## 1 Introduction

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 [1] and the observation of the quantization of the associated magnetic moment by Stern and Gerlach in 1922 [2] led to the understanding of the anomalous Zeeman effect and the fine structure splitting of spectral lines. Dirac [3] 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 [4]: 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 [5], 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 [6]

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 [11] 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.