Dynamical Theory of X-Ray Diffraction — I. Perfect Crystals
The geometrical, or kinematical theory, of diffraction considers that each photon is scattered only once, and that the interaction of X-rays with matter is so small that it can be neglected. It can therefore be assumed that the amplitude incident on every diffracting centre inside the crystal is the same. The total diffracted amplitude is then simply obtained by adding the individual amplitudes diffracted by each diffracting centre, only taking into account the geometrical phase differences between them. The result is that the distribution of diffracted amplitudes in reciprocal space is the Fourier transform of the distribution of diffracting centres in physical space. The integrated reflected intensities calculated this way are proportional to the square of the structure factor and to the volume of crystal bathed in the incident beam. The diffracted intensity according to the geometrical theory would therefore increase to infinity if the volume of the crystal was increased to infinity, which is of course absurd. The theory only works because the strength of the interaction is very weak and if it is applied to very small crystals. The geometrical theory presents another drawback: it gives no indication as to the phase of the reflected wave. This is due to the fact that it is based on the Fourier transform of the electronic density limited by the external shape of the crystal. This is not important when one is only interested in measuring the reflected intensities. For perfect or nearly perfect crystals, and for any problem where the phase is important, such as the case of multiple reflections, interference between coherent blocks, standing waves, etc., even for thin or imperfect crystals, a more rigorous theory which takes into account the interactions of the scattered radiation with matter should be used; this is the dynamical theory.
KeywordsDynamical Theory Spherical Wave Dispersion Surface Thin Crystal Entrance Surface
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