Wichmann-Kroll potential

This potential and its effect on the energy levels were first considered in [87]. Since each external Coulomb line brings an extra factor Za the energy shift generated by the Wichmann-Kroll potential increases for large Z. For practical reasons the effects of the Uehling and Wichmann-Kroll potentials were investigated mainly numerically and without expansion in Za, since only such results could be compared with the experiments. Now there exist many numerical results for vacuum polarization contributions. In accordance... 

The only other contribution of order a(Za) connected with the radiative insertions in the external photons is produced by the term trilinear in Za in the Wichmann-Kroll potential in Fig. 3.16. One may easily check that the first term in the small momentum expansion of the Wichmann-Kroll potential has the form [87, 92]... 

There are two contributions of order a Zay m to the energy shift induced by the Uehling and the Wichmann-Kroll potentials (see Fig. 3.10 and Fig. 3.16, respectively). Respective calculations go along the same lines as in the case of the Coulomb-line corrections of order a Zay considered above. 

This contribution is very small and it is clear that at the present level of experimental accuracy calculation of higher order contributions of the Wichmann-Kroll potential is not necessary. 

The representation (1.18) implies a subtraction scheme for calculating the finite part of the Wichmann-Kroll potential and the vacuum polarization charge density It was first considered by Wichmann and Kroll (1956). A detailed discussion of the evaluation of this contribution for high-Z nuclei of finite extent is presented in Soff and Mohr (1988) and Soff (1989). A special application of the computed vacuum polarization potential to muonic atoms has been presented in Schmidt et al. (1989). 

Given the Wichmann-Kroll density we can calculate first the contribution to the vacuum polarization potential and then the corresponding energy shift. The energy correction associated with the Wichmann-Kroll potential caused by the density (1.20) is usually expressed in terms of a function Hwk- Again for bound ns states we may write similarly to Equation (1.17)... 

The separation of the loop also implies a separation of the corresponding potential (24) into the Uehling potential and the Wichmann-Kroll potential, as the higher orders of the Za expansion were first considered by Wichmann and Kroll in 1956... 

A somewhat similar approach can also be used for the mixed self energy - vacuum polarization diagrams of Fig. 13. The detailed evaluation of these graphs is presented by Lindgren et al. [59] and we report only the result of their calculation here, which for the lsi/2-state of uranium yields 1.12 eV in the Uehling approximation (no Wichmann-Kroll vacuum polarization potential included in the Dirac equation) and 1.14 eV by taking into account the Wichmann-Kroll potential also [7]. 

Contribution of the Wichmann-Kroll diagram in Fig. 3.16 with three external fields attached to the electron loop [26] may be considered in the same way as the polarization insertions in the Coulomb potential, and as we will see below it generates a correction to the Lamb shift of order a Za) m. 

A convenient representation for the Wichmann-Kroll polarization potential was obtained in [13]... 

Practical calculations of the Wichmann-Kroll contribution are greatly facilitated by convenient approximate interpolation formulae for the potential in (7.25). One such formula was obtained in [14] fitting the results of the numerical calculation of the potential from [28]... 

For the potential correction, no Uehling-like contribution exists for a homogeneous external magnetic field [31,32], and the remaining Wichmann-Kroll part can be written as [40]... 

The function H can be divided into a Uehling potential part and the higher-order remainder + -i called the Wichmann-Kroll part... 

An experimentally not yet proven effect is the r-dependance of the Wichmann-Kroll charge distribution. The potential corresponding to the a Za) charge density (the second loop in the expansion of Fig. 6) is predicted to diminish as [32,40]... 

Corrections to Fyp nlj, aZ) of order (aZ) with n > 1 have been considered by Wichmann and KroU [18] and lead to an expansion similar to that given above for the self-energy. Coefficients of higher-order terms in the Uehling and Wichmann and Kroll potentials are given, for example, in Ref. [16]. 

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