Restoration of pseudo-spin symmetry in N = 32 and N = 34 isotones described by relativistic Hartree-Fock theory

The restoration of pseudo-spin symmetry (PSS) along the \begin{document}$ N = 32 $\end{document} and \begin{document}$ N =34 $\end{document} isotonic chains and the physics behind are studied by applying the relativistic Hartree-Fock theory with the effective Lagrangian PKA1. Taking the proton pseudo-spin partners \begin{document}$ (\pi2s_{1/2},\pi1d_{3/2}) $\end{document} as candidates, the systematic restoration of PSS along both isotonic chains is found from sulphur (S) to nickel (Ni), while an obvious PSS violation from silicon (Si) to sulphur is discovered near the drip lines. The effects of the tensor force components are investigated, introduced naturally by the Fock terms, which can only partially interpret the systematics from calcium to nickel, whereas they fail for the overall trends. Further analysis following the Schr?dinger-like equation of the lower component of Dirac spinor shows that contributions from the Hartree terms dominate the overall systematics of the PSS restoration. Such effects can be self-consistently interpreted by the evolution of the proton central density profiles along both isotonic chains. In particular, the PSS violation is found to tightly relate to the dramatic changes from the bubble-like density profiles in silicon to the central-bumped ones in sulphur.