Lamb shift polarization effect

VP vacuum polarization SE self-energy part of the Lamb shift LS = VP + SEE Lamb shift RC nucleus recoil correction, polarization Relativistic PT accounts for the main relativistic and correlation effects HOPT higher-order PT contributions. Data are from refs [1-10]. 

We have seen above that calculation of the corrections of order a"(Za) m (n > 1) reduces to calculation of higher order corrections to the properties of a free electron and to the photon propagator, namely to calculation of the slope of the electron Dirac form factor and anomalous magnetic moment, and to calculation of the leading term in the low-frequency expansion of the polarization operator. Hence, these contributions to the Lamb shift are independent of any features of the bound state. A nontrivial interplay between radiative corrections and binding effects arises first in calculation of contributions of order a Za) m, and in calculations of higher order terms in the combined expansion over a and Za. 

Before the individual parts of this function are discussed, the energy eigenvalue will be considered. The ground state energy g of the helium atom is just the energy value for double-ionization which can be determined accurately by several different kinds of experiments. Before the experimental value can be compared with the calculated one, some small corrections (for the reduced mass effect, mass polarization, relativistic effects, Lamb shift) are necessary which, for simplicity, are... 

To overcome this limitation will require the measurement of the Lamb shift (the 2S-2P energy difference) in muonic hydrogen. Here the main QED contribution is vacuum polarization, for which calculations are now available at a precision level of 10-6 [11,12,13,14]. Because the effect of the finite proton size contributes as much as 2% to the pp Lamb shift, a precise measurement of the shift will provide an accurate value of the proton radius. The knowledge of the proton radius has intrinsic interest as a fundamental quantity, and is important in other measurements. A measurement of rp at 0.1% precision will permit QED calculations of bound systems to be compared with the ep experiments at a precision level of fewxlO-7 gaining an order of magnitude over the present limits. 

Table 5. Lamb shift contribution for the ground state of 208Pb81+ i0n (in eV). The notations are the same as in Table 4. The finite nuclear size correction is calculated for a Fermi distribution with (r2 1,/2 = 5.505 0.001 fm. The SESE (a) (irred) correction is obtained by an interpolation from the known values for Z = 70, 80,92. The inaccuracy of the Uehling approximation for VPVP (f) and S(VP)E corrections is neglected. The zero value presented for the nuclear polarization is due to the cancellation of the usual nuclear polarization [35] with the mixed nuclear polarization (NP)-vacuum polarization correction [36]. The latter effect arises when the nucleus interacts with a virtual electron-positron pair. For lead, due to the collective monopole vibrations, specific for this nucleus, mixed NP-VP effect becomes rather large. Therefore, the nuclear polarization effects which otherwise limit very precise Lamb shift predictions are almost completely negligible for 208Pb, making this ion especially suitable for the most precise theoretical predictions...

The reaction of Eq. (3.6.15) is also possible in the reverse direction, even if relatively infrequent this is particle-antiparticle pair creation. This possibility is what underlies the idea of vacuum polarization and small effects, like the Lamb shift in atomic spectra. Positrons are not that rare Many radioactive nuclei decay by positron emission—for instance, sodium-22 ... 

The term Lamb shift of a single atomic level usually refers to the difference between the Dirac energy for point-like nuclei and its observable value shifted by nuclear and QED effects. Nuclear effects include energy shifts due to static nuclear properties such as the size and shape of the nuclear charge density distribution and due to nuclear dynamics, i.e. recoil correction and nuclear polarization. To a zeroth approximation, the energy levels of a hydrogen-like atom are determined by the Dirac equation. For point-like nuclei the eigenvalues of the Dirac equation can be found analytically. In the case of extended nuclei, this equation can be solved either numerically or by means of successive analytical approximation (see Rose 1961 Shabaev 1993). 

As shown in O Fig. 28.3, in the upper region of the X-ray transitions, the particle may interact with and thus probe the electronic shell, while in the lower one it already probes the atomic nucleus as its wave function partially penetrates it. The region between is also interesting as those transitions are weakly influenced both by the remaining or recombined electrons and by the field of the nucleus there one can get information on the properties of the captured particle or test the effects of quantum electrodynamics. As an example, O Fig. 28.7 presents (Horvath and Lambrecht 1984) the various energy terms in the 5g 9/2 4f 7/2 transition in muonic lead here the contribution of the Lamb shift, which is the dominant such term in ordinary atoms, is quite small as compared to the other vacuum polarization effects. 

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