Nuclear and QED corrections to the bound-electron $g$ factor

We calculate nuclear shape and quantum electrodynamic corrections to the $g$ factor of a bound electron~[1,2]. These theoretical studies are motivated by the current improvement of experimental possibilities: on the one hand, in a recent Penning trap measurement~[2], the $g$ factor of $^{28}\rm{Si}^{13+}$ has been determined with an unprecedented $5 \cdot 10^{-10}$ relative uncertainty. A novel experimental technique will further improve accuracy to the $10^{-11}$ level. On the other hand, experiments with ions as heavy as $^{238}\rm{U}^{91+}$ will be performed soon at the HITRAP-FAIR facility. For such heavy ions, nuclear effects play an important role. The leading relativistic nuclear deformation correction has been derived analytically and also its influence on one-loop quantum electrodynamic terms has been evaluated. We present results for medium- and high-$Z$ hydrogenlike ions, which become significant already for mid-$Z$ ions, and for very heavy elements it even reaches the $10^{-6}$ level, as we show in~[1].\\[4pt] [1] J.~Zatorski, N.~S.~Oreshkina, C.~H.~Keitel, and Z.~Harman, Phys. Rev. Lett., in press; arXiv:1110.3330 \newline [2] S.~Sturm, A.~Wagner, B.~Schabinger, J.~Zatorski, \textit{et al.}, Phys. Rev. Lett. \textbf{107}, 023002 (2011).