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

TL;DR

This work develops a macroscopic α-cluster description of $^{24}\mathrm{Mg}$ in a $\mathcal{D}_{4h}$-symmetric square bipyramid and analyzes its rotational–vibrational spectrum within the LO Watson Hamiltonian. By classifying normal modes under $\mathcal{D}_{4h}$ and constructing non-factorized rovibrational states, it derives comprehensive electromagnetic selection rules for $E$ and $M$ multipoles, including the prohibition of $E1$ within the model and the occurrence of $M1$ channels in singly excited bands. The results explain the observed singly excited rotational bands, connect specific modes to cluster-decay channels, and provide predictive guidance for intraband/interband transitions against empirical data. The framework offers a path to higher-order (NLO) corrections and to exploring possible doubly-excited bands near decay thresholds.

Abstract

In the framework of a macroscopic $α$-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 $6α$ 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 $3α$ 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.

Figures (2)

• Figure 1: Equilibrium $\alpha$-cluster configuration of ^24Mg in the intrinsic reference frame (left) and with the underlying microscopic structure in terms for protons (red) and neutrons (blue) with realistic charge radii (right). The structure parameters, highlighted in red, are evaluated at (2.38, 3.72) fm, corresponding to a prolate shape, in good agreement with the measured charge radius of the $0_1^+$ state and the electric quadrupole moment of the $2_1^+$ state. The charge distribution of the $\alpha$-particles is assumed to be pointlike.
• Figure 2: Normal vibrations of a square bipyramidal configuration with $\mathcal{D}_{4h}$ symmetry. The oriented segments with arrows denote the displacements of the $\alpha$-clusters with respect to their equilibrium positions.