$SU(6)$ limit of the quasiparticle-phonon nuclear model

It is shown that the Hamiltonian of the quasiparticle-phonon model has an $SU(6)$ limit, i.e., under certain restrictions it can be represented as a rotational invariant constructed from the generators of the $SU(6)$ algebra closed by the collective phonon operators of the random phase approximation and their commutators. This makes it possible to obtain microscopic expressions for the parameters of the quadrupole phonon model. For a number of zinc isotopes the maximal number of bosons is calculated microscopically and the fulfillment of the restrictions that lead to the $SU(6)$ limit is verified.