Interrelationship of Isospin and Angular Momentum

L. Zamick; A. Z. Mekjian; S. J. Lee

paper

Interrelationship of Isospin and Angular Momentum

arXiv:nucl-th/0402089

Abstract

It is noted that the simple interaction in isospin variables $a (1/4 - t (i) \cdot t (j))$ , in a single $j$ shell calculation, can also be written with angular momentum variables. For the configuration $(j^{2}) J_{A}$ for even $J_{A}$ the isospin is one; for odd $J_{A}$ it is zero. Hence the above interaction can also be written as $a (1 - (- 1)^{J_{A}}) /2$ . For the I=0 state of an even-even Ti isotope with $n$ neutrons, the hamiltonian matrix element of this interaction is $⟨ [∣ J^{'} J^{'}]_{0} ∣ H ∣ [J J]_{0} ∣ / ⟩ a = (n + 1) δ_{J J^{'}} - (n + 1) (j^{n} J j ∣} j^{n + 1} j) (j^{n} J^{'} j ∣} j^{n + 1} j)$ . The eigenvalues of this interaction can be found by using the isospin form of the interaction. They are $(n + 1) a$ for $T = ∣ N - Z ∣/2$ and zero for $T = ∣ N - Z ∣/2 + 2$ . One can apply this to some extent to obtain the number of pairs of nucleons with given total angular momentum $J_{A}$ in a given Ti isotope.

7 pages