G-factor of the bound electron

In contrast to the study of the Lamb shift and hyperfme structure, it is possible to perform experiments on the g factor of the bound electron in different hydrogen-like ions with about the same accuracy. The experiment [1] is now in progress and some other hydrogen-like ions can be measured soon. This provides a possibility to learn about the bound g factor as a function of the nuclear charge Z and the nuclear mass number A. The ions under study [1] must have spinless nuclei and so they have the most simple level scheme. 

Another metrological application of simple atoms is the determination of values of the fundamental physical constants. In particular, the use of the new frequency chain for the hydrogen and deuterium lines [6] provided an improvement of a value of the Rydberg constant (Roc)- But that is not the only the constant determined with help of simple atoms. A recent experiment on g factor of a bound electron [27,11] has given a value of the proton-to-electron mass ratio. This value now becomes very important because of the use of photon-recoil spectroscopy for the determination of the fine structure constant [41] (see also [8])-... 

The g Factor of a Bound Electron in a Hydrogen-Like Atom... 

An important feature of the study of the g factor of a bound electron at Z = 20 — 30 is also the possibility to learn about higher-order two-loop corrections, which are one of the crucial problems of bound state QED theory. Below we discuss in detail the present status of theory and experiment. We consider a new opportunity to precisely test bound state QED and to accurately determine two fundamental constants the electron-to-proton mass ratio and the fine structure constant. 

We follow the notation of our paper [4] and present the g factor of a bound electron in a hydrogen-like ion in the form... 

Precision studies of the g factor of a bound electron in simple atoms were started with experiments on hydrogen [11] and its comparison with deuterium [12,13,14] and tritium [15], as well as on the helium ion [16]. In particular in case of low Z the result for the non-trivial bound-state term... 

Concluding our consideration we would like to underline, that the study of the g factor of a bound electron [1] offers a new opportunity for us to precisely test bound state QED theory and to determine two important fundamental constants the fine structure constant a and the electron-to-proton mass ratio m/mp. The experiment can be performed at any Z with about the same accuracy [1] and one can expect new data at medium Z which will allow to verify the present ability to estimate unknown higher-order corrections (i. e. theoretical uncertainty) in both low-Z and high-Z calculations. 

The IS — 2S transition obeys the selection rule AF = 0, Am = 0 and is almost field-independent However, the g-factor for the bound electron is slightly less than for free space due to relativistic effects, and this gives the transition a small first-order field dependence. In the IS state the g-factor is g(lS) = g0 (1 - a2/3) [10]. The relativistic term is proportional to the binding energy so that g(2S) = ge (1 - a2/12). Thus, the field-dependence of the transition IS- 2S, (F=l,m=l AF=0,Am=0) leads to a frequency shift... 

For a free electron, the Dirac theory yields a value gfree = 2. Corrections to the free-electron g-factor are essentially due to the interaction with the free radiation field. The determination of the g-factor of a bound electron interacting with an additional, homogeneous magnetic field Bext via measurements of the induced energy shift on atomic levels is one of the basic experiments for testing QED. The deviation (g — 2) can be measured rather precisely. 

The experimental result obtained for a -factor measurement of a single ion is presented in Figure 7.8. The horizontal axis shows the ratio of the microwave frequency wmw to the cyclotron frequency >c, from which theg-factor of the bound electron can be obtained. The experimental g-factors of the hydrogen-like ions I2c5-i- and [11] agree within the uncertainties, which are dominated... 

We have performed an experiment to measure the g factor of the electron bound to a Carbon nucleus in a Hydrogen-like C5+ ion [9]. As shown below, the result of our measurement represents a significant test of bound state QED contributions and also accounts for the recoil correction from the finite mass of the carbon nucleus. The experiments are performed on single C5+ ions confined in a Penning ion trap at low temperatures, almost completely isolated from the environment. As outlined in the last paragraph the extension of our experiments to other highly charged systems opens a number of possibilities for future measurements of fundamental quantities such as the electrons mass or the fine structure constant. 

Recently a bound electron g factor in the hydrogen-like carbon ion was measured very accurately by the Mainz-GSI collaboration [1]. We consider here this new opportunity to test bound state QED and to determine precisely some fundamental constants from the study of the bound electron g factor. 

Below we discuss in detail a number of problems arising due to the precision study of the bound electron g factor. 

We summarize in Table 5 all results of different auxiliary experiments [12,11,16,17,18,19] important for comparison of the bound electron g factor hydrogen and deuterium atoms and the helium-3 ion. 

The precise measurement of the g-factor of the electron bound in a hydrogen-like ion is a sensitive test of bound-state Quantum Electrodynamics (QED) at high fields. Highly charged ions are ideal for such studies, because the electromagnetic... 

HGURE 7.7 Schematic of the double Penning trap setup used for g-factor measurements of the bound electron. 

FIGURE 7.8 Larmor resonance of the g-factor measurement of the bound electron in C + pO]. 

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