Counting pairs in ${}^{44}$Ti with various interactions

Abstract

We count the number of pairs in the single $j-$shell model of ${}^{44}$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 ${}^{44}$Ca. In that case (T=2), the magnitude of $D(JJ)$ is the same as that of a corresponding two particle fractional parentage coefficient. In counting the number of pairs with an even angular momentum $J_A$ we find a new relationship 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 that there are less ($J=1,T=0$) pairs than on the average. Relative to this 'no-interaction case' we find for the most realistic interaction used that there is an enhancement of pairs with angular momentum $J_A=0,2,1$ 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, only the ($J=1,T=0$) pair state is lowered and where both are equally lowered.

17 pages