Formulations for Increasingly Disordered Bodies

Starting with crystals, the reciprocal lattice is made of dots. It is empty outside. As atomic arrangement disorder appears and increases, an increasing blurring occurs 

Because atoms do not occupy periodic positions in real space, there is no unit cell repeated N times. As a consequence, atoms cannot be defined by their coordinates relative to the origin of the unit cell, but by a single vector r issued from an arbitrary origin. Real space is now defined by a continuous function p(r), the electronic density at the apex of r, which replaces the electronic density inside an atom (or the electrical potential KyJ.  

P(x) is the radial distribution function (RDF) as introduced in Section I.I.2.I.4. It is the product of electronic densities attached to two points of the real space separated by vector x, with x being the distance between a pair of atoms in a set of atoms arranged disorderly. P(x) is thus a probability it is a generalization of the Patterson function of a crystal [10]. It corresponds to the Fourier transform of I(s). Reciprocally to Equation (1.8), 

At last, and whatever the wavelength, the atomic distribution in real space is not directly accessible. In addition, because now reciprocal space corresponds to a continuous content, I(s) should be recorded in the whole reciprocal space. This is impossible because the pattern is limited on one side by the width of the incident beam (the zero 9 angle is not available) on the other side, the wide angles are never accessible beyond 2/X. These two conditions amount to introducing a mask in the diffraction pattern, responsible for many artifacts. 

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