$2_1^{+}$ and $2_2^{+}$ states in collective nuclei as multiple $Q$-phonon excitations

In the framework of the $Q$-phonon scheme and the interacting boson model a description of the $2_1^{+}$ and $2_2^{+}$ states is obtained. The wave vectors are written as an expansion in multiple $Q$-phonon excitations of the ground state. It is shown that the convergence of this expansion is fast and the main component already provides a good approximation of the wave vector. The accuracy of such a leading-order description exceeds 90% of the wave vector. By taking into account two $Q$-phonon configurations the accuracy of the description of the $2_1^{+}$ and $2_2^{+}$ states is increased to 98%. This is true for all collective nuclei with the exception of some rotors where the first excited $K=0\mathrm{}$ band lies below or very close to the γ band and, due to band mixing, the wave vector of the $2_2^{+}$ state contains higher $Q$-phonon excitations. An application to some decay transitions is given.