General Conditions of Diffraction by a Crystal

Let us consider an infinite periodic lattice, the nodes of which are being occupied by identical atoms. The lattice is defined by repetition of its unit cell built on ox, oy, oz. The unit vectors are a, b,c. The origin 0 is arbitrarily chosen on a node of the lattice. Any atom A is defined by its coordinates with respect to O as ua + vb + WC, uvw are integers varying between zero and infinity (the unit cell is devoid of motif). By limiting this set to N unit-cells containing a motif, a single crystal is created and localized in real space. 

As this crystal interacts with a planar incident wave X having a direction S, (unit vector), apart of the wave is absorbed, a part is scattered with a change of X, and another part is scattered without change of X. 

The change of X (incoherent scattering or Compton effect for x-rays) is due to an inelastic collision between the incident wave (photons, x-rays, or electrons) and the atom electronic cloud. The incoherent scattering is independent of the atom positions. The atoms of the target are thus incoherent sources of wavelets, not able to produce interference. 

If the incident wave is an X-ray beam, the interaction with the crystal (i.e., what the incident beam sees ) is the electronic density p(r) in the atom volume (p(r) is nil outside). If the incident wave is an electron beam, the interaction happens both with electrons and the nucleus (i.e., the wave sees the electrical potential of the atom). The atomic scattering factor/ is proportional to (Z - fx.Ray)- 

Doing this is equivalent to creating a reciprocal space formed by a lattice occupied by hkl nodes. They are defined by their coordinates 

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