# 1998 International Union of Crystallography Journal of Synchrotron Radiation Printed in Great Britain ± all rights reserved ISSN 0909-0495 # 1998 1383 J. Synchrotron Rad. (1998). 5, 1383±1389 Extracting Dynamic Information from EXAFS: Simultaneous Analysis of Multiple Temperature-Dependent Data Kelly A. Daly and James E. Penner-Hahn* Department of Chemistry, The University of Michigan, Ann Arbor, MI 48109, USA. E-mail: jeph@umich.edu (Received 23 August 1997; accepted 20 March 1998 ) A new approach to the extraction of dynamic information from extended X-ray absorption ®ne- structure (EXAFS) spectra has been developed. With this method, a complete set of temperature- dependent spectra are ®t simultaneously to one of a variety of pair-distribution functions. Distributions are calculated in r-space using the appropriate absorber±scatterer pair potential. The temperature-dependent EXAFS spectra are calculated by summing k-space models over a range of distances and angles weighted according to the relative contribution of each geometry to the distribution. This approach allows re®nement of data using a full multiple-scattering analysis with only modest computational time. Keywords: EXAFS; temperature-dependent analysis; anharmonicity. 1. Introduction Over the last twenty years, extended X-ray absorption ®ne structure (EXAFS) has come to be recognized as one of the premier tools for determining local atomic structure. For most applications, the information of interest is the static structure. In this case, the EXAFS can be de®ned as k F k k Z 1 0 gr exp 2r k  sin2kr  'k r 2 dr; 1 where (k) is the fractional modulation in the absorption coef®cient above the edge, k is the photoelectron wave- vector, F(k) is the theoretical backscattering amplitude, r is the scatterer distance, (k) is the photoelectron mean free path, '(k) is the phase shift encountered by the photo- electron on passing through the potentials of the absorber and scatterer, and g(r) is the radial distribution function (Teo, 1986). The EXAFS equation can be simpli®ed by assuming small disorder in the system (Beni & Platzman, 1976). This allows the pair-distribution function to be approximated as a Gaussian distribution, giving the more familiar k X s N s S 2 0 A s F k=kr 2 as  exp2r=k  exp2k 2  2 as  sin2kr as  ' as k; 2 where N s is the number of scatterers, S 2 0 is an inelastic loss term, A s (k) is the backscattering amplitude and  2 as is a disorder term also known as the Debye±Waller factor, and the sum is taken over every shell of scatterers. In (2), the Debye±Waller factor gives a measure of the disorder in the absorber±scatterer interaction. Disorder can arise either from static disorder (i.e. from a range of bond lengths within a single shell of scatterers) or from the dynamics of the system (i.e. the motion of the absorber± scatterer pair). From temperature-dependent EXAFS measurements it is possible, at least in principle, to extract information about the dynamics of a system. In cases where the disorder (static, dynamic or both) is large, analysis using the traditional EXAFS equation (2) can lead to errors in bond length, coordination number and  2 as as a result of breakdown of the small disorder assumption (Hayes et al. , 1978; Eisenberger & Brown, 1979; Crozier & Seary, 1980; Balerna & Mobilio, 1986). Large disorder can occur in a variety of situations but is parti- cularly likely to be important in temperature-dependent measurements where the temperature is high compared with the characteristic energy of an absorber±scatterer interaction. This frequently results in broad anharmonic pair-distribution functions that cannot be described by (2). As part of a study of dinuclear metal clusters, we have developed a computational approach to extract dynamic information from EXAFS in cases where anharmonicity is important. 2. Previous analyses 2.1. Temperature dependence Many approaches have been used to describe tempera- ture-dependent effects in EXAFS. For in®nite lattice systems, the temperature-dependent mean-square disorder  2 as  is often calculated from the phonon distribution. Two of the models commonly used to describe the distribution of phonon modes are the Einstein model (one character- istic phonon frequency, ! E ) and the correlated Debye model (a distribution of frequencies from 0 to ! D ) (Beni & Platzman, 1976; Sevillano et al. , 1979). These approaches