Interrelationship of Isospin and Angular Momentum

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 $⟨[|J^{\prime }{J}^{\prime }{]}_0|H|[JJ{]}_0|/⟩a=(n+1){\delta }_{J{J}^{\prime }}-(n+1)(j^{n}Jj|\}{j}^{n+1}j)(j^n{J}^{\prime }j|\}{j}^{n+1}j)$. 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.

Comments: 7 pages