(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 Ri on the radial distribution function graph should be a Gaussian peak centered at Ri, 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 Ri, the nearby shells’ detectability by scattering is weakened in turn, and the peak intensity corresponding to Ri 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 Ri obtained from the RSF graph, we can get the single-shell xi(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.
2.2.3 X-ray Absorption Near-Edge Structure
The NEXAFS stands for near-edge X-ray absorption fine structure. Technically, the NEXAFS is a synonym for the XANES. In practice, the term NEXAFS is generally used only for low-energy edges, typically those below 1000 eV.
As discussed in the previous section, the formal theories of the XANES and EXAFS are essentially the same and both are given by the Fermi’s golden rule. When an effective single-particle description of the spectrum is reasonable, this leads to
(2.12)
Several of the approximations appropriate for the EXAFS regime (beyond about 20–30 eV above the edge) are not valid in the near-edge regime, with some of these related to the reduction of the many-body formulation to an effective single-particle description. For example, in the highly correlated systems such as transition metal oxides and f-electron systems, many-body effects can change the qualitative behavior of the near-edge spectrum while the main effect on the EXAFS region is simply an overall reduction in the amplitude of the fine structure, which is taken into account by the
Many-body effects can be traced down to the different energy of photoelectron. The photoelectron with a larger kinetic energy is less affected by the neighboring coordinating atom. Under normal circumstances, it is only scattered by the neighboring coordinating atom. However, if the kinetic energy of the photoelectron is very small, it will be scattered many times by an unknown neighboring coordinating atom scattering. This is the biggest difference between the simplified models of the EXAFS and XANES. Based on single scattering, the EXAFS can generally only give average structural information. The multiple scattering signal that occurs on the high-energy side of the XANES region records the superposition of the scattered waves when scattered by more than one neighbor atom. Therefore, it can reflect the three-dimensional coordination environment of the absorbing atom, combined with the relevant information of the transition, and provide strong evidence to judge the absorption atomic coordination geometry.
The EXAFS has limitations. At high temperature, taking in situ reaction conditions as an example, it is difficult to analyze the EXAFS under such conditions [108]. The XANES is highly sensitive to the local symmetry of the short-range order of absorbing atoms, and the short-range order of matter still exists at high temperatures. Therefore, the XANES is widely applicable. In principle, the XANES can distinguish mixed systems. The reason is that the characteristic of the XANES spectrum is fingerprint authentication, and a mixture of multiple systems can be distinguished.
Although the central atoms are completely different, the lines and shapes of oxides and fluorides with the same short-range order structure in the multiple scattering zone are the same. This has been confirmed by a large number of experimental spectra. This is to identify the coordination geometry of the central atom. At present, the identification of this part of the spectrum is mainly based on experience and comparison with the standard samples.
The EXAFS is also less sensitive to the nonspherical details of the potentials, and a simple overlapped atomic muffin tin potential is adequate for most practical calculations. On the other hand, near-edge spectra can be quite sensitive to the details of charge transfer and changes in Fermi level due to the solid-state effects. Thus, the use of self-consistent potentials and often nonspherical symmetry are essential for accurate calculations of the XANES. Finally, calculations of the single-particle Fermi golden rule must be treated differently in the near-edge region because the path expansion detailed in the equation often fails to converge (or converges very slowly) for low-energy photoelectrons. This slow convergence is caused by two factors. First, the inelastic mean free path becomes large for low energy electrons so that very long paths must be included in the expansion. Second, large angle scattering amplitudes are not small at low energies, so that the XANES signal is not dominated by the nearly linear scattering paths, and all multiple scattering paths must be considered.
2.2.4 Application in Amorphous Nanomaterial Characterization
For the study of the atomic local environment, XAFS is one of the most powerful tools for structural characterization. Because the X-ray absorption spectrum and the coordination structure around the atom have a fingerprint-like correspondence, it can accurately study the structural parameters such as the oxidation state, coordination relationship, bond length, and chaos of the atom to be measured. Of note, the experimental observation is in an atomic short-range scale, which does not reflect whether the sample structure has a long-range order or not. In the following section, we will present some research on amorphous structure characterization by using the XAFS.
Zhang et al. used electrocatalysts operando XAFS to identify the active sites in NiFe PBAs during the OER process [109]. They discovered that the NiFe-PBA decomposed and transferred to amorphous nickel hydroxide with Fe disappearing in the decomposition (as shown in