SHELL MODEL INVESTIGATION OF SOME p AND sd -SHELL NUCLEI WITH HARMONIC OSCILLATOR AND SKYRME INTERACTIONS

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INTRODUCTION
The investigation of nuclei has more access neutrons, than what is known as "neutron-rich nuclei" far from the line of stability is important for understanding nuclear structure.Exotic features differ from those of stable nuclei and deserve experimental and theoretical exploration.One of the most striking characteristics of neutron-rich nuclei is nuclear deformation, which may be studied both theoretically and empirically using electromagnetic properties like electric quadrupole (Q) moments and magnetic dipole (μ) moments.Using a microscopic particle vibration model, Sagawa and Asahi [1] investigated the N/Z dependence of the quadrupole polarization charges of C isotopes.The single-particle wave functions and gigantic quadrupole resonances are approximated using the Hartree-Fock and random-phase approximations.The polarization charges of nuclei with a high N/Z ratio experienced a significant quenching.Cohen and Kurath [2] model properly explain the features of low energy p shell nuclei, but it fails to represent the form factors of higher momentum transfer.Radhi et al. [3][4][5][6] have previously stated that the CP effects must be taken into account for nuclei in the p shell and sd shell to improve form factor calculations.Taihua Heng et al. [7] used the ab initio no-core full configuration NCFC technique to explore the characteristics of 7 Li with the NNLO opt chiral nucleon-nucleon and JISP16 interactions, as well as 7 Be with the JISP16 interaction.They calculated observables like energy spectra, proton point radii at the root mean value, transitions, and electromagnetic moments.Zheng et al. performed calculations based on large-basis shell model without core calculations for p-shell nuclei using six main shells (from 1s to 3p-2f-1h) [8,9].All nucleons are active in these calculations, according to Zheng et al., If computer resources are restricted, we have to adopt a truncated calculation without core with some freezing orbits, in which only a few ℏ excitations of the lowest unperturbed configurations are evaluated can be used.The result will converge and approximate that of the full no-core calculations as the number of ℏ increases.In the work of Navratil et al. [10] it is found that the predicted rate of the E2 transition with 4ℏ space for 6 Li is weaker than 6ℏ space prediction.Majeed et al. [11][12][13] revealing that the form factor calculation on a large basis of the shell model was used to analyze nuclei in the p, sd, and Fp shells, including the contribution of high-energy configurations beyond the p, sd, and Fp shell space model space is essential to be considered to consider the effect of the core polarization contribution arises from the closed core.
In this study, the longitudinal charge and C2 form factors for 9 Be and 28 S exotic nuclei utilizing the shell model calculations by considering the major shells 1s, 1p, 2s-1d, 2p-1f, with partially inert core using the model space spsdpf with wbt effective interaction using the code NushellX@MSU [14].Since the space configuration is a very large dimension, the shell model calculations in the whole spsdpf space are not possible, therefore a truncation is required in the valence space.We will consider the (0 + 2)ℏ truncated spsdpf model space (ms) calculation.The form factors will be calculated by considering the residual interactions, between harmonic oscillator (HO) and Skyrme effective interaction (Sk35−Skzs * ) [15] with using the Tassie and Bohr-Mottelson collective models.The core polarization (CP) will be considered by evaluating the suitable effective charges at zero photon point.Theoretical results of C0 and C2 were compared with the available measured data for each studied case for the studied nuclei.

METHOD DETAILS
Tassie and Bohr-Mottelson models were adopted along with Harmonic Oscillator potential and Skyrme effective interaction (Sk35−Skzs * ) with suitable parametrization were adopted.As mentioned in Ref. [16], the neutron and proton effective charges for the model of Tassie are predicted by using microscopic perturbation calculations that include intermediate (1p-1h) excitations from the orbit contribution MS and core to include higher orbits with ℏ excitations.The effective proton and neutron charges for Bohr-Mottelson were estimated using the equations [17]  ( ) = ( ) + ( ) where In terms of transition charge density, the element matrix of Coulomb can be represented as the sum of the MS and CP elements [18] where the momentum transfer is q and  () is spherical Bessel function.For the initial () and final () nuclear states, the nucleons charge density F of the transition is described using the one-body density matrix [18]  , ( where  stands for (  ) the s.p. states and () is the index which refers to either neutrons or protons.The transition density for the CP valence model is given by [18] ∆ () =   , () +   , () ,  are the charges associated with the neutron and protons to account for polarization.
The CP for Tassie model transition density is provided by The charge density of the ground state is [18] At the photon point, the proportionality constant  is given by the matrix elements of gamma transitions (),  =  ℏ ⁄ , where  is the energy due to excitation [18] The matrix of gamma transition elements can be described as MS matrix elements with effective charges.
Equating Eq. ( 7) with Eq. ( 8) yields the constant of proportionality N by means of the effective charges.A detailed discussion of these above-mentioned models for effective neutrons and protons [18] | ( Skyrme energy is produced through Skyrme interaction.According to E Skyrme is a two-body density-dependent interaction that represents central spin-orbit and tensor components in coordinate space and replicates the strong force in the particle-hole channel [19] here  ⃗ = ⃗ ⃗ and  is the force of Skyrme parameters.The operators the relative momentum wavevectors for the two nucleons acting to the right and left, with coordinate ́ respectively.When the nuclear structure data is fitted, the variables t 0 , t 1 , t 2 , t 3 , x 1 , x 2 , x 3 , and W 0 are free parameters taken to describe the components of various interactions.The Pauli matrices ( ), the spin-exchange operator, and the delta function of Dirac ( ⃗ +  ⃗ ) are all mathematical terms.The energy of the total density of the nucleus is expressed as follows in the (SHF) model [19]: Here  refers to energy of kinetic motion,  refers to Skyrme energy which include time even (nucleon  , kinetic energy  and orbital-spin ℌ ⃗ and the odd-time (current  ⃗ , spin  ⃗ , and the vector of the kinetic energy  ⃗ ) both densities and the pairing energy  and the mass at the center  .The  label refers to neutrons or protons.The Skyrme type of parametrization is taken to perform the calculations is (Sk35−Skzs * ) [16].
The "mean square charge radius" is expressed by the formula [16] Here  is the radius comes from the distribution of the proton point of the nucleus,  and  are the charge radii of the free proton and neutron, and the final term is known as the Darwin-Foldy term (0.033 fm 2 ).

RESULTS AND DISCUSSION
The longitudinal Coulomb charge | ()| and |(2, )| form factors for the 9 Be and 28 Si nuclei have been calculated by considering a truncated spsdpf model space with wbt effective interaction [20] with (0 + 2)ℏ.The C0 and C2 from factors calculated for the ground state 3/2 -, the 5/2 -(2.429MeV) and 7/2 -(6.380MeV) for 9 Be, while the ground state 0 + and 2 + (1.779 MeV) state for 28 Si nucleus.In all proceeding figures (see Fig. 1, panels a, b, c, and d), the dotted grass green and magenta curves display the results of the calculations of the Tassie and Bohr-Mottelson models using the valence model (MS) calculations only, while the solid red and blue curves show the calculations of Tassie and Bohr-Mottelson models including the core polarization effect by means of effective charge of protons and neutrons.

Be nucleus
The nucleus 9 Be is a neutron-halo with 4 protons and 5 neutrons considered as (4+n) and it is stable with the ground state is 3/2 -.The effective charge at the zero-photon point considered in this work is taken from Ref. [21].The longitudinal Coulomb charge | ()| and |(2, )| Form factors for the ground state have been calculated  = 3 2 ⁄ ,  = 1 2 ⁄ of the 9 Be as displayed in Figure 1 panels (a, b, c and d) by utilizing the Tassie and Bohr-Mottelson collective models.Due to large number of dimensions to be used in the model space spsdpf, a truncation has to be used, therefore the truncation is taken as (0 + 2)ℏ following the restriction adopted in Ref. [16].
The state (3/2 -1/2) at 0.000 MeV (g.s) The C0 and C2 form factors and their sum (C0+C2) for the ground state 3/2 -are depicted in Fig. 1 , b, c, and d.The calculations were performed by using harmonic oscillator (HO) and Skyrme effective interactions (Sk35−Skzs * ) for both Tassie and Bohr-Mottelson models.Panels a and b represent the Tassie model with the Harmonic Oscillator potential (HO) and Skyrme effective interaction (Sk35−Skzs * ), while in c and d in Fig. 1, represent Bohr-Mottelson model Tassie model with the Harmonic Oscillator potential (HO) and Skyrme effective interaction (Sk35−Skzs * ).The effective charges for neutron and proton are 0.70e and 1.075e, respectively [21], while for Bohr-Mottelson estimated using Eq.1 as 0.17 and 0.4 for the proton and neutron respectively.The observed data from Ref. [22] for this state form factor.The data could not be reproduced in all regions of the momentum transfer using bare model space computations.Introducing the effective charge in both Tassie and Bohr Mottelson with account for the core polarization effect makes  The state (5/2 -1/2) 2.429 MeV Figure 2 shows the calculation of the form factor for C2 for the state (5/2 -1/2) at E x =2.429 MeV.The bare model space calculation for both Bohr-Mottelson collective model and Tassie model by using the Harmonic Oscillator potential (HO) and Skyrme effective interaction (Sk35−Skzs * ) underestimates the measured data in all momentum transfer regions.The measured data for this state are taken from [22][23][24].The form factor of Bohr-Mottelson model reproduces the high q values of the effective charge to consider the CP effects, explains the data up  ≥ 1.5 .The state (7/2 -1/2) 2.429 MeV The longitudinal C2 form factor for (7/2 -1/2) at E x =6.380 MeV.The calculation of the model space for the Tassie and Bohr-Mottelson models at the peak value, it underpredicts the data by nearly a factor of three.The measured data are scattered therefore Introducing the effective charge in both Tassie and Bohr Mottelson to account for the core polarization effect aren't able to reproduce the measured data for these states.The measured data are taken from [22] (Fig. 3). 28Si nucleus The nucleus 28 Si is even-even nucleus with 14 protons and 14 neutrons and it is stable with the ground state is 0 + .The longitudinal Coulomb charge | ()| and |(2, )| form factors are calculated for the transitions to the ground state  = 0 ,  = 0 of the 28 Si as demonstrated in Figure 4 by utilizing the Tassie and Bohr-Mottelson collective models.Due to large number of dimensions to be used in the model space spsdpf, a truncation has to be used, therefore the truncation is taken as (0 + 2)ℏ following the restriction model from Ref. [16].
The state (0 + 0) 0.000 MeV (g.s) The C0 form factor for the ground state 0 + is displayed in Fig. 4. The calculations were performed by using Harmonic oscillator (HO) and Skyrme effective interactions (Sk35−Skzs * ) for both Bohr-Mottelson collective model and Tassie model.In both Tassie and Bohr Mottelson models, the effective charge is included to account for the core polarization effect.The measured data is taken from [25].The theoretical calculations of the model space only don't reproduce the measured data for all q values.The inclusion of the core polarization in the Bohr Mottelson model harmonic oscillator (HO) describes the experimental data very well up to momentum transfer  ≥ 1.5 and is able to reproduce the measured data very well and agrees with previous theoretical work in Ref. [6].

EEJP. 2
(2023) Sarah M. Obaid, Shaimaa A. Abbas, et al. a remarkable improvement especially for Bohr Mottelson calculations shown in panel (a) where the solid bule line which is (C0+C2) form factor matches the measured data in all momentum transfer regions using HO as the residual effective interaction.

Figure 1 .
Figure 1.Longitudinal C0 and C2 form factors and their sum C0+C2 for the Tassie and Bohr-Mottelson models by using the Harmonic Oscillator potential (HO) and Skyrme effective interaction (Sk35−Skzs * ) in comparison to the measured data [22].

Figure 2 .
Figure 2. The C2 longitudinal form factors for the Tassie and Bohr-Mottelson models by using the Harmonic Oscillator potential (HO) and Skyrme effective interaction (Sk35−Skzs * ) in comparison to the measured data [23, 24].

Figure 3 .
Figure 3.The C2 longitudinal form factors for the Tassie and Bohr-Mottelson models by using the Harmonic Oscillator potential (HO) and Skyrme effective interaction (Sk35−Skzs * ) in comparison to the measured data [23, 24].

Figure 4 .
Figure 4. Longitudinal C0 form factors for the Tassie and Bohr-Mottelson models by using the Harmonic Oscillator potential (HO) and Skyrme effective interaction (Sk35−Skzs * ) in comparison to the measured data [25].

Figure 4 .
Figure 4. Longitudinal C2 form factors for the Tassie and Bohr-Mottelson models by using the Harmonic Oscillator potential (HO) and Skyrme effective interaction (Sk35−Skzs * ) in comparison to the measured data [25].CONCLUSION The longitudinal charge | ()| and |(2, )| form factors the nuclei ( 9 Be and 28 Si) in the p-and sd-shells by utilizing the shell model calculations by considering the major shells 1s, 1p, 2s-1d, 2p-1f, including a partially inert core using the model space spsdpf with wbt effective interaction using the code NushellX@MSU were conducted.The residual interactions used in the calculation of the form factors are harmonic oscillator (HO) and Skyrme effective interaction (Sk35−Skzs * ) by employing Tassie and Bohr-Mottelson collective models.The result of the form factors with (0 + 2)ℏ shell is not able to reproduce the data for all momentum transfer regions for both nuclei under study.Introducing the effective charges in both Tassie and Bohr Mottelson to account for the core polarization effect makes a remarkable improvement especially for Bohr Mottelson calculations.