Calculations of The Shell Model for 27 Mg Isotope

Using shell-model calculations


INTRODUCTION
The shell model (SM) is the primary theoretical framework utilized by both experimentalists and theoreticians when investigating nuclear structures [1][2][3]. This microscopic model of the atomic nucleus is the most comprehensive one currently available [4]. In the atomic SM, electron motion is governed by a Coulomb potential emanating from the nucleus. In contrast, nucleons are responsible for generating the potential that governs their motion. The nucleons in this model move in orbits at regular intervals and interact via a simple average potential with an amplitude of about 1fm [5].
In a mean-field (MF) picture, nuclei are seen as separate particles, and their interactions are limited to the average MF potential that the other nucleons put on them. In this model, nucleons live in discrete states (also called "orbits") that are bound solutions of the MF potential. The many-body function of the whole system is a product of these singleparticle states, and this product is antisymmetric [6,7]. MF can be used to figure out the wave function of a single particle. It can be done in either a non-relativistic or a relativistic way. This phenomenological method has been used to study the structure of the nucleus for many years, and the results have been pretty good. Assuming a Schrodinger equation with central and spin-orbit terms, as well as other possible interactions like spin-spin interactions. Because theoretical research can explain experimental data on electromagnetic transitions and nuclear energy levels, this study will explain the experimental data available for the magnesium-27 isotope in terms of nuclear energy levels and electromagnetic transitions. It will also explain how the effect of core polarization changes the values of the reduced transition probabilities.
We have performed the large scale shell model calculations for the 27 Mg within the model spaces (zbme), psdpf, sdpf, and sd by using OXBASH code [8]. The inert cores of these model spaces are, respectively, 12 C, 4 He, and 16 O. The positive parity states are calculated within ℏ configurations of sd and psdpf model spaces. The recent development of Magilligan and Brown (USDC/I) [9] for the USD interaction is used in the calculation of the positive parity states, as well as USDB interaction [10]. To calculate the negative-parity states, zbme and psdpf spaces are used, and the adopted interactions are respectively, Reehal-Wildenthal (REWILE) [11] and Warburton-Brown (WBT) [12]. In the calculation of reduced transition probability, core polarization effects are introduced such that 1p-1h configurations up to ℏ are taken into account. The ℏ configurations within psdpf space are obtained from the distribution of nucleons within sd shells, where the sd part of the WBT interaction is Wildenthal (W) interaction as used by Warburton and Brown [12]. Since p 1/2 sd space allows for up to 4 nucleons to jump from 1p 1/2 orbit to 1d 5/2 , 2s 1/2 and 1d 3/2 orbits, the positive-parity states are obtained through partial ℏ configurations, in the presence of REWILE interaction. For the negative-parity states, two model spaces are considered, p 1/2 sd and psdpf. For psdpf space, the complete ℏ configurations are made from all allowed excitations of one nucleon from 1p to 2s1d shells or the excitation of one nucleon from 2s1d to 2p1f shells.
In OXBASH code, the WBT interaction within psdpf model space is performed by using the restriction ℏ on the space spsdpf. On the other side, the negative-parity states are made in p 1/2 sd space through partial ℏ configurations. Also, OXBASH code is used to generate One-Body Density Matrix elements (OBDMs) which are important in the calculations of the reduced transition probabilities ( ) between nuclear shell model states.
In the calculation of the electric transitions and the magnetic transitions , core-polarization effects are introduced through microscopic theory that include excitations of nucleons from the core obits (of each model space) into higher shells up to ℏ excitations outside model space, with ℏ [13]. The core orbits are 1s and 1p for sd model space, 1s 1/2 and 1p 3/2 for zbme model space and 1s 1/2 for psdpf model space. The results of that calculated within model space are denoted as MS, while that incorporated core-polarization effects are denoted as MS+CP.

THEORY
The reduced probability for the electromagnetic transition operator ( for electric or magnetic operators) between the initial (i) and final (f) nuclear states of spin and isospin and , respectively, is given by [14]: isospin. , and . The T can take on two possible values: T=0 is called an isoscalar and T=1 is called an isovector. These values represent the different possible states that a particle can have under isospin symmetry. The concept of isospin is important because it allows physicists to describe the behavior of particles that have similar interactions but different masses or charges. By considering particles with different isospin values. The total isospin of a proton-neutron system can either be T=1 if the two isospins are aligned and T=0 if they are antialigned. The triple-bar matrix element is used to indicate the reduction in spin and isospin spaces. The reduced many-particle matrix elements are written in terms of reduced single-particle one, as [13] ( ‖| |‖ )

∑ ‖| |‖
where and denote the quantum numbers of single-particle states (including isospin), and the OBDM are defined by [13] ( The inclusion of core-polarization effects on the one-body transition operator, through first-order perturbation theory, in the presence of the residual interaction V res will separate the reduced single-particle matrix elements into three parts [13] ‖| |‖ The operator Q is the projection operator onto the space outside the model space. E i,f are the initial and final states energies. The first term is due to model space, while the second and third terms are due to core-polarization effects. The core-polarization terms can be evaluated in terms of the matrix elements of residual interaction and the transition operator by introducing intermediate particle | 〉 and hole | 〉 states, and using some Racah algebra [13,15] The triple-bar single-particle matrix elements are written in terms of double-bar one, by [13] 〈 ‖| |‖ 〉 where T denotes the isospin, for a proton and for a neutron. The single-particle energies in the denominator of eq.(5) are calculated by [13] ( ) The reduced matrix elements of the electric and magnetic operators are given, respectively, by [13] and Where the notation ⟨ | | ⟩ represents the matrix element between two single-particle states, characterized by their quantum numbers ( ) and ( ), and refers to the radial part of the final and initial wave function respectively. the g-factors are for the proton and the neutron respectively, and the radial integral involving harmonic oscillator radial wave functions are defined as The adopted residual interaction is the Modified Surface Delta Interaction (MSDI). the strength parameter of the MSDI model represents the strength of the short-range delta potential that describes the strong interaction between nucleons when they are very close to each other in the atomic nucleus and are set to be , where A is the atomic mass [13,16].

RESULTS AND DISCUSSION
Both the experimental and the theoretical level schemes of the 27 Mg nucleus are depicted in Figure 1. The calculations for the p 1/2 sd shell model are carried out by the interaction of 15 nucleons, which consist of (6 protons + 9 neutrons). Positive energy levels were obtained by sharing only sd orbital nucleons, while negative parity-states were obtained by sharing one nucleon jump from the P to sd orbital since we restricted the transition to ℏ . The same constraint is applied to the space of the psd model, so it is certain that the production of levels with positive parity will be in the same distribution as the space of the p 1/2 sd model. However, it is noted that the levels produced from the space of the psd model were the best result when compared with other calculations and are closest to the experimental values. In general, we note that the calculated excitation energy levels within the sd model are in agreement with the approved experimental values [17,18]. It is noted that the energy level is an uncertain experimental value that appears at 3.109 MeV; in our calculation, the appears at 3.33 MeV, 3.03 MeV, and 2.79 MeV within effective interactions (SDPFMUPN, USDB/C/I, and PSDWBT) respectively. In addition to that, these values were recently found by a research paper [19], with . e experimentally and .
e theoretically. This convergence of values suggests this value is likely to be adopted as The negative-parity states that were produced within the model spaces used gave a somewhat satisfactory description, and this can be seen through what was shown in the states that carry negative parity in Table 1  The negative parity states in the sdpf model space were high value compared to the remainder of the model spaces (as shown in Table 1) and the reason was due to the adoption of the proton-neutron formalisms, during which it restricted the movement of neutron within the sd orbital, while we allowed one proton to jump between sd and pf orbitals (sd) 3 (pf) 1 , it was noticed that most of the negative cases were formed from the configuration maxing for proton orbitals and the presence it with a higher percentage at p(1f 7/2 ) or p( 2 p 1/2 ).  [21] was high, especially in (case1), compared to the calculation of the CP effect.

CONCLUSION
The shell model with configurations interaction is applied on 27 Mg isotope to produce the positive-and negative-parity states and calculate the reduced transition probabilities between them. To achieve this task, three model spaces are used: sd, zbme and psdpf. The universal SD interactions, USDB, USDC and USDI reproduce comparable results for the positive-parity states. In zbme space, the 1p 1/2 orbit is added to the 2s1d shells, and thus the negative-as well as positive-parity states are produced. The psdpf space, is the most suitable space to produce all intruder negative-parity states in sd-shell nuclei, due to full ℏ excitations. Thus, positive-and negative-parity states in exotic 27 Mg isotopes are very well reproduced by calculations achieved within zbme and psdpf spaces in the presence of Reehal-Wildenthal and WBT interactions, respectively.
In the model-dependent calculations of reduced transition probabilities, effective charge and effective g-factors are usually used. Instead of using those free parameters, we depend, in the present work, on microscopic theory without using any free parameter to calculate B(EL) and B(ML). In core-polarization effects, intermediate particlehole excitations are taken into account in the presence of residual interaction. The inclusion of microscopic core-polarization effects, in the presence of MSDI as residual interaction, enhance the results of B(E2) and shift them closer to the experimental data, while B(E1), B(M1) and B(M2) are not.