Shell Model for Study Quadrupole Transition Rates in B2 in Some Neon Isotopes in sd-shell with Using Different Interactions

Solving the nuclear many-body problem is a fundamental task in nuclear structure studies. The spherical shell model has continually been a reliable tool when comparing with experimental observables. In practical shell model calculations, the valence space is limited within one or several adjacent major shells. The bulk of its wave function is presumably contained in this restricted configuration space [1]. Nuclear shell model is one of the most powerful tools for giving a quantitative interpretation to the experimental data. The two main ingredients of any shell model calculations are the N-N interaction and the configuration space for valence particles. In principle one can either perform shell model calculations with realistic N-N interaction in unlimited configuration space or with renormalized effective interaction limited configuration space [2]. Shell model calculations are carried out within a model space in which the nucleons are restricted to occupy a few orbits. If appropriate effective operators are used taking into account the effect of the larger model space, the shell model provides a reasonable description of these observables [3]. The calculations of shell model, carried out within a model space in which the nucleons are restricted to occupy a few orbits are unable to reproduce the measured static moments or transition strengths without scaling factors. Calculations of transition strengths using the model space wave function alone are inadequate for reproducing the data. Therefore, effects out of the model space, which are called core polarization effects, are necessary to be included in the calculations [4]. A study of nuclei in the sd shell can thus lead to a better understanding between a microscopic description of the nucleus (shell model) and a macroscopic (collective) description [5]. The sd-shell nuclei are considered as an inert 16O core and the valence nucleons are distributed in 1d5/2, 2s1/2 and 1d3/2 shell. Higher configurations can be included through perturbation theory, where particle-hole excitations are allowed from the core and the valence nucleons to all allowed orbits with nћω excitations. The number n depends on the convergence of the calculations. The deformation can be investigated experimentally and theoretically, through their electromagnetic transitions. The general trend of the 2+ excitation energy E ( 1 2 ) and the reduced electric quadrupole transition strength between the first excited 2+ state and the 0+ ground state, B (E 2, 1 1 0 2 + + → for even-even nuclei are expected to be inversely proportional to one another [6]. States of mixed configurations the situation differs in the valence shell sd shell model for N (neutron) > 8 and p (proton) > 8). Figure 1 indicates how nucleons move via the nucleon–nucleon interaction. The occupancy pattern of nucleons over different orbits is called configuration [7]. *Corresponding author: Ahmed H. Ali, PhD in Theoretical Nuclear Physics and Medical Physics, College of Medicine, University of Fallujah, Iraq, Tel no: + 9647815262642; E-mail: dr.ahmedphysics@uofallujah.edu.iq.


Introduction
Solving the nuclear many-body problem is a fundamental task in nuclear structure studies. The spherical shell model has continually been a reliable tool when comparing with experimental observables. In practical shell model calculations, the valence space is limited within one or several adjacent major shells. The bulk of its wave function is presumably contained in this restricted configuration space [1]. Nuclear shell model is one of the most powerful tools for giving a quantitative interpretation to the experimental data. The two main ingredients of any shell model calculations are the N-N interaction and the configuration space for valence particles. In principle one can either perform shell model calculations with realistic N-N interaction in unlimited configuration space or with renormalized effective interaction limited configuration space [2]. Shell model calculations are carried out within a model space in which the nucleons are restricted to occupy a few orbits. If appropriate effective operators are used taking into account the effect of the larger model space, the shell model provides a reasonable description of these observables [3]. The calculations of shell model, carried out within a model space in which the nucleons are restricted to occupy a few orbits are unable to reproduce the measured static moments or transition strengths without scaling factors. Calculations of transition strengths using the model space wave function alone are inadequate for reproducing the data. Therefore, effects out of the model space, which are called core polarization effects, are necessary to be included in the calculations [4]. A study of nuclei in the sd shell can thus lead to a better understanding between a microscopic description of the nucleus (shell model) and a macroscopic (collective) description [5]. The sd-shell nuclei are considered as an inert 16 O core and the valence nucleons are distributed in 1d 5/2 , 2s 1/2 and 1d 3/2 shell. Higher configurations can be included through perturbation theory, where particle-hole excitations are allowed from the core and the valence nucleons to all allowed orbits with nћω excitations. The number n depends on the convergence of the calculations. The deformation can be investigated experimentally and theoretically, through their electromagnetic transitions. The general trend of the 2 + excitation energy E ( 1 2 + ) and the reduced electric quadrupole transition strength between the first excited 2 + state and the 0 + ground state, B (E 2, 1 1 0 2 + + → for even-even nuclei are expected to be inversely proportional to one another [6]. States of mixed configurations the situation differs in the valence shell sd shell model for N (neutron) > 8 and p (proton) > 8). Figure 1 indicates how nucleons move via the nucleon-nucleon interaction. The occupancy pattern of nucleons over different orbits is called configuration [7].
The reduced electric transition probability from j i to j f be defined as [8]:

Results and Discussion
The calculation of the reduced electric transition probability B(E2) from the ground 0 + state to the first excited 2 + state for some neon even-even 18,20,22,24,26,28 Ne isotopes and which were performed by using equation (6). The one body transition matrix element (OBTM) values were obtained by the shell model calculations that performed via the computer code NuShellX [10] MSU and using different interactions such as USDB (Universal sd-shell interaction B) [11], USDA interaction (Universal sd-shell interaction A) [11] and Bonn-A interaction [12]. The reduced quadrupole transition probability is calculated using different effective charges such as conventional effective charges (CEF) [13], Bohr-Mottelson effective charges (B-M) [9,14] and standard effective charges (ST) e p =1.36 and e n =0.45 [14,15]. The radial wave functions for the single-particle matrix elements were calculated with the harmonic oscillator (HO) potential with size parameters for each isotope are calculated as 1 1 0 2 + + → with ħω=45A -1/3 -25A -2/3 as shown in Table 1 [16].
The presented results for B(E2) values in this work were compared with the available experimental values give in reference [17].

USDB Interaction
Reduced transition probabilities in units of e 2 fm 4 are calculated for Neon Ne isotopes (Z=10) with mass number A=18, 20, 22, 24, 26, 28 and with neutron number N=8, 10,12,14,16,18, respectively. Shell model calculations in sd model space and USDB interaction [11] was used to generate the OBTM elements for the ground state with J=0 and excited state with J=2. The harmonic oscillator size parameter b [16] was calculated for each isotope and tabulated in Table 1. All isotopes in the present work composed of the core 16 Table 1 and plotted in Figure 2a as a function of neutron number N and mass number A in comparison with the experimental values [17]. The Bohr-Mottelson effective charges (B-M) [9] were calculated for 18,20,22,24,26,28 Ne isotopes as shown in Table 1, Conventional effective charges (CEF) [13] which are for proton 1.  Table 1 and plotted in Figure 2b which shows an inverse relation between the excitation energy and transition rate B(E2) [5]. Theoretical values overestimate the experimental values where the excitation energy for some isotopes were high when fill orbit such as N=14 and 16    [17]. Calculations B(E2) using USDB interaction [11] and set effective charges, conventional effective charges (CEF) ep=1.3 and en=0.5 [13], Bohr-Mottelson effective charges (B-M) [9], and standard effective charges (ST)ep=1.36 and en=0.45 [15].

USDA Interaction
Reduced transition probabilities in units of e 2 fm 4 are calculated for Neon Ne isotopes (Z=10) with mass number A=18, 20, 22, 24, 26, 28 and with neutron number N=8, 10,12,14,16,18, respectively. Shell model calculations in sd model space and USDA interaction [11] was used to generate the OBTM elements for the ground state with J=0 and excited state with J=2. The harmonic oscillator size parameter [16] was calculated for each isotope and tabulated in Table 1. All isotopes in the present work composed of the core 16 Table 2 and plotted in Figure 3a as a function of neutron number N and mass number A in comparison with the experimental values [17]. The Bohr-Mottelson effective charges (B-M) [9] were calculated for 18,20,22,24,26,28 Ne isotopes as shown in Table 2, Conventional effective charges (CEF) [13] which are for proton 1.      [17]. Calculations B(E2) using USDA interaction [11] and set effective charges, conventional effective charges (CEF) ep=1.3 and en=0.5 [13], Bohr-Mottelson effective charges (B-M) [9], and standard effective charges (ST)ep=1.36 and en=0.45 [15].  Table 2 and plotted in Figure  3b which shows agreement theoretical values with experimental values except the excitation energies of 18,24,28 Ne isotopes. For magic number N=8, the B(E2) value is lower than those of N ≤ 18, which corresponds to a maximum value of the excitation energy. The excitation energy is decreasing when N=12 to become minimum. The excitation energies will increase for 24,26 Ne when N=14, 16 to become maximum because the neutrons in 24 Ne fill the 0d 5/2 orbit and in 26 Ne fill the 1s 1/2 orbit. The excitation energy is decrease when N=18 to become minimum because neutrons in 28 Ne not fill 0d 3/2 orbit. There are Similarities in the behavior of the excitation energies with USDB interaction and of the excitation energies with USDA interaction as shown in Figure 2b.

SDBA interaction
Reduced transition probabilities in units of e 2 fm 4 are calculated for Neon Ne isotopes (Z=10) with mass number A=18, 20, 22, 24, 26, 28 and with neutron number N=8, 10,12,14,16,18, respectively. Shell model calculations in sd model space and SDBA interaction [12] was used to generate the OBTM elements for the ground state with J=0 and excited state with J=2. The harmonic oscillator size parameter [16] was calculated for each isotope and tabulated in Table 1. All isotopes in the present work composed of the core 16 Table 3 and plotted in Figure 4a as a function of neutron number N and mass number A in comparison with the experimental values [17]. The Bohr-Mottelson effective charges (B-M) [9] were calculated for 18,20,22,24,26,28 Ne isotopes as shown in Table 3, Conventional effective charges (CEF) [13] which are for proton 1.   Table 3 and plotted in plotted in Figure 4b which shows the theoretical values agree to the experimental values, except the excitation energies of 18,24,28 Ne isotopes.
For magic number N=8, the B(E2) value is lower, which corresponds to a maximum value of the excitation energy. The excitation energy is decreasing when N=12 to become minimum and the excitation energy will increase when N=14, 16 because the neutrons in 24,26 Ne isotopes fill the 0d 5/2 orbit and the 1s 1/2 orbit, respectively. The excitation energy is decrease when N=18 because the neutrons not filled the d 3/2 orbit as shown in Figure 4b [18].