Degeneracies when T=0 Two Body Interacting Matrix Elements are Set Equal to Zero : Talmi's method of calculating coefficients of fractional parentage to states forbidden by the Pauli principle

Abstract

In a previous work we studied the effects of setting all two body T=0 matrix elements to zero in shell model calculations for ${}^{43}$Ti (${}^{43}$Sc) and ${}^{44}$Ti. The results for ${}^{44}$Ti were surprisingly good despite the severity of this approximation. In this approximation degeneracies arose in the T=1/2 I=$(1/2{)}_1^{-}$ and $(13/2{)}_1^{-}$ states in ${}^{43}$Sc and the T=1/2 $I=(13/2{)}_2^{-}$, $(17/2{)}_1^{-}$, and $(19/2{)}_1^{-}$ in ${}^{43}$Sc. The T=0 $3_2^{+}$, $7_2^{+}$, $9_1^{+}$, and $10_1^{+}$ states in ${}^{44}$Ti were degenerate as well. The degeneracies can be explained by certain 6j symbols and 9j symbols either vanishing or being equal as indeed they are. Previously we used Regge symmetries of 6j symbols to explain these degeneracies. In this work a simpler more physical method is used. This is Talmi's method of calculating coefficients of fractional parentage for identical particles to states which are forbidden by the Pauli principle. This is done for both one particle cfp to handle 6j symbols and two particle cfp to handle 9j symbols. The states can be classified by the dual quantum numbers ($J_{\pi },J_{\nu }$).