Bethe logarithm

In Bethe theory the shell correction ALsheii is conveniently defined as the difference between the stopping number LBom in the Born approximation and the Bethe logarithm LBethe —in (2mv /I). Fano [12] wrote the leading correction in the form... 

Shell corrections can also be evaluated without recourse to an expansion in powers of v, but existing calculations such as Refs. [13,14] are based on specific models for the target atom and, unlike equation (19), do not end up in expressions that would allow to identify the physical origin of various contributions. It is clear, however, that orbital motion cannot be the sole cause of shell corrections The fact that the Bethe logarithm turns negative at 2mv /I< 1 cannot be due to the neglect of orbital motion but must be of a purely mathematical nature. Unfortunately, the uncertainty principle makes it impossible to eliminate orbital motion in an atom from the beginning. 

The Bethe logarithm is formally defined as a certain normalized infinite sum of matrix elements of the coordinate operator over the Schrodinger-Coulomb wave functions. It is a pure number which can in principle be calculated with arbitrary accuracy, and high accuracy results for the Bethe logarithm can be found in the literature (see, e.g. [13, 14] and references therein). For convenience we have collected some values for the Bethe logarithms [14] in Table 3.1. 

This part is easily done, but the Bethe logarithm (3(nLS), representing the emission and absorption of virtual photons, is much more difficult to calculate. It is... 

Table 6. 1/n expansion coefficients bi for the Bethe logarithms of helium. The coefficients di and da give the finite mass correction due to mass polarization effects on the wave function. See Eqs. (16) and (17)... 

The closest theoretical result, the Unified theory [68], differs by more than 300 times the experimental uncertainty. This discrepancy should be partially removed by analysis including an estimate of the order (Za)4a2mec2 relativistic term and a complete calculation of the two-electron Bethe-logarithm [92]. The 14,i5N5+ 21S o — 23Pi isotope shift was measured to be —1.6623(10) cm-1, in fair agreement with an estimate based on [68]. The hyperfine corrected 3Pq —3Pi... 

The Bethe logarithms ln/co(n, L) in (11) are given with very high accuracy in [8] ... 

Table 1. Contributions to the positronium energy levels in units of cR.oo- In order to keep the size of the table as small as possible the following contributions which must be added to all energy levels have been omitted the lowest order contribution — and the contribution with the Bethe logarithm — lnfco(n, L)...

Here ln[fco(o)/Ry] is the nonrelativistic Bethe logarithm [10] and is the only quantity in Eq. (1) that depends on the whole wave function and requires significant computational efforts. Recently, following Schwartz (see [11]) a new method has been elaborated that is applicable to both one- and two-electron systems of an arbitrary angular momentum [12]. The final accuracy of expansion (1) is a5 In a 1.0 10-10. 

The Bethe logarithm values for the parent and daughter states were obtained with sufficiently high accuracy,... 

The total relativistic and QED energy shift for many-electron atoms consists of two parts. The first part contains the Bethe logarithm and the other is the average value of some effective potential. Throughout the exact nonrelativistic (Schrddinger) wave functions for the many-electron atom are used. The energy shift is [50] ... 

The expression (204) incorporates the one-electron corrections of order (aZ) eo that follow from the Dirac equation for the electron in the field of the nucleus. These corrections are included in the potentials f/jo. The Breit operator (164) is also included in the expression for Uik, the corresponding corrections are of order a Zeo- The radiative (order a aZ) eo) and nonradiative (order Q (aZ)eo) QED corrections are also incorporated in Eq(204). The radiative corrections, discussed in Sec.3.1 are included in the Bethe logarithm term and in the potential (7[Pg.452]

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