In non-relativistic quantum mechanics, neglecting the magnetic scattering terms, the total atomic scattering amplitude, , for photons with energy and with incident and scattered momenta and , is the sum of the non-resonant Thomson scattering amplitude, , and the ``anomalous'' scattering amplitude, (see Fig. 3).
The Thompson and anomalous scattering amplitudes are given, in terms of the
classical single electron scattering amplitude, , by
[12, 14, 15, 16]
The self-energy corrections that produce the Lamb-shift and the linewidth of the resonant term are shown explicitly .
The Thomson amplitude is a scalar which depends on the photon momentum transfer, , and on the photon polarization factors, , 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, and , and also depends on the photon energy, E. Thus, in general, 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 . It has been established experimentally, however, that the and dependencies of anomalous scattering are often small, and the full photon energy- and momenta-dependent is conventionally  approximated by its momenta-independent forward scattering limit, denoted . Consequently, the total atomic scattering amplitude, f, depends on the photon energy, E, via its f' and f'' terms, and on the wavevector transfer, , via its term.