Spin-tensor decomposition:
A useful tool for shell model effective interaction

The spin-tensor decomposition is employed to construct a new interaction, named CKHeN, for 0p-shell. This new interaction is used to calculate the ekective single-particle energies of π0p1/2 and π0p1/2orbitals in Li isotopes, and the level structures of 7,8,9Li isotopes. The calculated level structures are found in good agreement with experimental data.


Introduction
Spin-tensor decomposition (STD) is a useful tool to decompose the model-space dependent shell model effective two-nucleon interaction into its central, spin-orbit and tensor force structure [1]. For last one and half decades, it has been used with the aim to understand the role of different components of two-nucleon interaction in the shell evolution in neutron-rich nuclei [2][3][4]. It has also been used to show why microscopic shell model interactions fail to describe the shell evolution in neutron-rich nuclei [3]. In this study, we use STD for the CK(8-16) interaction derived for 0p-shell nuclei, and examine the properties of its total spin (J) averaged proton-neutron central and tensor force matrix elements;V πν j j = where, sum runs only over the Pauli principal allowed J values.
For bare-tensor force, Otsuka et. al., [6], showed that proton-neutron interactionV πν j j corresponding to proton spin-up orbital ( j > = l + 1/2) and neutron spin-down orbital ( j < = l − 1/2) (or vice-versa) is attractive, whereas, if both proton and neutron orbitals are spin-up (or spindown), the interaction is repulsive. It is also demonstrated that bare-tensor force matrix elements barely change and hold its nature after dealing with short-range repulsion part of two-nucleon problem and in-medium effects [7]. Furthermore, the numerical analysis shows that tensor interaction of well-established shell model effective interaction, e.g., USDB, has same nature as for bare-tensor force.
The proton-neutron central componentV πν of shell model effective interaction is found to possess strongorbital node (nl) and weak-spin (j) dependency [8]. It means that proton-neutron central force matrix elements V πν ) corresponding to proton lorbital and neutron l -orbital are nearly same. The strong- * e-mail: pawan.kumar@iitrpr.ac.in  [9] using the spin-exchange zero-range δ potential.
In Fig. 1, we show proton-neutron central and tensor matrix elements of the CK(8-16) interaction. Here, central force matrix elements do not have similar strength which manifest that central component of CK(8-16) lacks weakspin dependency. Further, tensor force matrix elements are present with opposite nature than its regular nature. In present work, we construct a new interaction in which these discrepancies are not present.

Spin-tensor decomposition and New effective interaction -CKHeN
Spin-tensor decomposition: Nucleons are intrinsic spin 1/2 fermions; therefore, the interaction between two-nucleon can be written as the linear sum of scalar product of configuration space operator Q and spin space operator S of where rank k = 0, 1 and 2 represent central, spin-orbit and tensor force, respectively. Using the LS -coupled twonucleon wave functions, the matrix element for each V k can be calculated from matrix element V [10].
CKHeN interaction: To construct a new effective interaction for 0p-shell, we have considered the single-particle matrix elements and isospin T = 1 two-body matrix elements of the interaction developed in Ref. [11], and T = 0 two-body matrix elements of the CK (8-16) interaction. The two-body matrix elements of this integrated interaction, mainly diagonal matrix elements, have been modified to gain common features for central and tensor matrix elements. Note that the spin-tensor decomposition has been performed at each step of modification to check the properties of central and tensor force matrix elements. The final interaction is named CKHeN, and its proton-neutron central and tensor force matrix elements are shown in Fig. 1. In this new interaction, central force matrix elements are present with good weak-spin dependency, and tensor force matrix elements are present with its characteristic properties.

Shell model calculations
The CKHeN interaction consists of p 3/2 and p 1/2 orbitals for protons and neutrons above 4 He core, and is tested for the effective single-particle energies (ESPEs) of π0p 3/2 and π0p 1/2 orbitals in Li isotopes and the level structures of 7,8,9 Li isotopes. The ESPE (ε ) of πj orbital in Li isotopes is given by [8]; where, ε j is unperturbed single-particle energy of orbital j, and n j is number of neutrons in orbital j . The level structures of 7,8,9 Li isotopes are calculated using shell model code-NUSHELLX@MSU [12]. The calculated ESPEs and level structures are shown in Fig. 1 and 2, respectively. The results obtained using the CK(8-16) interaction are also shown in these figures. With the CKHeN interaction, the energy gap between spin-orbital partners, π0p 3/2 and π0p 1/2 , in Li isotopes is found to remain nearly constant (see Fig. 1). This is similar to the observation seen in F and Sc isotopes where the energy gap between π0d 5/2 and π0d 3/2 orbitals, and π0 f 7/2 and π0 f 5/2 orbitals, respectively, remains nearly constant [8,13]. However, with the CK(8-16) interaction, the energy gap between spin-orbital partners in Li isotopes increases when neutrons occupy ν0p 3/2 orbital, which is not consistent with the systematic.
In Fig. 2, the CKHeN interaction is shown to reasonably predict the experimental low-lying states of 7,8,9 Li relative to the CK(8-16) interaction.

Summary
In this work, spin-tensor decomposition is employed to discuss the discrepancies of CK(8-16) interaction and to construct a new effective interaction, named CKHeN, for 0p−shell. The new interaction is tested for Li isotopes and found reasonably predicting their spectroscopic properties.