Nuclear Structure Effects in the $\gamma$ Decay of Isobaric Analog States in ${\mathrm{Mg}}^{24}$, ${\mathrm{Al}}^{27}$, and ${\mathrm{Si}}^{28}$

The ${\mathrm{Na}}^{23}(d, n)$, the ${\mathrm{Mg}}^{26}(d, n)$, and the ${\mathrm{Al}}^{27}(d, n)$ reactions were used to populate isobaric analog states. Their $\gamma$ decays were measured with a Ge(Li) detector (i) in coincidence with the neutron groups, (ii) in singles arrangements using a bare detector, and (iii) in a three-crystal configuration. The observed branching ratios were compared to the approximate selection rules operating in the rotational and the Hartree-Fock model. It is found that the 4.23-MeV state in ${\mathrm{Mg}}^{24}$ has a wave function with admixtures of angular momentum projection $K=0\mathrm{}$ and $K=2\mathrm{}$. This fact supports the model proposed by Kelson et al. which gives an ellipsoidally symmetric deformation to ${\mathrm{Mg}}^{24}$. The observed transitions in ${\mathrm{Si}}^{28}$ indicate that the analog of the ground-state doublet in ${\mathrm{Al}}^{28}$ has an oblate deformation.