Symplectic group structure of the ${}^{48}\mathrm{Cr},{}^{88}\mathrm{Ru}$, and ${}^{92}\mathrm{Pd}$ ground states

The ground states of ${}^{48}\mathrm{Cr},{}^{88}\mathrm{Ru}$, and ${}^{92}\mathrm{Pd}$ are studied in the $1f_{7/2}$ or $1g_{9/2}$ shell model with effective interactions from the literature. They are found to be composed, quite independently of the shell and the interaction, roughly of 75% of $(s,t)=(0,0)$ and 25% of $(s,t)=(4,0)$, where $s$ is the seniority and $t$ the reduced isospin. Other irreps of the symplectic group $\mathrm{Sp}(2j+1)$, where $j$ is the single-nucleon angular momentum, make only very small contributions. The state $\chi$ obtained by antisymmetrization and normalization of the ground state in the stretch scheme of Danos and Gillet [ and , Phys. Rev. 161, 1034 (1967)] has a very different structure where the $\mathrm{Sp}(2j+1)$ irreps other than $(s,t)=(0,0)$ and $(4,0)$ contribute 20% and 41% for $j=7/2$ and $9/2$, respectively. The contributions of $\chi$ and the $s=0$ state to the calculated states are about equal for ${}^{48}\mathrm{Cr}$. For ${}^{88}\mathrm{Ru}$ and ${}^{92}\mathrm{Pd}$ the $s=0$ state is unambigously a better approximation to the calculated states than $\chi$. A state ${\chi }^{\prime }$ obtained by antisymmetrization and normalization of the product of two stretch-scheme ground states of the system with two valence nucleons or nucleon holes of each type has much larger overlaps with the calculated ground states than $\chi$ but a deviating $\mathrm{Sp}(2j+1)$ decomposition.