The $g_{9/2}$ shell of identical particles is the first one for which one can have seniority-mixing effects. We consider three interactions: A delta interaction that conserves seniority, a quadrupole-quadrupole ($Q·Q$) interaction that does not, and a third one consisting of two-body matrix elements taken from experiment (${}^{98}\mathrm{Cd}$) that also leads to some seniority mixing. We deal with proton holes relative to a $Z=50,N=50$ core. One surprising result is that, for a four-particle system with total angular momentum $I=4$, there is one state with seniority $v=4$ that is an eigenstate of any two-body interaction—seniority conserving or not. The other two states are mixtures of $v=2$ and $v=4$ for the seniority-mixing interactions. The same thing holds true for $I=6$. Another point of interest is that, in the single-j-shell approximation, the splittings $\Delta E=E(I_{max})-E(I_{min})$ are the same for three and five particles with a seniority conserving interaction (a well-known result), but are equal and opposite for a $Q·Q$ interaction. We also fit the spectra with a combination of the delta and $Q·Q$ interactions. The $Z=40,N=40$ core plus $g_{9/2}$ neutrons (Zr isotopes) is also considered, although it is recognized that the core is deformed.

• Received 2 December 2005

DOI:https://doi.org/10.1103/PhysRevC.73.044302