States of Nuclear Quadrupole Vibrations and the Two-Phonon Triplets

A complete classification of the states of an $n$-dimensional isotropic harmonic oscillator is explicitly given in terms of a product of phonon creation operators applied to the "vacuum." The states are labeled by the phonon number and a set of integers associated with rotation subgroups of $U_n$. For the three-dimensional case it corresponds to the Bargmann-Moshinsky solution. For $n=5\mathrm{}$ (nuclear quadrupole vibrations), the solution contains Rakavy's seniority. In this case, appropriate linear combinations of the above states are constructed in order to get eigenstates of definite angular momentum. Anharmonic terms of fourth order in the creation and destruction operators breaking down the $U_5$ symmetry are proposed, in order to account for the existence of the two-phonon triplets. Results of the numerical application for a number of even-even nuclei are given.