What can we learn from the Interacting Boson Model in the limit of large boson numbers ?

Over the years, studies of collective properties of medium and heavy mass nuclei in the framework of the Interacting Boson Approximation (IBA) model have focused on finite boson numbers, corresponding to valence nucleon pairs in specific nuclei. Attention to large boson numbers has been motivated by the study of shape/phase transitions from one limiting symmetry of IBA to another, which become sharper in the large boson number limit, revealing in parallel regularities previously unnoticed, although they survive to a large extent for finite boson numbers as well. Several of these regularities will be discussed. It will be shown that in all of the three limiting symmetries of the IBA [U(5), SU(3), and O(6)], energies of 0+ states grow linearly with their ordinal number. Furthermore, it will be proved that the narrow transition region separating the symmetry triangle of the IBA into a spherical and a deformed region is described quite well by the degeneracies E(02 ) = E(6 + 1 ), E(03 ) = E(10 + 1 ), E(0 + 4 ) = E(14 + 1 ), the energy ratio E(6 + 1 )/E(0 + 2 ) turning out to be a simple, empirical, easy-to-measure effective order parameter, distinguishing between firstand second-order transitions. The energies of 0+ states near the point of the first order shape/phase transition between U(5) and SU(3) will be shown to grow as n(n+3), where n is their ordinal number, in agreement with the rule dictated by the relevant critical point symmetries studied in the framework of special solutions of the Bohr Hamiltonian. The underlying dynamical and quasi-dynamical symmetries are also discussed. Collective phenomena in atomic nuclei are described in terms of two complementary models, the algebraic Interacting Boson Approximation (IBA) model [1], and the geometrical collective model [2,3]. In the former, s and d bosons (bosons of angular momentum 0 and 2 respectively) are used, while in the latter the collective variables β (the ellipsoidal deformation) and γ (a meaPreprint submitted to Elsevier 28 July 2008