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 ⁴⁴Ti for various interactions. For a state of total angular momentum I, the wave function can be written as ψ = ∑_{JP JN} D(J_P J _N)[(j²)_JP(j²)_JN] _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 ⁴⁴Ca. In that case (T = 2), the magnitude of D(J J) 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 ⁴⁶Ti and ⁴⁸Ti by noting that the unique state with isospin |T_z| + 2 is orthogonal to all the states with isospin |T_z|.