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The Original Mixed Symmetry States - $6^{+}_{1}$ and $6^{+}_{2}$ in $^{48}$Ti

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 $μ$ 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.