Competition between $T=1$ and $T=0$ pairing in $pf$-shell nuclei with $N=Z$

The pairing correlation energy for two-nucleon configurations with the spin-parity and isospin of $J^{\pi }=0^{+}$, $T=1$ and $J^{\pi }=1^{+}$, $T=0$ are calculated with $T=1$ and $T=0$ pairing interactions, respectively. To this end, we consider the $\left(1f2p\right)$-shell-model space, including single-particle angular momenta of $l=3$ and $l=1$. It is pointed out that a two-body matrix element of the spin-triplet $T=0$ pairing is weakened substantially for the $1f$ orbits, even though the pairing strength is much larger than that for the spin-singlet $T=1$ pairing interaction. In contrast, the spin-triplet pairing correlations overcome the spin-singlet pairing correlations for the $2p$ configuration, for which the spin-orbit splitting is smaller than that for the $1f$ configurations, if the strength for the $T=0$ pairing is larger than that for the $T=1$ pairing by 50$\%$ or more. Using the Hartree-Fock wave functions, it is also pointed out that the mismatch of proton and neutron radial wave functions is at most a few percent level, even if the Fermi energies are largely different in the proton and neutron mean-field potentials. These results imply that the configuration with $J^{\pi }=0^{+}$, $T=1$ is likely in the ground state of odd-odd $pf$-shell nuclei even under the influence of the strong spin-triplet $T=0$ pairing, except at the middle of the $pf$ shell, in which the odd proton and neutron may occupy the $2p$ orbits. These results are consistent with the observed spin-parity $J^{\pi }=0^{+}$ for all odd-odd $N=Z$ $pf$-shell nuclei except for ${}_{29}^{58}$Cu, which has $J^{\pi }=1^{+}$. The magnetic moment of a $(J^{\pi },T)=(1^{+},0)$ state is also discussed in order to show a manifestation of the change of the shell-model scheme from $jj$ coupling to $LS$ coupling.

Figure 1

Single-particle energies of the proton and neutron orbits in ${}^{56}$Ni, obtained with the Skyrme interaction SLy4.

Figure 2

Pairing correlation energies for lowest ($J^{\pi }=0^{+}$,$T=1$) and ($J=1^{+}$,$T=0$) states with $l=3$ and $l=1$ configurations as a function of scaling factor $f$ for $T=0$ pairing. The strength of the spin-singlet $T=1$ pairing interaction is fixed to be $G^{(T=1)}=24/A$ MeV with a mass $A=56$, while the strength for the spin-triplet $T=0$ pairing, $G^{(T=0)}$, is varied with the factor $f$ multiplied to G${}^{(T=1)}$.

Figure 3

Magnetic moment $\mu$ for lowest $(J^{\pi },T)=(0^{+},1)$ state with the $l=1$ configuration as a function of scaling factor $f$ for $T=0$ pairing. The strength of the spin-singlet $T=1$ pairing interaction is fixed to be $G^{(T=1)}=24/A$ MeV with a mass $A=56$, while the strength for the spin-triplet $T=0$ pairing, $G^{(T=0)}$, is varied with the factor $f$ multiplied to G${}^{(T=1)}$. The spin $g$ factor $g_s$ is renormalized by 0.7 as the effective operator in the results shown by the dashed line, while the bare $g_s$ factor is used in the results shown by the dotted line. The solid line denotes the experimental data of ${}^{58}$Cu taken from Ref. [25].