Skyrme-Rpa Description of Spin-Flip M1 Giant Resonance

The spin-flip $M1$ giant resonance is explored in the framework of the random-phase-approximation (RPA) on the basis of the Skyrme energy functional. A representative set of eight Skyrme parametrizations ($\mathrm{SkT}6$, $\mathrm{SkM}*$, $\mathrm{SLy}6$, $\mathrm{SG}2$, $\mathrm{SkO}$, $\mathrm{SkO}'$, $\mathrm{SkI}4$, and $\mathrm{SV}$-bas) is used. Light and heavy, spherical and deformed nuclei ($^{48}\mathrm{Ca}$, $^{158}\mathrm{Gd}$, $^{208}\mathrm{Pb}$, and $^{238}\mathrm{U}$) are considered. The calculations show that spin densities play a crucial role in forming the collective shift in the spectrum. The interplay of the collective shift and spin-orbit splitting determines the quality of the description. None of the considered Skyrme parametrizations is able to describe simultaneously the $M1$ strength distribution in closed-shell and open-shell nuclei. It is found that the problem lies in the relative positions of proton and neutron spin-orbit splitting. This calls for a better modeling of the tensor and isovector spin-orbit interaction.