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Form of the Thomson and anomalous amplitudes

In non-relativistic quantum mechanics, neglecting the magnetic scattering terms, the total atomic scattering amplitude, tex2html_wrap_inline1390, for photons with energy tex2html_wrap_inline1392 and with incident and scattered momenta tex2html_wrap_inline1394 and tex2html_wrap_inline1396, is the sum of the non-resonant Thomson scattering amplitude, tex2html_wrap_inline1398, and the ``anomalous'' scattering amplitude, tex2html_wrap_inline1400 (see Fig. 3).

The Thompson and anomalous scattering amplitudes are given, in terms of the classical single electron scattering amplitude, tex2html_wrap_inline1402, by [12, 14, 15, 16]

The self-energy corrections that produce the Lamb-shift tex2html_wrap_inline1404 and the linewidth tex2html_wrap_inline1406 of the resonant term are shown explicitly [15].

The Thomson amplitude is a scalar which depends on the photon momentum transfer, tex2html_wrap_inline1408, and on the photon polarization factors, tex2html_wrap_inline1410, but is independent of the photon energy. The Thomson amplitude is proportional to the Fourier transform of the atom's electronic charge distribution. In contrast, the anomalous amplitude depends separately on the incident and scattered wavevectors, tex2html_wrap_inline1394 and tex2html_wrap_inline1396, and also depends on the photon energy, E. Thus, in general, tex2html_wrap_inline1400 is a tensor which depends on the matrix elements between the ground state and the virtual intermediate states, and is not proportional to the Fourier transform of the total or subshell charge density [17]. It has been established experimentally, however, that the tex2html_wrap_inline1394 and tex2html_wrap_inline1396 dependencies of anomalous scattering are often small, and the full photon energy- and momenta-dependent tex2html_wrap_inline1422 is conventionally [14] approximated by its momenta-independent forward scattering limit, denoted tex2html_wrap_inline1424. Consequently, the total atomic scattering amplitude, f, depends on the photon energy, E, via its f' and f'' terms, and on the wavevector transfer, tex2html_wrap_inline1434, via its tex2html_wrap_inline1398 term.