Model-independent extraction of the shapes and Fourier transforms from patterns of partially overlapped peaks with extended tails

This work presents a technique for extracting the detailed shape of peaks with extended, overlapping tails in an X-ray powder diffraction pattern. The application discussed here concerns crystallite size broadening, though the technique can be applied to spectra of any origin and without regard to how the profiles are to be subsequently analyzed. Historically, the extraction of profile shapes has been difficult due to the complexity of determining the background under the peak, resulting in an offset of the low-frequency components of the Fourier transform of the peak known as the `hook' problem. The use of a carefully considered statistical weighting function in a non-linear least-squares fit, followed by summing the residuals from such a fit with the fit itself, allows one to extract the full shape of an isolated peak, without contributions from either the background or adjacent peaks. The extracted shape, consisting of the fit function recombined with the residuals, is not dependent on any specific shape model. The application of this to analysis of microstructure is performed independently of global parametric models, which would reduce the number of refined parameters; therefore the technique requires high-quality data to produce results of interest. The effectiveness of the technique is demonstrated by extraction of Fourier transforms of peaks from two sets of size-broadened materials with two differing pieces of equipment.