Calculation of the Magnetic Dipole and Electric Quadrupole Moments of some Sodium Isotopes using Shell Model with Skyrme Interaction

Article Info. In the present work, the magnetic dipole and electric quadrupole moments for some sodium isotopes have been calculated using the shell model, considering the effect of the two-body effective interactions and the single-particle potentials. These isotopes are; 21 Na (3/2+), 23 Na (3/2+), 25 Na (5/2+), 26 Na (3+), 27 Na (5/2+), 28 Na (1+) and, 29 Na (3/2+). The one-body transition density matrix elements (OBDM) have been calculated using the (USDA, USDB, HBUMSD and W) two-body effective interactions carried out in the sd -shell model space. The sd shell model space consists of the active 2 s 1/2 , 1 d 5/2, and1 d 1/2 valence orbits above the inert 16 O nucleus core, which remains closed. Skyrme interaction was implemented to generate the single-particle matrix elements with Hartree-Fock approximation and compared with those of harmonic oscillator and Wood-Saxon potentials. From the outcome of our investigation, it is possible to conclude that the shell model calculations with Skyrme-type interaction give a reasonable description for most of the selected Na isotopes. No significant difference was noticed for the magnetic dipole moments and electric quadrupole moments with experimental data, where all signs for the experimental data are reproduced correctly.


Introduction
The magnetic dipole (μ) and electric quadrupole (Q2) moments are considered one of the most important ways to learn about the nuclear structure and obtain dense and deep information like deformations, charge, and nuclear moments in this field [1]. The magnetic dipole moments of nuclei are sensitive to the orbits occupied by the valence nucleons. They offer a perfect test of the purity of a specific configuration mixing shell model [2]. The electric quadrupole and magnetic dipole moments can assist us in understanding some aspects of the nuclear structure.
On the other hand, electromagnetic moments are considered one of the most basic probes for obtaining information on the nuclear structure across the entire nuclear chart [3,4]. Nuclear deformation can affect the shell structure of single-particle levels in a spherical potential [5]. The shell model has been successful in describing nuclei near magic numbers. However, it fails to describe the structure of nuclei with many valence nucleons outside closed shells, where the residual interactions between such nucleons' deformation are the most significant and essential consequences [6]. This model has shortcomings, particularly true for heavier nuclei [7]. We have already seen that the shell model does not predict magnetic dipole moments or the spectra of excited states very well [8]. The nuclear shell model has been applied to the properties of nuclei near stability and how its applications are extended to nuclei out to the proton and neutron drip lines. The nuclear electric quadrupole moment is considered a probe for the deviation of nuclear charge from spherical symmetry, which gives a valuable measure of how the core is polarized, especially if the valence nucleons are neutrons that do not directly participate in the electric quadrupole moment [9]. According to the convention, the value of electric quadrupole moments is positive if the ellipsoid is prolate and negative if it is oblate. It is the simplest deformation and one of the significant quantities that limit the shape of the nucleus [10].
The present work aims to calculate the magnetic dipole and electric quadrupole moments for some sodium isotopes using the shell model, considering the effect of the two-body effective interactions and the single-particle potentials. The sd shell model space will be used for this purpose, consisting of the active 2s 1/2 , 1d 5/2 and1d 1/2 valence orbits above the inert 16 O nucleus core, which remains closed.

Theoretical consideration
The electric pole operator ( ) possesses parity and the electric pole moment should vanish for odd values of L. The intrinsic is defined with respect to the charge distribution axis of symmetry. The sum of the products of the one-body density matrix (OBDM) times the single-particle matrix elements is the nuclear matrix element of the electromagnetic ( ̂) and electron scattering ( ̂) operators between the initial I and final (f) nuclear states for certain multipolarity [11]: The electromagnetic operator ( ̂) and the electron scattering operator ( ̂) are represented by the ̂o perator. k represents the single-particle states (nlj). The OBDM is determined by: All the quantum numbers required to differentiate the states are included in I and f. In terms of the M1 operator, the magnetic dipole moment is defined as [12]: where ⟨ ‖ ̂ ‖ ⟩ is the operator of the magnetic transition, and is the nuclear magneton 0.1051 efm, with the proton mass. The orbital and spin free nucleon g factors g(free) are: = 1, = 5.585 for proton and = 0, = -3.826 for neutron [12].
The electric quadrupole moment is defined in terms of the E2 operator as: where | ⟩ includes all the quantum numbers needed to differentiate the nuclear states in the initial and final nuclear states. The Skyrme potential is employed for the central potential; it is a two-body interaction. In the Hartree-Fock (HF) approximation, it may be used to construct a one-body potential [13]. It is a mean-field potential, and it is designed to mimic the realistic nucleon-nucleon (and nucleon-nucleon-nucleon) forces by providing the mean-field due to all the nucleons that make up the nucleus. The Skyrme potential can be written as [14]: 2  2  1  1 2  0  0  12  1  12  12   3  12  2  2  12  3  12  0  12 1  2   2  1  2  1  2  12 where and ̂, ̂ are the relative momentum operators with ̂ acting on the right, while ̂ is the operator acting on the left and given by: and ̂ is the spin-exchange operator that is given as:

Result and discussion
In the present work, the nuclear shell model calculations are performed using the NuShellX@MSU shell model code [15]. It uses a J -coupled proton-neutron basis, and Jscheme matrix dimensions of up to 100 million can be considered. The radial wave functions of the single-particle matrix elements were calculated using a two-body Skyrme interaction potential and the HO and WS. Discussion of the results will be divided into three sections; the first is devoted to the magnetic dipole moment, the second section examines the electric quadrupole moments, followed by the conclusions section.

Magnetic dipole moment
The calculation of magnetic dipole moments of Na isotopes in sd model space using four two-body effective interactions; USDA, USDB, HBUMSD and W, with free nucleon g factors. The calculated results are presented in Table 1 compared to the experimental data from Ref. [16,17]. The previously displayed values show that the calculated magnetic dipole moments are in good agreement with the experimental data provided [16,17]. For more illustration, the comparison of the calculated μ moments with the corresponding experimental data is depicted in Figs. 1 and 2. It is obvious that the HBUMSD interaction gives results that are broadly consistent with experimental data. 12 1 2 () rr   Table 1 To achieve the better results, the applicability of the present method is tested in calculating the moments using different single-particle potentials and two-body effective interaction HBUMSD. Namely, the harmonic oscillator (HO), Woods Saxon (WS), and Skyrme Hartree-Fock (SHF) with SLy4 parameterization. From Table 2 and Figs. 3 and 4, it is clear that the calculated results using these types of potentials are in good agreement with the experimental data of magnetic dipole moments. In contrast, 26 Na isotope gives the best and closest result to the experimental data [16,17].

Electric quadrupole moments
The electric quadrupole moments of Na isotopes are calculated and shown in Table 3 and Figs. 5 and 6. The electric quadrupole moments for isotopes are calculated with twobody effective interactions of sd model space MS and compared with the available experimental data from Ref [16,17]. It is clear that the calculations with sd wave functions reasonably explain the experimental data and predict the correct sign of the prolate deformation for the isotopes 21 Na, 23 Na, and 29 Na and oblate deformation for 26 Na and 27 Na isotopes. The USDA and USDB interactions failed to reproduce the correct sign for the 25 Na isotope. As in the case of magnetic dipole moments, to achieve the best results, we test the applicability of the present method in calculating the electric quadrupole moments using different single-particle potentials, namely; the HO, WS, and SLy4. As shown in Table 4 and Figs. 7 and 8, the calculated Q2 results are not very sensitive to changing the potentials compared to the experimental data [16,17], where we have good agreement.

Figure 6: The calculated electric quadrupole moments Q2 (e 2 fm 2 ) vs mass number (A) for Na-isotopes using different two-body effective interactions compared to the experimental data of
Refs. [16,17].

Conclusions
The magnetic dipole and electric quadrupole moments for Na isotopes have been calculated in sd shell model space, considering the sensitivity for changing the two-body effective interactions and the single-particle potentials. All the calculated results have also been compared with available experimental data. From the outcome of our investigation, it is possible to conclude that the shell model calculations in sd model space with Skyrme type interaction give a reasonable description for most of the selected Na isotopes. No significant difference was noticed for the magnetic dipole moments μ and electric quadrupole moments Q2 with experimental data, where all signs for the experimental data are reproduced correctly, where it was found that 26 Na isotope gives the best result and the closest to the experimental data. It is obvious that the HBUMSD interaction gives results that are broadly consistent with experimental data. The present work provides more information on previously known spectroscopic properties by implementing the shell model calculations with mean-field potential.