Xanes and the Determination of Bond Angles

It is argued that conventional weak scattering probes provide no direct information beyond the radial distribution function. In contrast, X-ray Absorption Near Edge Structure is produced by the multiple scattering of low energy electrons and has sensitivity to bond angles and symmetry of the environment 1. WEAK SCATTERING AND PAIR CORRELATIONS The traditional probe of structure is a wave that is weakly scattered by the material under consideration,such as X-rays or neutrons. Weak scattering has the great virtue of a simple interpretation. For example, a plane wave, incident on the solid is scattered to an outgoing wave, where A(g) = s fs(g) exp(iq.Rs), ( 3 ) and R is the position of the çth atom, withscatteriqfactor fS; ki and k ar8 the incident and scattered wave vectors respectively. Thë egperiment observes an expression in which the atomic coordinates occur in pairs, and only in pairs. Such an expression can only ever tell us about the pair correlation function in the material but in ordered, crystalline materials this information is usually sufficient to give us the higher order correlation functions by implication / 1 / 2 / . In the case of an orientated crystal measurement of IAl for a full set Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985911 C9-94 JOURNAL DE PHYSIQUE of 3D vectors, q, provides enough information to fix the higher order correlations. By contrast, in a liquid there is much less information available because the scattering is on average isotropie so that ( A ( ~ is a function only of (q(. Occasionally this provides enough information to determine al1 correlation functions, but in most instances this is not the case. In this sense disordered materials differ in a fundamental way £rom their ordered counterparts. Their structure is much more subtle, as indicated by their finite entropy content, and demands new probes to reveal its details to us. II. MULTIPLE SCATTERING A PROBLEM FOR THE THEORIST When scattering is strong, the probe can be expected to interact with more than one scattering centre in the solidland equation ( 3 ) must be generalised to ~ ~ , ( k ~ , & ~ i = C exp(iSf.Rsi fs(kf,ki) exp(ik. .R ) + -1 -S exp(ikf.Rs) fs Gst ft exp(iki.Rt) + exp(ikf.Rs) fs GsU fU Gut ft exp(iki.Rt) (plus higher order terms) (6 The new ingredients in this formula are (i) the scattering factors, f, are now generally complex, reflecting multiple scattering events 'within individual atoms. (ii) The scattered amplitude, A , is no longer a function of ( & -Li( aloneysbut depends on kf and k . independentl5. -1 (iii) Several atomic coordinates appear in a typical term through the propagators, G, which represent the wave travelling between atoms. This last point becomes more explicit if we take the case of point scatterers for which so that the overall dependence on atomic coordinates of the last term in equation (6) would be As the wave explores the solid, scattering from successive atoms, it measures the path length through the phase which it acquires. We might imagine that in principle we could find al1 possible path lengths for multiple scatterings in the medium by Fourier transforming the scattered intensities. This information could be interpreted in terms of pair correlations. However the program is not a practicable one, partly because the propagators, G, are rarely simple phase factors, and partly because the individual atomic scattering factors, f, introduce further complications which are difficult to correct for. The only practical way of proceeding that has been used to interpret multiple scattering data has been to postulate a trial structure, calculate the implied spectrum, compare with experiment, and then successively refine the structure. The passage from trial structure to calculated spectrum is non-trivial and is one of the limiting factors to interpretation of data which in principle contain the information we require. One multiple scattering probe that has been much used in structure determination is the low energy electron/3/. Traditionally low energy electron diffraction (LEED) has been used as a surface structure probe. Here the strong scattering interaction is required to give surface sensitivity, and the multiple scattering processes are regarded as something of a nuisance. LEED experiments are now routinely interpreted to give surface structures, and essentially the same theory can be used in another context to give higher order correlation functions. X-ray absorption near edge structure (XANES, or NEXAFS as some cal1 it) /1/4/ results £rom the ejection of an electron £rom an inner core of the absorbing atom. The ability of the electron to escape £rom the atom determines the absorption cross section and in the near edge region, where the electron has, Say, less than 50eV of kinetic energy, multiple scattering is dominant; in contrast to the region of extended X-ray absorption fine structure (EXAFS) /5/6/, more than 50 eV above the edge, in which cross sections are small enough for most multiple scattering processes to be neglected. Here we have a probe which can be tuned to the environment of a particular atom by selecting the appropriate absorption edge. The ejected electron having a finite elastic lifetime explores only the irnmediate surroundings of that atom before it vanishes in incoherent processes which in turn limits the number of multiple scattering events that we have to interpret. Fig. 1 The relative absorption coefficient for X-rays in the vicinity of the iron K edge. The division of the spectrum into XANES and EXAFS is somewhat arbitrary. K4Fe(CN)6.3H20 and K3Fe(CN)6. Ar: l