Shell-Model Study of Neutron Rich Even-Even 52,54,56Cr Isotopes

Fouad A. Majeed1 and Abeer J. Ejam2

1Department of Physics College of Education for Pure Sciences,
University of Babylon, Babylon- Iraq.
fouadalajeeli@yahoo.com

2Department of Physics College of Education for Pure Sciences,
University of Babylon, Babylon- Iraq.
abeer_ejam@yahoo.com

Abstract: Large-scale shell model calculations were performed to study the low lying energies, binding energy and the reduced transition probabilities between energy levels for even-even 52,54,56Cr isotopes by employing effective interactions GXPF1, GXPF1A, FPD6 and KB3G near the closed core 48Ca with model space (HO). The results of calculation were compared with the recent experimental data and good global agreement were obtained for all nuclei under study and the best results achieved by using the effective interaction GXPF1A.
Keywords: Gamma transitions, excitation energies, Shell model


Introduction

The neutron-rich fp-shell nuclei far from the valley of stability are of particular interest in recent experimental and theoretical studies [1]. To provide a satisfactory description of these nuclei, the challenge remains to understand what mechanisms cause changes in nuclear shell structure as neutron number increases in nuclear systems. Theoretical calculations have questioned the persistence of the traditional magic numbers, which have been known to exist in stable nuclei [2].
Right from the time the shell model (SM) was established as the most basic model in nuclear structure, the nuclei in the 2p 1f region have been the focus of attention for various theoretical investigations based on the practical assumption of a Ca inert core within the domain of the shell model. The complexities arising due to the large number of valence particles outside the Ca core have hindered most of such theoretical attempts.
Only very recently some calculations in the framework of the projected Hartree-Fock (PHF), band mixing, and the highly truncated shell model have been reported for some of the nuclei in this region [1].
The nuclear shell model, a sophisticated theory based on the configuration interaction method, has been one of the central theoretical tools for understanding a wealth of data from nuclear structure experiments. Due to the rapid growth in the dimensionality of the Hilbert space with increasing degrees of freedom, we have to work within a reduced Hilbert space, the so-called model space. Accordingly, we use an effective interaction which is tailored to the chosen model space. This effective interaction forms an essential input to all shell-model studies. Equipped with modern sophisticated effective interactions, the shell model has successfully described many properties of nuclei [3].

The aim of the present work is to study the nuclear structure of even-even 52,54,56Cr isotopes by means of large-scale shell model calculations by employing GXPF1, GXPF1A, FPD6 and KB3G effective interactions and compare the theoretical results with the most recent experimental data.
Shell Model Calculation

The independent-particle Hamiltonian of an A-particle system can be written in terms two-particle interactions as [4] ,
H=?_(k=1)^A?T_k + ?_(k=1)^A??_(l=k+1)^A??W((r_k ) ? ,(r_l ) ? ) (1)?
where W((rk) ? ,(rl) ? ) is the two-body interaction between the kth and lth nucleons. Choosing an average potential U(rk), the Hamiltonian becomes [4] ,
H=?_(k=1)^A??[T_k ?+U(r_k)]+ ?_(k=1)^A??_(l=k+1)^A??W((r_k ) ? ,(r_l ) ? )-?_(k=1)^A?U(r_k ) (2)?
where the first term is identical to the independent-particle Hamiltonian, and the second and third account for the deviation from independent particle motion, known as the residual interaction. Separating the summations into core and valence contributions, eqn.(2) can be re-written [4],H = ?????????? + ??1 + ??2 + V??1 ,??2 (3) In the above equation, ?????????? contains all of the interactions of nucleons making up the core, ??1 and ??2 are the single-particle contributions from particles 1and 2, and ????1 ,??2 is the residual interaction describing all interactions between particles 1 and 2 as well as any interaction with core nucleons. Inserting this form of the Hamiltonian into the Schr?dinger equation yields an analogous expression for the energy [4],



??=??????????+??1+??2+ ???,??????1 ,??2 ???,?? (4)
Here, ?????????? is the binding energy of the core nucleus, ??1 and ??2are defined as the single-particle energies of orbitals outside the core, and ???,?? ????1 ,??2 ???,?? is the residual interaction which needs to be defined by theory. It is important to note that the energy given by eqn. (4) is for pure configurations only. In principle, any close-lying state with the same total angular momentum J and total isospin ? will mix. The mixed eigenstates are given by linear combinations of the unperturbed wave functions [4] ,

(????,??)??= ??=1????????( ???,??)?? (5)
where g is the number of configurations that mix and the label p =1,2,..., g. The coefficients akp fulfill the condition [4],
??=1??|??????|2=1 (6)
Inserting eqn. (5) into the schr?dinger equation gives,
??(????,??)??=????(????,??)?? (7)
which leads to a system of linear equations
[4].

3. Results and Discussions

3.1 Excitation Energies

The core are taken at 48Ca for all nuclei under study with 4,6 and 8 valence nucleons for 52Cr, 54Cr and 56Cr respectively.
Figure 1 present the compassion between our theoretical work and the experimental for 52Cr isotope. Our theoretical work predicts the 2+ state at 1.483 MeV, 1.286 MeV, 1.381 MeV and 0.830 MeV by employing the effective interactions GXPF1, GXPF1A, FPD6 and KB3G respectively. By comparing them with experimental value 1.434 MeV we obtained the best agreement at GXPF1 interaction. For 4+ the theoretical results are 2.277 MeV, 2.081 MeV, 2.125 MeV and 1.430 MeV by employing the same effective interactions respectively. We can notice from the figure that the experiment value at 4+ 2.369 MeV has a good agreement with GXPF1 interaction like 2+ state.


Fig. 1. Comparison of the experimental excitation energies taken from Ref [5] with the present theoretical work using GXPF1, GXPF1A, FPD6 and KB3G effective interactions.

Figure 2 presents the calculated excitation energy levels for 54Cr with 6 particles outside core. Our calculations predicts the values 1.075 MeV, 0.939 MeV, 0.973 MeV and 0.667 MeV for 2+ state by employing GXPF1, GXPF1A, FPD6 and KB3G effective interactions respectively which compared with experimental value at 0.834 MeV. The 4+ state is predicted with values 2.128 MeV, 1.885 MeV, 1.941 MeV and 1.431 MeV by employing the same effective interactions respectively and compared with experimental value 1.823 MeV. The best agreement with experimental value is achieved at GXPF1A interaction for both 2+ and 4+ states.