Competition between isoscalar and isovector pairing correlations in $N=Z$ nuclei

We study the isoscalar $(T=0)$ and isovector $(T=1)$ pairing correlations in $N=Z$ nuclei. They are estimated from the double difference of binding energies for odd-odd $N=Z$ nuclei and the odd-even mass difference for the neighboring odd-mass nuclei, respectively. The empirical and BCS calculations based on a $T=0$ and $T=1$ pairing model reproduce well the almost degeneracy of the lowest $T=0$ and $T=1$ states over a wide range of even-even and odd-odd $N=Z$ nuclei. It is shown that this degeneracy is attributed to competition between the isoscalar and isovector pairing correlations in $N=Z$ nuclei. The calculations give an interesting prediction that the odd-odd $N=Z$ nucleus ${}^{82}\mathrm{Nb}$ has possibly the ground state with $T=0$.

Figure 1

The experimental odd-even mass differences ${\Delta }_n^{\left(3\right)}(Z=\text{even},Z+1)$ (solid diamonds) in odd-mass nuclei with $N=Z+1$, and the pairing gaps (open circles) obtained by the BCS calculations. The solid curve is $5.18A^{-1∕3}$ and the dashed curve denotes $12A^{-1∕2}$.

Figure 2

The $pn$ pairing gaps estimated from the double differences of experimental binding energies. The solid circles denote the $T=0$ $pn$ pairing gap, and the solid triangles the $T=1$ $pn$ pairing gap. The odd-even mass differences in odd-mass nuclei with $N=Z+1$ are shown by the open squares. The dashed curve is the half of the $T=0$ pairing force strength $k^0$.

Figure 3

The energy difference between the $T=0$ and $T=1$ states in odd-odd $N=Z$ nuclei. The experimental values of the differences are denoted by solid diamonds. The open squares present the values estimated from the experimental odd-even mass differences in Fig. 1 and the $T=0$ pairing force strength $k^0$. The dashed line is $k^0-10.4A^{-1∕3}$.

Figure 4

The calculated energy differences between the lowest $T=0$ and $T=1$ states in even-even (upper plots) and odd-odd (lower plots) $N=Z$ nuclei. The solid diamonds are the same as Fig. 3. The open circles denote the energy differences obtained by the BCS calculations. The dashed curve is $2a\left(A\right)∕A$.