How XAFS works

X-ray absorption fine structure (XAFS) spectroscopy is one of the most powerful tools we have for mapping local structure. In this technique, we probe a sample with x-rays that are tuned to the energy of a core electron shell in the element we wish to study. We monitor how many x-rays are absorbed as a function of their energy. If taken with sufficient accuracy, the spectrum exhibits small oscillations that are the result of the local environment's influence on the target element's basic absorption probability. From the spectrum, we can extract the distances between the absorber and its near-neighbor atoms, the number and type of those atoms, and the oxidation state of the absorbing element-all parameters that determine local structure. By selecting a different x-ray energy, we can obtain this information for any element in the sample.

An example of the absorption spectrum in the x-ray region is shown in the figure at right. The steplike rises occur where the x-ray energy has come into resonance with a core electron shell of one of the elements in the sample (plutonium in this example), exciting the electron into the continuum. Because of the shape of the spectral feature, the data are referred to as an absorption edge. For the most part the edges are widely separated and the target element is selected simply by scanning over an appropriate energy range. Following the edge, the absorbance decreases monotonically with increasing x-ray energy as the x-ray penetration depth becomes larger.

The x-ray absorption near-edge structure (XANES) shows a large peak, beyond which the x-ray absorption fine structure (XAFS) oscillations are readily observable against the spline polynomial (blue) that is used to approximate the smooth, atomic-like absorption above the edge.

Fine structure is observed when the spectrum is expanded past a specific edge. The x-ray absorption near-edge structure (XANES) region occurs as peaks and shoulders over a 20 to 30 electronvolt-wide region immediately past the edge onset. The fine structure on the high-energy side of the edge that damps out over several hundred electronvolts is termed x-ray absorption fine structure (XAFS). This fine structure in both the XANES and the XAFS regions is well understood and enables XAFS to be applied to the determination of chemical speciation and local structure.

Beyond the edge region the XAFS fine structure occurs as a series of oscillations superimposed upon what would be the smooth absorbance of the isolated atom. The origin of this fine structure arises from interference between the outgoing photoelectron wave and the portions of this wave backscattered off of neighboring atoms. The modulation of the interference condition with the change in x-ray energy results in oscillatory fine structure contributed by each neighboring atom.

The modulation in the XAFS is described by a single-scattering formulation that for the chemist contains several structurally significant metrical parameters. These parameters (noninclusive) include the number of neighbor atoms of the same atomic number at the same distance from the absorbing atom (i.e., a shell of neighbor atoms); a Z-dependent, per-atom backscattering amplitude function for that shell; the pair-wise Debye-Waller factor; and a phase-shift characteristic of the particular absorber-scatterer pair.

When x-ray energy E is absorbed by a central atom (blue), a photoelectron wave is propagated outward. This outgoing wave is backscattered off neighboring atoms (red and green) and interferes with itself at the origin. Left: Because of the ratio of the absorber-neighbor distance to the photoelectron wavelength, it is interfering constructively with respect to the first shell (red) and destructively with the second shell (green). Right: When x-ray energy E+ delta is absorbed by a central atom (blue), the photoelectron wavelength is changed and the interference conditions are altered. As the x-ray energy is scanned, the oscillation interference conditions result in oscillation in the absorption cross section, X(k).

Mathematically, the scattering contributions can be summed over all of the shells and result in the composite XAFS spectrum. These metrical parameters are extracted from the data via nonlinear least-squares curve fits. A model consisting of a number of neighboring shells of atoms (supplemented by multiple scattering paths when necessary) is first devised. The XAFS is the sum of the individual waves from these shells, calculated from the single-scattering equation. The source of the requisite phases and amplitudes for the fitting has evolved from the spectra of structurally analogous standard compounds to tabulated ones to very accurate ab initio calculations for arbitrarily arranged clusters of atoms that can be quite close to the final structure.

Phase shifts and amplitudes are unique to the different elements, with enough difference with increasing Z to allow the type of element to be identified to ± 3-4 in principle. The structural parameters, e.g., R, N, and sometimes , are allowed to float until the least-squares difference between the data and the fit are minimized. Additional chemical information such as relationships between various parameters or shells or a permissible range for a parameter are introduced as constraints.

The view of XAFS as a superposition of sine waves dictates Fourier analysis, converting from X(k) to X(R) by Fourier transformation. Since this converts each wave into a peak, X(R) is related to the population-weighted average radial structure function around the absorber and the modulus does suggest a pair distribution around the absorber. For this reason, the Fourier transform representation (X(R)), and usually just the modulus, is most often used in figures despite most of the analysis actually occurring in k space (X(k)). The data are frequently shown as weighted spectra (X³(k)) to (over)emphasize scattering contributions at longer distances.

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