EIGEN-SYSTEM ANALYSIS OF X-RAY-DIFFRACTION PROFILE DECONVOLUTION METHODS EXPLAINS ILL-CONDITIONING

Several deconvolution methods common in X-ray diffraction profile stud ies have been examined using an eigen-system analysis in which an erro r-bound function is used to represent the maximum difference between t he solution and true specimen profiles. This approach quantifies the s ources of misfitting and ill-conditioning that appear in the solution profile and expresses them as a function of the control parameter for a particular method. A simulation of an instrument-broadened profile o verlaid with random noise and background signals was used to evaluate the error-bound function for the iterative and constrained deconvoluti on methods, and the properties of the error-bound function were relate d to the features of the solution profile for each method. This analys is quantifies the terms that contribute to the ill-conditioning of the solution profile. It shows that, even for optimum values of the contr ol parameters, positivity is not preserved and spurious oscillations a re present in the solution profile.