Electromagnetic Selection Rules for $^{24}\mathrm{Mg}$ in a $6\alpha$ Cluster Model with $\mathcal{D}_{4h}$ Symmetry

In the framework of a macroscopic $\alpha$-cluster model, the structural properties and the spectroscopy of the $^{24}\mathrm{Mg}$ nucleus are investigated. Special attention is devoted to the electromagnetic selection rules imposed by the point-symmetry group $\mathcal{D}_{4h}$ that leaves invariant the adopted $6\alpha$ equilibrium configuration, a square bipyramid. The analysis entails the application of group-theoretical identities and character tables, in a way familiar to quantum chemists. The results show that the occurrence of interband E0, E2, and M1, M2, M3 transitions is strictly regulated by the transformation properties of the excited vibrational modes to which the states in the process belong. Unlike the $^{12}\mathrm{C}$ nucleus in the $\mathcal{D}_{3h}$-symmetric $3\alpha$ arrangement, M1 transition channels are active between states corresponding to a single quantum of vibrational excitation. Conversely, the measured E1 strengths in the $^{24}\mathrm{Mg}$ spectrum are attributable to the excitation of single-nucleon degrees of freedom, as E1 transitions are forbidden by the model. The present investigation is a only part of a wider work, encompassing the spectrum and the whole electromagnetic properties of this nucleus in the considered $\mathcal{D}_{4h}$-symmetric configuration, in preparation.