Determining the radial pair distribution function from EXAFS spectra by use of the Landweber iteration method

The Landweber iteration method (1) is used to construct the radial pair distribution function (PDF) from an EXAFS spectrum. The PDF as solution of the fundamental EXAFS integral equation is determined in a stable way without additional supplementary conditions due to the semi-convergent behaviour of the Landweber iteration.
For the correct determination of the PDF it is important to know the energy shift dE0. If dE0 is unknown then usually it is determined by a shell fit of the spectrum before an inversion method is applied. If the shell fit model is incorrect then the determined dE0 is erroneous and an inverse method would give a wrong PDF. This leads in turn to the conclusion that the recent inversion procedures are only applicable if the structural model for the investigated system is already known; hence they are not “stand-alone” methods.

We developed an algorithm based on the Landweber iteration for deriving the appropriate dE0. The performance was tested by several theoretical and experimental EXAFS spectra of metal hydrates (U(IV), Np(IV), Th(IV), Cm(III)). For the estimation of the optimum number of iterations we adopted the L-curve concept which was recently used for the determination of the optimal regularization parameter in Tichonov’s method [2]. The examples show that the Landweber iteration in combination with the simultaneous determination of dE0 and the L-curve concept for the estimation of the optimum number of iterations is a robust way to calculate the PDF. The approach enables the calculation of the PDF even for strong overlapping shells like for Cm(III) hydrate.

(1) Landweber, L. Am. J. Math. 73, 615-624 (1951).
(2) Kunicke, M., et al. Physica Scripta T115, 237–239 (2005).