Two-electron Lamb shift

The uncertainty of theoretical calculations including the estimation of missing or uncalculated terms has been receiving increasing scrutiny as techniques have advanced. One of the most recent two-electron Lamb shift calculations by Persson et al. [9] estimates missing correlation effects in QED contributions at 0.1 eV for all elements or 20 ppm of transition energies in medium Z ions. In earlier work, Drake [4] claimed uncertainty for Z = 23 was < 0.005 eV or 1 ppm of helium-like resonance lines due to uncalculated higher order terms. Some of the latest theoretical calculations for the w transition in medium Z ions are summarized in Table 3. 

As a simple alternative calculation, we have estimated the two-electron Lamb shift for hehum-like ions of Z less than 32 by extrapolating the results of Persson et al. for Z = 32-92 using a power law fit (2e QED = aZ ). The Z power dependence is interestingly found to be 6 2.5. Derived results for Z = 18 and... 

For example one has in the mass difference from Cu-like to Ni-like Pb a QED effect in the order of 2 eV [28], corresponding to around 10 of the ion mass difference. For the Li-like to He-like Pb mass difference it is around 30 eV [29] and already 1.5 x 10 . For H-like Pb one calculates around 245 eV [30] QED effect in the Is state, which makes a mass contribution of 1 ppb. In the Is Lamb shift is 500 eV which corresponds to a relative mass shift of SE/M 2 ppb. So, measuring the relative mass difference between and and between and to 10", which should not be too difficult, one would have the one- and two-electron Lamb-shift to s 2 eV, i.e. a higher accuracy then present day X-ray spectroscopic measurements. 

Thus, for Fe, the Lamb shift contributes about 2% of the total energy difference. We first discuss below the nonrelativistic and relativistic contributions to the energies, and then turn to a discussion of the two-electron Lamb shift. 

Comparison between scaled theoretical and experimental one-electron Lamb shifts, expressed as deviations from the theoretical mjjn. The upper horizontal line for each ion is Erij son s theory and the lower horizontal line is Mohr s theory and S is the average of the two. The experimental data are labelled by the method of measurement according to (0) microwave resonance (a) anisotropy measurement (+) quench-rate measurement ( ) laser resonance. The dashed lines are the Borie corrections to the theoretical values. (From Drake )... 

Soon after the Schrodinger equation was introduced in 1926, several works appeared dealing with the fundamental problem of the nuclear motion in molecules. Very soon after, the relativistic equations were introduced for one-and two-electron systems. The experiments on the Lamb shift stimulated... 

The two-loop electron polarization contribution to the Lamb shift may be calculated exactly like the one-loop contribution, the only difference is that one has to use as a perturbation potential the two-loop correction to the Coulomb... 

Unlike the Lamb shift, the hyperfine splitting (see Fig. 8.1) can be readily understood in the framework of nonrelativistic quantum mechanics. It originates from the interaction of the magnetic moments of the electron and the nucleus. The classical interaction energy between two magnetic dipoles is given by the expression (see, e.g., [1, 4])... 

The muon is about two hundred times heavier than the electron and its orbit lies 200 times closer to the nucleus. The nuclear structure effects scale with the mass of the orbiting particle as m3R2 (for the Lamb shift It is a characteristic value of the nuclear size) and as m R2 (for the hyperfine structure), while the linewidth is linear in m. That means, that from a purely atomic point of view the muonic atoms offer a way to measure the nuclear contribution with a higher accuracy than normal atoms. However, there are a number of problems with formation and thermalization of these atoms and with their collisions with the buffer gas. 

The latter presents the largest sources of uncertainty in the theory of the muo-nium hfs interval, positronium energy spectrum and the specific nuclear-structure-independent difference for the hfs in the helium ion. The former are crucially important for the theory of the Lamb shift in hydrogen and medium-Z ions, for the difference in Eq. (2) applied to the Lamb shift and hyperfine structure in hydrogen and helium ion, and for the bound electron (/-factor. In the case of high-Z, the Lamb shift, (/-factor and hyperfine structure require an exact treatment of the two-loop correction. 

QED contributions to the Lamb shift consist of electron self-energy and vacuum polarization terms. In one-electron atoms the former is both the larger and the more difficult to calculate and has been the focus of much recent theoretical work. Up to Feynman diagrams including two-loops the self-energy contribution to a hydrogenic energy level can be written as [32]... 

Fig. 1. Feynman diagrams representing various contributions to the Lamb shift. A solid line represents an electron, a wavy line a virtual photon and a cross denotes exchange of a Coulomb photon (a) Leading self-energy term (b) One-loop vacuum polarisation term. The loop represents a virtual electron-positron pair (c) Some diagrams contributing to the two-loop binding correction...

This paper describes the progress of a laser resonance experiment which aims to measure the Lamb shift in hydrogenic silicon with an accuracy that will allow it to test the two-loop binding corrections mentioned above. This in turn should allow the viability of calculable frequency standards, based on transitions in lower-Z one-electron systems such as hydrogen and He+, to be assessed. Following a review of some theoretical contributions to hydrogenic energy levels, the details of the laser resonance experiment are outlined. 

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