Energy expressions for $n=3$ and $4$ systems in a single-$j$ shell

For systems with three and four fermions within a single-$j$ shell, analytical expressions for the state energies are presented from a decomposition of the angular momentum. In some important cases the expressions acquire a very simple form. The expression may help us in understanding the structure of isomeric states. The decomposition also makes it possible to construct the algebraic condition for conservation of seniority.

Figure 1

(Top) Expansion coefficients $C_J^I$ as a function of spin $I$ for yrast states of ${}^{96}\mathrm{Cd}$ in the $0g_{9/2}$ shell; (bottom) same as the top panel but for those of ${}^{43}\mathrm{Sc}$ (and ${}^{43}\mathrm{Ti}$) in the $0f_{7/2}$ shell.