Expressions for the number ofJ=0pairs in even-even Ti isotopes

Abstract

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 $\ensuremath{\Psi}=\ensuremath{\sum}_{{J}_{P}{J}_{N}}D({J}_{P}{J}_{N})[({j}^{2}){}_{{J}_{P}}({j}^{2}){}_{{J}_{N}}]{}_{I}$, 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 ($I=0,\phantom{\rule{0.3em}{0ex}}T=0$) and one with ($I=0,\phantom{\rule{0.3em}{0ex}}T=2$). 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 ($J=1,\phantom{\rule{0.3em}{0ex}}T=0$) 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 ($J=0,\phantom{\rule{0.3em}{0ex}}T=1$) pair state is at lower energy, interactions where only the ($J=1,\phantom{\rule{0.3em}{0ex}}T=0$) pair state is lowered, interactions where both are equally lowered, and the $Q\ifmmode\cdot\else\textperiodcentered\fi{}Q$ 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 $|{T}_{z}|+2$ is orthogonal to all the states with isospin $|{T}_{z}|$.