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

Abstract

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=∑JPJND(JPJN)[(j2)JP(j2)JN]I, where D(JPJN) is the probability amplitude that the protons couple to JP and the neutrons to JN. 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(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 46Ti and 48Ti by noting that the unique state with isospin |Tz|+2 is orthogonal to all the states with isospin |Tz|.