Interrelationship of Isospin and Angular Momentum

It is noted that the simple interaction in isospin variables $a (1/4 - t(i)\cdot t(j))$,inasingle$j$ 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 $a(1-(-1{)}^{J_A})\mathrm{/}2$. For the I=0 state of an even-even Ti isotope with $n$ neutrons, the hamiltonian matrix element of this interaction is $\bra [J’J’]_0 |H| [JJ]0\ket/a = (n+1) \delta{JJ’}

• (n+1) (j^n Jj|} 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 $T=\mathrm{\mid }N-Z\mathrm{\mid }\mathrm{/}2$ and zero for $T=\mathrm{\mid }N-Z\mathrm{\mid }\mathrm{/}2+2$. 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.