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To clearly elucidate with the term XAFS, we can start with XAS. When X-ray passes through a sample thickness of d, its intensity I₀ will decay to I because of the absorption of the sample, so the X-ray absorption coefficient of the sample, μE, can be defined as:

(2.1)

The X-ray absorption spectrum is to measure the change curve of X-ray absorption coefficient with X-ray energy. After the absorption edge, there will be a series of oscillations. This kind of small structure is generally a few percent of the absorption cross section, that is, XAFS.

The XAS encompasses both the EXAFS and XANES, where the terms refer, respectively, to the structure in the X-ray absorption spectrum at high and low energies relative to the absorption edge with the crossover typically at about 20–30 eV above the edge [104].

Formally, the base theory of the XAFS is Fermi’s golden rule (Figure 2.8) [105]:

(2.2)

However, it requires a summation over the exact many-body final states |F>| and its energy E_F. The normalized fine structure in the XAS in terms of the oscillatory contributions from near-neighbor atoms can be described as:

(2.3)

where μ₀(E) is the absorption coefficient in the case of isolated atoms, without considering the smooth absorption background caused by scattering, and μ(E) is the experimentally measured absorption coefficient with neighboring atoms.

Figure 2.8 The spectra and origin of XANES and EXAFS.

Incorporating many-body effects can follow the two-step approach. The first step is the production of the photoelectron, by photoexcitation from a certain core state, with one-body absorption μ⁽¹⁾(E). The second is the effect of the inelastic losses and secondary excitations, which can be represented by an energy-dependent “spectral function” A(E, E′), which subsequently broadens and shifts the spectrum. Incorporation yields an exact representation of the many-body XAS in terms of a convolution

(2.4)

In the theoretical description, the EXAFS can be expressed as the sine function of the photoelectron wave vector. We can use 2.4 to convert photoelectron energy E (eV) to wave vector .

(2.5)

The EXAFS is caused by the single scattering of the emitted electromagnetic wave by the neighboring atoms. In other words, if there is no neighbor atom, the EXAFS of the isolated atom will only approximate a straight line. Because of the neighboring atoms, the outgoing photoelectron wave is blocked by the neighboring atoms to scatter, and the scattering wave interacts with the original outgoing wave – interference. The change reflected in the absorption coefficient is the oscillating structure superimposed on the smooth background, which is the EXAFS [106].

It is generally believed that the three major contributions to the EXAFS application are the introduction of Fourier transform (FT), synchrotron radiation, and the progress in theoretical parameters calculation. The FT is a mathematical and physical method that decomposes a complex wave into the sum of sine waves of different frequencies. The EXAFS is an oscillating structure superimposed on the smooth background μ₀. In the early 1970s, Sayers et al. creatively proposed that the oscillating structure of EXAFS is due to the scattering of photoelectrons from the center by the neighboring shells, which is a superposition of the multi-shell sine waves [107]:

(2.6)

Because it is a superposition of sine waves, they proposed that χ(k) can be decomposed by FT to obtain an independent sine wave function x_i(k) for each shell. In addition, the relevant structural information can be solved; thus, they create a precedent for the EXAFS to determine the structure of matter. Remarkably, based on this principle, they came up with a widely accepted expression:

(2.7)

In fact, this formula is the product of the amplitude and the sine function. In other words, the expression of the EXAFS oscillation of a single shell can be written as:

(2.8)

This is the theoretical description in the form of sine waves, where ϕ_i(k) is the phase shift and is the amplitude, usually expressed as the product of the amplitude function and a series of correction terms.

(2.9)

Where N_i is the near-neighbor coordination number of the ith shell, R_i is the shell spacing, F_i(k) is the scattering amplitude, λ is the mean free path, exp(−2R_i/λ) is the attenuation caused by photoelectrons on amplitude, σ² is the Debye–Waller factor, is the amplitude attenuation caused by thermal vibration, and is the attenuation factor in the fitting technique.

The amplitude function F_i(k) in the above formula can be considered as a known term, the same as the optoelectronic attenuation term exp(−2R_i/λ) and . These factor needs to be calibrated according to the standard sample. In addition, Sayers et al. also proposed to perform FT on the EXAFS oscillation to obtain the radial structure function (RSF) so as to obtain the single-shell information:

(2.10)

The concept of the central absorbing atoms is important for the XAFS. It changes the focus of people’s habit of understanding the structure of matter. The central absorbing atom is relative to the neighboring atoms, all of which counting into particles. In a three-dimensional particle system, the neighboring particles of any particle can be found from the shell with a volume of 4πR³dR. The formula is:

(2.11)

where R is the shell density, N is the particle density, and P(R) is the radial distribution function.

Because of the thermal motion of the particles, what we observe at R_i on the radial distribution function graph should be a Gaussian peak centered at R_i, and its peak area is the coordination number N. In the EXAFS, limited by the mean free path of the emitted photoelectrons, as R increases from R_i, the nearby shells’ detectability by scattering is weakened in turn, and the peak intensity corresponding to R_i in the distribution function shares the same pattern. What one actually gets is the RSF ρ(R). Its physical meaning is like the P(R). It should be noted that because of the scattering phase shift, the R value observed on the RSF ρ(R) is slightly smaller than its true value.

Using the Fourier filtering on the Gaussian peak with the center of R_i obtained from the RSF graph, we can get the single-shell x_i(k), which can be substituted into Eq. (2.7). At last, we can acquire the coordination number (N), shell distance (R), and Debye–Waller factor (σ²).

We have already seen what information about the structure of the material the EXAFS can give us, including the coordination number (N), shell distance (R), and Debye–Waller factor (σ²). This structural information plays an important role in clarifying the microscopic composition of matter. In addition, these structural factors we obtained are a short-range order state of the internal particle arrangement, which can be used not only for crystals but also for amorphous materials. The EXAFS is a strong and powerful tool to investigate amorphous materials.