The magnetic moment of the electron, expressed by the g-factor in units of the Bohr magneton, is a key quantity in the theory of quantum electrodynamics (QED). Experiments using single particles confined in Penning traps have provided very precise values of the g-factor for the free electron as well as the electron bound in hydrogen-like ions. In this paper the status of these experiments is reviewed. The results allow testing calculations of higher order Feynman diagrams. Comparison of experimental and theoretical results for free and bound particles show no discrepancy within the limits of error, thus representing to date the most sensitive test of QED. Moreover, the g-factor provides a unique access to fundamental constants, as e.g. the electron mass or the fine structure constant.
The magnetic moment associated with the spin of a particle or an atom has played a central role in the development of atomic theory and quantum electrodynamics. The concept of the spin quantum number in units of introduced by Goudsmit and Uhlenbeck in 1925  and the observation of the quantization of the associated magnetic moment by Stern and Gerlach in 1922  led to the understanding of the anomalous Zeeman effect and the fine structure splitting of spectral lines. Dirac  showed that the spin is a purely relativistic effect and that the electron's magnetic moment is the Bohr magneton . A dimensionless quantity g scales the magnetic moment μ of any particle with spin s in units of :
From the Dirac equation follows that for the free electron. For an atom with orbital angular momentum L, a total Spin S, and total angular momentum J the g-factor can be expressed by the Landé formula
Numerous measurements of magnetic moments of neutral atoms have been performed using the Rabi-technique : A Stern-Gerlach force acting on an atomic beam in an inhomogeneous magnetic field separates different spin states in space. Transitions between different Zeeman substates are induced in a homogeneous magnetic field region and are monitored by a deflection in a second inhomogeneous B-field. With the knowledge of the magnetic field strength the g-factor can be derived. A particularly interesting result from measurements on atomic hydrogen was the fact that the g-factor of the electron deviates from the Dirac value of 2 by about 1 part in a thousand. This result, together with the discovery of the Lamb shift in hydrogen , triggered the development of the theory of quantum electrodynamics (QED), which describes the electromagnetic interaction by exchange of virtual photons. The explanation of the deviation of g from 2 by an additional term 
with the fine structure constant , was one of the first successes of the new theory. Increasing experimental accuracy required higher order calculations. The comparison of experimental and theoretical results on the g-factor of the electron still represents to date the most stringent test of QED theory [7, 8]. This is particularly true for the electron in free space or bound in simple atomic systems, while in complex atoms relativistic effects can not be calculated sufficiently accurately. Soon after the Stern-Gerlach experiment it was proven by different authors that the technique used in their experiment can not be applied to charged particles [9, 10]. This was circumvented by Dehmelt  who coined the term “continuous Stern-Gerlach effect”: A charged particle, suspended in a Penning ion trap with superimposed inhomogeneous magnetic field experiences a force acting on the magnetic moment of the particle. The sign of this force depends on the spin direction. It adds to the trapping force of the electric field of the Penning trap and leads to a spin dependence of the particles oscillation frequency which serves to monitor induced spin flips. In the experiments to determine the g-factor of a trapped particle spin flips are induced by a microwave field at the Larmor precession frequency . All cases of interest so far dealt with particles where . The magnetic field strength is calibrated by the cyclotron frequency of the particle with charge q and mass M. The g-factor is then determined by the ratio of the two measured frequencies and the charge-to-mass ratio of the electron to the particle under investigation:
In this article we summarize the present status of our knowledge of the electrons g-factor. It serves as test of quantum electrodynamics for free and bound electrons and as test of fundamental symmetries under charge, space, and time reversal. All relevant experiments are performed on charged particles confined in Penning ion traps. Therefore we start our discussion with a brief description of this device and related techniques for single particle operation.
2 The Penning trap
In the ideal Penning trap [12, 13], a homogeneous magnetic field is superimposed to an electrostatic quadrupole potential. In good approximation this potential can be created by a ring electrode and two endcaps of hyperbolical shape as shown in Fig. 1. A voltage U0 applied between the ring and endcaps creates a potential
r0 is the ring radius and 2z0 the minimal distance between the endcaps. The square dependence of the potential on the coordinates leads to linear forces on a charged particle inside the device. While axial confinement is provided by a properly chosen sign of U0, the radial force leads to particle loss. This is circumvented by a magnetic field B oriented along the z-axis of the trap. The equations of motion of a single particle of charge q and mass M in the Penning trap can be solved analytically. The result is a superposition of three uncoupled harmonic oscillations, the axial, the perturbed cyclotron, and the magnetron motion with respective frequencies , , and :
Here, is a geometrical factor that describes the dimension of the trap. Some useful relations between these frequencies are:
The latter equation is known as “invariance theorem” . It holds also when the trap potential deviates from the ideal form by a tilt of the magnetic field direction with respect to the trap's axis or by an ellipticity of the trap's electrodes, while the individual frequencies are shifted.
Typically we have a hierarchy of frequencies . A stability criterion follows from requirement that the expression in Eq. (7) has to evaluate to a real number: or equivalently .
2.1 Trapped particle detection
Trapped particles can be detected both destructively and non-destructively. For the destructive detection they are extracted from the trap and counted in a particle detector. This method is being used successfully in precision mass spectrometry using Penning traps [15, 16]. Non-destructive detection can be realized either by laser-induced fluorescence when the particle has a suitable  energy level scheme, or by the image currents induced in the trap electrodes by the ion's oscillation [18-21]. Since this method is applied in all experiments discussed below, we will exclusively consider it here.
We take the axial motion as example. The induced image charge in the endcap electrodes generates a current between any two electrodes. An inductance L, which connects one electrode to ground as shown in Fig. 2, forms a resonant circuit together with a parasitic capacitance C and the effective damping resistance . The current through the circuit increases the noise amplitude across the impedance at the ion's oscillation frequency. When the thermal noise of the impedance is kept sufficiently small by cooling to low temperatures, a Fourier analysis of the noise spectrum exhibits a maximum at the ion's oscillation frequency. Ideally, the amplifier exerts negligible back-action onto the resonant circuit. In practice, recent developments have yielded amplifier systems that cut down the undesired noise current leaking from the primary transistor stage largely and which feature a noise voltage of only 400 pV/. Combined with superconducting tank circuits, which can reach Q-values in excess of 80,000, equivalent to a parallel resistance of more than [22, 23], these amplifiers allow to sample the thermal noise of the tank circuit with excellent signal-to-noise ratio.
2.2 Ion cooling
The detection circuit serves also for ion cooling: When the ion's oscillation frequency is kept continuously in resonance with the circuit by adjustment of the trapping voltage the induced image currents raise the circuit's temperature. The excess energy is dissipated to the environment until thermal equilibrium between the ion and the circuit is reached. The time constant τ for exponential energy loss is given by 
where is the impedance of the circuit and D is the effective electrode distance of the pick-up electrode. Figure 3 shows an example of cyclotron mode cooling of a single 28Si13 + ion.
The presence of the trapped ion is then monitored by a minimum in the Fourier transform. This can be understood when we consider that the equivalent circuit of the oscillating ion is a series resonance circuit . Since there is very little damping of the ion oscillation the quality factor of this circuit is very high. The series resonance shortcuts the thermal noise at the ion's oscillation frequency. Figure 4 shows a signal from a single trapped 28Si13 + ion in thermal equilibrium with the detection circuit at 4 K. In a similar way the perturbed cyclotron oscillation can be detected when the traps ring electrode is divided into two segments and the detection circuit attached between the segments.
Provided that the effective voltage noise of the cryogenic amplifier is significantly lower than the thermal noise of the tank circuit, it is possible to cool the ion below the temperature of the environment. To this end, part of the detected noise is suitably phase-shifted and fed back to the resonator, canceling a fraction of the thermal noise (see figure 5) . As a result, both the effective resistance as well as the effective noise of the resonator can be adjusted arbitrarily, within the limits of the available signal-to-noise ratio. With this active noise feedback technique, sub-Kelvin temperatures can be reached .
2.3 Measurement of the oscillation frequencies
The dip in the noise spectrum of the axial resonator as shown in Fig. 4 is used to measure the axial oscillation frequency. The shape of the noise spectrum is proportional to the real part of the impedance given by the superposition of the response function of the equivalent series circuit of the ion superimposed to the noise spectrum of the detection circuit:
with and being the respective equivalent inductance and capacitance of the particle oscillating at . is the effective resistance of the circuit at the resonance frequency ω0, linked to its quality factor Q by .
The width of the ion's axial resonance is determined by the coupling to the detection circuit and the harmonicity of the trap potential. It is given by 
and can be used as a measure of the number N of trapped particles . The statistical uncertainty of the ion's axial oscillation frequency is of the order of a few 10 mHz in a total frequency of several 100 kHz in case of highly charged atomic ions.
As stated above, the perturbed cyclotron frequency can in principle be measured in a similar way when a high quality tuned circuit is attached between two segments of the ring electrode. Since tuning of the magnetic field to a value at which the cyclotron frequency matches the resonance of the circuit is difficult, the cyclotron detector can be fine-tuned with a varactor  diode. Alternatively, the perturbed cyclotron oscillation can be excited by a radiofrequency field of the respective frequency. This leads to an excess noise in the detector circuit, as shown in Fig. 6. Although the statistical uncertainty is only in the 10−10 range, the required oscillation amplitude causes a systematic shift of the line center when the magnetic field is not perfectly homogeneous and eventually from the relativistic mass increase. This problem can be circumvented when one couples the cyclotron mode to the axial one by an additional rf field at the difference frequency . The coupling  leads to a modulation of the axial oscillation amplitude with a frequency Ω, which depends on the amplitude of the coupling field:
It describes a symmetric splitting of the axial motion into two frequencies with a difference of 2Ω. In case of non-resonant coupling, i.e. when the coupling frequency is detuned by a frequency δ relative to the sideband frequency, the frequency components are given by
where A is a measure in units of frequency for the strength of the coupling field amplitude. This behavior is in analogy to the avoided crossing of a near-resonantly driven quantum-mechanical two-level system . Figure 7 shows the splitting of the motional frequency into two components. From a measurement of the split and unsplit frequencies the size of the detuning and thus the perturbed cyclotron frequency can be obtained. Similarly the magnetron frequency can be obtained through coupling to the axial motion.
The accuracy of the frequency measurements depends on a number of factors: The width of the axial resonance is determined by the coupling strength to the attached detection circuit, on the stability of the trapping voltage during the sampling time, and on the degree to which anharmonicities of the trap potential can be canceled.
The influence of voltage variations during the sampling time of the axial oscillation can be significantly reduced by detecting the cyclotron frequency directly rather than through the dressed axial frequencies. Simultaneously, the cyclotron energy can be kept small when using a novel method which allows to detect the phase of the cyclotron motion at energies below the detection threshold [28, 29]. The cyclotron oscillation is excited by a pulse, followed by a free oscillation period during which the phase evolves freely. After a given time the cyclotron oscillation is coupled to the axial one by a short pulse at their sum frequency. This pulse can be used to imprint the original phase of the cyclotron motion on the resulting axial motion, which then can be detected. Figure 8 illustrates the timing sequence. Because of the high signal amplitude this method reduces the time needed for the cyclotron measurement by more than one order of magnitude, reducing the uncertainty from instabilities of trap parameters tremendously. It has been applied recently in an experiment on 28Si13 + .
The accuracy of the perturbed cyclotron frequency is of greatest importance for the precise determination of the free particle's cyclotron frequency as required for calibration of the magnetic field. Using the method mentioned above typical statistical uncertainties of a single measurement are in the 10−10 range. This requires, however, a temporal stability of the magnetic field in the same order of magnitude. Superconducting magnets as used in all g-factor experiments show variations of the magnetic field B of about per hour. Temperature and pressure stabilization can improve on this by more than one order of magnitude . In any case it is mandatory to perform the frequency measurements as fast as possible in order to reduce uncertainties.
3 Theoretical background
3.1 The free electron/positron
The theory of quantum electrodynamics is based on the exchange of virtual photons between particles and fields. It can be represented by Feynman diagrams. The basic diagrams are shown in Fig. 9. From these one can construct higher orders by repetitive application. The evaluation of these diagrams for an electron in a magnetic field leads to a deviation of the g-factor from the Dirac value . The theory enables us to calculate the deviation as finite radiative corrections. The result can be represented by a series expansion with the fine structure constant as expansion parameter:
Each of the expansion coefficients corresponds to the evaluation of a number of Feynman diagrams with n lepton-photon vertices. The number of contributing diagrams grows rapidly with the loop order, making calculation of the highest-order terms a tour de force. Additionally, small contributions ahadronic and aweak from hadronic and weak interaction contribute to the g-factor. The most recent values of the coefficients , taken from  are
The coefficients C2, C4 and C6 have been calculated analytically while the values for C8  and C10  have been obtained by numerical methods. ahadronic has also been obtained by T. Kinoshita and M. Nio .
The most precise value of α, , obtained largely independent of QED theory arises from photon recoil measurements  leading to a theoretical value for the g-factor anomaly :
3.2 The single bound electron
For a single electron, bound to an atomic nucleus of charge Z, additional effects contribute to the g-factor. The Dirac equation in the Coulomb potential of a point-like nucleus of infinite mass can be solved analytically and leads to a g-factor 
The fractional changes in g range from 10−7 for light nuclei to 10−3 for the heaviest ones.
The strong electric field which the electron experiences at the distance of the Bohr radius from the nucleus is by many orders of magnitude larger than available in any laboratory and needs modifications of the formalism of QED in order to account for such an environment. The corresponding Feynman graphs of first order in α, taking care of the binding potential, are shown in Figure 10. Formally, a similar expansion series as in the free electron case can be used for calculation. The coefficients , however, now depend on Z:
The new expansion parameter is, at least for large Z, no longer small compared to unity and many more orders would have to be calculated in order to match the precision obtained in the free electron case. A summary of calculations of expansion coefficients and their results is given in .
More reasonable is a non-perturbative approach. The electron propagator as derived from the Dirac equation includes the full interaction with the binding potential of the nucleus in all orders of the parameter . The complex expression of the propagator, however, inhibits an analytical solution and requires numerical evaluation. So far, only the diagrams of first loop order have been evaluated in with this propagator, whereas the second order diagrams have been evaluated only in an expansion in orders of . The uncertainty arising from uncalculated contributions of higher order in the nuclear binding potential dominates the theoretical error budget.
In addition to the QED modification, nuclear size and structure as well as recoil corrections add to the g-factor. They have been calculated by several groups with uncertainties ranging from 10−12 for low-Z ions up to 10−5 for hydrogen-like uranium [39, 41, 42]. Figure 11 illustrates the size of all contributions for low numbers of the nuclear charge Z. Table 1 presents the complete theoretical results for selected examples which have been subject to experimental investigations.
Table 1. Complete theoretical results for selected examples which have been subject to experimental investigations
The first trap experiments to determine the g-factor of the free electron were performed at the University of Bonn in 1968 and 1969 [48, 49]. A cloud of about 105 electrons was used which were spin polarized by spin exchange collisions with a polarized sodium beam passing through the trap. For detection of microwave induced spin flips inelastic collisions with the sodium atoms were used. The small spin dependence of the inelastic collision cross section led to a modification of the energy distribution of the electron cloud which was analyzed. The accuracy of these experiments was limited to in the g-factor by instabilities in the electron cloud produced by the Coulomb interaction between the electrons as well as by trap imperfections. Much higher precision was obtained in a series of experiments at the University of Washington using a single electron confined in a Penning trap. The trap used hyperbolical contours for ring and endcaps with compensation electrodes in between to minimize trap imperfections.
The key element to detect induced spin transitions was the development of the “continuous Stern-Gerlach effect” by H. Dehmelt : When the magnetic field of the Penning trap is made inhomogeneous, the trapped particle having the magnetic moment experiences a force in the axial direction . This force adds or subtracts to the electric trapping force, depending on the spin direction and leads to a change in the axial oscillation frequency. This can be measured and serves to determine the spin orientation.
When the magnetic field inhomogeneity is symmetric around the trap center, e.g. by introducing a ferromagnetic ring in the center plane of the trap as shown in Figure 12, the magnetic field can be expanded in a series around the origin and reads in first order in the axial direction:
In this case the difference in the axial frequency for the two spin directions is given by
where M is the particles mass. In the case of the Washington experiment on free electrons one obtains with T/mm2 and MHz a value of Hz. In order to measure such a small frequency difference the trap parameters need to be extremely stable.
Spin flips were induced by microwaves around the Larmor precession frequency (nominally 160 GHz at T) introduced into the apparatus and the number of axial frequency jumps were recorded. These have a maximum around . is calibrated by the cyclotron frequency of the electron and the g-factor is determined through .
A significant gain in precision was obtained by driving the so-called anomaly frequency at which the spin direction and the cyclotron energy are changed simultaneously, see Fig. 13.
The experiment was performed on electrons as well as on positrons, yielding for the g-factor anomaly a, defined as , the values :
The main limitations in accuracy arose from the asymmetry of the line shape of the anomaly resonance due to the Boltzmann distribution of the electron oscillation amplitudes in the inhomogeneous magnetic field. Moreover shifts of the resonance frequency from the modification of the vacuum field by the trap electrodes were difficult to calculate considering the hyperbolic electrodes as a microwave cavity.
The experiment was significantly improved by G. Gabrielse and coworkers at Harvard who replaced the hyperbolical trap by a cylindrical one. Considering the trap as a microwave cavity this allowed better calculation of the electric field distribution and reduced the uncertainty arising from cavity shifts. Most important, however, was the reduction of the environmental temperature to below 100 mK. The electron loses energy from its cyclotron oscillation by synchrotron radiation. It has been shown that at equilibrium with the environment only the lowest quantum state of the motion is occupied . Then any ambiguity from the Boltzmann distribution of the cyclotron mode energy is removed and the linewidth is significantly reduced. The most recent result of the experiment for the g-factor anomaly a is 
This is in very good agreement to the theoretical value
4.2 The electron bound in hydrogenlike ions
4.2.1 The cylindrical Penning trap
Experiments to determine the g-factor of the electron bound in hydrogen-like ions have been performed at the University of Mainz [63-65]. They follow similar principles as for the free electron's g-factor outlined in 'The free electron/positron'. However, in these experiments the precision of the determined g-factor would not benefit from a measurement of the anomaly frequency due to the large mismatch of the ion's cyclotron frequency and the electron's spin precession frequency. In order to compensate for that, a number of significant modifications have been introduced to meet the requirements for high-precision measurements on these systems.
The trap used in these experiments is a cylindrical five electrode trap , consisting of a central ring electrode, two endcaps, and guard electrodes placed between them (Fig. 14). The potential inside this trap with a voltage U applied between ring and endcaps  can be described by a series expansion in which only even terms are non-zero because of the inherent rotational and mirror symmetry of the device.
Terms with exhibit an undesired dependence of the ion's oscillation frequency on its amplitude. Thus it is of utmost importance to minimize those terms to avoid systematic frequency shifts. A rigorous calculation of the resulting frequency corrections was carried out by different authors [13, 59]. Using two additional electrodes placed between the ring and endcaps allows the application of an additional voltage to these correction electrodes. In general it will be possible to choose a voltage such that . The dimensionless quantity T is called the tuning ratio. Moreover, in the process of trap optimization it is very convenient to make independent of the correction voltage and thus the tuning ratio T. Using the Green's function of conducting cylinders, it is possible to calculate the coefficients for a given set of electrode dimensions. After freely choosing the trap radius, it is generally possible to make and simultaneously by adjusting the ring- and correction electrode lengths and the tuning ratio T. The resulting trap configuration is called compensated and orthogonal .
The trap used for the experiments on the highly charged ions has an identical inner diameter of 7 mm for all electrodes. The central ring, the correction electrodes, and the endcaps have a length of 0.92 mm, 2.85 mm, and 6.8 mm, respectively, with gaps of 0.14 mm in between. The magnetic field is 3.76 T and we apply typically 7.5 V between ring and endcaps. For these parameters a single 28Si13 + ion has (nominal) oscillation frequencies MHz, kHz, and kHz.
4.2.2 Triple-trap configuration
The detection of the bound electron's spin direction requires the use of an inhomogeneous magnetic field as outlined in 'The free electron/positron'. The inhomogeneity is provided when the central ring electrode is made from ferromagnetic material. We use nickel and obtain a magnetic field inhomogeneity of 10 mT/mm2. Thermal fluctuations of the ion's oscillation amplitude in the inhomogeneous field  will lead to broadening and asymmetric line shapes of the motional as well as of the spin precession resonances and thus limit the precision of the g-factor determination. In order to circumvent this limitation we added a second trap with identical dimensions but without ferromagnetic material. We call the first trap, used to analyze the bound electrons spin direction, “Analysis Trap” (AT), and the second one “Precision Trap” (PT). The two traps are separated by five transport electrodes with a total length of 2 cm. By variation of the voltages applied to these electrodes the ion can be moved between the two traps.
A third trap, consisting of three electrodes, is added for charge breeding of the highly charged ion under investigation in an EBIT-like manner . A sketch of this triple-trap configuration is shown in Fig. 15, a more detailed description is given in .
The triple trap and the attached electronics for ion detection is located in a sealed copper housing at the center of a superconducting magnet. The box is in thermal contact to a liquid helium bath. The low temperature serves for a vacuum below 10−16 mbar by cryopumping. This leads to a virtual absence of charge exchange collisions which would limit the availability of the ion under investigation. In fact storage times of many months have been obtained for single hydrogenlike ions up to 28Si13 + . The low temperature reduces also the thermal noise of the amplifiers allowing single ion detection with a very good signal-to-noise ratio as demonstrated in Figures 4 and 6. Figure 16 shows the complete setup.
4.3 Experimental procedure
After preparation and cooling of a single ion its spin direction is determined in the AT as described in 'The free electron/positron'. Owing to the significantly higher mass of the ion compared to the free electron, the strength of the magnetic bottle has to be increased by more than two orders of magnitude in order to produce an appreciable shift of the axial frequency with the spin-state. In the triple-trap configuration the nickel ring electrode of the Analysis Trap serves to produce an inhomogeneity of mT/mm2. This leads for the case of 28Si13 + to a difference in the axial oscillation frequency of 240 mHz in 411 kHz for the two spin directions. The detection of such small frequency differences produced by an induced spin flip requires an extremely stable voltage source and a superb constancy of the motional amplitudes. Figure 17 demonstrates that the observed frequency jumps are consistent with the expectation and can be unambiguously distinguished from noise.
After preparation in the AT the ion is transportet to the PT where a microwave field is used to induce spin flips during a given period of typically 90 s. It is transported back to the AT and by the same procedure as described above it is determined whether a spin flip has taken place. This sequence is repeated many times with variation of the microwave frequency near the expected Larmor resonance frequency of (nominally) 105 GHz. Simultaneously with the attempts to induce spin flips the cyclotron frequency is measured. The number of successful spin flips vs. the microwave frequency gives a resonance curve as shown Figure 18. We plot the ratio Γ of the microwave frequency and the cyclotron frequency as abcissa. Γ is independent of magnetic field fluctuations in first order. The g-factor is determined from the center frequency of this resonance using the cyclotron frequency for B-field calibration. With and we get
From Eq. (25) we obtain the g-factor of the bound electron taking the mass of the electron and of the ion under investigation from the literature.
Experiments have been performed on 3 different hydrogen-like ions, 12C5 + , 16O7 + , and 28Si13 + . The results are listed in Table 2. The experimental values for the measured frequency ratio are subject to systematical shifts and uncertainties. They are dominated by a shift of the perturbed cyclotron frequency from image charges induced in the trap electrodes by the oscillating ion. A calculation for a cylindrical trap geometry  yields
where a is the radius of the ring electrode. It changes by for 28Si13 + with an estimated uncertainty of 5% of this value. Taking the slits between the individual electrodes into account decreases this correction by about 10% Other potential shifts such as line asymmetries caused by the residual magnetic field inhomogeneity in the precision trap or relativistic shifts have been considered and found to be much smaller. The final results are listed in Table 2. They all agree with the theoretical predictions.
The importance of the results as test of BS-QED calculations is illustrated in Figure 19 where sizes of different contributions are plotted and compared to the experimentally achieved uncertainty for the case of 28Si13 + . Here, the 2-loop contributions of order become visible for the first time. An upper bound of even higher orders in , , can be extracted using the tabulated value of the mean squares radius of 28Si for the calculation of the nuclear size correction, leading to .
An extension of these experiments to lithiumlike systems introduces the relativistic interaction of the valence electron with the 1s core. A comparison of the prediction with the experimentally determined value then yields a stringent test of the quantumelectrodynamical calculations in the three electron system . Provided that the theoretical evaluation can appropriately account for the three electron interaction, the g-factor of the valence electron can be used to extract information about the size and the structure of the nucleus, which is an important prerequisite for the interpretation of results in heavy, hydrogenlike ions.
Table 2. Experimental and theoretical g-factors for different hydrogen-like ions. The first error for gexp is the combined statistical and systematical uncertainty, the second one is the uncertainty of the electron mass taking its value at the time of the measurement
As seen from Table 2, the uncertainties in gexp and gth in case of C5 + and O7 + are dominated by the finite knowledge of the electron mass. Trust in the experimental and the theoretical values for the g-factor as well as in the determination of the respective ion masses allows adjusting the electron mass in order to have agreement between both results. Using the available data of the theory at the time of the measurements it has led to an electron mass (in atomic units) of 
This is a factor of 6 more precise than the previous most accurate value based on comparison of the cyclotron frequencies of both particles in a Penning trap . Higher accuracy in the electron mass using future g-factor determinations is at hand since by now the theoretical values for C5 + and O7 + have reduced error bars by calculation of higher order BS-QED contributions  and technical progress in the g-factor experiment have been successfully tested in case of Si13 + . We expect a reduction of the present error bar in by at least one order of magnitude.
The presently achieved uncertainty and prospects for future improvements may also allow for a determination of the fine structure constant α. The largest α-dependent contribution to the bound electron g-factor comes from the solution of the Dirac equation as given in Eq. (16). Assuming that all other contributions are small and sufficiently well tested, an uncertainty in the g-factor determination will lead to an uncertainty in the fine structure constant of
This might indicate to use a hydrogen-like system with high Z in order to achieve the highest accuracy. However, as evident from Figure 11 the nuclear size and structure corrections increase rapidly in the high-Z region and their accurate theoretical calculations is very difficult. To this end, Shabaev et al.  have pointed out that by a comparison of g-factors in H- and B-like ions in high-Z systems the nuclear structure corrections can be determined with sufficient accuracy making a precise determination of α possible, provided that the theoretical prediction of the g-factor of the B-like system can be pushed to accuracies comparable to the H-like system.
The presently most accurate values of the fine structure constant α come from the g-factor of the free electron  and from photon recoil measurements  which have quoted uncertainties of 0.8 ppb and 2.4 ppb, respectively. In order to achieve a similar uncertainty by g-factor measurements of bound electron systems the experimentally required precision in case of Pb81 + is of the order of . This is within reach using technical improvements which are being developed in our laboratory. However, to improve the theoretical prediction of the g-factors of both H- and B-like heavy ions is a considerable challenge.
6 Implications for fundamental symmetries
CPT invariance implies that the magnetic moments of particles and antiparticles are identical. This has been confirmed to 2 parts in 10−12 in a comparison of the g-factors of electron and positron .
A similar test in a hadronic system may be performed by the g-factors of proton and antiproton. The magnetic moment of the proton is presently known to . Using similar experimental methods as described for hydrogen-like ions for g-factor determination of the proton an improvement by more than one order of magnitude seems possible. The small size of the proton's magnetic moment, however, makes it technically more challenging. Experiments are under way at the University of Mainz  and at Harvard University , as well as at CERN for the antiproton [73, 74]. Both have very recently produced first data and an improvement of the relative precision to the anticipated 10−9 is foreseeable for the close future. First single spinflips of a single proton have been observed by both teams [75, 76].
A future extension of these experiments to antiprotons will require injection of low-energy antiprotons from an accelerator, similar as performed in anti-hydrogen experiments at CERN. It is planned to have a low-energy antiproton facility at GSI/Darmstadt  where such experiments may be performed.
7 Nuclear physics implications
Assuming that the BS-QED contributions to the g-factor of hydrogen-like ions are tested with sufficient accuracy the remaining uncertainty concerns the nuclear size and structure effects. Their calculation requires the knowledge of the root-mean-square nuclear radius . A comparison of the experimental and theoretical g-factor can then be used to determine . This has been tested in 28Si13 + . Although the result of fm is much less accurate than the value obtained from scattering experiments, fm , it can be considered as proof-of-principle to demonstrate the possibility of nuclear radius determination. Even more detailed insight into the structure of the nucleus becomes accessible through the detailed calculations of nuclear shape effects on the g-factor of hydrogen-like ions, which have been performed recently .
8 Future experiments
The method of g-factor determination described above can in principle be applied to any charged particle without loss of accuracy. Since the bound-state QED contributions as well as the nuclear size and structure corrections increase with the nuclear charge it is obvious that high-Z hydrogen-like ions will provide a more sensitive test of the corresponding calculations. This calls for theoretical consideration of higher order BS-QED terms. Experimentally a limitation of the present setup is given by the fact that the production of high-Z ions by electron-impact ionization requires electron energies of many keV, which represents a technical problem. Thus injection of ions from outside will be required. Corresponding experiments are planned at the heavy ion facility GSI where highly charged ions up to hydrogen-like uranium are produced and confined in a storage ring at high energies. It is planned to extract these ions, slow them down and inject then into a Penning trap for further investigation. A description of this HITRAP project can be found in .
Apart from hydrogen-like ions the investigation of lithium-like ions is of interest. As pointed out by Shabaev et al.  the difference of the g-factors in a H- and Li-like isotope reduces the uncertainty in the nuclear structure contribution and the BS-QED part can be tested more rigorously. However, the relativistic interaction of the valence electron with the 1s core plays a role. A comparison of the prediction [81, 82] with the experimentally determined value then yields a stringent test of the quantum-electrodynamic calculations in the three electron system . Provided that the theoretical evaluation can appropriately account for the three electron interaction, the g-factor of the valence electron can be used to extract important information about the size and the structure of the nucleus, which is an important prerequisite for the interpretation of results in heavy, hydrogen-like ions.
The experiments so far are restricted to isotopes without nuclear spin I. When applied to odd isotopes having a nuclear magnetic moment the method described in this article can be applied to determine the -factor
The -factor can be taken from even isotopes since the isotopic dependence is small . Then the nuclear -factor can be extracted. The shielding of the nuclear magnetic moment has been calculated including relativistic, nuclear, and QED effects . Comparison of the -factors in H-like ions with those obtained from neutral atoms or singly charged ions would test for the first time calculations of diamagnetic shielding factors in multi-electron systems. This would resolve discrepancies between different calculations of this quantity appearing in the literature .
An alternative method of g-factor determinations in high-Z hydrogen-like ion has been proposed by W. Quint et al. . The hyperfine splitting of several of those systems is in the optical region allowing laser induced transitions. In the magnetic field of a Penning trap the electronic and nuclear g-factors as well as the hyperfine coupling constants can be derived from the transition frequencies.
This work was supported by the Max-Planck Society, the Helmholtz Alliance HA216/EMMI and by the EU (ERC Grant No. 290870 - MEFUCO).
Sven Sturm has worked at ISOLDE/CERN for his diploma thesis. He then joined the group of Klaus Blaum at the University of Mainz, where he performed the g-factor experiment for highly-charged ions. After graduating in 2012, he moved to the Max-Planck Institute for Nuclear Physics in Heidelberg as a group leader where he is currently setting up a next-generation g-factor experiment.
Günter Werth graduated in 1969 at the University in Bonn. After a postdoctoral stay at the NASA Goddard Space Flight Center from 1972–1973 he returned to Germany for his habilitation. As professor at the University of Mainz he led a group for precision measurements in traps and pioneered amongst others the g-factor determination of highly-charged ions. He retired in 2004.
Klaus Blaum graduated in 2000 at the University in Mainz. After a postdoctoral position at GSI, Darmstadt, he went to CERN to lead the ISOLTRAP experiment. In 2004 he became head of a Helmholtz Research Group in Mainz. Since 2007 he is director at the Max-Planck Institute for Nuclear Physics in Heidelberg and head of the Cooled and Stored Ions Division. His main research focus is on precision measurements of atomic and nuclear ground state properties as well as tests of fundamental symmetries using Penning traps.