International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. C, ch. 9.2, pp. 759760

Statistical distribution of stacking faults in closepacked structures introduces disorder along the stacking axis of the closepacked layers. As a result, one observes on a singlecrystal diffraction pattern not only normal Bragg scattering near the nodes of the reciprocal lattice of the average structure but also continuous diffuse scattering between the nodes owing to the incomplete destructive interference of scattered rays. Just like the extra polytype reflections, the diffuse streaks are also confined to only those rows for which h − k 0mod3. A complete description of the real structure of such onedimensionally disordered polytypes requires knowledge of the average structure as well as a statistical specification of the fluctuations due to stacking faults in the electrondensity distribution of the average structure. This cannot be accomplished by the usual consideration of the normal Bragg reflections alone but requires a careful analysis of the diffuse intensity distribution as well (Pandey, Kabra & Lele, 1986).
The first step in the structure determination of onedimensionally disordered structures is the specification of the geometry of stacking faults and their distribution, both of which require postulation of the physical processes responsible for their formation. An entirely random distribution of faults may result during the layerbylayer growth of a crystal (Wilson, 1942) or during plastic deformation (Paterson, 1952). On the other hand, when faults bring about the change in the stacking sequence of layers during solidstate transformations, their distribution is nonrandom (Pandey, Lele & Krishna, 1980a, b, c; Pandey & Lele, 1986a, b; Kabra, Pandey & Lele, 1986). Unlike growth faults, which are accidentally introduced in a sequential fashion from one end of the stack of layers to the other during the actual crystal growth, stacking faults involved in solidstate transformations are introduced in a random space and time sequence (Kabra, Pandey & Lele, 1988b). Since the pioneering work of Wilson (1942), several different techniques have been advanced for the calculation of intensity distributions along diffuse streaks making use of Markovian chains, random walk, stochastic matrices, and the Paterson function for random and nonrandom distributions of stacking faults on the assumption that these are introduced in a sequential fashion (Hendricks & Teller, 1942; Jagodzinski, 1949a, b; Kakinoki & Komura, 1954; Johnson, 1963; Prasad & Lele, 1971; Cowley, 1976; Pandey, Lele & Krishna, 1980a, b). The limitations of these methods for situations where nonrandomly distributed faults are introduced in the random space and time sequence have led to the use of Monte Carlo techniques for the numerical calculation of pair correlations whose Fourier transforms directly yield the intensity distributions (Kabra & Pandey, 1988).
The correctness of the proposed model for disorder can be verified by comparing the theoretically calculated intensity distributions with those experimentally observed. This step is in principle analogous to the comparison of the observed Bragg intensities with those calculated for a proposed structure in the structure determination of regularly ordered layer stackings. This comparison cannot, however, be performed in a straightforward manner for onedimensionally disordered crystals due to special problems in the measurement of diffuse intensities using a singlecrystal diffractometer, stemming from incidentbeam divergence, finite size of the detector slit, and multiple scattering. The problems due to incidentbeam divergence in the measurement of the diffuse intensity distributions were first pointed out by Pandey & Krishna (1977) and suitable correction factors have recently been derived by Pandey, Prasad, Lele & Gauthier (1987). A satisfactory solution to the problem of structure determination of onedimensionally disordered stackings must await proper understanding of all other factors that may influence the true diffraction profiles.
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