The interpretation of X-ray diffraction intensity distributions from small distorted crystals. Definition of the average coherently scattering region

X-ray diffraction patterns from a polycrystalline material or a small distorted crystal do not always contain enough information for the determination of the (average) geometry and the degree of distortion of the crystal(s). To avoid the consequent difficulties in the interpretation of diffraction patterns from such small distorted crystals the author introduces for the material the average lattice function (x), that is, a repartition function. When the diffraction spots from the material can be measured separately, the (x) function belonging to that material has certain special properties. For that case the average coherently scattering region can be defined in terms of (x). It is shown that the intensity distribution in such a diffraction spot can be described with the aid of the product of C(x) and φ (x, Δx), where C(x) is the average form function of the coherently scattering regions, φ (xm, Δx), the special value of φ (x, Δx), is one peak of the quasiperiodic function (x) around the mth lattice point, xm, of the average lattice. The vector x represents the distance between two arbitrary unit cells in the structure; Δx = x−xm. Consequently the line profiles in a powder diffraction pattern can be described with the aid of C(t). φ(tm, L), the product of the projections of C(x) and φ (xm, Δx) on the perpendicular to the reflecting planes; L, t and t_m are the projections of Δx, x and xm respectively. It is shown that the condition φ(t_m, L) = 0 for |L| ≥ ½d (d is the interplanar spacing) holds when the diffraction lines can be measured separately. A criterion for the applicability of the Warren-Averbach method is given.