Charge radii of thallium isotopes near the N=126 shell closure

The changes in the mean-squared charge radius of ${}^{209}\mathrm{Tl}^g$ ($N=128$) and ${}^{207}\mathrm{Tl}^m$ ($N=126$) relative to ${}^{205}\mathrm{Tl}$ have been measured for the first time using the in-source laser resonance-ionization spectroscopy technique with the Laser Ion Source and Trap (LIST) at ISOLDE (CERN). The application of the LIST suppresses the dominant background from isobaric francium isotopes and allows access to thallium nuclides with $A⩾207$. The characteristic kink in the charge radii at the $N=126$ neutron shell closure, as well as the odd-even effect similar to that in the adjacent bismuth, lead, and mercury isotopic chains, have been observed. The self-consistent theory of finite Fermi systems based on the energy density functional by Fayans et al. reproduces the behavior of charge radii in these isotopic chains near $N=126$. The comparison with calculations in the framework of the relativistic mean field (RMF) approach is also presented. In the case of the Fayans functional it is a specific form of pairing interaction with the dependence on the density gradient that is essential to provide agreement with the experimental charge radii. In particular, the kink is reproduced without the inversion of $g_{9/2}$ and $i_{11/2}$ neutron single-particle states, which is a prerequisite to correctly describe the kink in the RMF models.