Use of pattern decomposition to study microstructure: practical aspects and applications

Abstract

Abstract Since diffraction-line broadening is influenced by structural imperfections and the small size of crystallites (i.e. domains over which diffraction is coherent), powder diffraction is a valuable method for the determination of these properties. The theory of diffraction by imperfect solids was well established half a century after the introduction of X-ray diffraction and many applications were reported from powder diffraction data. These were essentially based on a Fourier analysis of diffraction-line broadening and, to a lesser extent, on an interpretation of the integral breadth, when an approximate characterization was sufficient (see Wagner, 1966). The detailed information, apparent crystallite size, cell-column length distribution and microdistortion, extracted from the theory based on Fourier series, has been found to be attractive and has often influenced the choice of the method for ‘precise’ line-broadening analyses, in spite of constraints (e.g. well-resolved lines are required) and propagation of errors during the treatment (Chapter 26; Young et al.,1967; Delhez et al.,1982). Experimental ‘crystallite size’ distributions are of low precision (see Bertaut, 1950; Le Bail and Louer, 1978) and their interpretation is complex, except in a few cases, such as in the direction perpendicular to uniformly-shaped disk or platelet crystallites (Le Bail and Louer 1978, 1980). As far as ‘size’ effect is concerned, Fourier apparent size ‘may appear to be of no advantage since a simple measurement of the line widths leads to another average function of the diameters’ (Guinier, 1963). Indeed, the theory of diffraction based on the integral breadth is well established (Chapter 5) and shows that average values of sizes and structural imperfections can be extracted from powder diffraction data. The general use of this approach was limited, in the past, by severe approximations (e.g. the correction of the instrumental contribution was based on simple assumptions about line shapes). The recent introduction of pattern modelling techniques, combined with the use of the properties of the Voigt function (Langford, 1978), has contributed to a renewal of interest in the use of the integral breadth for line-broadening analyses. The procedure presents the advantage of speed and convenience over the elaborate Fourier analysis, which may be a serious benefit when a number of samples for the same material have to be studied. Based on all information available in the diffraction pattern, the method is appropriate for characterizing anisotropic microstructural properties.