Regularization method and determination of structure of disordered materials

One of the major applications of EXAFS spectroscopy is the determination of the structure of disordered materials. It is well known that the Radial Distribution Function (RDF) is a key element in describing these systems. For simple systems, the conventional methods of EXAFS RDF determination (Fourier methods, ratio-cumulant method, nonlinear least-square fitting, splice method) have proven to be reliable, but they encounter many difficulties when applied to disordered systems. Most importantly, none of the above methods can be considered a "general approach" to the problem, and most of them require substantial user intervention in the analysis process. Direct methods of determining the RDF from the data are needed.; The method of Tikhonov regularization was introduced to EXAFS as a solution for the above problems nearly two decades ago, but it has not gained widespread use despite its appeal from a formal point of view. Regularization methods are based only on the general physical properties of the distribution function; they are free of any "model dependent" assumptions for fitting, and they enable one to develop a step-by-step procedure for RDF determination. Our method employing Friedman regularization allows us to address shortcomings of previous approaches. Introduction of the General Discrepancy Method allows our program to automatically choose the regularization parameters; to determine error bars for the RDF; and to self consistently and rigorously account for errors introduced by not only the experimental data but also the theoretical model (amplitudes and phases). Using the Friedman regularization method we also have solved the long-standing problem of data filtering in EXAFS. The methods are illustrated on both extreme test cases and practical examples.