What is exafs?

Extended X-ray absorption fine structure (EXAFS), along with X-ray absorption near edge structure (XANES), is a subset of X-ray absorption spectroscopy (XAS). Like other absorption spectroscopies, XAS techniques follow Beer's law. The X-ray absorption coefficient of a material as a function of energy is obtained using X-rays of a narrow energy resolution are directed at a sample and the incident and transmitted x-ray intensity is recorded as the incident x-ray energy is incremented.

When the incident x-ray energy matches the binding energy of an electron of an atom within the sample, the number of x-rays absorbed by the sample increases dramatically, causing a drop in the transmitted x-ray intensity. This results in an absorption edge. Every element has a set of unique absorption edges corresponding to different binding energies of its electrons, giving XAS element selectivity. XAS spectra are most often collected at synchrotrons because of the high intensity of synchrotron X-ray sources allow the concentration of the absorbing element to reach as low as a few parts per million. Absorption would be undetectable if the source is too weak. Because X-rays are highly penetrating, XAS samples can be gases, solids or liquids.

EXAFS spectra are displayed as plots of the absorption coefficient of a given material versus energy, typically in a 500 – 1000 eV range beginning before an absorption edge of an element in the sample. The x-ray absorption coefficient is usually normalized to unit step height. This is done by regressing a line to the region before and after the absorption edge, subtracting the pre-edge line from the entire data set and dividing by the absorption step height, which is determined by the difference between the pre-edge and post-edge lines at the value of E0 (on the absorption edge).

The normalized absorption spectra are often called XANES spectra. These spectra can be used to determine the average oxidation state of the element in the sample. The XANES spectra are also sensitive to the coordination environment of the absorbing atom in the sample. Finger printing methods have been used to match the XANES spectra of an unknown sample to those of known "standards". Linear combination fitting of several different standard spectra can give an estimate to the amount of each of the known standard spectra within an unknown sample.

X-ray absorption spectra are produced over the range of 200 – 35,000 eV. The dominant physical process is one where the absorbed photon ejects a core photoelectron from the absorbing atom, leaving behind a core hole. The atom with the core hole is now excited. The ejected photoelectron's energy will be equal to that of the absorbed photon minus the binding energy of the initial core state. The ejected photoelectron interacts with electrons in the surrounding non-excited atoms.

If the ejected photoelectron is taken to have a wave-like nature and the surrounding atoms are described as point scatterers, it is possible to imagine the backscattered electron waves interfering with the forward-propagating waves. The resulting interference pattern shows up as a modulation of the measured absorption coefficient, thereby causing the oscillation in the EXAFS spectra. A simplified plane-wave single-scattering theory has been used for interpretation of EXAFS spectra for many years, although modern methods (like FEFF, GNXAS) have shown that curved-wave corrections and multiple-scattering effects can not be neglected. The photelectron scattering amplitude in the low energy range (5-200 eV) of the photoelectron kinetic energy become much larger so that multiple scattering events become dominant in the XANES (or NEXAFS) spectra.

The wavelength of the photoelectron is dependent on the energy and phase of the backscattered wave which exists at the central atom. The wavelength changes as a function of the energy of the incoming photon. The phase and amplitude of the backscattered wave are dependent on the type of atom doing the backscattering and the distance of the backscattering atom from the central atom. The dependence of the scattering on atomic species makes it possible to obtain information pertaining to the chemical coordination environment of the original absorbing (centrally excited) atom by analyzing these EXAFS data.

Since EXAFS requires a tunable x-ray source, data are frequently collected at synchrotrons, often at beamlines which are especially optimized for the purpose. The utility of a particular synchrotron to study a particular solid depends on the brightness of the x-ray flux at the absorption edges of the relevant elements.

XAS is an interdisciplinary technique and its unique properties, as compared to x-ray diffraction, have been exploited for understanding the details of local structure in:

XAS provides complementary to diffraction information on peculiarities of local structural and thermal disorder in crystalline and multi-component materials.

The use of atomistic simulations such as molecular dynamics or the reverse Monte Carlo method can help in extracting more reliable and richer structural information.

EXAFS is, like XANES, a highly sensitive technique with elemental specificity. As such, EXAFS is an extremely useful way to determine the chemical state of practically important species which occur in very low abundance or concentration. Frequent use of EXAFS occurs in environmental chemistry, where scientists try to understand the propagation of pollutants through an ecosystem. EXAFS can be used along with accelerator mass spectrometry in forensic examinations, particularly in nuclear non-proliferation applications.

A very detailed, balanced and informative account about the history of EXAFS (originally called Kossel's structures) is given by R. Stumm von Bordwehr.[1] A more modern and accurate account of the history of XAFS (EXAFS and XANES) is given by the leader of the group that developed the modern version of EXAFS in an award lecture by Edward A. Stern.[2]

To represent the EXAFS region from the whole spectrum, a function χ can be roughly defined in terms of the absorption coefficient, as a function of energy:

\[\chi (E)=\frac{\mu (E)-\mu _{o}(E)}{\Delta \mu _{o}}\]

Here, the subtracted term represents removal of the background and the division by Δμo represents the normalization of the function, for which the normalization function is approximated to be the sudden increase in the absorption coefficient at the edge. When interpreting data for the EXAFS, it is general practice to use the photoelectron wave vector, k, which is an independent variable that is proportional to momentum rather than energy.

We can solve for k by first assuming that the photon energy E will be greater than E0 (the initial X-ray absorption energy at the edge). Since energy is conserved, excess energy given by E - E0 is conserved by being converted into the kinetic energy of the photoelectron wave. Since wavelengths are dependent on kinetic energies, the photoelectron wave (de Broglie wavelength) will propagate through the EXAFS region with a velocity of ν where the wavelength of the photoelectron will be scanned. This gives the relation, (E - E0) = meν2/2. One of the identities for the de Broglie wavelength is that it is inversely proportional to the photoelectrons momentum (meν): λ=h/meν. Using simple algebraic manipulations, we are able to obtain the following:

\[ k = \dfrac{2\pi}{\lambda} = \dfrac{2\pi m_{e}\nu}{h} = \left[\dfrac{8\pi^2m_{e}(E-E_o)}{h^2}\right]^{\frac{1}{2}} \]

To amplify the oscillations graphically, k is often plotted as k3

Now that we have an expression for k, we begin to develop the EXAFS equation by using Fermi's Golden Rule. This rule states that the absorption coefficient is proportional to the square of the transition moment integral, or ||2 , where i is the unaffected core energy level before it interferes with the neighboring atoms, H is the interaction, and f is the final state in which the core energy level has been affected and a photoelectron has been ejected. This integral can be related to the total wavefunction Ψk, which represents the sum of all interacting waves, from the backscattering atoms and the absorbing atom. The integral is proportional to the total wavefunction squared, which refers to the probability that the photoelectron is found at the atom where the photon is absorbed, as a function of radius. This wavefunction describes the constructive/destructive nature of the wave interactions within it, and varies depending on the phase difference of the waves interacting. This phase difference can be easily expressed in terms of our photoelectron wave number and R, the distance to the innershell from the wave, as 2k/R. In addition, another characteristic of this wave interaction is the amplitude of waves backscattering to the central atom. This amplitude is can provide coordination number as well since it is thought to be directly proportional to the number of scatters.

The physical description of all these properties is given in a final function for χ(k), called the EXAFS equation:

Looking at the qualitative relationships between the contributions to this equation gives an understanding of how certain factors can be extracted from this equation. In this equation, f(k) represents the amplitude and δ(k) represents the phase shift. Since these two parameters can be calculated for a certain k value, R, N, and σ are our remaining unknowns. These unknown values represent the information we can obtain from this equation: radius, coordination number(number of scattering atoms), and the measure of disorder in neighboring atoms, respectively. These are all also properties of the scattering atom.

We can also see the terms in this equation represented in a typical EXAFS spectrum: the sine term gives the origin of the sinusoid shape, with a greater phase shift making the oscillations greater. In addition, this oscillation depends on the energy and on the radial distance. The Debye-Waller Factor explains the decay with increasing energy, as well as increasing disorder. This factor is partially due to thermal effects. In addition, we deduce the reason why EXAFS does not work over long distances (up to 4-5 Å): the term Rj-2 causes the expression to decrease exponentially over large values of Rj (larger distances between absorbing and scattering atoms), making EXAFS much weaker over long-distances as opposed to short ranged neighboring atoms.