International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. C. ch. 7.4, pp. 659-660
|
The Compton effect is a relativistic phenomenon and it is accordingly more satisfactory to start from this basis, i.e. the Klein & Nishina (1929) theory and the Dirac equation (see Jauch & Rohrlich, 1976
). In second-order relativistic perturbation theory, there is no overt separation of
and
terms. The inclusion of electron spin produces additional terms in the Compton cross section that depend upon the polarization (Lipps & Tolhoek, 1954a
,b
); they are generally small at X-ray energies. They are of increasing interest in synchrotron-based experiments where the brightness of the source and its polarization characteristics compensate for the small cross section (Blume & Gibbs, 1988
).
Somewhat surprisingly, it is the spectral distribution, d2σ/dΩ dE2, rather than the total intensity, dσ/dΩ, which is the better understood. This is a consequence of the exploitation of the Compton scattering technique to determine electron momentum density distributions through the Doppler broadening of the scattered radiation [see Cooper (1985) and Williams (1977
) for reviews of the technique]. Manninen, Paakkari & Kajantie (1976
) and Ribberfors (1975a
,b
) have shown that the Compton profile – the projection of the electron momentum density distribution onto the X-ray scattering vector – can be isolated from the relativistic differential scattering cross section within the impulse approximation. Several experimental and theoretical investigations have been concerned with understanding the changes in the spectral distribution when electron binding energies cannot be discounted. It has been found (e.g. Pattison & Schneider, 1979
; Bloch & Mendelsohn, 1974
) that, to a high degree of accuracy, the spectral distribution is merely truncated at energy transfers
.
This has led to the suggestion that the incoherent intensity can be obtained by integrating the spectral distributions, i.e. from
Unfortunately, this requires the Compton profile of each electron shell as input [Compton line shapes have been tabulated by Biggs, Mendelsohn & Mann (1975)] for all elements.
Ribberfors (1983) and Ribberfors & Berggren (1982
) have shown that this calculation can be dramatically simplified, without loss of accuracy, by crudely approximating the Compton line shape. Fig. 7.4.3.2
shows the incoherent scattering from aluminium, modelled in this way, and compared with experiment, Waller–Hartree theory, and an exact integral of the truncated impulse Compton profile.
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