X-ray diffraction broadening analysis

Abstract

The microstructure is very important in research aimed to the development of new materials. The microstructural parameters, crystallite size, crystallite size distribution, crystallite strain, dislocation density and stacking fault probability, play a major role in physical and chemical properties of the material. These parameters can be determined by a proper analysis of X-ray diffraction line profile broadening. The observed XRD line profile of the studied sample, <em>h</em>(<em>ε</em>), is the convolution of the instrumental profile, <em>g</em>(<em>ε</em>), inherent in diffraction, and pure diffraction profile, <em>f</em>(<em>ε</em>), caused by small crystallite (coherent domain) sizes, by faultings in the sequence of the crystal lattice planes, and by the strains in the crystallites. That is, <em>f</em>(<em>ε</em>) is the convolution of the crystallite size/faulting profile, <em>p</em>(<em>ε</em>), and the strain profile, <em>s</em>(<em>ε</em>). The derivation of <em>f</em>(<em>ε</em>) can be performed from the measured <em>h</em>(<em>ε</em>) and <em>g</em>(<em>ε</em>) by the Fourier transform method, usually referred to as the Stokes method. That method does not require assumptions in the mathematical description of <em>h</em>(<em>ε</em>) and <em>g</em>(<em>ε</em>). The analysis of <em>f</em>(<em>ε</em>) can be done by the Warren-Averbach method, which is applied to the Fourier coefficients obtained by the deconvolution. On the other hand, simplified methods (which may bypass the deconvolution) based on integral widths may be used, especially in studies where a good relative accuracy suffices. In order to obtain the relation among integral widths of <em>f</em>(<em>ε</em>), <em>p</em>(<em>ε</em>) and <em>s</em>(<em>ε</em>), one assumes bell-shaped functions for <em>p</em>(<em>ε</em>) and <em>s</em>(<em>ε</em>). These functions are routinely used in the profile fitting of the XRD pattern and in the Rietveld refinement of the crystal structure. The derived crystallite size and strain parameters depend on the assumptions for the profiles <em>p</em>(<em>ε</em>) and <em>s</em>(<em>ε</em>). Integral width methods overestimate both strain and crystallite size parameters in comparison to the Warren-Averbach-Stokes method. Also, the crystallite size parameter is more dependent on the accuracy, with which the profile tails are measured and how they are truncated, than it is the strain parameter. The integral width also depends on the background level error of the pure diffraction profile. The steps and precautions, which are necessary in order to minimize the errors, are suggested through simple examples. The values of the crystallite size and strain parameters, obtained from integral widths derived by the Stokes deconvolution, are compared with those which followed from the Warren-Averbach treatment of broadening. Recent approaches in derivation of microstructure are also mentioned in short.