Expressions for the number of $J=0$ pairs in even-even Ti isotopes

We count the number of pairs in the single-j-shell model of ${}^{44}\mathrm{Ti}$ for various interactions. For a state of total angular momentum I, the wave function can be written as $\Psi =\sum _{J_P{J}_N}D(J_P{J}_N)[(j^2){}_{J_P}(j^2){}_{J_N}]{}_I$, where $D(J_P{J}_N)$ is the probability amplitude that the protons couple to $J_P$ and the neutrons to $J_N$. For $I=0$ there are three states with ($I=0,T=0$) and one with ($I=0,T=2$). The latter is the double analog of ${}^{44}\mathrm{Ca}$. In that case ($T=2$), the magnitude of $D(\mathit{JJ})$ is the same as that of a corresponding two-particle coefficient of fractional parentage. In counting the number of pairs with an even angular momentum J, we find a new relationship is obtained by diagonalizing a unitary nine-j symbol. We are also able to get results for the “no-interaction” case for $T=0$ states, for which it is found, e.g., that there are fewer ($J=1,T=0$) pairs than on the average. Relative to this no-interaction case, we find that for the most realistic interaction used there is an enhancement of pairs with angular momentum $J=0,2,1$, and 7, and a depletion for the others. Also considered are interactions in which only the ($J=0,T=1$) pair state is at lower energy, interactions where only the ($J=1,T=0$) pair state is lowered, interactions where both are equally lowered, and the $Q·Q$ interaction. We are also able to obtain simplified formulas for the number of $J=0$ pairs for the $I=0$ states in ${}^{46}\mathrm{Ti}$ and ${}^{48}\mathrm{Ti}$ by noting that the unique state with isospin $\left|T_z\right|+2$ is orthogonal to all the states with isospin $\left|T_z\right|$.