Relativistic Hartree-Fock-Bogoliubov model for axially deformed nuclei

Abstract

Background: At the forefront of nuclear science, unstable nuclei are of special significance, in which the deformation and pairing correlations play an essential role in determining the nuclear structure. Under the mean field approach, the pairing and deformation effects have been uniformly considered within the deformed nonrelativistic Hartree-Fock-Bogoliubov model and the deformed relativistic Hartree-Bogoliubov (D-RHB) model. However, due to the limitation of the Hartree approach, an important ingredient of the nuclear force---the tensor force that contributes via the Fock diagram---is missing in the D-RHB model. Recently, the axially deformed relativistic Hartree-Fock (D-RHF) model was established for deformed nuclei, in which the $\ensuremath{\pi}$-pseudovector ($\ensuremath{\pi}$-PV) coupling can take into account the tensor force effects naturally. While limited by the BCS pairing, it cannot be safely applied to describe unstable nuclei.Purpose: The aim of this work is to develop the axially deformed relativistic Hartree-Fock-Bogoliubov (D-RHFB) model for the reliable description of a wide range of unstable nuclei, by utilizing the spherical Dirac Woods-Saxon (DWS) base.Method: Staring from the Lagrangian density that is based on the meson-propagated picture of nuclear force, the full Hamiltonian, that contains both mean field and pairing contributions, is derived by quantizing the Dirac spinor field in the Bogoliubov quasiparticle space, and the expectation with respect to the Bogoliubov ground state gives the full energy functional. As an extension of the D-RHF model, the degree of freedom associated with the $\ensuremath{\rho}$-tensor ($\ensuremath{\rho}\text{\ensuremath{-}}\mathrm{T}$) coupling is implemented, and incorporating with the Bogoliubov scheme the finite-range Gogny force D1S is utilized as the pairing force. Moreover, qualitative analyses on the nature of the $\ensuremath{\pi}$-PV and $\ensuremath{\rho}\text{\ensuremath{-}}\mathrm{T}$ couplings are presented to better understand their enhancements on the deformation effects.Results: Space convergence related to the spherical DWS base is confirmed for the D-RHFB model by taking light nucleus $^{24}\mathrm{Mg}$ and mid-heavy one $^{156}\mathrm{Sm}$ as candidates. Compared to light nuclei, extraordinarily more negative energy states are necessitated to keep the expansion completeness on the spherical DWS base for mid-heavy and heavy nuclei, due to the enhanced correlations between the expansion components with large $\ensuremath{\kappa}$ quantity, as indicated by the nature of the $\ensuremath{\pi}$-PV and $\ensuremath{\rho}\text{\ensuremath{-}}\mathrm{T}$ couplings. Furthermore, because of the enhanced deformation effects by the $\ensuremath{\pi}$-PV and $\ensuremath{\rho}\text{\ensuremath{-}}\mathrm{T}$ couplings, the RHF Lagrangian PKA1 presents a deeper bound ground state for $^{24}\mathrm{Mg}$ than the other selected Lagrangians, in addition to predicting a fairly deep bound local minimum with large oblate deformation.Conclusions: The axially deformed relativistic Hartree-Fock-Bogoliubov model has been established with confirmed convergence checks. The effects of the $\ensuremath{\pi}$-PV and $\ensuremath{\rho}\text{\ensuremath{-}}\mathrm{T}$ couplings, coupled with nuclear deformation, are analyzed in both light and mid-heavy nuclear systems, and are expected to be manifested in a wide range of unstable nuclei.