Isospin symmetry breaking and large-scale shell-model calculations with the Sakurai-Sugiura method

Abstract

Abstract. Recently isospin symmetry breaking for mass 60-70 region has been investigated based on large-scale shell-model calculations in terms of mirror energy differences (MED), Coulomb energy differences (CED) and triplet energy differences (TED). Behind these investigations, we have encountered a subtle problem in numerical calculations for odd-odd N = Z nuclei with large-scale shell-model calculations. Here we focus on how to solve this subtle problem by the Sakurai-Sugiura (SS) method, which has been recently proposed as a new diagonalization method and has been successfully applied to nuclear shell-model calculations.The isospin symmetry breaking is one of the current topics in nuclear structure physics. An asymmetry between spectra of isospin analogue states with mirror pair nuclei has been extensively investigated in the upper sd− and the lower f p−shell regions [1], and in mass 60-70 region [2][3][4][5][6]. This asymmetry, that is, isospin symmetry breaking arises partly due to the Coulomb force and partly due to the strong nucleon-nucleon interaction.To analyze this isospin symmetry breaking, mirror energy differences (MED) [7], Coulomb energy differences (CED)[8] and triplet energy differences(TED) [7] have been discussed. The MED is a measure of charge symmetry breaking in an effective interaction and is defined aswhere E x J,T T z is the excitation energies of analog states with spin J and isospin T, T z . The TED of T = 1 states in triplet nuclei is a measure of the charge-independence breaking and is defined asFor the TED, we have to evaluate the energy of the T = 1 and T z = 0 state for odd-odd N = Z nuclei precisely, employing shell-model calculations with isospin non-conserving interactions [5,6]. This calculation involves a rather subtle problem in numerical calculations, especially for large-scale shell-model calculation with the widely used Lanczos method.For the p f -shell or larger shells, we need largescale shell-model calculations where the M-scheme (shellmodel space with definite total magnetic quantum number) is often used. Third component of isospin, T z , is also given, choosing proton and neutron numbers. As the shellmodel interaction is rotationally invariant, the obtained eigenstates naturally have a good total angular momentum. For isospin-conserving interactions, they also have a good total isospin. This property is often utilized by taking a state with good total angular momentum and isospin as an initial one of the Lanczos method.Here we consider isospin symmetry breaking with large-scale shell-model calculations. Once the shell-model interaction contains several isospin symmetry breaking terms, total isospin is no longer a conserved quantity. This non-conservation is not, however, quantitatively significant [5] but it brings numerical difficulty for Lanczos diagonalizations, especially for odd-odd N = Z nuclei with large-scale shell-model calculations.Without isospin symmetry breaking, the low-lying states of odd-odd N = Z nuclei have T = 0 or T = 1. As the ground state or l...