STUDY OF THE NUCLEAR STRUCTURE FOR SOME NUCLEI USING SELF-CONSISTENT RPA CALCULATIONS WITH SKYRME-TYPE INTERACTION

In the present research, some static and dynamic nuclear properties of the closed-shell nuclei; 58 Ni, 90 Zr, 116 Sn, and 144 Sm have been studied using the Random Phase Approximation (RPA) with different Skyrme parameterizations, particularly SyO-, Sk255, SyO+, SLy4, BSk17, and SLy5. In particular, in studies of static properties such as single-particle radial nuclear densities for neutrons, protons, mass, and charge densities with their corresponding root mean square (rms) radii, All the obtained results agreed well with the relevant experimental data. Concerning the dynamic nuclear properties such as, the excitation energy, transition density, and giant resonance modes for the excitation to the low-lying negative partite excited states 1 – , 3 – , 5 – , and 7 – have also been studied. The obtained results that estimates of RPA with Skyrme-type interactions are a good way to describe the properties of the structure of even-even, closed-shell nuclei.


INTRODUCTION
The structure of the atomic nucleus is dictated by the interactions of its constituents. Understanding the behavior of multidegree-of-freedom systems in terms of the fundamental interactions between their constituent components is one of the most difficult challenges in theoretical physics. The theoretical models need to be greatly simplified to find a solution to the problem of many-body in non-relativistic quantum mechanics, which governs self-bound composite systems [1]. When it comes to explaining the structure and dynamical features of nuclei, this is precisely the case [2][3][4]. A nuclear multi-body issue, which occurs in many disciplines of physics, is notoriously difficult to answer precisely. There are a number of approximate techniques for dealing with such systems. The independent particle approximation is used in the Simple Shell Model (SM), which ignores all correlation effects. The nuclear SM is one of the most important tools for understanding the structure of atomic nuclei and their radioactive decay stability. It is, like the atomic SM, is a microscopic approach that calculates the interaction of individual nucleons with a mean-field potential created by each of the system's constituent nucleons [5]. The nuclear shell model is based on the premise that each nucleon (proton or neutron) travels in the average field independently to the first order. RPA based on Hartree-Fock (HF) has been particularly successful in describing collective motion phenomena in nuclei at the microscopic level [6]. The HF and RPA nuclear structure approaches are widely regarded as highly successful in designing the ground states and excitations of spherical nuclei with effective interactions that are density-dependent [7]. The categorization of excitation modes is contingent upon the process by which a nucleus is excited [8].
In the present, the static and dynamic nuclear properties of some closed-shell nuclei, such as 58 Ni, 90 Zr, 116 Sn, and 144 Sm nuclei, were investigated using a basis on Skyrme-type interactions and the self-consistent HF-RPA, SyO+, SyO-, Sk255, SLy4, BSk17, and SLy5. In particular, the charge, neutron, proton, and mass densities associated with root mean square radii (rms) In addition, excitation energy, transition densities, and strength functions for the transition to negative parity states are also included. The estimated result will be compared with the available experimental data. In these nuclei, giant resonance (GR) modes and transition densities were also detected, indicating that the nucleus is moving collectively.

THEORETICAL CONSIDERATION Hartree-Fock Method
The "self-consistent field" approach was invented by Hartree, which he developed shortly after the formulation of the Schrödinger equations, to approximate the calculation of the wave functions and energy of atoms using the wave function of a single particle, which may be thought of as a combination of many-body wave functions [9]. Using variational principles, the HF equations can be obtained [10]. The variational principle is a method for calculating the wave functions and bound-state energies of a time-independent Hamiltonian [11]. Calculations for closed-shell nuclei were performed using the density-dependent nucleon-nucleon interaction of Skyrme [12]. Which of the following is regarded as a Slater determinant for the single-particle state: where is the set of space r  ,  spin, and q isospin coordinate, A is the number of sets.
Regarding the HF wave function, the predicted value of the whole Hamiltonian is used to estimate the ground-state energy in the HF approach [11].
With the addition of a non-local term, it resembles the ordinary one-body Schrödinger equation. The ground state Slater determinant, also known as the solution of the wave function matrix of a single particle, may be used to derive a sequence of one-body densities.

Skyrme-Type Interaction
The Skyrme interaction was developed for nuclear structure computations based on the idea that the energy functional could be rescaled and written as a zero-range expansion. As a result, the HF equations may be inferred easily by exchanging terms with the same mathematical structure as direct terms. Since Brink and Vautherin's pioneering work [13]. The Skyrme interaction is defined as the sum of two and three body components [14]. It is possible to express the interaction between two nucleons with r1and r2 spatial coordinates in the standard form [15]: where, 12  = 1 2 ( ) r r     is the interaction Dirac delta function [16], are the Pauli spin matrices and P   = (1+ 1   . 2   )/2 is the spin exchange operator, while k   and k  are the operators for relative momentum. k' acting on the right and k' acting on the left. They are defined as follows: The free parameters t 1 , t 0 , t 3 , t 2 , x 0 , x 1 , x 2 , x 3 , and W 0 , to produce the experimental data obtained from a limited number of nuclei, including binding energies as well as rms radii, are combined with the theory of nuclear matter properties.

Skyrme Energy Density
The Skyrme energy density functional, which is a function of various densities, can be calculated using the conventional Skyrme force [15]. The integral of the Hamiltonian density H can be used to calculate the probability value for the total Hamiltonian (H) of the Slater determinant: The Hamiltonian operator can be expressed as a sum of terms representing various aspects of the force [15]:

Skyrme-Hartree-Fock Equations
It's difficult to represent the system's energy as a function of the standard density matrix using the Skyrme interaction because of the dependence on J. In this situation, the energy must be adjusted with single-particle wave functions, with the additional stipulation that the φ k is normalized [17].
where the coefficients of the variation are, assuming a spherical closed-shell nucleus. As a result, the SHF radial equations can be deduced [18] which give can be deduced.
( 1 3 ) The Random Phase Approximation RPA is commonly regarded as a suitable method for studying nuclear structure since it provides an effective formalism for describing numerous giant resonances or excitations caused by low-lying particle holes, which are examples of nuclear excitations [19]. This theory can be used to describe the properties of low-energy excited states [20]. The straightforward HF Slater-determinant of the Schrödinger equation for this situation is [21].
The so-called creation operators † v Q can be employed to express the where 0 , the ground state, is also known as the vacuum.
In the RPA, we can generate and annihilate p-h pairs; the real ground state is not merely the HF vacuum [21]. † † Hermitian conjugation yields the following annihilation operator: The state's excitation energy v and the correlation matrix are combined into one matrix equation [21].

EEJP. 4 (2022)
Noor M. Kareem, Ali A. Alzubadi If the size of the p-h configuration space is N, the matrices elements. The sub matrices Band A are formed from the particle-hole configurations, where matrix A is Hermitian and the correlation matrix B is symmetric [22]. With the use of a multipole, transition density characterizes the connection between each excited state and the ground state. A transition is a generated amplitude for the operator. It is possible to compute its transition density's radial component. It has the definition below [23]: Below is the description of the density of state radial transition [23].
Excitation characteristics such as volume or surface type can be revealed through transition densities. The transition density for neutrons and protons is given as it can be either an isoscalar (IS) (T = 0) or an isovector (IV) (T = 1) [23]: In general, spherical harmonics related sum rule features can be enlarged in IS and IV external fields (F). It is possible to compute it using [23]. It can be calculated from: The IS and IV giant resonances' strength function distributions S(E) have been discovered to be extremely sensitive to nuclear matter's physical characteristics. The strength function of low-lying energy could be calculated by using the following: When the sum rule characteristics of generic IS and IV external fields are connected [23]. We can calculate the energy moments by integral the strength function, S(E).
where E1 and E2 are the excitation energy limits. The energy moments can be calculated using the formula integrating the strength function S(E). These densities can be used for calculating the rms radius of neutrons, protons, charge, and mass distributions [13], [24]: The proton radius is used to determine the charge radius [22].
where the proton's rms charge radius is 0.8 fm. The neutron skin thickness t is calculated using the difference in rms radii between the proton and the neutron. This difference provides information on the structure of nuclei [13].

RESULTS AND DISCUSSION The Nuclear Ground State Properties
The ground state nuclear properties of an HF, such as the charge density distribution rms, and the binding energies per nucleon and neutron skin thickness, were calculated and compared using several Skyrme parameterizations: Sk255, BSk17, SyO+, SyO-, SLy4 and SLy5. These parameterizationsare described in tabulated 1. Tables (2)(3)(4)(5) show the calculated static nucleon properties for the closed-shell nuclei chosen using these parameterizations in conjunction with experimental data from Refs. [25,26]. In the microscopic model, the radius values have risen from 3.7 fm (for 58 Ni) to 4.9 fm (for 144 Sm). And the binding energies per nucleon are in the range (8.68.26) MeV. This means that the binding energy per particle is nearly 8 MeV. From the tabulated results, it is clear that charge rms radius values in BSk17 parameterization are the closest to experimental data for 58 Ni and 116 Sn , followed by the results of SyO+ for 90 Zr and SLy5 for 144 Sm.
The nuclear charge densities provide a description of a nucleus' interior structure. Fig. 1 depicts the estimated HF radial charge density distribution for a single particle for 58 Ni, 90 Zr, 116 Sn, and 144 Sm, respectively. A comparison was made with the available experimental data 58 Ni, 90 Zr, 116 Sn, and 144 Sm, respectively. A comparison was made with the experimental data [27,28] except for 116 Sn, for which no available experimental data exists. The 58 Ni results are in complete good agreement with experimental data using SLy5, SLy4, BSk17, SyO-and SyO+ parameterizations. However, the Sk255 results are comparable to experimental evidence but do not fit within the nucleus. For 90 Zr, the SLy4, BSk17, SLy5, and Sk255 parameterizations coincide quite well. However, the results for SyO+ and SyO-are close to the experimental data but don't fit inside the nucleus. For 116 Sn Sk255, BSk17, SyO+, SLy5, SLy4, and SyO-parameterizations, although the findings are comparable to experimental data, they do not fit inside the nucleus. Consideration must also be given to the fact that nucleons possess intrinsic electromagnetic structure. Therefore, the proton, neutron, and mass density profiles must be computed. The profile of every nuclear density type of each of the chosen nuclei is shown in Fig. 2. Examining these curves demonstrates that the nuclear charge distributiondensity is depends on the rms radius, which rises as the number of protons in each nucleus increases. As the nucleus radius rose for 144 Sm, the density distributions for mass, charge, proton, and neutron increased. While in 58 Ni, 90 Zr, and 116 Sn, the charge density distributions gradually declined as the nucleus' radii shrank. From Tables (2)(3)(4)(5), it is evident that the energy per nucleon for all nuclei match with experimental data, particularly when the SLy5 and SLy4 parameterizations are used. In Figs. (3, 4), the theoretical rms charge radii and the binding energies per nucleon for closed-shell nuclei are compared with experimental data from Refs. [25], [26]. Very good agreement with experiments is seen, especially when Skyrme parameterization is used.     The Excited States Properties RPA calculations were widely utilized to describe the transition densities of the proton and neutron as well as GRs modes. This technique classifies collective resonances of various multipolarities, like the giant dipole photon absorption cross-section multipole at rank 2, referred to as isoscalar giant dipole resonance (ISGDR), rank 3 is giant octupole resonance (GOR), rank 5 is giant triacontadipole resonance (GTR), and rank 7 is giant octacosahectapole resonance (GOCR). In addition, the relation between the IV and IS strength functions and also the excitation energy functions has been defined, as have multipole excitations characterized by p-h pairings in which the particles and hole states are slightly below and above the Fermi surface. For optimum results, SLy5 and SLy4 parameterizations were chosen for this purpose based on the best binding energy estimates compared with experimental data [26].   The energy of excitation was compared with experimental data taken from Ref. [33] Compared to the existing experimental data, the estimated excitation energy of the negative parity excited states is qualitatively consistent with the results of the experiments ( 90 Zr has state 3 -(2.747875MeV), 116 Sn has states 3 -(2.66159MeV) and 7 -(2.90885MeV), and 144 Sm has state 5 -(2.82517MeV)). There is a small difference between theoretical values and experimental data for ( 58 Ni with states 3 -(4.971MeV) and 5-(5.5890MeV)), ( 90 Zr with states 5 -(2.319000MeV) and 7 -(4.037507MeV)), ( 116 Sn with state 1 -(6.020MeV)), and ( 144 Sm with state 7 -(3.12407MeV)) and a significant difference for ( 58 Ni with state 7 -(6.00847MeV)) and ( 90 Zr with state 1 -(6.020 MeV)). The associated transition densities effectively represent density oscillations around the ground state configuration. It is a crucial characteristic that sheds light on the nature of nuclear excitations in excited states, which determine protons and neutrons responses to external disturbance to obtain a simple estimation of the collectivity shown by the various dipole responses. It is shown that protons and neutrons can vibrate either in the skin mode's IS phase, where protons oscillate against neutrons, or in phase IV, which is the opposite (pn mode). Furthermore, the IV characterization is crucial for describing the motion of the neutron and proton within the nucleus, as it appears the peaks for proton and neutron transition densities appear to be positive for both protons and neutrons. The quantum numbers identifying these p-h pairs' SP states are identical. The electric multipole excitations (Jπ = 1-, 3-, 5-, 7-) emphasize this section of the current work. GR states are classified using the spin S, isospin T transmitted and total angular momentum J, as a result of nuclear ground state excitation. The scope of this study was confined to electric IS resonance with S = 0 and T = 0.Electric IV resonance takes place when S = 1 and T = 1.    116 Sn, and 144 Sm nuclei. The ISGDR is significantly fragmented compared to ISGTR, ISGOR, and ISGOCR, which appear to be less fragmented. This was reflected in IVGTR, IVGOR, IVGDR, and IVGOCR. This indicates that the collective strength of high multipolarity GRs is distributed, and the quantity of collective strength falls as the multipolarity J of the GRs [32]. It is important to note that the IS mode is produced when the neutrons and protons vibrate in phase, while the IV mode is produced when they vibrate in the opposite phase. The higher modes of strength functions for higher modes emerge clearly using RPA approach calculations, which explored upper-level excitation and yielded the collective nuclear motion of nucleons, energy, spin, and parity, all properties of exciting levels. This emphasized the significance of calculating GR strength functions using completely self-consistent HF-based RPA. Collective motion was believed to be a common characteristic of quantum many-body systems. It is believed that the distribution of transition strengths of a low external field that excites the nucleus exhibits resonance peaks that correspond to the elementary vibrational collective modes. Tables (6-9) summarize the overall calculated results of the EWSR m1 and RPA m1 using SLy5 parameterizations. The total EWSR of strengths is growing compared to those computed using RPA. Therefore, m1 includes those from ph strengths. The D.C (represents the double commutator which gives EWSR). The deviations of m1 (RPA) from m1 (EWSR) from R values are minor for all collective modes, indicating high precision for self-consistent HF-RPA calculations. It is also worth mentioning that it is the most widely used. The GDR is a well-studied GR wherein protons vibrate collectively against neutrons. This, in the end, shows how strongly neutrons and protons interact inside the nucleus, which is a key part of nuclear interaction.

CONCLUSIONS
The present research has led us to conclude that self-consistent RPA calculations are based on Skyrme effective nucleon-nucleon interaction, and these calculations give a significant description of the ground state properties for the closed-shell for 58 Ni, 90 Zr, 116 Sn, and 144 Sm. Considering the binding energy, neutron, proton, and neutron skin thickness, and charge radii, remarkable correlations were found with experimental data. For the static nuclear properties, the SLy4 and SLy5 parameterizations give the best results in compression with experimental data. Regarding dynamic properties, the RPA method is an efficient tool for describing the higher modes resulting from the collective motion of nucleons within the nucleus. It can also be used to describe the transition. density for proton . and neutron modes, in addition to GRs modes. The RPA method is the simplest theory that predicts the main features of giant resonance (GR) with a high degree of accuracy.