Triplet-odd pairing in finite nuclear systems: Even-even singly closed nuclei

Background: The appearance of the pairing condensate is an essential feature of many-fermion systems. There are two possible types of pairing: spin-singlet and spin-triplet. However, an open question remains as to whether the spin-triplet pairing condensate emerges in finite nuclei.

Purpose: The aim of this work is to examine the coexistence of the spin-singlet and spin-triplet like-particle pairing condensates in nuclei. We also discuss the dependence on the type of pairing functional.

Method: The Hartree-Fock-Bogoliubov calculations with a Skyrme $+$ local-pair energy-density functional (EDF) are performed to investigate the pairing condensate in the spherical ground states of Ca and Sn isotopes.

Results: The spin-singlet pair EDF induces not only the spin-singlet but also the spin-triplet pairing condensates due to a strong spin-orbit splitting. By discarding the spin-orbit EDF, only the spin-singlet pairing condensate appears. The spin-triplet pair EDF, however, induces the spin-orbit splitting and accordingly the spin-singlet pairing condensate.

Conclusions: The spin-orbit splitting plays an essential role in the coexistence of the spin-singlet and spin-triplet pairing condensates in nuclei.

Figure 1

Spin-singlet and triplet pairing components of neutrons, $S_{{\rho }_n}$ and $S_{J_n}$, respectively, in the Ca and Sn isotope chains.

Figure 2

The neutron local pair density ${\tilde{\rho }}_n\left(r\right)$ and the radial component of the neutron tensor pair density ${\tilde{J}}_{nr}\left(r\right)$ of ${}^{42}\mathrm{Ca}$ and ${}^{56}\mathrm{Ca}$ and the contributions from $1f_{7/2}$ and $1f_{5/2}$ orbits.

Figure 3

Spin-singlet and spin-triplet pairing components, $S_{{\rho }_n}$ and $S_{J_n}$, calculated in the Ca isotopes by changing the coupling constant of the spin-orbit EDF. The labels 1.0 LS, 0.5 LS, and no LS represent the calculations with the original spin-orbit EDF, the reduced one multiplied by a factor of 0.5, and the calculation without the spin-orbit EDF, respectively.

Figure 4

Pairing gap $|{\mathrm{\Delta }}_n|$ and pairing energy $E_{\mathrm{pair},n}^{S=0,1}$ calculated for the singlet- and triplet-pair EDFs in the Ca isotopes by changing the coupling constant of the spin-orbit EDF.

Figure 5

Occupation numbers of $pf$-shell orbits in Ca isotopes calculated with the singlet- and triplet-pair EDFs.

Figure 6

Two-neutron separation energies and moments of inertia of the neutron pairing rotation in the Ca isotopes calculated for the singlet- and triplet-pair EDFs. The inertia calculated with the triplet-pair EDF at $N=30$ ($-17.86{\mathrm{MeV}}^{-1}$) is not shown in the figure.