Scattering and diffraction. Kinematic model
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Kinematic model for  X-ray diffraction

A material sample is primarily regarded as a distribution of electrons which is peculiar to each case. The interaction of electromagnetic radiations with matter is modeled considering radiation as particles (photons), or as waves. We are dealing here with a typical ondulatory phenomenon. In fact, in addition to the well known experiment of Max von Laue, the original experiments (Friedrich, Knipping and Laue, 1912, in CuSO4 crystals) were the proof that X-rays are waves and that crystals are structured in three-dimensional lattices with periodic distances in the range of X-ray wavelengths.

Waves scatter whenever there is a change in the incident wave front due to a discontinuity in the medium in which they propagate (an astronaut does not see the "blue sky"). If the phase relationship between waves scattered by the discontinuity remain constant, the waves combine in a cooperative and coherent manner, producing interferences, known as diffraction. However, in the scattering process the phase relations occur randomly and are not maintained over time.

The theoretical model of matter-wave interaction is provided by the four Maxwell equations, the equation of charge continuity and the two equations that characterize the materials (the electric and magnetic polarization and the dielectric and permeability constants).

Although this model describes the phenomenon macroscopically, it also applies to the atomic scale for electronic distributions weakly bound in the atom and which move at speeds lower than light. In addition, the nuclei are regarded as massive, and therefore fixed (Born approximation). Furthermore, it is supposed that the material sample extends indefinitely, that its dielectric constant is independent of time, that the permeability of the material is close to the unit, and that both constants are homogeneous and isotropic.

Basically, the kinematic conditions are summarized considering that the incident wave is hardly modified by the material as it passes through. Therefore, the scattered wave is seen as a small perturbation due to an interaction of a very weak magnitude. More specifically:

Distribution of single crystal micro-blocks in a crystal sample suitable for implementing the kinematic model. As in many other aspects of Crystallography, we must reach a compromise... The orientations must not be totally random, but without reaching the perfect alignment of the micro-blocks. This is why we say that the sample must be "perfectly imperfect" .

Kinematic model for a single crystal

Continuing with the above, for the structural analysis of samples using kinematic X-ray diffraction, we consider that:

<structure>  (over time and unit-cells)   =
 <lattice & symmetry> (according to distortions, vibrations ...)   +
<motif>  (according to vibrations, orientations, disorder ...)
(symbols < > mean "average")

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