Statistical theory of X-ray scattering from crystals of finite size with pure displacement disorder in one dimension

The statistical theory of X-ray scattering from a crystal that is disordered in one dimension is discussed. The state of disorder is characterized by probabilities that different types of scattering elements occupy specified positions. The effect this disorder has on the corresponding ensemble average axial reflections is determined. Finite size is explicitly accounted for. Motivated by recent experiments on water intercalation into thin, lyotropic multilayers, special consideration is given to systems whose components differ with respect to size, but not with respect to their scattering factors. Focusing on this case of pure displacement disorder in a binary mixture, a representation of the scattering function in closed form is derived. The case of a random binary mixture leads to the results of Hendricks & Teller [J. Chem. Phys. (1942), 10, 147- 167], and Méring [Acta Cryst. (1949), 2, 371-377], while in the presence of nearest-neighbor correlations, a connection is established with the theory of Kakinoki & Komura [J. Phys. Soc. Jpn (1952), 7, 30-35]. In extension of their treatment, systems with non-stationary transition probabilities are investigated. The effects on the scattering function of constraining composition fluctuations are also studied. Particular characteristics of the scattering function are displayed and discussed.