X-ray diffraction by a one-dimensional paracrystal of limited size

An explicit equation for X-ray diffraction by a finite one-dimensional paracrystal is derived. Based on this equation, the broadenings of reflections due to limited size and disorder are discussed. It depicts the paracrystalline diffraction over the whole reciprocal space, including the small-angle region where it degenerates into the Guinier equation for small-angle X-ray scattering. Positions of diffraction peaks by paracrystals are not periodic. Peaks shift to lower angles compared to those predicted by the average lattice constant. The shifts increase with increasing order of reflections and degree of disorder. If the heights and widths of the paracrystalline diffraction are treated as reduced quantities, they are functions of a single variable, N1/2 g. The width of the first diffraction depends mostly on size broadening for a natural paracrystal, where N1/2 g = 0.1-0.2. A method to measure the paracrystalline disorder and size using a single diffraction profile is proposed based on the equation of paracrystal diffraction. An initial value of size may be obtained using the Scherrer equation, that of the degree of disorder is then estimated by the alpha * law. Final values of the parameters are determined through least-squares refinement against observed profiles. An equation of diffraction by a polydisperse one-dimensional paracrystal system is presented for 'box' distribution of sizes. The width of the diffraction decreases with increasing breadth of the size distribution.