Odd\_J Pairing Interaction

We consider in the g9/2 shell an interaction which acts only when a neutron and proton act in a state with J=Jmax=2j =9. We uses the abbreviated notation for a unitary 9j-symbol U(Jx Jp Jn J)= $<$(jj)9 (jj)Jx | (jj)Jp (jj)Jn $>$J . The Pauli principle demands that J p and Jn are both even. The matrix element of the hamiltonian is E(9) {*} SJx U(Jx, Jp Jn J) U(Jx Jp\textquoteright{} Jn\textquoteright{} J). For J=0 and 1 the Hamiltonian is a single separable term and the lowest eigenfunctions are the components of unitary 9j symbols, $\surd{2}$ U(9 Jp Jn 0) for J=0 and 2 U(8 Jp Jn 1) for J=1. These states have isospin T=0 . For J=2 and higher the Hamiltonian is no longer separable but there still some simple states. For J=2 there is a T=1 state 2U(8 Jp Jn 2 ) and for J=3 T=0 , 2U (7 Jp Jn 3) . For all these Jx serves as a good quantum number. The 2 lowest J=2 T=0 states are admixtures of $\surd{2}$ U(9 Jp Jn 2) and 2 U(7 Jp Jn 2) but the coupling is so weak that these are almost separate eigenstates with quantum numbers Jx=9 and Jx=7 respectively. The coupling matrix element is -1/2 U (9 9 7 2)= 0.00009113. The normalizations of the 2 admixed states are respectively such that N-2 =1/2- U(9 9 9 2) = 0.499993950935 and 1/4- 1/2 U(7 9 7 2)= 0.250376267385.