International
Tables for Crystallography Volume I Xray absorption spectroscopy and related techniques Edited by C. T. Chantler, F. Boscherini and B. Bunker © International Union of Crystallography 2024 
International Tables for Crystallography (2024). Vol. I. ch. 5.4, pp. 645652
https://doi.org/10.1107/S1574870721010922 Multiplescattering EXAFS analysis^{a}Department of Materials Science and Chemical Engineering, Stony Brook University, 100 Nicolls Road, Stony Brook, NY 11794, USA Advances in multiplescattering theory play a key role in the ongoing progress of analysis and modelling methods of extended Xray absorption fine structure (EXAFS). This chapter will focus on the history and development of the multiplescattering theory, introduce dataanalysis and modelling strategies that employ multiplescattering contributions to EXAFS, and discuss their applications to a variety of systems. Keywords: multiple scattering; EXAFS; FEFF. 
Singlescattering analysis of extended Xray absorption finestructure (EXAFS) spectroscopy limited researchers to the first and, rarely, the second nearestneighbouring shells of atomic species around the Xray absorbing atoms. The inability to `peek' beyond the first shell is not much of a limitation for many systems, notably those that are strongly disordered and/or of low dimensionality, where only the first peak in rspace can be reliably analyzed. In the case of relatively ordered materials, in particular those possessing relatively open lattices [for example f.c.c. metals (Alayoglu et al., 2009; Frenkel, 1999; Rehr & Albers, 1990), alkali halide salts (Frenkel et al., 1993, 1995) and some perovskites with formula BX_{3} (Balerna et al., 1991; Kuzmin & Purans, 1993; Kuzmin et al., 1993)], strong multiplescattering contributions to EXAFS account for a large portion of the spectrum (Lee & Pendry, 1975; Rehr & Albers, 1990) and are comparable with singlescattering contributions. Therefore, multiplescattering contributions have to be included in EXAFS calculations and analysis for reliable structural refinement beyond the first nearestneighbouring shell whenever the experimental data quality allows such analysis. The role of multiplescattering events has been the subject of extensive and longterm discussions, stimulating the development of theories and dataanalysis techniques,
The multiplescattering effect occurs when the photoelectron wave is scattered more than once by surrounding atoms before returning to the Xray absorbing atom. Multiplescattering (MS) paths differ in the number of legs and in the type of scattering geometry. For example, singlescattering paths have two legs. Doublescattering paths (for example MS1 and MS4–MS6 in Fig. 1) have three legs, and triplescattering paths (MS2 and MS3 in Fig. 1) have four legs. Multiplescattering paths are very important in the Xray absorption nearedge structure (XANES) region, because in this region a photoelectron with low kinetic energy has a long mean free path that permits the contribution of extensive multiplescattering events to the XANES structure (PennerHahn, 2001). Multiplescattering paths can be extraordinarily important when the atoms are present in a linear or nearly linear arrangement. We consider a threeatom (ABC) system (Teo, 1981; Teo & Lee, 1979) where a photoelectron is emitted from atom A and is scattered first by atom B and then by atom C before returning to atom A, as seen in Fig. 2 using oxygen as an example. The calculated scattering amplitude shows a dependence on the value of the angle β and the wavenumber k. For all k values, the scattering amplitude at atom B reaches a maximum of F ≃ 1.7–1.8 at β = 0°, while F ≃ 1 at β ≃ 30°, which corresponds to focused (or forwardscattering) paths (MS1, MS2 and MS3 in Fig. 1) and nearfocused paths that contribute significantly to the EXAFS.

Representative singlescattering (SS) and multiplescattering (MS) paths. Adapted with permission from Frenkel et al. (2001). Copyright (2001) American Chemical Society. 
This chapter will address the basics of the EXAFS theory behind the multiplescattering contributions, introduce strategies for their quantitative analysis, offer ideas for modelling of local structure based on the multiplescattering contributions, and present examples and applications.
The multiplescattering theory was rapidly developed from the early 1970s and many efforts have been made to solve this problem using different approaches (Ashley & Doniach, 1975; Lee & Pendry, 1975; Schaich, 1973). In the work of Lee and Pendry, the formalism of singlescattering and multiplescattering paths was developed using a wavefunction approach. They also demonstrated that the Green's function method (Ashley & Doniach, 1975; Schaich, 1973) yields identical results. The major convenience of the multiplescattering theory for dataanalysis applications is that the multiplescattering paths are classified by the total path lengths (or, for better compatibility with the realspace distances, by the half path lengths). Hence, they contribute to different spectral regions in rspace, depending on their half path length, just like the singlescattering paths, and thus can be interpreted and analyzed using the same Fourier transform methods. The multiplescattering expansion in terms of the oneelectron Green's function in real space, known as the realspace multiplescattering (RSMS) approach, significantly simplified the problem by avoiding the explicit calculation of final states and eigenvalues (Rehr & Albers, 2000; Rehr et al., 2002). A great research effort has been dedicated to improving the realspace fullpotential multiplescattering theory (Natoli et al., 1986, 2012). Other significant improvements in the theoretical calculation of EXAFS (Rehr & Albers, 2000) include the spherical wave approximation for scattering terms and the effective treatment of intrinsic losses, manybody and exchange correlations that are used to more accurately describe EXAFS. More recently, the Rehr–Albers approach, which is based on separable representation of Green's function propagators (Rehr & Albers, 1990), was developed to overcome the computational difficulties in the multiplescattering expansion. With these advances, the multiplescattering formulation of EXAFS can now be regarded as a fairly well understood problem.
The XAFS theory generally starts from Fermi's golden rule, where the contribution to the Xray absorption coefficient from a core state is related to the transition rate of the electrons between the initial and final states,ψ_{i} and ψ_{f} represent the wavefunctions of the excited electron for the initial and final eigenstates of the effective oneelectron Hamiltonians, with energies E_{i} and E_{f}, respectively. ψ_{f} is calculated using the finalstate oneparticle Hamiltonian (`finalstate rule'). The finalstate oneparticle Hamiltonian consists of the kinetic energy of photoelectrons, the Coulomb potential () and the photoelectron selfenergy [Σ(E)]. The muffintin approximation used in EXAFS calculations refers to a spherical scattering potential centred on each atom and having a constant value in the interstitial region between atoms. With the separation of the potential to local scattering potentials from each atom, the muffintin approximation gives , with a finite range (defined by the muffintin radius) of scattering potentials. It has been demonstrated that the muffintin approximation is adequate for the theoretical simulation of EXAFS and for the analysis of experimental data. p is the momentum operator of the electron. A is the vector potential of the incident electromagnetic field and
Employing the dipole approximation [exp(iκ · r) ≅ 1], equation (1) becomesConsidering real space, G(r, r′; E) has the spectral representationwhere Γ is the corehole lifetime, and when Γ → 0^{+}Finally, the Xray absorption coefficient is rewritten aswhere E = E_{i} + ℏω and θ_{Γ}(E − E_{F}) is a broadened step function at the Fermi energy with a nonzero value only when E > E_{F}.
The Green's function can be separated into intraatomic contributions from the central (absorbing) atom G^{c} and multiplescattering contributions from the environment G^{sc}, so that G = G^{c} + G^{sc}. G^{sc} can be expressed as a sum over all multiplescattering paths. Using multiplescattering expansion in terms of the freeparticle Green's function G_{0} and scattering T matrix, this yieldsIn equation (6), G_{0} is a spherical wave, for which it is known that the longer scattering paths oscillate with higher frequency but have smaller amplitude than the shorter paths. Tmatrix elements t_{Li,L′i′} = δ_{i,i′}δ_{L,L′}t_{l} include all repeated scatterings within a given atomic cell. The index i denotes the atomic site in the cluster and l is the orbital angular momentum index.
An important advance in multiplescattering theory is the approach developed by Rehr & Albers (1990, 2000) that is based on a rapidly convergent separable representation of the electron propagator, which permits fast, accurate calculations of any multiplescattering path.
For an Nleg path that can be either single or multiple scattering, the XAFS amplitude is expressed asM_{l} is a termination matrix for the final state of angular momentum, F^{i} is the scattering matrix at site i, ρ_{i} = p(R_{i} − R_{i−1}), is the photoelectron momentum measured with respect to the muffintin zero (in Rydberg atomic units), is a manybody reduction factor and is the meansquare variation in total path length R_{total}.
Finally, the standard EXAFS equation, originally proposed by Sayers, Stern and Lytle (Sayers et al., 1971), can be recast using the effective scattering amplitude f_{eff}, In this equation, k is the photoelectron wavenumber and f_{eff} is the effective scattering amplitude. R is defined as the effective path length and equals half the total path length (R_{total}/2) instead of the interatomic distance in singlescattering theory. N is the degeneracy of the scattering paths. σ^{2} is known as the EXAFS Debye–Waller factor that reduces the intensity of EXAFS oscillations at high k as a consequence of disorder in interatomic distances. It is the meansquare deviation of the effective path length (R_{total}/2), which includes effects due to thermal variations and possible structural disorder. Φ_{k} is a phase function that takes into account the varying potential field along which the photoelectron moves. λ_{k} is the energydependent XAFS mean free path. The amplitudereduction factor describes the intrinsic losses upon excitation which arise due to manybody effects in the photoabsorption process.
In recent decades, there have been a number of major developments in the theoretical calculation of XANES and EXAFS, in part due to advances in multiplescattering theory. The FEFF code (FEFF3 through FEFF9; Rehr & Albers, 2000; Rehr et al., 1991, 2009; Kas et al., 2024), named after the effective scattering amplitude f_{eff}, is one of the most widely used theoretical calculation packages. In FEFF, the multiplescattering calculation is based on the separable representation of the photoelectron propagator that sums over scattering paths with no conceptual distinction between single and multiplescattering paths. The utilization of the efficient multiplescattering path filters and a fast pathgeneration and sorting algorithm (Zabinsky et al., 1995) in the FEFF program that eliminates paths with negligible contribution solves the pathproliferation problem.
The FEFF approach is based on the realspace multiplescattering (RSMS) method within the quasiparticle picture. The RSMS approach has been widely used in calculation of the XANES and EXAFS region. The earliest version of FEFF (Rehr et al., 1986, 1991) only included ab initio singlescattering XAFS calculations and was subsequently improved to involve multiplescattering effects (Rehr & Albers, 1990; Rehr et al., 1991). The latest version 9 (Ahmed et al., 2012; Rehr et al., 2009, 2010; Kas et al., 2024) features efficient ab initio models including a manypole model of the selfenergy, inelastic losses and multipleelectron excitations, a linear response approach for the core hole and a Lanczos approach for Debye–Waller effects, and yields improved calculations of both EXAFS and XANES. It is also able to calculate EELS and NRIXS spectra, which were not available in previous versions.
EXAFS data analysis can be performed by a number of software packages. Here, we use FEFF and IFEFFIT as examples to illustrate the analysis procedure. To generate the path list with the evaluated path parameters as implemented in FEFF, the procedure starts by constructing a model of atomic positions in three dimensions (Bunker, 2010; Newville et al., 1995). The scattering paths are then constructed. The number of paths is restricted by the `path filters', so that only multiplescattering paths with an amplitude larger than a given cutoff value are retained. The physically equivalent paths are sorted in order of the increased halfpath length and are filtered. For each path, the values of the parameters (referred to in the standard EXAFS equation) effective scattering amplitude f_{eff}, mean free path λ_{k} and phase shift are obtained from the FEFF calculation for each value of the kgrid. The changes in energy origin ΔE_{0}, , ΔR, σ^{2} and the third and fourth cumulants of the effective pair distribution function are available for analysis as adjustable (or fixed) parameters for each path to be used in the fitting of experimental data. FEFF calculations face a complication when dealing with some statistically or thermally disordered systems with many nonequivalent atomic sites that make it impractical to include distant multiplescattering paths due to the large number of fitting parameters. Thus, an alternative `direct modelling' approach, in which EXAFS spectra are simulated on the basis of moleculardynamics simulations, has been utilized to overcome this limitation (D'Angelo et al., 2002; Kuzmin & Chaboy, 2014; Vila et al., 2008).
Following the FEFF calculation, the output of the FEFF program can be incorporated into an independent dataanalysis program (i.e. IFEFFIT) for data fitting. The experimental data first have to be converted from the measured μ(E) to χ(k) by normalization and background subtraction. The experimental χ(k) in the range of interest is then Fourier transformed in a selected krange window, so that information outside the range of interest can be ignored. In this way, the frequency of the oscillations in EXAFS can be quantitatively related to the distances between the absorbing atom and the atoms within a given coordination shell around it, as shown in Fig. 3. A weighting factor (either k, k^{2} or k^{3}) is applied to emphasize (or deemphasize) a particular part of the krange in the data. The Levenberg–Marquardt nonlinear leastsquares algorithm is employed in most of the commonly used programs to fit the theoretical EXAFS spectrum to the experimental data.
As mentioned in Section 1, amongst the various types of multiplescattering paths, the forwardscattering (also known as focusing or shadowing) paths are particularly important in EXAFS data analysis. Forward scattering refers to scattering along a collinear pathway in the forward direction (with zero scattering angle). In fact, the effective path length R = R_{path}/2 for a forwardscattering path is the same as for the corresponding singlescattering path connecting the end atoms of the linkage. The EXAFS Debye–Waller factor is defined as the meansquare deviation of the halflength of the photoelectron path from the average. It accounts for the radial disorder, but is much less sensitive to variations in the scattering angle (Kuzmin & Chaboy, 2014; Kuzmin et al., 1993). In materials with longrange periodicity, it can be related to the local displacements u_{i} from the average lattice sites. For any scattering path, the EXAFS Debye–Waller factor can be expressed as (Frenkel, 2015; Shanthakumar et al., 2006)where, following the notation of Poiarkova & Rehr (1999), each leg of the path connecting instantaneous atomic positions is given by the vector r_{ii+} = R_{ii+} + u_{i+} + u_{i}, as shown in Fig. 4. i+ indicates the nextneighbour atom to i in the direction of the path and R_{ii+} and u_{i} correspond to the average leg vector and atomic displacement vector, respectively.
For a forwardscattering path, the corresponding Debye–Waller factor is given by Thus, the path length and Debye–Waller factor of the forwardscattering path in fitting EXAFS can be defined as the same value as the corresponding singlescattering path. In addition, it is also observed that in a highly symmetrical structure the path degeneracy changes the importance of paths even for the weaker nonlinear scattering. As seen in the case of metallic copper (Rehr & Albers, 1990), triangular paths with large degeneracy are also important and can be comparable in importance to the forwardscattering paths. In some cases of highly disordered systems, such as asymmetrically distorted nanoparticles, equation (10) is no longer valid (Frenkel, 2015). To solve this problem, direct modelling approaches based on densityfunctional theory and molecular dynamics have been developed (D'Angelo et al., 2002; Price et al., 2012; Tse, 2002; Vila et al., 2008; Yancey et al., 2013).
In order to evaluate the quality of fitting results, the R factor and reduced χ^{2} () are often used. The R factor is an indicator of the percentage misfit to the data. It has to be interpreted carefully by checking whether the fitted parameters are physically meaningful. The reduced χ^{2} is given by . Here, where [χ_{data}(k_{i}) − χ_{model}(k_{i})] is the difference between the experimental data and calculation at each point i, and ɛ_{i} is the rootmeansquare uncertainty of experimental data χ_{data}(k_{i}). The number of degrees of freedom in the fit ν = N_{idp} − N_{var}, where N_{var} is the number of variables evaluated in the fit. The number of relevant independent data points is N_{idp} ≃ (2/π)ΔRΔk [ΔR is the fitting range in Rspace and Δk is the krange of the experimental χ(k)]. The reduced χ^{2} allows a comparison of the quality between different fits for the same data because the value decreases when the fitting improves.
Alternative methods to the pathbypath approach adopted by FEFF have also been developed, for example EXCURVE (Gurman et al., 1984, 1986; Feiters et al., 2024) and GNXAS (Filipponi et al., 1995, 2024; Westre et al., 1995). The name GNXAS is derived from g_{n} (the nbody distribution function) and Xray absorption spectroscopy (Filipponi et al., 1995; Westre et al., 1995). EXCURVE (exact, curvedwave approach) was the first code for EXAFS analysis, and has more recently been updated to the DL_EXCURV package. It comes with a ligand database and is well suited to users from the biological community. The GNXAS approach performs an `nbody decomposition' of the Green's function. All of the multiplescattering contributions to the absorption cross section σ for a given n atoms under consideration are accounted for. The EXAFS signal is expanded in terms of the irreducible nbody signals γ^{(n)} that are directly calculated using a muffintin potential and advanced models for the energydependent exchange–correlation selfenergy. For a twoatom system, the γ^{2} function includes the single, triple and all successive odd orders of scattering contributions between the absorber and scatterer, which are equivalent to the sum of all of the filtered paths calculated by FEFF. Practically, the lower order (n ≤ 4) γ^{(n)} within the first few coordination shells are usually accounted for. EXAFS data are then fitted using the calculated γ^{(n)} modified by the parameters of the coordination environment. This expansion is found to have a better convergence rate than the MS series because each γ^{(n)} signal accounts for an infinite number of MS terms.
EXAFS data analysis using multiplescattering theory has been widely applied for the structural characterization of various material systems in order to solve many different problems such as the size and shape of nanoparticles (Araujo et al., 2008; Frenkel, 1999; Witkowska et al., 2007), buckling angles of mixed salts (Frenkel et al., 1993, 1994, 1995), the local structure of highT_{c} superconductors (Han et al., 2002; Haskel et al., 2000, 2001; Sahiner et al., 1999), structural phase transitions in perovskites (HanskePetitpierre et al., 1991; Ravel et al., 1998; Rechav et al., 1993; Shanthakumar et al., 2006), structural disorder of oxide cathode materials (Giorgetti et al., 2006; Greco et al., 2014) and determination of the geometry of proteins (Chen et al., 2003; Immoos et al., 2005; Rich et al., 1998).
The work on resolving the structures (size and shape) of a number of platinum and platinumbased bimetallic nanoparticles by including the multiplescattering contributions in EXAFS data analysis is a good example of the application of multiplescattering analysis of EXAFS to nanoparticles. One study of carbonsupported Pt and PtRu nanoparticles (Frenkel, 1999) demonstrated that it is possible to analyze the structure (shape and size) of carbonsupported Pt and heteroatomic PtRu nanoparticles by taking multiplescattering paths into account in EXAFS analysis. The reliability of size and shape determination from EXAFS analysis increases if the coordination numbers are obtained from higher shells. This study shows that a common strategy for analyzing monometallic and bimetallic nanoparticles involves first obtaining the amplitudereduction factor from a fit to a bulk standard (for example metal foil) and then fixing it in subsequent fits to the nanoparticle data to obtain coordination numbers. In this paper, the Pt foil data were well fitted through 6 Å in Rspace by adding the first five shells of single scattering and three nonlinear multiplescattering paths in fitting. For the Pt nanoparticles, the multiplescattering analysis allows the measurements of coordination numbers within the first four singlescattering paths and one nonlinear multiplescattering path. As a test, the Pt nanoparticles were proved to have the shape of a hemispherical cuboctabedron with a size of 15–20 Å. The latter analysis of carbonsupported PtRu nanoparticles is performed by including four shells of single scattering and the most dominant collinear double and triplescattering paths. It also constrains the heterometallic bonds to have the same bond lengths and disorder parameters as viewed from either metal absorption edge. Furthermore, the coordination numbers of heterometallic bonds were fixed according to the element composition. The obtained information on local environments shows a marked preference for segregation of Pt atoms to the particle surface in the fully reduced particles. The carbonsupported PtRu nanoparticles were shown to have adopted a hemispherical cuboctahedron f.c.c. structure with an average diameter of about 15 Å. In combination with electron microscopy and electron diffraction, the EXAFS analysis of Pt and PtRu nanoparticles, considering multiplescattering contributions, provides structural information with higher accuracy than only using the firstshell EXAFS analysis.
Similarly, in the case of Pt nanoparticles on a γAl_{2}O_{3} support (measured under a 2.5% CO/97.5% He flow at room temperature; Frenkel et al., 2014), the peak positions are correlated with the pair distribution function peaks that correspond to the first, second, third etc. coordination shells, although for higher order shells such determination is difficult due to the contribution of multiplescattering paths in the same rrange as singlescattering paths of the same length (Fig. 4). In the EXAFS analysis of 4.0 nm Ru@Pt coreshell nanoparticles (NPs; Alayoglu et al., 2009), the structural parameters of the Pt shell were extracted beyond the first coordination shell by including the multiplescattering contributions in the FEFF analysis of Ru@Pt NPs, while for the Ru K edge such a higher order contribution is absent due to the poorly ordered core structure (Fig. 5). The EXAFS results revealed that the Ru@Pt coreshell NPs consist of a highly disordered Ru core and a relatively crystalline Pt shell, which is consistent with XRD and TEM studies.
Besides the size and shape determination of nanomaterials, the multiplescattering contributions from forward or nearly collinear scattering paths in EXAFS were also used to reveal the r.m.s. buckling angle of mixed salts from the average NaCl structure (Frenkel et al., 1993, 1994, 1995). In the perfect NaCl structure, the collinear doublescattering (DS) and triplescattering (TS) paths in the distant R range, containing the first nearest neighbour as the focusing atom, are enhanced compared with the singlescattering path with the same path length. This effect is very sensitive to the angular deviation from collinearity in these paths and can be used to measure it as long as it is smaller than 20°. In a mixture A_{x}B_{1−x}C of salts AC and BC with composition x and choosing C as the absorbing atom, the rootmeansquare (r.m.s.) values of the buckling angles, described as and , could be determined from fits that employ the DS and TS paths. The forwardscattering amplitude F of the focusing atom in these paths has a maximum when the atoms are collinear (Θ = 0), as demonstrated above. Polynomial expansion of F(k, Θ) has its lowest order of Θ^{2} in the vicinity of Θ = 0. Averaging over the total number of absorbing atoms, one obtainsThe forwardscattering amplitude F(k, 0) for the intervening atom in the threeatom linkage is determined by FEFF calculations on the ordered structure of the corresponding pure salts. F(k, Θ) is calculated by FEFF for DS paths with the focusing atom moving out of the collinear position according to the predefined Θ values. The curvature coefficient b(k) is then found by the fit to equation (11) in a small Θ range (up to 20°). One can approximate the process further by fixing the value of b(k) at a constant; Θ^{2} can then be refined as a new amplitude correction in the fit of EXAFS data. Applying the EXAFS fitting using the above strategy, the r.m.s. buckling angles in the mixed salts Rb_{0.76}K_{0.24}Br and RbBr_{0.62}Cl_{0.38} were obtained as 7–9° (Frenkel et al., 1994).
Angulardependent XAFS measurements were used to study local disorder in the structure of hightemperature superconductors (Han et al., 2002; Haskel et al., 2000, 2001). One of the challenges is to determine the orientation of CuO_{6} octahedra, which differs between the bulk structures of different phases in Li_{2}CuO_{4} materials [for example lowtemperature tetragonal (LTT), lowtemperature orthorhombic (LTO) and hightemperature tetragonal (HTT)]. Using the case of La_{2−x}Ba_{x}CuO_{4} (x = 0.125, 0.15) (Haskel et al., 2000) as an example, the local tilt angle of CuO_{6} octahedra was determined by the buckling angle of the collinear multiplescattering path La–O–Cu obtained from analysis of the La Kedge EXAFS data. The buckling angles obtained from the EXAFS data fitting were the same as in the LTT configuration and were found to be nearly temperatureindependent. These results suggest that the local tilts of CuO_{6} octahedra remain the same as in the LTT phase, which is evidence of a significant order–disorder contribution to the mechanism of phase transitions in La_{2−x}Ba_{x}CuO_{4} (x = 0.125, 0.15).
Another good example is the multiplescattering EXAFS analysis of tetraalkylammonium (TAA) manganese oxide colloids (Ressler et al., 1999). Two series of colloidal TAA MnO_{x} prepared with tetrapropylammonium (TPA) and tetraethylammonium (TEA) cations were studied. EXAFS analysis was carried out to distances of 6 Å around the central Mn atom using the singlescattering, collinear forwardscattering, selected triangular and higher order scattering paths. Differences in the refined scattering shell distances between the TPA and TEA series suggest a structureinfluencing effect of the two ammonium ions. Bond angles between neighbouring MnO_{6} octahedra were determined from the amplitude dependence of a collinear Mn–Mn–Mn forwardscattering path. Based on the obtained bond angles and the assumption of a certain degree of ordering of trivalent and tetravalent manganese in the MnO_{x} layers, three distinct 2D structures are proposed: one for birnessite, one for TAA sols and one for TAA gels.
The multiplescattering theory of EXAFS has rapidly been developed in the past few decades. The computational codes and software based on multiplescattering theory have become very efficient and useful tools for calculating and interpreting EXAFS data. The science community in many fields, including catalysts (Frenkel, 2012, 2015; Frenkel et al., 2014), batteries (Giorgetti, 2013), biology (Charnock, 1995; Parsons et al., 2002), semiconductor heterostructures (Boscherini, 2008), ferroelectric materials (Cabrera, 2011) and so on (Wende, 2004; Kuzmin & Chaboy, 2014; Sun et al., 2013), have greatly benefited from the utilization of multiplescattering analysis of EXAFS. However, in some cases of lowsymmetry or amorphous systems, it is not necessary to apply the multiplescattering analysis of EXAFS since the multiplescattering contributions are very weak and can be neglected. As the EXAFS information is limited to the nearest environment of absorbing atoms, complementary characterization techniques, such as Xray diffraction and electron microscopy, are usually employed in conjunction with XAFS to provide 3D structural information on the materials.
As mentioned earlier, although multiplescattering theory in EXAFS has been well established, there is potential to improve the accuracy of calculation in general; for example, corrections to the muffintin potential approximation and a better treatment of inelastic losses and disorder may potentially enhance the accuracy of calculation (Rehr & Albers, 2000).
Funding information
The authors gratefully acknowledge support of this work by NSF Grant No. CHE1413937.
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