Progress in quantitative XRD profile analysis has been closely linked with instrumental improvements which include higher intensity fine focus diffractometers equipped with highly monochromatic radiation, greatly improved detectors and labor saving automation which allows to examine smaller and more subtle profile changes. With high speed desk computers now commonly available in most laboratories, one can make full use of recent analytical advances beyond the classical approaches to attain new levels of understanding.
The classical Fourier series as described by Warren is well know and minimizes the need for {\em a priori} assumptions. But, in dealing with a statistical treatment of strain and particle size, it never makes the connection with dislocation models. One must be content with mean square strain, over a range of column distances, and the mean column length as a final description of the highly deformed state. The model is based on the assumption that particle size broadening may be defined to include any sample broadening that is independent of order in a multiple order profile analysis, while strain broadening varies smoothly with order. Particle size is taken to be a measure of the average column length for a system of subgrains having small angular or positioning shifts at their boundaries, which result in a loss in diffraction coherency. Dislocation fields are assumed to act continuously and elastically across sub-boundaries. This is the fundamental strength and limitation of the model.
The Fourier series which describes measured profiles contains products of coefficients which makes the form equivalent to a multiple convolution process between particle size, strain and instrumental components of broadening. Typically, coefficients corresponding to the shortest column distances contain errors resulting from estimates of background and overlapping tails, while at increasing distances from the mean column length instrumental errors become more evident. Attempts to synthesize full sets of experimental Fourier coefficients to test how well a series fits the measured profile data are troubled by noise, uncertainties in the background determination and profile truncation.
However, it has been demonstrated several times that the mean square strain decreases monotonically according to a simple relationship and particle size broadening is well determined by the mean column height and the variance of the height distribution. Both findings have been incorporated into calculating exact analytical profiles of specimen broadening from highly deformed materials. This can be done without degrading the basic purity of the original Fourier series. These analytical functions require two strain variance parameters, the average column height, and the variance of column length to describe two orders of profiles. Corresponding instrumental functions require four additional parameters to describe profiles from a pair of orders. A total of eight parameters are required to define the shapes of experimental curves in a two order analysis. This may be compared with 40-100 Fourier coefficients in a comparable two order Fourier analysis.
Strains and particle size parameters have been inter related with dislocation density and correlation distance by a special series expansion. This includes three parameters from the analytical formulation and two parameters from the dislocation approach. Dislocation density is found to vary directly with the distance dependent mean square strain parameter and scales inversely with the average column height. Dislocation correlation distance is expressed relative to the average column height. Analytical functions have also been extended to include stacking faults. With these functions diffraction from columns having different orientations relative to the stacking fault plane, but included in the same (hkl) multiplicity, can be displayed as their sum or be examined as separate profiles.
Numerically calculated analytic functions can be generated quickly enough to use non linear least squares optimization routines for finding the parameters mentioned above. Likewise, misfit surfaces can be constructed by stepping parameters within their planes away from their minimum point. A broad misfit surface would suggest a greater uncertainty of fitted parameters.
Further development of the method in order to include more general cases than the model which is basis of the Warren-Averbach analysis depends on the availability of an analytic function with a few free microstructural parameters describing sufficiently well the corresponding diffraction profile.
1. C.R. Houska \& T. Smith: J. Appl. Phys. 52,(1981) 748-754.
2. S. Rao \& C.R. Houska: Acta Cryst. A42 (1986) 6-13.
3. S. Rao \& C.R. Houska: Acta Cryst. A42 (1986) 14-19.
4. S. Rao \& C.R. Houska: Acta Cryst. A44} (1988) 1021-1028.