical harmonic.The total form factor is equal to the addition of the longitudinal and transverse relations:

(b) Hartree-Fock plus BCS calculations
The self-consistent mean field based on HF plus BCS calculations are designed to describe the structure of nuclei and study the evolution shapes, using the Skyrme forces performed to study the transitional in density shape where the pairing correlation have been taken into account.HF method is probably the best method for anticipating the total binding energies and single particle energies of closed shell nuclei [14].Also SHF is a useful tool because this force is central and has zero-range interactions [15].Nuclei is a quantum many-body system exhibiting the quadrupole collectivity associated with the shape of the mean field.The collective degree of freedom is associated with the measure of the operator Q .
The formula for the resulting BCS many-body state is [16].
Where 0 is the particle-vacuum state, is the Fermion production operator âα + in state α ψ , and is the time-reversed partner to state α .The local density of nucleons is defined as [16] ( ) ( ) The local nucleon density is defined as [16] ( ) ( ) The total energy consists of ( ) ( ) where E Culomb is the Coulomb energy the pairing energy is where dV represents the volume element in whole three-dimensional space and is the fundamental charge e. with e 2 =1.43989MeV.fm, and q ξ is the pairing density, w α is a soft pairing space cut-off.The variables s ϵ ±1 represent the spinor component of the wave functions.
The pairing energy includes the parameter ρ 0,pair that controls the equilibrium between volume and surface pairing.Deformation of the nucleus is defined as the departure from spherical symmetry about the center of mass (c.m), which is quantified by the electric quadrupole moment.Hence, the most significant moments are center-of-mass moments [16] ( ) ( ) In terms of the spherical quadrupole moments, the anisotropic combinations can be quantified ( ) the quadrupole deformation parameter

III. RESULTS AND DISCUSSION
In this study, the OBDM elements for low-lying positive parity  states were calculated using the sd-shell model space.For negative parity  states, the spsdpf no core with (0, 1) ћω restriction and the zbme shell model spaces have been employed.Using the NuShellX@MSU code [17], all calculations were performed.As we mentioned previously, the Skyrme interaction was used to calculate the radial wave functions of the single-particle matrix elements, from which a one-body potential in HF theory with SLy4, parametrization can be generated in addition to the harmonic oscillator (HO) and Wood-Saxson potentials.For 20,22 Ne isotopes, using USDC [18] two-body effective interaction in the calculation of the OBDM and SLy4 parameterization yields root mean square (rms) charge radii of 2.954 and 2.9525 fm, which are in good agreement with the experimental values 3.005 and 2.9525 fm [19].The calculated binding energies are 150.15 and 168.84 MeV, which are in reasonable agreement with the experimental values of 160.64 and 177.76 MeV [20].The nuclear magnetic dipole moment is (1.076 and 0.780) nm, in good agreement with experimental values of +1.08 and +0.65 nm [19], and the electric quadrupole moment (Q2) is -14.31, and -14.03 e 2 .fm 2 , the experimental -23(3), -19(4) e 2 .fm 2 [19].All these results together match the experimental value.
Also, we have used the code SkyAx [16] is a highly optimized two-dimensional HF+ Bardeen-Cooper-Schrieffer (BCS) code is using for computing ground states and deformation energy surfaces for axially symmetric deformed nuclei.The calculated results will be discussed in three sections.The first will focus on the Excitation Energies and reduce transition probability, the second on electroexcitation Form Factor, and the third on structure densities in two dimensions (Contour Lines), which will be given.

Excitation Energies and reduce transition probability
The excitation energies and reduced transition probabilities for low-lying positive and negative parity states in 20,22 Ne isotopes are calculated and tabulated in Table 1 and compared with the corresponding experimental data.The OBDM

EEJP. 2 (2023)
Omar A. Alswaidawi, Ali A. Alzubadi elements for positive parity states were calculated using sd MS with the last updated two-body effective interactions USDE [21] and USDC [18].In contrast, for negative ones, we have used the spsdpf (no core with one ћω restriction) and zbme model spaces with WBP and REWILE effective interactions, respectively.The majority of estimated excitation energies in all model spaces closely match experimental data.
[19], except for a few levels.Those levels with deviations exceeding 1.5 MeV include 2 at 7.833 MeV for 20 Ne and 4 at 6.345 MeV for 22 Ne.The negative-parity energy levels are accurately predicted using zbme MS.However, for spsdpf MS, the excitation energy state 1 at 7.119 MeV is overestimated by the experimental data.The coupling between the states, such as the 1p 1/2 holes coupled to the 1d 5/2 neutron (1p-1h), may account for these discrepancies.The additional pairing correlation and a change in the proton-neutron interaction, which results in a more considerable quadrupole deformation energy, reduce the energy of the 1p-1h state [8].Fig. 1 illustrates the extent of convergence between theoretical calculations and experimental data.
The calculated reduced transition probabilities B(EL) for the low-lying positive and negative parity states in 20 Ne and 22 Ne isotopes are presented in Table 2.The discrepancies with experimental data regarding the energy of transitions from these states might be due to the possible admixture of states involving neutron and proton excitations.From a general point of view, the B(EL) values for the transitions of low excitation energies agree reasonably well with experimental data [20,[22][23][24].The B(E2) transition rates are slightly larger than the experimental data except for 2 , where the agreement is quite good for 2 in 22 Ne using the USDE interaction.The slight difference between the experimental data and the theory could be because the quadrupole vibration was not taken into account.Regarding B(E3) and B(E1) transition probabilities in 22 Ne, the calculated results are in poor agreement with the experimental results.

Electroexcitation form factor
The nuclear structure can only enter the cross-section through longitudinal (FC), transverse (FE), and magnetic (FM).These form factors are functions of the momentum transfer q only.The FC and FT form factors correspond to fields parallel and perpendicular to the direction of momentum transfer, respectively [25].|F T (q)| 2 , is influenced by both magnetic and electric currents, whereas |F L (q)| 2 is caused solely by the electric Coulomb field.
A. Elastic electron scattering form factor (  =  ) Fig. 2 (a) and (b) depicts the calculated elastic longitudinal C0 electroexcitation form factors for the ground state (GS) of 20,22 Ne isotopes in the sd-shell MS wave functions employing SLy4, HO, and WS parameterization compared with experimental data from Ref. [22,26].Observable agreement exists in the momentum transfer region of q (0-1.3)fm -1 , based on experimental data.[22,23] using all the single particle potentials.Although we have not been changing parameters, they were altered to accommodate the experimental electron scattering data.The WS potential agrees satisfactorily with experimental data, except 2 , at (7.422MeV) higher than experimental data at all momentum transfer points.   Ne at (4.2477MeV) and in 22 Ne at (3.357 MeV).It is obvious that the longitudinal form factor is dominated by electric hexa transition C4 components and in reasonable agreement with experimental data in light of the fact that parameters were not changed to fit the experimental electron scattering data [23,24].

Negative parity states
Based on the results obtained in showing the sensitivity of the effect of changing the single particle potentials.The longitudinal form factors were calculated, considering the effect of the two-particle interactions in improving the convergence with the practical values.Fig. 6 shows the calculated total form factors for the transition to the negativeparity state1 _ , (5.787 MeV) and 3 _ , (5.624 MeV) for 20 Ne isotope compared with experimental data [23], which refer to the total sum of (1 _ ,3 _ ) states together.The WBP interaction predicts a rapid increase in low-lying C3 concentration.While C1 form factor result under the experimental data.On the contrary, the REWILE result is in reasonable agreement with experimental results for the C3 form factor, in contrast to the C1 result, which was based on experimental data.   Ne isotope for the transitions 1 , (6.689 MeV) and 3 , (5.910 MeV) states.The WBP prediction agrees qualitatively with experimental data in all momentum transfer regions of these data.Also, it can be observed that the REWILE prediction for the longitudinal C1 form factors under estimate the experimental result.The longitudinal C1 experimental data reveals an additional maximum form factor.The theoretical C1 contribution does not include this maximum.From the our previous calculates EEJP. 2 (2023) we noted that WBP is the best in approximation with experimental data, thus means the contribution valance nucleon in 1p 3/2 state is accountable for the substantial C1, C3 strength.3. The Quadrupole Deformation using BCS Calculation Fig. 8 and 9 shows the potential energy curve of 20, 22 Ne isotopes (on the left) as a function of the quadrupole deformation parameter β2.The corresponding neutron and proton structure densities are also displayed (on the right).Following the color code, the red and blue colors correspond to the high density (~0.08 fm -3 ) and low density (~0.02 fm -3 ), respectively.The two local minima in Fig. 8 are predicted as (a) β 2 =-0.154 and (b) β 2 =0.406. 20Ne have a stable quadrupole deformation where the high neutron and proton density distribution is in the center for the two regions (a) and (b), as expected ( 20 Ne have the same numbers of protons and neutrons), where there is no effect of n-p pairing.For the 22 Ne isotope, the two local minima in Fig. 9 are predicted as (a) β 2 =-0.204 and (b) β 2 =0.404.We can notice that the stability decreases with decreasing central proton density because of the two neutrons added to the last state, which increases the effect of n-p pairing.IV.CONCLUSION In this study, the nuclear structure of 20,22 Ne isotopes was investigated in the framework of the shell model and BCS calculation with Skyrme parametrization.In this context, excitation energies and the corresponding reduced transition probabilities, the elastic and inelastic electroexcitation form factors for positive and negative parity states in the momentum-transfer range 0.0 <q< 3.0 fm -1 , and quadrupole deformation parameter are discussed.Additionally, the work inspects the effect of three single-particle potentials; in particular, HO, WS and SLy4 parameterizations.It can be concluded that, on the whole, the impression of using different model spaces has fairly well reproduced the experimental data for positive and negative parity states and is not sensitive for changing the single particle potentials rather than the two-body effective interactions.The most important limitation lies in the fact that the strong collective feature of the nuclei in this mass region and internally consistent interaction formulation.

Figure 1 .
Figure 1.Theoretical excitation energies in MeV states vs. experimental for the different transition to low-lying excited states using USDE, WBP, and REWILE two-body effective interactions.The experimental data are taken from Ref. [19].

Fig. 5 (
Fig.5(a) and (b) show the calculated inelastic longitudinal C4 electroexcitation form factor of the transition to the 4 + state in20 Ne at (4.2477MeV) and in22 Ne at (3.357 MeV).It is obvious that the longitudinal form factor is dominated by electric hexa transition C4 components and in reasonable agreement with experimental data in light of the fact that parameters were not changed to fit the experimental electron scattering data[23,24].

Figure 6 .
Figure 6.Total form factor for the transition -11(5.578)and -31(5.621MeV)using SLy4 parameterization in comparison with experimental data taken from Ref [26]Fig.7(a), and (b) show the calculated longitudinal C1, C3 form factors in22 Ne isotope for the transitions 1 , (6.689 MeV) and 3 , (5.910 MeV) states.The WBP prediction agrees qualitatively with experimental data in all momentum transfer regions of these data.Also, it can be observed that the REWILE prediction for the longitudinal C1 form factors under estimate the experimental result.The longitudinal C1 experimental data reveals an additional maximum form factor.The theoretical C1 contribution does not include this maximum.From the our previous calculates

Figure 8 .Figure 9 .
Figure 8. Left, the potential energy curve of 20 Ne as a function of the quadrupole deformation parameter.The neutron and proton structure densities corresponding to the two local minima, marked a, and b are shown in the right panel Alzubadi, https://orcid.org/0000-0002-7226-1141

Table 1 .
Excitation energies in MeV for a different transition to excited states using two-body interactions USDE, USDC, WBP, and REWILE.The experimental data taken from Ref.[19]