X-ray diffraction effects by non-ideal crystals of biotite, muscovite, montmorillonite, mixed-layer clays, graphite, and periclase*

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Abstract
A Fourier-transform computer program has been developed to calculate the x-ray diffraction effects expected from non-ideal crystals. The program evaluates the function [unk], where G ( HKL ) is the Fourier transform at a particular H , K , L coordinate in reciprocal space, f n is the atomic scattering factor of the n th atom, and x , y , z its coordinates in direct space. Diffraction profiles were calculated for biotite, muscovite, and montmorillonite crystallites 2, 3, 4, 5, 6, 8, 10, 20 and 30 layers thick, and for various models of graphite and periclase. The apparent basal spacings given by finite muscovite, montmorillonite, graphite, and periclase models deviate from the true values as the platelets become thinner. Diffraction profiles were also calculated for 30-layer packets of random mixed-layer muscovite-montmorillonite clays. The migration curves of the (001), (002), (003), (004) and (005) muscovite maxima and of the (001), (002), (003), (004), (005), (006), (007) and (008) montmorillonite maxima, obtained from those calculations, are in general agreement with the Hendricks-Teller formulation, although certain new details appear giving a more quantitative estimate of the diffraction effects. Diffraction profiles also give a very accurate measure of line broadening. Evaluation of the profiles for graphite gives the following relationship between half-width line broadening and crystal thickness: 2(sin θ ′−sin θ ″) = 0.90 λ / t , where t is the thickness of the crystal and θ ′ and θ ″ are, respectively, the values of theta at the large- and small- angle sides of the peak.