EXAFS Analysis: Fundamentals & Advances

• EXAFS is a quantitative spectroscopic method that probes local atomic arrangements, coordination numbers, and lattice disorder through oscillatory x-ray absorption modulations.
• It integrates advanced techniques such as quantum-chemical simulations, molecular dynamics, and Bayesian inference for precise data processing and parameter extraction.
• Practical analysis requires careful preprocessing, normalization, and optimally fitting experimental data with simulation and model-based approaches to resolve multiple atomic shells.

Extended X-ray Absorption Fine Structure (EXAFS) analysis is a quantitative spectroscopic approach for probing local atomic arrangements, chemical identity, and lattice disorder in condensed matter systems. EXAFS interrogates the oscillatory modulations in x-ray absorption just above a K- or L-edge, arising from the interference of ejected photoelectrons with those scattered by neighboring atoms. Analysis of these modulations enables direct extraction of interatomic distances, coordination numbers, disorder parameters, and, with advanced treatments, local lattice distortions or vibrational dynamics. Modern EXAFS methodologies now integrate quantum-chemical simulation, molecular dynamics, advanced statistical inference, and machine-learning-enhanced workflows, adapting to the structural complexity found in alloys, nanomaterials, amorphous phases, and multicomponent functional materials.

1. Theoretical Foundations and EXAFS Equation

EXAFS is fundamentally described by the single-scattering formula: $$\chi(k) = \sum_{j} \frac{N_{j} S_{0}^{2}}{k R_{j}^{2}} f_{j}(k) \exp[-2R_{j}/\lambda(k)] \exp[-2\sigma_{j}^{2}k^{2}] \sin[2kR_{j} + \delta_{j}(k)]$$ where $k = \sqrt{2m_e(E-E_0)}/\hbar$ is the photoelectron wavevector, $N_{j}$ is the coordination number, $S_{0}^{2}$ is the amplitude reduction (many-body effects), $f_{j}(k)$ is the backscattering amplitude, $R_{j}$ is absorber–scatterer distance, $\lambda(k)$ is the mean free path, $\sigma_{j}^{2}$ is the Debye–Waller factor representing atomic disorder, and $\delta_{j}(k)$ is the total phase shift.

For complex materials, models are extended to incorporate multiple-scattering, non-Gaussian disorder via cumulant expansion: $$\langle e^{2ik\Delta R} \rangle = \exp[2ikC_{1} - 2k^{2}C_{2} + (4/3)ik^{3}C_{3} - (4/3)k^{4}C_{4} + ...]$$ where $C_n$ are cumulants (mean, variance, skewness, kurtosis) of the pairwise distance distribution (Kuzmin et al., 2020). This provides a powerful route to describe anisotropic, anharmonic, or highly disordered systems (Olovsson et al., 2016).

2. Data Processing, Extraction, and Fitting Protocols

Preprocessing begins with measurement of the absorption coefficient $\mu(E)$, normalization, and background subtraction (typically via low-order polynomial or spline functions above/below the edge), followed by conversion to $\chi(k)$ using the relation: $$\chi(E) = [\mu(E) - \mu_{0}(E)] / \Delta\mu_{0}(E_{0})$$ The energy is mapped to $k$, and the oscillatory part $\chi(k)$ is isolated and weighted ($k^{n}$, $n=2-3$) to optimize signal-to-noise at high-$k$. Fourier transformation of $k^{n}\chi(k)$ yields $\chi(R)$, where peaks correspond approximately (shifted by a phase factor) to absorber–scatterer shell radii.

EXAFS analysis software, such as EDA (Kuzmin, 2021), Athena/Artemis (Demeter suite), and Larch, perform extraction, filtering, fitting (via non-linear least squares with FEFF-generated amplitudes/phases), cumulant model regression, regularization-based radial distribution function reconstruction (EDARDF), and MD-EXAFS averaging.

Key fitting parameters per shell are $N_j$, $R_j$, $\sigma_j^{2}$, $S_{0}^{2}$, the edge shift $\Delta E_0$, additional cumulants for anharmonicity, and energy-dependent mean free paths. The number of independent parameters supported by a given $k$, $R$ window is governed by the Nyquist criterion $N_{\rm ind} \approx (2\Delta k \Delta R)/\pi$ (Haddad et al., 9 Sep 2025).

3. Advanced Modeling: Atomistic Simulations and Bayesian Approaches

Where conventional EXAFS fitting is challenged by disorder or limited prior structural knowledge, atomistic simulation methods are employed:

• Ab initio and classical MD-EXAFS: AIMD or MD trajectories generate ensembles of atomic configurations, from which instantaneous pair distances $R_j(t)$ yield the full radial distribution function $g_j(R)$ and Debye–Waller factor $$\sigma_{j}^{2} = \langle(R-\langle R \rangle)^{2}\rangle$$ (Hönicke et al., 2 Sep 2025, Jonane et al., 2018). Averaged spectra from many snapshots reproduce temperature-dependent EXAFS, enabling validation of interatomic potentials and decomposition into bulk, defective, or surface contributions.
• Reverse Monte Carlo (RMC-EXAFS): Random displacements in large supercells are accepted/rejected to minimize residuals between computed and experimental configuration-averaged EXAFS in $k$, $R$, or wavelet domains. This captures disorder, short-range correlations, and shell overlap without assumed functional forms for $g(R)$ (Kuzmin et al., 2020, Jonane et al., 2018).
• Bayesian sparse modeling: Recent methods apply Bayesian inference to select basis sets (Fourier, advanced Fourier) and optimize RDF and Debye–Waller parameters directly from $\chi(k)$, leveraging physical priors on shell occupancy ($N(R) \sim R^{2}$), posterior uncertainties, and marginal likelihood (Bayesian free energy) for model selection (Igarashi et al., 2021, Iesari et al., 2021, Haddad et al., 9 Sep 2025).

4. Specialized Applications and Challenges in EXAFS Analysis

Multicomponent and High-Entropy Materials

EXAFS is a powerful probe for short-range order in CCAs and high-entropy alloys, but the analysis is hindered by path combinatorics, low Z-contrast among component elements, and parameter degeneracies. Multi-edge joint fitting with symmetry/self-consistency constraints, global fits across composition/temperature series, and multimodal data fusion (PDF+EXAFS, DFT-priors, EXELFS mapping) improve identifiability (Joress et al., 2023, Kuzmin, 8 Nov 2024). Benchmark suites and machine learning–accelerated workflows remain areas of active development.

Thermometry and Lattice Dynamics

Temperature-dependent EXAFS harnesses the sensitivity of the Debye–Waller factor to thermal disorder. The correlated Debye model computes temperature-dependent mean-square relative displacement as: $$\sigma^{2}(T) = \frac{3\hbar}{2\mu\omega_{D}} \int_{0}^{\theta_{D}/T} x \coth(x/2)\left[1 - \frac{\sin(xu)}{xu}\right] dx$$ where $\mu$ is reduced mass, $\omega_D$, $\theta_D$ Debye parameters, and $u$ relates to shell distance. EXAFS-based thermometry achieves 5–10% accuracy for $T \gtrsim \theta_{D}/3$, with application to nanoscale temperature mapping in bcc/fcc metals (Kuzmin et al., 25 Oct 2024).

Nanostructures, Amorphous and Doped Materials

EXAFS quantifies short-range order, site occupancies, and local distortion in alloys, nanowires, doped oxides, and amorphous phases (e.g., Er:AlNO, Gd:SrTiO$_3$, GaN nanowires) (Katsikini et al., 2019, Mesquita et al., 2020, Parida et al., 2021). Atomistic approaches (DFT, stochastic quenching, RMC) are necessary for model selection and accurate reproduction of shell splitting, coordination statistics, and the impact of defect chemistry.

5. EXAFS in Ultrafast and Laboratory-Scale Spectroscopies

Modern developments enable EXAFS analysis on single-shot, femtosecond timescales using laser-plasma wakefield accelerators (Kettle et al., 2023) and X-ray Free Electron Lasers (XFELs) (Harmand et al., 2020). Laboratory-based Rowland-circle spectrometers now deliver high-fidelity K-edge EXAFS measurements for 3d metals, lanthanides, and actinides, with $k$-space coverage up to 12 Å⁻¹ and energy resolution $\leq$0.4 eV (Jahrman et al., 2018). While photon flux remains lower than at synchrotrons, benchtop platforms enable routine measurements, battery electrode characterization, and development of high-throughput analytical workflows.

Ultrafast EXAFS platforms employ concatenated single-shot spectra, energy-dispersive CCD spectrometers, and robust normalization/background subtraction strategies to extract structure over $k$-ranges suitable for resolving multiple atomic shells with fs time resolution, enabling studies of warm dense matter and dynamic phase transformations.

6. Practical Recommendations, Limitations, and Future Directions

Robust EXAFS analysis demands precise preprocessing (background, normalization, $k$-conversion), careful choice of $k$, $R$, and weighting windows, and judicious parameter constraints to avoid degeneracy and overfitting. The Bayesian Information Content Criterion (BFI) now supersedes the traditional Shannon–Nyquist rule, accounting for parameter correlations and prior ranges, explicitly defining the number and nature of structural features supported by the data (Haddad et al., 9 Sep 2025).

Atomistic and statistical simulation approaches are now routine for complex, disordered, or multicomponent systems, yielding realistic representations of disorder, shell overlap, and vibrational dynamics over extended length scales. Joint multimodal data fusion, automated model selection, and physically motivated priors (e.g., from MD, DFT, experimental PDFs) push the boundaries of quantitative EXAFS, while the integration of laboratory and ultrafast spectroscopies makes high-throughput and time-resolved measurements possible beyond synchrotron facilities.

EXAFS continues to evolve as an indispensable probe of local structure, especially where alternative techniques (diffraction, PDF) fail to provide element-specific or environment-resolved insight. Ongoing developments in advanced modeling, inference, and instrumentation broaden its applicability to next-generation materials science, catalysis, energy storage, and quantum functional materials.