A possibility for a determination of the fine structure constant in experiments on the bound-electron $g$-factor is examined. It is found that studying a specific difference of the $g$-factors of B- and H-like ions of the same spinless isotope in the Pb region to the currently accessible experimental accuracy of $7\times {10}^{-10}$ would lead to a determination of the fine structure constant to an accuracy which is better than that of the currently accepted value. Further improvements of the experimental and theoretical accuracy could provide a value of the fine structure constant which is several times more precise than the currently accepted one.

The $g$ factor of lithiumlike silicon ${}^{28}\mathrm{Si}^{11+}$ has been measured in a triple-Penning trap with a relative uncertainty of $1.1\times {10}^{-9}$ to be $g_{\exp }=2.0008898899(21)$. The theoretical prediction for this value was calculated to be $g_{\mathrm{th}}=2.000889909(51)$ improving the accuracy to $2.5\times {10}^{-8}$ due to the first rigorous evaluation of the two-photon exchange correction. The measured value is in excellent agreement with the theoretical prediction and yields the most stringent test of bound-state QED for the $g$ factor of the $1s^{2}2s$ state and the relativistic many-electron calculations in a magnetic field.

The magnetic moment μ of a bound electron, generally expressed by the g-factor μ=−g μ_Bs ħ⁻¹ with μ_B the Bohr magneton and s the electron’s spin, can be calculated by bound-state quantum electrodynamics (BS-QED) to very high precision. The recent ultra-precise experiment on hydrogen-like silicon determined this value to eleven significant digits, and thus allowed to rigorously probe the validity of BS-QED. Yet, the investigation of one of the most interesting contribution to the g-factor, the relativistic interaction between electron and nucleus, is limited by our knowledge of BS-QED effects. By comparing the g-factors of two isotopes, it is possible to cancel most of these contributions and sensitively probe nuclear effects. Here, we present calculations and experiments on the isotope dependence of the Zeeman effect in lithium-like calcium ions. The good agreement between the theoretical predicted recoil contribution and the high-precision g-factor measurements paves the way for a new generation of BS-QED tests.

The relativistic nuclear recoil, higher-order interelectronic-interaction, and screened QED corrections to the transition energies in Li-like ions are evaluated. The calculation of the relativistic recoil effect is performed to all orders in $1/Z$. The interelectronic-interaction correction to the transition energies beyond the two-photon-exchange level is evaluated to all orders in $1/Z$ within the Breit approximation. The evaluation is carried out employing the large-scale configuration-interaction Dirac-Fock-Sturm method. The rigorous calculation of the complete gauge invariant sets of the screened self-energy and vacuum-polarization diagrams is performed utilizing a local screening potential as the zeroth-order approximation. The theoretical predictions for the $2p_j$-$2s$ transition energies are compiled and compared with available experimental data in the range of the nuclear charge number $Z=10$–60.

A rigorous evaluation of the two-photon exchange corrections to the hyperfine structure in lithiumlike heavy ions is presented. As a result, the theoretical accuracy of the specific difference between the hyperfine splitting values of H- and Li-like Bi ions is significantly improved. This opens a possibility for the stringent test of the many-electron QED effects on a few percent level in the strongest electromagnetic field presently available in experiments.

The magnetic-dipole transition probabilities between the fine-structure levels $\left(1s^{2}2s^{2}2p\right)$ ${}^{2}P_{1∕2}–{}^{2}P_{3∕2}$ for B-like ions and $\left(1s^{2}2s2p\right)$ ${}^{3}P_1–{}^{3}P_2$ for Be-like ions are calculated. The configuration-interaction method in the Dirac-Fock-Sturm basis is employed for the evaluation of the interelectronic-interaction correction with negative-continuum spectrum being taken into account. The $1∕Z$ interelectronic-interaction contribution is derived within a rigorous QED approach employing the two-time Green function method. The one-electron QED correction is evaluated within framework of the anomalous magnetic-moment approximation. A comparison with the theoretical results of other authors and with available experimental data is presented.