International
Tables for Crystallography Volume H Powder diffraction Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk © International Union of Crystallography 2018 |
International Tables for Crystallography (2018). Vol. H, ch. 3.3, p. 267
Section 3.3.5.1. Crystallite size broadening^{a}Advanced Photon Source, Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, IL 60439–4814, USA |
The reciprocal space associated with an ideal large crystal will consist of a periodic array of infinitely sharp δ functions, one for each of the structure factors, as expected from the Fourier transform of the essentially infinite and periodic crystal lattice. For real crystals, this limit is reached for crystal dimensions exceeding circa 10 µm. The Fourier transform of a crystal lattice that is smaller than this will show a profile that follows the form described by the sinc(x) = sin(πx)/πx function. Any dispersion in the crystal sizes in a powder sample will smear this into a form intermediate between a Gaussian and a Lorentzian, which is well described by either a Voigt [equation (3.3.9)], a pseudo-Voigt [equation (3.3.8)] or the less-useful Pearson VII [equation (3.3.7)] function. The physical process used to form the powder will influence the details of the size distribution; usually this will approximate a log-normal distribution and the resulting peak-shape contribution from crystallite size effects will be largely Lorentzian with a width Γ_{SL}. Predominantly Gaussian size broadening can only occur if the size distribution is very tightly monodisperse. Then, for isotropic crystal dimensions this broadening is uniformly the same everywhere in reciprocal space; e.g. Δd* = constant ≃ 1/p, where p is the crystallite size. Transformation via Bragg's law to the typical measurement of a powder pattern as a function of 2θ gives this Lorentzian width asexpressed in degrees and the Scherrer constant, K, which depends on the shape of the crystallites; e.g. K = 1 for spheres, 0.89 for cubes etc. (see Table 5.1.1 in Chapter 5.1). A similar expression for the crystallite size from a neutron TOF experiment iswhere C is defined by equation (3.3.21). In some cases the crystallites have anisotropic shapes (e.g. plates or needles), in which case the peak broadening will be dependent on the respective direction in reciprocal space for each reflection. Many Rietveld refinement programs implement various models for this anisotropy.