Low-energy quadrupole states in neutron-rich tin nuclei

We present a study on the isoscalar quadrupole strength in tin nuclei, focusing mainly on the low-energy region. The calculations are performed using the Skyrme type energy density functionals within the fully self-consistent quasiparticle random phase approximation, allowing for a good description of the experimental data for the first 2$^+$ state and the isoscalar giant quadrupole resonance. It is found that the first $2^+$ state and the low-energy quadrupole states between 3 and 6 MeV display an opposite behavior with increasing neutron number. While the strength of the first $2^+$ state decreases, some excited states start to accumulate between 3 and 6 MeV, and increase their strength with increasing neutron number. This low-energy region between 3 and 6 MeV is quite sensitive to the changes in the shell structure with increasing neutron number. In particular, between $^{116}$Sn and $^{132}$Sn, the filling of the neutron orbitals with large values of $j$, has an important impact on the low-energy region. Our analysis shows that the low-energy states have a non-collective character, except the first 2$^+$ state. In addition, the states in the low-energy region above 5 MeV display an interesting pattern: with the increase of the neutron number, their strength increases and their nature changes, namely they switch from proton excitations to neutron-dominated one. We conclude that the low-energy quadrupole states between 3 and 6 MeV can provide information about the shell evolution in open-shell nuclei.