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Wichmann-Kroll

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... [Pg.54]

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]... [Pg.58]

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. [Pg.73]

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. [Pg.75]

Wichmann-Kroll Electron-Loop Contribution of Order a(Za) m... [Pg.141]

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. [Pg.141]

A convenient representation for the Wichmann-Kroll polarization potential was obtained in [13]... [Pg.141]

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]... [Pg.142]

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]... [Pg.610]

It is not enough to consider the free vacuum polarization. The relativistic corrections to the free vacuum polarization in Eqs. (11-14) are of the same order as the so-called Wichmann-Kroll term due to Coulomb effects inside the electronic vacuum-polarization loop. To estimate this term we fitted its numerical values from Ref. [17], which are more accurate for some higher Z 30, by expression... [Pg.641]

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). [Pg.46]

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)... [Pg.47]

The function H can be divided into a Uehling potential part and the higher-order remainder + -i called the Wichmann-Kroll part... [Pg.89]

The Uehling part, Gy)l Za), can be calculated numerically to any precision, with the result shown in Table 2 [51]. The Wichmann-Kroll part Gyp (. a) is small, with the leading term given by [52, 53, 54]... [Pg.89]

Fig. 6. Expansion of the vacuum polarization loop into different powere of Za. The first term of the expansion corresponds to the Uehling contribution, the remaining terms are known as Wichmann-Kroll contribution. Fig. 6. Expansion of the vacuum polarization loop into different powere of Za. The first term of the expansion corresponds to the Uehling contribution, the remaining terms are known as Wichmann-Kroll contribution.
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... [Pg.133]

Fig. 9. Wichmann-Kroll vacuum-polarization charge density and the in-... Fig. 9. Wichmann-Kroll vacuum-polarization charge density and the in-...
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]... [Pg.136]

Fig. 10. Contributions of self energy, vacuum polarization, and finite size to the Lamb shift of the lsi/2-state. For the energy values a dimensionless quantity similar to F Za) in eq. (10) is used. The vacuum polarization contribution is separated into Uehling and Wichmann-Kroll parts. The negative of the Uehling contribution is shown. For high Z, the finite nuclear size becomes a major contribution to the total Lamb shift. Also the Wichmann-Kroll part becomes more important for large Z. Fig. 10. Contributions of self energy, vacuum polarization, and finite size to the Lamb shift of the lsi/2-state. For the energy values a dimensionless quantity similar to F Za) in eq. (10) is used. The vacuum polarization contribution is separated into Uehling and Wichmann-Kroll parts. The negative of the Uehling contribution is shown. For high Z, the finite nuclear size becomes a major contribution to the total Lamb shift. Also the Wichmann-Kroll part becomes more important for large Z.
These shifts amount to —1.67 xeV for the Uehling part and 3.51 reV for the a Za) Wichmann Kroll part in the 4/5/2-state which is the lowest lying state showing this... [Pg.136]

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]. [Pg.142]

Vacuum polarization Uehling-like loop correction Uehling corr. of wave function Wichmann-Kroll corr. of wave f. 0.0093 0.0260 -0.0007... [Pg.156]


See other pages where Wichmann-Kroll is mentioned: [Pg.54]    [Pg.58]    [Pg.70]    [Pg.75]    [Pg.142]    [Pg.143]    [Pg.156]    [Pg.610]    [Pg.641]    [Pg.655]    [Pg.610]    [Pg.641]    [Pg.655]    [Pg.45]    [Pg.47]    [Pg.47]    [Pg.134]    [Pg.135]    [Pg.136]    [Pg.153]    [Pg.153]   
See also in sourсe #XX -- [ Pg.45 ]




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