The Original Mixed Symmetry States - $6_1^{+}$ and $6_2^{+}$ in ${}^{48}$Ti

Abstract

The $6_1^{+}$ and $6_2^{+}$ in ${}^{48}$Ti form a nearly degenerate doublet. In a single j shell calculation with the matrix elements from experiment the $6_1^{+}$ changes sign under the interchange of protons and neutron holes (odd signature) while the $6_2^{+}$ does not (even signature). As a consequence the calculated B(E2) $6_1^{+}\to 4_1^{+}$ is much stronger than the $6_2^{+}\to 4_1^{+}$ and the Gamow-Teller matrix element to the $6_2^{+}$ state vanishes. When using some popular interaction e.g. FPD6 in single j shell the ordering of the even signature and odd signature states gets reversed, so that the Gamow-Teller matrix element to the $6_1^{+}$ state vanishes and the $6_2^{+}\to 4_1^{+}$ E2 transition is the strong one. When configuration mixing is introduced, the E2 transition $6_2^{+}\to 4_1^{+}$ persists in being large. However the Gamow-Teller strengths reverse, with the large matrix element to the $6_1^{+}$ state in agreement with experiment. Static properties $\mu$ and Q for the two $6^{+}$ states are also considered. The experimental B(E2)'s from the $6^{+}$ states to the $4_1^{+}$ state are not well known.