Further evidence for three-nucleon spin-orbit interaction in isotope shifts of $Z=\mathrm{magic}$ nuclei

It was pointed out [Phys. Rev. C \textbf{91}, 021302(R)] that the isotope shifts of the Pb nuclei, the kink at $N=126$ in particular, can be well described by the Hartree-Fock-Bogolyubov calculations if a density-dependent LS interaction derived from the $3N$ interaction is incorporated. Effects of the density-dependence in the LS channel on the isotope shifts are extensively investigated for the Ca, Ni and Sn isotopes, using the semi-realistic M3Y-P6 interaction and its LS modified variant M3Y-P6a, as in the Pb case. It is found that almost equal charge radii between $^{40}$Ca and $^{48}$Ca are reproduced, as well as the isotope shifts in a long chain of the Sn nuclei, owing to the density-dependence in the LS channel. A kink is predicted at $N=82$ for the isotope shifts of the Sn nuclei, in clear contrast to the interactions without the density-dependence.