research papers 26 # 2003 International Union of Crystallography  Printed in Great Britain ± all rights reserved J. Synchrotron Rad. (2003). 10, 26±42 X-ray absorption spectroscopy: state-of-the-art analysis C. R. Natoli, a * M. Benfatto, a S. Della Longa b and K. Hatada a a Laboratori Nazionali di Frascati, INFN, CP13, 00044 Frascati, Italy, and b Universita' dell'Aquila, Via Vetoio, loc. Coppito II, 67100 L'Aquila, Italy. E-mail: natoli@lnf.infn.it State-of-the-art techniques for analysing X-ray absorption spectra are reviewed, with an eye to biological applications. Recent attempts to perform full spectral ®tting of the XANES energy region and beyond for the purpose of structural analysis have met with encouraging success. The present paper analyses the theoretical motivations behind this success and indicates routes for future improvements. The theoretical background is not entirely new, although the point of view is, and some sections and appendices present material that the authors believe has never been published before. The aim of this paper is to provide a theoretical analysis that is as self-contained as possible. Keywords: X-ray absorption spectra; XANES; biological applications. 1. Introduction In the past 20 years, X-ray absorption spectroscopy (XAS) from inner-shell electrons has proved to be an invaluable tool in the study of the electronic and structural properties of condensed matter systems and, in particular, of the active sites of metalloproteins. In spectroscopic analyses, it has been common practice to treat sepa- rately the near-edge region (conventionally from below the edge up to 30±50 eV above it) and the high-energy part of the spectrum (above 30±50 eV), the so-called EXAFS (extended X-ray absorp- tion ®ne structure) region. The motivation behind this separation is entirely empirical, in that the extraction of a structural `signal' from the absorption spectrum via subtraction of an approximate atomic background can be performed with a certain con®dence and relia- bility only in the EXAFS region, whereas the low-energy part cannot be adequately background subtracted. There exist several analysis packages based on spherical-wave multiple-scattering theory (MST) and the complex optical potential of the Hedin±Lundqvist type (Hedin & Lundqvist, 1969, 1971). These packages are able to reproduce satisfactorily the EXAFS signal [for a review see Rehr & Albers (2000)]. The same codes even offer the possibility of ®tting the entire spectrum; however, it does not seem that this potential has been seriously exploited as yet. Some of the factors that have probably deterred the various practitioners of MST from extending the ®tting procedure used in the EXAFS region to the edge region, via the calculation of the total cross section, are well known: the inadequacy of the muf®n-tin approximation to the potential at low photoelectron energies, the lack of a satisfactory description of the screening and relaxation processes following the sudden creation of the core hole, and the need to include electronic correlations and, in particular, two-electron excitations (since they change the slope of the background absorption). Therefore, the analysis of the near-edge region has remained at a semi-quantitative level. Nevertheless, quantitative analysis of the X-ray near-edge struc- ture (XANES) in order to obtain structural and electronic informa- tion can be very relevant in many ®elds of scienti®c application, like extra-dilute systems, surface spectra, real-time measurements of dynamic systems, trace-element analysis, the local investigation of materials under extreme conditions and especially biological systems (enzymes), where the low S/N ratio and the weak scattering power of the light elements constituting the organic material limit the k-range of the available experimental data. In all these instances, the EXAFS region of the spectrum cannot be adequately exploited, since the usable data are very often below 200 eV from the absorption edge. Very recently, a method for performing full spectral ®tting of the XANES energy region and beyond (up to 200 eV above the edge) for the purpose of structural analysis has been proposed and applied to the K-edge spectra of a number of transition metal compounds, both organic and inorganic (Benfatto & Della Longa, 2001; Della Longa et al., 2001). The encouraging success of this attempt requires a reconsideration of the theory with a view to understanding why the method works and highlighting areas of possible improvement. The present formulation re¯ects our long experience of the performance of multiple-scattering theory with a complex optical potential and might not duly acknowledge other groups' contribu- tions to the subject. We apologize if this is the case. 2. Theoretical background In this section we shall present the derivation of the photoemission cross section for the ejection of a photoelectron of ®nal momentum k and kinetic energy k 2 along the direction ^ k and illustrate the reduc- tion of this many-body problem to an effective one-particle problem with complex energy-dependent optical potential. This process of reduction will help us to understand the validity of the necessary approximations to the optical potential and will give us guidance in choosing among various approximation schemes. The photoabsorp- tion cross section is nothing more than the integration of the photoemission cross section over all the emission angles and all the ®nal channels (elastic plus inelastic) with the same ®nal energy. The reason we treat both cases together is duplex. Firstly, the mathema- tical formalism is the same, as is the reduction process to a one- particle problem; secondly, and more importantly, on purely physical grounds we can think of photoabsorption as a kind of photoemission that has the same electron source (the photoabsorber), in which the detector, instead of being outside the measured system, coincides with the source. The validity of this assumption will be apparent from the mathematical formalism in the following. By treating photo- absorption and photoemission together, we can judge the sensitivity to structural details of a particular potential in absorption by looking at its performance in photoelectron diffraction. In photoabsorption, because of the obvious impossibility of controlling the measured variables (except total energy), we are obliged to sum over all ®nal states at a given energy. 2.1. Reduction of the many-body problem to an effective one-particle problem: the photoemission case The photoemission cross section in the many-body case for the ejection of a photoelectron of ®nal momentum k and kinetic energy k 2 along the direction ^ k can be written in the dipole approximation as d!=d ^ k  4 2  0 h - !  N k    "  P N i1 r i  N g            2 ; 1