Degeneracies whenT=0two body interaction 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 [S.J.Q. Robinson and Larry Zamick, Phys. Rev. C 63, 064416 (2001)] we studied the effects of setting all two body $T=0$ matrix elements to zero in shell model calculations for ${}^{43}\mathrm{Ti}$ ${(}^{43}\mathrm{Sc})$ and ${}^{44}\mathrm{Ti}.$ The results for ${}^{44}\mathrm{Ti}$ were surprisingly good despite the severity of this approximation. In single-j shell calculations ${(f}_{7/2}^{n})$ degeneracies arose between the $T=\frac{1}{2}$ $I=(\frac{1}{2}{)}_{1}^{\ensuremath{-}}$ and $(\frac{13}{2}{)}_{1}^{\ensuremath{-}}$ states in ${}^{43}\mathrm{Sc}$ as well as the $T=\frac{1}{2}$ $I=(\frac{13}{2}{)}_{2}^{\ensuremath{-}},$ $(\frac{17}{2}{)}_{1}^{\ensuremath{-}},$ and $(\frac{19}{2}{)}_{1}^{\ensuremath{-}}$ in ${}^{43}\mathrm{Sc}.$ For ${}^{44}\mathrm{Ti}$ the $T=0$ states ${3}_{2}^{+},$ ${7}_{2}^{+},$ ${9}_{1}^{+},$ and ${10}_{1}^{+}$ are degenerate as are the ${10}_{2}^{+}$ and ${12}_{1}^{+}$ states. 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 the vanishing $6j$ and $9j$ symbols. In this work a simpler, more physical method is used. This is Talmi's method of calculating coefficients of fractional parentage (cfp) for identical particles to states which are forbidden by the Pauli principle. This is done for both the one particle cfp to handle $6j$ symbols and the two particle cfp for the $9j$ symbols. From this we learn that the common thread for the angular momenta I for which the above degeneracies occur is that these angular momenta cannot exist in the calcium isotopes in the ${f}_{7/2}$ shell. There are no $T=\frac{3}{2}$ ${f}_{7/2}^{3}$ states with angular momenta $\frac{1}{2},$ $\frac{13}{2},$ $\frac{17}{2},$ and $\frac{19}{2}.$ In the same vein there are no $T=2{f}_{7/2}^{4}$ states with angular momenta 3, 7, 9, 10, or 12. For these angular momenta, all the states can be classified by the dual quantum numbers ${(J}_{\ensuremath{\pi}}{,J}_{\ensuremath{\nu}}).$