Electron lines

Auger Electrons. The fraction of the holes in an atomic shell that do not result in the emission of an x-ray produce Auger electrons. In this process a hole in the 4h shell is filled by an electron from theyth shell, and the available energy is transferred to a kth shell electron, which in turn is ejected from the atom with a kinetic energy = E — Ej —. Usually, the most intense Auger electron lines are those from holes in the K shell and involve two... 

If a paramagnetic salt is cooled, eventually a temperature is reached where the magnetic moments of the electrons line up. This temperature is known as the Curie point or the Neel point, depending upon whether the spins line up parallel below this temperature as in (b), with the moments pointing in the same direction to reinforce one another and produce ferromagnetism, or the moments line up in opposite directions as in (c) so that they cancel and antiferromagnetism occurs. 

T.M. Brown, R.H. Friend, I.S. Millard, D.J. Lacey, T. Butler, J.H. Burroughes, and F. Cacialli, Electronic line-up in light-emitting diodes with alkali-halide/metal cathodes, J. Appl. Phys., 93 6159-6172 (2003). 

Radiative Insertions in the Electron Line and the Dirac Form Factor Contribution... 

Calculation of the contribution of order a Za) induced by the radiative photon insertions in the electron line is even simpler than the respective calculation of the leading order contribution. The point is that the second and higher order contributions to the slope of the Dirac form factor are infrared finite, and hence, the total contribution of order Za) to the Lamb shift is given by the slope of the Dirac form factor. Hence, there is no need to sum an infinite number of diagrams. One readily obtains for the respective contribution... 

This correction is generated by the sum of all possible radiative insertions in the electron line in Fig. 3.9. In the approach described above one has to calculate the electron factor corresponding to the sum of all radiative corrections in the electron line, make the necessary subtraction of the leading infrared asymptote, insert the subtracted expression in the integrand in (3.33), and then integrate over the exchanged momentum. This leads to the result... 

We have restored in (3.39) the characteristic factor 1/(1 — rn /MX which was omitted in (3.33), but which naturally arises in the skeleton integral. However, it is easy to see that an error generated by the omission of this factor is only about 0.02 kHz even for the electron-line contribution to the IS level shift, and, hence, this correction may be safely omitted at the present level of experimental accuracy. 

Insertion of One-Loop Electron Factor in the Electron Line and of the One-Loop Polarization in the Coulomb Lines... 

Direct substitution of the radiatively corrected electron factor C k) in the skeleton integral in (3.33) would lead to an infrared divergence. This divergence reflects existence in this case of the correction of the previous order in Za generated by the two-loop insertions in the electron line. The magnitude of this previous order correction is determined by the nonvanishing value of the electron factor C k) at zero... 

Diagrams with Insertions of Two Radiative Photons in the Electron Line... 

As we have already seen, contributions of the diagrams with radiative insertions in the electron line always dominate over the contributions of the diagrams with radiative insertions in the external photon lines. This property... 

A few comments are due on the magnitude of this important result. It is sometimes claimed in the literature that it has an unexpectedly large magnitude. A brief glance at Table 3.3 is sufficient to convince oneself that this is not the case. For the reader who followed closely the discussion of the scales of different contributions above, it should be clear that the natural scale for the correction under discussion is set by the factor 4a Za) / Trn )m. The coefficient before this factor obtained in (3.48) is about —1.9 and there is nothing unusual in its magnitude for a numerical factor corresponding to a radiative correction. It should be compared with the respective coefficient 0.739 before the factor Aa. Za) /n m in the case of the electron-line contribution of the previous order in a. 

Fig. 3.12. Nineteen topologically different diagrams with two radiative photons insertions in the electron line...

An effective method to separate contributions of low- and high-momenta avoiding at the same time the problems discussed above was suggested in [70, 71]. Consider in more detail the exact expression (3.56) for the sum of all corrections of orders a Zo) m (n > 4) generated by the insertion of one radiative photon in the electron line... 

We would like to emphasize two features of these results. First, the state dependence of the constant is very weak, and second, the scale of the constant is just of the magnitude one should expect. In order to make this last point more transparent let us write the total electron-line contribution of order a Za) to the 15 energy shift in the form... 

Note that the scale of this contribution is once again exactly of the expected magnitude, namely, this contribution is suppressed by the factor (o/tt) ln(Zo ) in comparison with the leading logarithm squared contribution of order a Za). Of course, the additional numerical suppression factor 8/27 could not be obtained without real calculation. The calculation in [95] was followed by a series of papers [96, 97, 98] where a subset of all diagrams with two-loop radiative insertions in the electron line were calculated numerically without expansion in Za. The results of these considerations were controversial, and the authors of [96, 98] even came to the conclusion that the leading logarithm contribution should be different from the result in (3.75). 

The values of the coefficients Bsn obtained above lead to a quite peculiar behavior of different contributions to the energy shifts. To get a better idea about relative magnitude of these electron line contributions of order a Za) let us write explicitly contributions to 15 and 25 states... 

Unlike the case of radiative insertions in the electron line, this time the coefficients before the logarithms do not demonstrate any peculiarities. The respective correction to the energy shift is about 2 kHz for the 15 state in hydrogen. 

Recent progress in calculation of nonlogarithmic coefficients Beo is reported in [80, 81]. The authors of these papers derived a general NRQED expression for the contribution of all diagrams (both with radiative insertions in the electron line and in the external photons) to the difference of the coefficients Bqo 1S) — Bso nS), which is necessary for calculation of the phenomenologically important energy difference Z = n AE nS) — AE lS). The accuracy of formulae obtained in [80, 81] allows one to calculate the a Za) contribution to the energy difference with accuracy about 0.1 kHz. 

Logarithmic Electron-Line Contribution 1 / 0) Erickson, Yennie (1965) [62, 63] [(1 - ... 

Leading Logarithmic Two-Loop Electron-Line Contribution ... 

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