Pair correlations in low-lying $T=0$ states of odd-odd nuclei with six nucleons

In this paper, we study pair correlations in low-lying $T=0$ states for two typical cases of odd-odd $N=Z$ nuclei. The first case is six nucleons in a single $j=9/2$ shell, for which we study the $S$-broken-pair approximation, the isoscalar spin-1 pair condensation, and the isoscalar spin-aligned pair condensation, with schematic interactions. In the second case, we study pair approximations and correlation energies for ${}^{22}\mathrm{Na}$, ${}^{34}\mathrm{Cl}$, ${}^{46}\mathrm{V}$, ${}^{62}\mathrm{Ga}$, and ${}^{94}\mathrm{Ag}$ in multi-$j$ shells with effective interactions. A few $T=0$ states are found to be well represented by isoscalar nucleon pairs. The isoscalar spin-aligned pairs play an important role for the yrast $T=0$ states with $I\sim 2j$ and $I\sim I_{\max }$ in ${}^{22}\mathrm{Na}$, ${}^{46}\mathrm{V}$, and ${}^{94}\mathrm{Ag}$. The overlap between the isoscalar $\mathcal{J}=1$ pair wave function and the shell-model wave function is around 0.5 for the $I=1,3$ states of ${}^{34}\mathrm{Cl}$ and the $I=1$ state of ${}^{94}\mathrm{Ag}$. The $I=9$ state of ${}^{62}\mathrm{Ga}$ is very well described by the isoscalar $\mathcal{J}=3$ pair condensation. The broken-pair approximation (which is similar to the 2-quasiparticle excitation of the isovector pair condensation) is appropriate for quite few states, such as the $I=1-3$ states of ${}^{34}\mathrm{Cl}$ and the $I=5$ state of ${}^{62}\mathrm{Ga}$. The correlation energies are presented in this paper. It is noted that the picture based on nucleon-pair wave functions is not always in agreement with the picture based on correlation energies.

Figure 1

Overlaps between the pair-truncated wave functions and the shell-model wave function for yrast $T=0$ states with spin $I=1,3,5,7,9$, for six nucleons in a single $j=9/2$ shell with a schematic Hamiltonian ${\widehat{H}}_{\mathrm{sch}}(a,b)=(1-a-b){\widehat{\mathcal{V}}}_{\mathcal{J}=0}+a{\widehat{\mathcal{V}}}_{\mathcal{J}=2j}+b{\widehat{\mathcal{V}}}_{\mathcal{J}=1}$, where $a$ and $b$ are adjustable parameters ranging from 0 to 1. The abbreviations “$S$-BP,” “SO,” and “SA” represent the $S$-broken-pair approximation, the isoscalar spin-1 pair condensation, and the isoscalar spin-aligned pair condensation, respectively.

Figure 2

Overlaps between the shell-model wave functions and pair-truncated wave functions for yrast $T=0$ states of odd-odd $N=Z$ nuclei, with effective interactions. “$S$-BP,” “$SD$-BP,” “SO,” and “SA” correspond to the $S$-broken-pair approximation, the $SD$-broken-pair approximation, the isoscalar $\mathcal{J}=1$ pair condensation, and the isoscalar spin-aligned pair condensation, respectively.

Figure 3

Yrast $T=0$-state energies of odd-odd $N=Z$ nuclei calculated with effective interactions. “SM” corresponds to the shell model, and “$S$-BP,” “$SD$-BP,” “SO,” and “SA” are the same as in Fig. 2.

Figure 4

Electric-quadrupole moment for yrast $T=0$ states of odd-odd $N=Z$ nuclei. “SM,” “$S$-BP,” “$SD$-BP,” “SO,” and “SA” are the same as in Fig. 2.

Figure 5

Correlation energies $E_{\mathrm{C}}^{(\mathcal{J},\mathcal{T})}$ [see Eq. (3)] in yrast $T=0$ states of odd-odd $N=Z$ nuclei.