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The pairing correlation energy for two-nucleon configurations with thespin-parity and isospin of $J^\pi=0^+$, $T$=1 and $J^\pi=1^+$, $T$=0 arecalculated with $T$=1 and $T$=0 pairing interactions, respectively. To thisend, we consider the $(1f2p)$ shell model space, including single-particleangular momenta of $l=3$ and $l=1$. It is pointed out that a two-body matrixelement 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 thespin-singlet $T$=1 pairing interaction. In contrast, the spin-triplet pairingcorrelations 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 thatfor the T=1 pairing by 50% or more. Using the Hartree-Fock wave functions, itis also pointed out that the mismatch of proton and neutron radial wavefunctions is at most a few % level, even if the Fermi energies are largelydifferent in the proton and neutron mean-field potentials. These results implythat the configuration with $J^\pi=0^+$, $T$=1 is likely in the ground state ofodd-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 protonand neutron may occupy the $2p$ orbits. These results are consistent with theobserved spin-parity $J^{\pi}=0^+$ for all odd-odd $pf$ shell nuclei except for$^{58}_{29}$Cu, which has $J^{\pi}=1^+$.

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