Electromagnetic Selection Rules for $$\varvec{^{12}}$$ 12 C in a 3 $$\varvec{\alpha }$$ α Cluster Model

Abstract

The recent successful application of the Algebraic Cluster Model to the energy spectrum of \(^{12}\)C has brought a new impetus on spectroscopy of this and other \(\alpha -\)conjugate nuclei. In fact, known spectral properties have been reexamined on the basis of vibrations and rotations of three \(\alpha \) particles at the vertexes of an equilateral triangle and new excited states have been measured that fit into this scheme. The analysis of this system entails the application of molecular models for rotational–vibrational spectra to the nuclear context and requires deep knowledge of the underlying group-theoretical properties, based on the \(\mathcal {D}_{3h}\) symmetry, similarly to what is done in quantum chemistry. We have recently analyzed the symmetries of the model and the quantum numbers in great depth, reproducing the all-important results of Wheeler and we have derived electromagnetic selection rules for the system of three \(\alpha \) particles, finding, for instance, that electric dipole E1 and magnetic dipole M1 excitations are excluded from the model. The lowest active modes are therefore E2, E3,\(\cdots \) and M2, M3, \(\cdots \) although there are further restrictions between certain types of bands. The selection rules summarized in the text provide a criterion for assigning of observed lines to the alpha cluster model or not and they might help to further unravel the electromagnetic properties of \(^{12}\)C. With the perspective of new facilities (such as ELI) where photo-excitation and photo-dissociation experiments will play a major role, a complete understanding of e.m. selection rules as a tool to confirm or disprove nuclear structure models, is mandatory.