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Extended X-ray Absorption Fine Structure
Beyond the edge region, 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 constructive and destructive interference between the outgoing photoelectron wave and the portions of this wave backscattered off of neighboring atoms as illustrated in (d). The modulation of the interference condition with the change in X-ray energy results in oscillatory fine structure contributed by each neighboring atom and depends on the electron wavelength and the distance to the backscattering atoms.
The modulation in the extended X-ray absorption fine structure (EXAFS) is described by a single-scattering formulation that 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 absorberscatterer pair.
Mathematically, the scattering contributions can be summed over all of the shells and result in the composite EXAFS 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 EXAFS 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 spectra of structurally analogous standard compounds to very accurate ab initio calculations for 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 σ, 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 EXAFS as a superposition of sine waves dictates Fourier analysis, converting from χ(k) to χ(R) by Fourier transformation (e). Since this converts each wave into a peak, χ(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 (χ(R)), and usually just the modulus, is most often used in figures despite most of the analysis actually occurring in k space (χ(k)). The data are frequently shown as weighted spectra (χ3(k)) to (over)emphasize scattering contributions at longer distances.
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An X-ray absorption spectrum over a 1,000-eV region shows the rising absorption edge, the X-ray absorption near-edge structure (XANES) that occurs within the first 20 to 30 eV of the edge, and the EXAFS oscillations that occur past the edge.
When X-ray energy is absorbed by a central atom (blue), a photoelectron wave is propagated outward. Thiis outgoing wave is backscattered off neighboring atoms (red and green) and interferes with itself at the origin. 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).
EXAFS Fourier transform (black line) and fi t (blue line) of the fi rst three shells of PuO2. The individual shells that make up the fit are shown with negative amplitudes and correspond to eight oxygen atoms at 2.34 Å, 10 plutonium atoms at 3.83 Å, and 23 oxygen atoms at 4.53 Å. The experimental data can be compared with expectations based on the crystal structure shown in (f).
The cubic crystal structure of PuO2 can be viewed from the perspective of the central plutonium atom (black) to understand the results of EXAFS studies. The central plutonium atom is surrounded by a shell of eight near-neighbor oxygen atoms (red) with a Pu-O distance of 2.33 Å. Next comes a shell of 12 neighboring plutonium atoms (grey) at 3.81 Å, followed by a shell of 24 oxygen atoms (pink) at 4.47 Å. ese three shells of atoms correspond to the three shells shown in the EXAFS Fourier transform spectrum (e).
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