Bound-electron g-factor measurements for the determination of the electron mass and isotope shifts in highly charged ions

In the context of this thesis the electron mass has been determined in atomic mass units with a relative uncertainty of 2.8.10^-11, which represents a 13-fold improvement of the 2010 CODATA value. The underlying measurement principle combines a high-precision measurement of the Larmor-to-cyclotron frequency ratio on a single hydrogenlike carbon ion ¹²C⁵⁺ with a very accurate g-factor calculation. Furthermore, this thesis contains the first isotope shift measurement of bound-electron g-factors of highly charged ions. Here, the g-factors of the valence electrons of the lithium-like calcium isotopes ⁴⁰Ca¹⁷⁺ and ⁴⁸Ca¹⁷⁺ have been measured with relative uncertainties of a few 10^-10, constituting a so-far unrivaled level of precision for lithium-like ions. These calcium isotopes provide a unique system across the entire nuclear chart to test the pure relativistic nuclear recoil effect. The corresponding and successfully tested theoretical prediction is based on bound-state quantum electrodynamics but goes beyond the standard formalism, the so-called Furry picture, where the nucleus is considered as a classical source of the Coulomb field. The three Larmor-to-cyclotron frequency ratios of ¹²C⁵⁺, ⁴⁰Ca¹⁷⁺ and ⁴⁸Ca¹⁷⁺ have been determined in sequence in a non-destructive manner on single trapped ions stored in a triple Penning trap setup. The cyclotron frequency is measured by a dedicated phase-sensitive detection technique while simultaneously probing the Larmor frequency. The spin-state of the bound valence electron is determined by the continuous Stern-Gerlach effect. In the very last part of this thesis, a new design of a highly compensated cylindrical Penning trap has been developed, which will be used in next generation's high-precision Penning trap experiments.