Approximate Estimation of Contributions to Pure X-Ray Diffraction Line Profiles from Crystallite Shapes, Sizes and Strains by Analysing Peak Widths

Abstract

A polycrystalline material may be considered as a set of crystallites. Since the crystallites have rather regular shapes, the assumption about the same shape is not far from physical reality for most polycrystals, especially powders. Such a system may be characterised in a statistical manner by two functions, the crystallite size distribution and the crystalline lattice strain distribution (for some materials other lattice distortions inside the crystallites, like stacking faults or dislocations, are to be considered additionally). The crystalline microstructure can be determined by investigating an X-ray diffraction pattern, what should be based on comparing an experimental pattern with a simulated one, derived from an appropriate physical model. Pure X-ray diffraction line profiles, containing information about crystalline microstructure, can be extracted from experimental data. An important step in analysing them is the separation of contributions from crystallite shapes and sizes and from strains, enabling the proper determination of both distributions together with the estimation of prevalent crystallite shape. A model of polycrystalline material combined with a description of X-ray diffraction on it, making such an analysis possible, is presented in this article. An approximate formula for separating both effects is based on results of computer simulation of pure X-ray diffraction line profiles from different crystalline powders, done under simplifying assumptions that the crystallites are prismatic or spherical, the size distribution is logarithmic-normal and the second-order strain distribution is normal.

Crystalline microstructure of sepiolite influenced by grinding

“…A similar model was proposed implicitly by Wilson (1962Wilson ( , 1963Wilson ( , 1970 but without a good method for finding microstructure characteristics. Kojdecki (1991) proposed a new model, together with a stable method for determining microstructure characteristics; a version of this approach, recently improved and updated, has made it possible to achieve convincing results (Kojdecki & Mielcarek, 2000, 2001Kojdecki, 2001Kojdecki, , 2004. Researchers active in the field of the diffraction analysis of the microstructure of materials generally accept crystallite shape, crystallite size distribution and crystalline lattice strain distribution as the principal microstructural characteristics of polycrystalline materials.…”

“…when both the line profiles, g from a standard sample (instrumental) and h from an investigated sample (experimental), are known; s is the reciprocal-lattice vector length, s ' 4 À1 # À # 0;hkl À Á cos # 0;hkl , is the X-ray wavelength and is a sufficiently large number (Wilson, 1963). In the vicinity of the Bragg angle, the pure X-ray diffraction line profile f (for the hkl reflection) from a crystal like that described above, with a volume-weighted crystallite size distribution v, and a second-order crystalline lattice strain distribution r, may be interpreted (up to an approximately constant multiplier) as (Wilson, 1963;Kojdecki, 1991Kojdecki, , 2004Kojdecki & Mielcarek, 2000;, under the assumption that the structure factor is constant for all crystallites. For assumed crystallite shape and fixed n, the function É hkl ðn; sÞ ¼ n À3 È hkl ðn; sÞ describes the pure diffraction line (hkl reflection) from a single crystallite (scattering X-rays coherently) with a perfect lattice and with a size characterized by the number n (taken with weight n À3 ); N must be sufficiently large [so that vðnÞ ¼ 0 for n > N].…”

“…is the interplanar distance for the ðhkl Þ planes (Kojdecki, 1991(Kojdecki, , 2004Kojdecki & Mielcarek, 2000).…”

“…with the same meaning of G (Wilson, 1962;Kojdecki, 2004) (a constant multiplier is omitted from both formulae). The FWHM (full width at half-maximum) in reflection angle scale of a profile from a prism is approximately…”

Crystal structure and microstructure of synthetic hexagonal magnesium–cobalt cordierite solid solutions (Mg2−2xCo2xAl4Si5O18)

“…A quantitative crystalline microstructure analysis of samples was performed by means of a mathematical model of polycrystalline material which has been discussed in detail earlier (Kojdecki, 2004;Kojdecki & Mielcarek, 2000;Kojdecki et al, 2005Kojdecki et al, , 2007Kojdecki, Ruiz de Sola et al, 2009). In this model a polycrystalline sample is interpreted as a statistical population of crystallites, which are assumed to be domains with perfect crystalline order inside, each of them scattering X-rays in a coherent way independently of one another, being separated by small-angle or large-angle boundaries and randomly oriented in space.…”