Neutron-proton pairing correction in the extended isovector and isoscalar pairing model

Abstract

An extended ${\mathrm{O}}_{\mathcal{S}T}$(8) model with multi-$j$ orbits is constructed based on the angular momentum decomposition with ``pseudo''-spin $\mathcal{S}$ for valence nucleons in a $j$ orbit. It is shown that the isovector $\mathcal{S}=0$ and $T=1$ pairs are exactly the $J=0$ and $T=1$ pairs in a given $j$ orbit, while the isoscalar $\mathcal{S}=1$ pairs are linear combinations of $J=\mathrm{odd}$ pairs, with which the pairing Hamiltonian can be used to estimate isovector and isoscalar pairing interactions. As an example of the model application, some low-lying $J={0}^{+}$ level energies of even-even and odd-odd $A=18$--28 nuclei up to the half-filling in the $ds$ shell above the $^{16}\mathrm{O}$ core are fit by the model and compared with the fitting results of the same Hamiltonian in the ${\mathrm{O}}_{LST}$(8) form. It has been verified from the fitting of both the models that the isoscalar pairing interaction can be neglected in the lower-energy part of the spectra of these $ds$-shell nuclei as far as binding energies and a few $J={0}^{+}$ excited levels of these nuclei are concerned. With the mean-field plus isovector pairing interaction only, neutron-neutron, proton-proton, and neutron-proton pairing contributions at the ground or the lowest $J={0}^{+}$ state of these nuclei are estimated. It is shown that the isovector $np$ pairing contribution to the binding in the odd-odd $N=Z$ nuclei is systematically larger than that in the even-even nuclei. Furthermore, the isoscalar $np$ pair content at the lowest $J={0}^{+}$ state of these nuclei is also estimated. In both the ${\mathrm{O}}_{\mathcal{S}T}$(8) and ${\mathrm{O}}_{LST}$(8) models, it is clearly shown that the isoscalar pair content in the lowest $J={0}^{+}$ state of the $N=Z$ and $N=Z\ifmmode\pm\else\textpm\fi{}2$ nuclei increases with increasing of the valence nucleons, especially in those even-even nuclei, which indicates the isoscalar pairing correlation to be of importance at low-lying states of $N=Z$ and $N=Z\ifmmode\pm\else\textpm\fi{}2$ nuclei, especially in those even-even nuclei with more valence nucleons up to the half-filling, even though the isoscalar pairing interaction is negligible.