The many relationships between the IBM and the Bohr model

Abstract

Relationships between the IBM-1 and the Bohr collective model are explored in which the states of the IBM in its various dynamical symmetry limits are identified with subsets of Bohr model states of corresponding dynamical symmetries. The maps of interest are ones which give the contractions of the IBM in the limit of large boson number. The known map from the IBM into the Bohr model gives a contraction appropriate for the U(5) dynamical symmetry limit. A new map is given consistent with a contraction of the O(6) dynamical symmetry of the IBM. This map gives an explicit identification of IBM states in an O(6) ⊃ O(5) basis with states of the Wilets–Jean model and makes it possible to benefit from the different but complementary perspectives of the two models. For example, it leads to explicit expressions for the matrix elements of an IBM O(6) irrep in terms of the known matrix elements of the corresponding Wilets–Jean model. The relationship also shows how to obtain the familiar rotor plus beta- and gamma-vibrational bands of the Bohr–Mottelson model, in the IBM, by the addition to an O(6) Hamiltonian of a scalar cubic in the quadrupole moment operators of the type considered recently by Van Isacker. The establishment of close relationships between the two models enables one to benefit from the different and complementary perspectives they afford. For example, recent developments of an algebraic version of the collective model has shown that the Bohr model has an SU ( 1 , 1 ) × O ( 5 ) dynamical group with representations ranging from those of a spherical vibrator to a beta-vibrational soft-gamma rotor of the Wilets–Jean limit and, with the addition of an interaction, to a rotor of the standard Bohr–Mottelson type with beta- and gamma-vibrational bands. A parallel range of results can now be obtained in the IBM, albeit with some differences, within the framework of an SU ( 1 , 1 ) × O ( 5 ) dynamical group.