Mean-field theory of nuclear stability and exotic point-group symmetries

We formulate the principles of the mean-field theory of nuclear stability employing the point-group and group-representation theories. The related point-group hierarchy of importance in the context of nuclear stability is constructed and discussed. We introduce the notion of the magic-number chains associated with each symmetry-in analogy with the spherical-symmetry nuclear magic numbers. To prepare the criteria for the experimental search of introduced symmetries, we examine the simplified collective rotation-vibration model whose Hamiltonian is invariant under the symmetries in question. We illustrate the construction of the solutions that form at the same time irreducible representations of the point groups in question-in view of formulating the experimental symmetry criteria through the application of the branching-ratio techniques. Since the criteria may involve very weak transitions whose experimental research may be at the limit of the present-day experiments, the desires may arise, as it was the case in the past, to replace the difficult experiments by an inadequate modelling. In this context, we present an alert: the use of oversimplified quantum mechanics exercises in place of experiments and/or microscopic theories is likely to produce meaningless results.