Deconvolution of instrument and Kα₂ contributions from X-ray powder diffraction patterns using nonlinear least squares with penalties

A new deconvolution method, tolerant of noise and independent of knowing the number of Bragg peaks present, has been developed to deconvolute instrument and emission profile distortions from laboratory X-ray powder diffraction patterns. Removing these distortions produces higher-resolution patterns from which the existence of peaks and their shapes can be better determined. Deconvolution typically comprises the use of the convolution theorem to generate a single aberration from instrument and emission profile aberrations and then the Stokes method to deconvolute the resulting aberration from the measured data. These Fourier techniques become difficult when the instrument function changes with diffraction angle and when the signal-to-noise ratio is low. Instead of Fourier techniques, the present approach uses nonlinear least squares incorporating penalty functions, as implemented in the computer program TOPAS-Academic. Specifically, diffraction peaks are laid down at each data point with peak shapes corresponding to either expected peak shapes or peak shapes narrower than expected; a background function is included. Peak intensities and background parameters are then adjusted to obtain the best fit to the diffraction pattern. Rietveld refinement of the deconvoluted pattern results in background parameters that are near identical to those obtained from Rietveld refinement of the original pattern. Critical to the success of the deconvolution procedure are two penalty functions, one a function of the background parameters and the other a function of the peak intensities. Also of importance is the use of a conjugate gradient solution method for solving the matrix equation Ax = b.