Relationship between the Bohr-Mottelson model and the interacting boson model

The interacting boson model was invented in two independent modes: The Schwinger mode using six bosons (s and five d bosons) and the Holstein-Primakoff mode using five quadrupole quasibosons. We show that the mathematical equivalence of the two modes can be used to define a number conserving quadrupole boson (the b boson). Two equivalent bases, the usual s-d basis and a new s-b basis are exhibited. By an exercise of (possibly objectionable) physical licence, the result can be interpreted as a proof of equivalence of interacting boson model I with the Bohr-Mottelson model. In the s-b basis, the Hamiltonian and other operators depend only on the b boson. In this form, all the topics usually associated with Bohr-Mottleson model can be discussed: Potential energy surface, shape parameters, vibrations vs rotations, etc. The precise relationship of our method to that employed in previous work is exposed. The latter is shown to correspond to the use of the Dyson generators of SU