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X-ray absorption fine structure

X-ray absorption fine structure (XAFS) is a spectroscopic technique that probes the local atomic and electronic structure surrounding specific elements in complex materials by analyzing the oscillatory features in X-ray absorption spectra at energies near and above the core-level binding energies of the absorbing atoms.^[1] These fine structures arise from the interference of photoelectrons ejected by incident X-rays with those scattered by neighboring atoms, providing element-specific information without requiring long-range crystalline order.^[2] XAFS encompasses two primary regions: the X-ray absorption near-edge structure (XANES), which spans approximately the first 50 eV above the absorption edge and reveals details about the absorber's oxidation state, coordination geometry, and electronic transitions, and the extended X-ray absorption fine structure (EXAFS), which extends beyond 50 eV and yields quantitative structural parameters such as interatomic distances (typically accurate to 0.02 Å), coordination numbers, and disorder in the local environment.^[1] The EXAFS signal, often expressed in terms of the photoelectron wave number k, is modeled using equations that account for backscattering amplitudes, phase shifts, and mean-free-path effects of the photoelectrons.^[2] Historically, the phenomena underlying XAFS were first observed in the 1920s,^[3] but quantitative analysis and widespread application emerged in the 1970s, driven by the development of high-brightness synchrotron radiation sources that enabled high-resolution measurements.^[4] Early advancements, such as those at Stanford Synchrotron Radiation Lightsource in 1974, transformed XAFS into a versatile tool for studying non-crystalline and dynamic systems.^[1] In practice, XAFS is performed using synchrotron beamlines equipped with monochromators for energy tuning and detectors for transmission or fluorescence modes, making it ideal for in situ studies under realistic conditions like varying temperature or pressure.^[2] Data analysis involves Fourier transforms to isolate coordination shells and nonlinear least-squares fitting to theoretical standards, often using software like FEFF or IFEFFIT.^[2] Applications of XAFS span diverse fields, including materials science for characterizing nanoparticles and thin films, catalysis to identify active sites and reaction intermediates, biology for metalloproteins like the Mn₄Ca cluster in photosystem II, and environmental science for trace element speciation in soils and waters.^[5] Its sensitivity to dilute species (down to 0.1 at.%) and ability to handle amorphous or disordered materials have made it indispensable for advancing understanding in these areas.^[1]

Fundamentals

Definition and Basic Principles

X-ray absorption fine structure (XAFS) is the fine oscillatory structure observed in X-ray absorption spectra (XAS) at energies near and above the core-level binding energies of an atom, arising from the interference between the outgoing photoelectron wave and the backscattered waves from surrounding atoms.^[6] This modulation in the absorption coefficient provides detailed information about the local chemical and physical environment of the absorbing atom.^[7] The basic principles of XAS involve the absorption of X-ray photons by core electrons in an atom, leading to their ionization via the photoelectric effect and the creation of a core hole.^[8] The resulting XAS spectrum is typically divided into three regions: the pre-edge area below the absorption edge, where absorption is relatively smooth; the edge itself, marked by a sharp increase in absorption; and the post-edge region, where the fine structure oscillations appear due to photoelectron scattering.^[6] Absorption edges correspond to specific core levels, such as the K-edge for 1s electrons or L-edges for 2p electrons, with their energies scaling approximately with the square of the atomic number Z minus a shielding constant σ, following Moseley's law: edge energy ≈ (Z - σ)^2 × Rydberg constant.^[9] This element-specific nature allows XAFS to probe individual atomic species selectively.^[10] High-resolution XAFS measurements rely on synchrotron radiation sources, which provide intense, tunable X-ray beams with energy resolutions down to ~1 eV, enabling precise scanning across absorption edges.^[6] Unlike long-range techniques like X-ray diffraction, XAFS yields element-specific local structural information around the absorbing atom—such as bond distances and coordination—within a radius of about 5 Å, independent of crystalline order.^[8]^[7]

Historical Development

The theoretical foundations of X-ray absorption fine structure (XAFS) were laid in the early 1930s through the work of Ralph de Laer Kronig, who predicted oscillatory fine structure in X-ray absorption spectra due to interference effects between the ejected photoelectron and the surrounding atomic potential in solids. Kronig's initial long-range order model, based on quantum mechanical considerations of crystal periodicity and Brillouin zone boundaries, was published in 1931, followed by refinements in 1932 that incorporated short-range scattering contributions from neighboring atoms. These predictions provided the first conceptual framework for understanding the modulation of absorption coefficients beyond the absorption edge, although experimental verification was limited by the resolution of available X-ray sources at the time. Experimental advancements in the 1960s marked the transition to more detailed observations of XAFS, facilitated by the emergence of synchrotron radiation sources such as the Synchrotron Ultraviolet Radiation Facility (SURF) at the National Bureau of Standards (now NIST). Researchers like Farrel W. Lytle utilized these tunable, high-brightness beams to record high-resolution absorption spectra, revealing asymmetric line shapes in the near-edge region attributable to Fano resonances arising from interference between discrete and continuum states. These studies, conducted around facilities like SURF starting in the early 1960s, demonstrated the sensitivity of fine structure to local atomic environments, laying groundwork for quantitative analysis despite challenges in data quality and interpretation. A pivotal milestone occurred in 1971 when Dale E. Sayers, Edward A. Stern, and Farrel W. Lytle formalized the extended X-ray absorption fine structure (EXAFS) technique through the application of Fourier analysis to absorption data, explicitly linking the observed oscillations to single-scattering events from nearest-neighbor atoms. This approach transformed EXAFS into a robust tool for determining interatomic distances and coordination numbers in noncrystalline materials, with initial demonstrations on systems like copper and germanium.^[11] In 1980, Antonio Bianconi introduced the term X-ray absorption near-edge structure (XANES) to describe the complex spectral features within about 50 eV of the edge, attributing them to multiple-scattering processes and shape resonances that probe shorter-range electronic and geometric structures.^[12] Post-1980s developments focused on ab initio theoretical modeling to enable parameter-free simulations of XAFS spectra. The FEFF code, initiated by John J. Rehr and Robert C. Albers in the late 1980s and refined through the 1990s, implemented real-space multiple-scattering Green's function methods for calculating scattering amplitudes across both EXAFS and XANES regions. Similarly, the GNXAS package, developed by Augusto Filipponi and collaborators in the early 1990s, extended these capabilities by incorporating full multiple-scattering theory and distribution functions for disordered systems. By the 2010s, integration of machine learning accelerated XAFS analysis; for instance, neural networks trained on large datasets have enabled rapid spectrum fitting and structural parameter extraction since around 2018, reducing computational demands while improving accuracy for complex materials like catalysts.

Theoretical Basis

X-ray Absorption in Isolated Atoms

The theoretical description of X-ray absorption spectroscopy (XAS) in isolated atoms begins with the application of Fermi's golden rule, which quantifies the transition rate from an initial core electron state to unoccupied final states upon absorption of an X-ray photon. The absorption coefficient \mu(E) is proportional to the sum over final states |f\rangle of the squared matrix element of the dipole operator between the initial core state |i\rangle and the final state, weighted by a delta function ensuring energy conservation:

\mu(E) \propto \sum_f |\langle f | \mathbf{e} \cdot \mathbf{r} | i \rangle|^2 \delta(E_f - E_i - \hbar \omega),

where \mathbf{e} is the polarization vector of the incident X-ray, \mathbf{r} is the position operator, E is the photon energy, and \hbar \omega = E. This expression captures the probability of photoexcitation in the electric dipole approximation, valid for the soft and hard X-ray regimes relevant to atomic inner-shell transitions.^[13] In isolated atoms, the initial state |i\rangle corresponds to a tightly bound core level, such as the 1s orbital for K-shell absorption, while the final states |f\rangle include both discrete bound orbitals below the ionization potential (IP) and continuum states above it. Transitions to discrete bound states produce sharp absorption lines due to the well-defined energy levels in the atomic potential, as seen in the pre-edge region of atomic spectra. Above the IP, the photoelectron enters the continuum, forming a smooth absorption edge where the density of states broadens the transitions. This dichotomy establishes the baseline atomic absorption profile, devoid of the modulations observed in condensed systems.^[13] In the X-ray energy regime (typically above 100 eV), the dominant absorption mechanism is the photoelectric effect, where the incident photon ejects a core electron, leaving a core hole. The atomic photoelectric cross-section \sigma near absorption edges scales approximately as \sigma \propto Z^4 / E^{3.5}, with Z the atomic number and E the photon energy; this dependence arises from the strong coupling of the X-ray field to high-Z inner shells and the rapid decrease with increasing energy due to the centrifugal barrier in the continuum wavefunction. This power-law behavior provides the smooth background absorption in isolated atoms, essential for normalizing spectra in more complex materials.^[14] Electric dipole selection rules govern the allowed transitions in atomic XAS, requiring a change in orbital angular momentum \Delta l = \pm 1 and spin \Delta s = 0, which determines the relative intensities of different edges. For K-edges (1s to np transitions), the promotion is to p-like continuum or bound states, yielding strong absorption; L-edges (2s or 2p to nd or ns) show split intensities due to the 2p_{1/2} (L_2) and 2p_{3/2} (L_3) spin-orbit components, with L_3 typically more intense owing to higher degeneracy. These rules explain the prominence of K-edges for elements with Z > 20 and the utility of L-edges for transition metals.^[15] Unlike in condensed matter, isolated atoms exhibit no extended fine structure oscillations in their absorption spectra because the ejected photoelectron propagates outward without backscattering from neighboring atoms, resulting in a featureless continuum above the edge.

Fine Structure in Condensed Matter Systems

In condensed matter systems, such as molecules and solids, the theory of X-ray absorption extends beyond the isolated atom case by incorporating the influence of the local atomic environment on the ejected photoelectron's final states. While isolated atoms exhibit a smooth absorption profile due to the absence of neighboring interactions, the presence of surrounding atoms introduces a local potential that perturbs the photoelectron wavefunction, leading to interference effects and oscillatory fine structure in the absorption coefficient μ(E). This modulation arises primarily from the interaction between the outgoing photoelectron and the potential created by nearby atoms, distinguishing condensed-phase spectra from the baseline atomic absorption.^[7]^[6] The physical origin of this fine structure lies in the wave-like nature of the photoelectron, which is ejected with kinetic energies typically in the range of 10-100 eV following core-level excitation. The outgoing photoelectron wave propagates outward from the absorbing atom and scatters off neighboring atoms, generating backscattered waves that interfere constructively or destructively with the original wave upon returning to the absorber. This interference produces periodic oscillations in μ(E) as a function of photon energy, with the oscillation period related to the interatomic distances. In multi-atom systems, these backscattering events dominate the fine structure formation, enabling the determination of local coordination geometry and atomic species around the absorber—a capability absent in isolated atomic absorption.^[7]^[16]^[6] The extent of the probed local environment is limited by the photoelectron's mean free path, the average distance it travels before undergoing inelastic scattering or other damping processes, which is approximately 5-10 Å for these kinetic energies. This short range ensures that XAFS is inherently sensitive to the immediate atomic coordination shell, typically encompassing a few nearest neighbors, without resolving longer-range crystallographic details. Thermal vibrations and structural disorder further influence the fine structure by introducing variations in interatomic distances, which dampen the oscillation amplitudes through phase smearing effects analogous to Debye-Waller factors in diffraction. These disorder contributions, quantified by the mean-square displacement of atoms, reduce the coherence of the interfering waves, particularly at higher temperatures or in amorphous systems, thereby broadening and attenuating the observed fine structure.^[7]^[6]^[16]

Spectral Regions

X-ray Absorption Near Edge Structure (XANES)

X-ray absorption near edge structure (XANES) refers to the oscillatory features in the X-ray absorption spectrum occurring from the absorption edge up to approximately 50 eV above it, arising primarily from multiple scattering of the photoelectron within small atomic clusters around the absorbing atom, typically spanning 0.2 to 2 nm in size.^[17]^[18] The term XANES was coined in 1980 by Antonio Bianconi to describe these strong absorption peaks and related structures sensitive to the local electronic environment.^[19] This region provides insights into short-range order and electronic properties that are distinct from longer-range structural information obtained elsewhere in the spectrum. Key features of XANES spectra include white lines, which are sharp intensity peaks immediately above the edge resulting from enhanced transitions to unoccupied electronic states, often reflecting changes in the local density of states.^[20] Edge shifts in XANES, where the absorption onset moves to higher energies with increasing oxidation state (up to about 5 eV per unit change), serve as a direct indicator of the chemical valence of the absorbing atom.^[21] These features are particularly pronounced in transition metal oxides, where XANES has been used to probe metal-insulator transitions, such as in vanadium dioxide (VO₂), revealing shifts in electronic occupancy during phase changes.^[22] Theoretically, XANES is described using full multiple scattering theory, which accounts for the photoelectron's interactions with surrounding atoms modeled via muffin-tin potentials that approximate the atomic potential as spherical within non-overlapping spheres.^[23] Cluster size plays a critical role, as resonances in the spectrum converge only when the cluster encompasses multiple coordination shells (typically 5–10 Å radius), ensuring accurate representation of scattering paths.^[24] Interpretation of XANES focuses on matching spectral shapes to simulated profiles to infer bond angles and coordination geometry, as multiple scattering paths are highly sensitive to the three-dimensional arrangement of nearest neighbors.^[25] Additionally, the spectral features project the partial density of unoccupied states onto the absorbing atom's symmetry, enabling analysis of electronic structure, such as hybridization effects in coordination complexes.^[18] For instance, pre-edge peaks in titanium K-edge XANES distinguish between octahedral and tetrahedral coordination by their intensity and position, reflecting selection rule violations due to distortions.^[26]

Extended X-ray Absorption Fine Structure (EXAFS)

The extended X-ray absorption fine structure (EXAFS) manifests as sinusoidal oscillations in the X-ray absorption coefficient, typically observed more than 50-100 eV above the absorption edge. These oscillations arise from the interference between the outgoing photoelectron wave ejected from the absorbing atom and the backscattered waves from surrounding atoms, enabling the probing of interatomic distances up to approximately 5 Å through single-scattering events.^[27]^[28] The quantitative description of EXAFS is provided by the EXAFS function \chi(k), expressed in the single-scattering approximation as:

\chi(k) = \sum_j \frac{N_j}{k R_j^2} S_0^2(k) |f_j(k)| e^{-2R_j / \lambda(k)} e^{-2\sigma_j^2 k^2} \sin[2k R_j + \phi_j(k)],

where k is the photoelectron wavevector, N_j is the number of scattering atoms in the j-th shell, R_j is the absorber-scatterer distance, \sigma_j is the root-mean-square disorder in that distance, f_j(k) is the backscattering amplitude, \phi_j(k) is the total phase shift, \lambda(k) is the mean free path of the photoelectron, and S_0^2(k) accounts for the amplitude reduction due to many-body effects. This formula captures the oscillatory modulation of the absorption cross-section, with the sinusoidal term reflecting the interference pattern and the exponential terms describing damping due to inelastic scattering and thermal disorder.^[29]^[28] The derivation of this expression begins with the plane-wave approximation for the photoelectron wavefunction, treating the ejected electron as a plane wave that is backscattered by neighboring atoms assumed to be point scatterers. The interference modulates the absorption probability, leading to \chi(k) as the normalized deviation from the smooth atomic background. Applying a Fourier transform to \chi(k) weighted by k^n (typically n=2 or 3) produces a radial distribution-like function, where peaks correspond to coordination shells around the absorber, facilitating extraction of structural parameters.^[30]^[28] While the plane-wave single-scattering model provides a foundational framework, more accurate treatments incorporate spherical-wave effects to account for the curvature of the photoelectron wavefront, improving reliability for higher coordination shells. Near the absorption edge, multiple-scattering contributions become significant, necessitating corrections to the single-scattering formula for precise analysis in the extended region. The term EXAFS was introduced in 1971 by Sayers, Stern, and Lytle, whose work demonstrated its utility for structural determination, achieving absorber-scatterer distance precision of approximately 0.01 Å in favorable cases.^[29]^[30]

Intermediate Region

The intermediate region in X-ray absorption fine structure (XAFS) spectroscopy encompasses the spectral area approximately 30–100 eV above the absorption edge, where multiple scattering and single scattering processes overlap substantially, resulting in complex oscillatory features that challenge straightforward interpretation.^[31] This transitional zone lies between the near-edge region dominated by multiple scattering and the extended region where single scattering prevails, with the overlap arising from the relatively low photoelectron energies that allow significant contributions from both mechanisms.^[31] Theoretically, this region is characterized by prominent low-order n-body multiple scattering paths involving 3–4 atoms, such as double backscattering, which extend beyond pairwise interactions and require advanced modeling beyond simple single-scattering approximations.^[31] Approaches like the GNXAS package address these complexities by performing full curved-wave multiple-scattering calculations of the XAFS cross-section, enabling refinement of structural parameters including n-body distribution functions for accurate spectral reproduction.^[32] Such methods decompose the signal into contributions from correlated atomic distributions, accounting for the interference between scattering paths that dominates at these energies.^[32] Key spectral features include smoother oscillations compared to the more pronounced modulations in the extended XAFS region, reflecting the damped interference from higher-order paths and thermal disorder effects.^[31] This region is especially sensitive to medium-range order (3–6 Å), providing insights into bonding angles and cluster geometries that are less accessible via edge or extended analyses alone, particularly in disordered systems like amorphous materials.^[31] Interpretation of the intermediate region presents notable challenges, as it is frequently overlooked or approximated by truncating data to either the near-edge or extended portions, due to the difficulty in disentangling overlapping scattering contributions without comprehensive cluster models.^[31] However, incorporating this region enhances the reliability of full-spectrum fits, serving as a critical test for theoretical validity in capturing the transition from electronic to geometric structural information.^[31] Overall, the intermediate region bridges the electronic-sensitive near-edge features with the atomic-distance-focused extended structure, offering complementary data on local coordination and medium-range correlations; for instance, it has been applied to probe cation environments in oxide glasses, revealing details of network polymerization and disorder.^[31]^[33] Similar utility extends to polymers, where multiple scattering analysis in this energy window elucidates ligand arrangements around metal centers in coordination complexes embedded in amorphous matrices.^[31]

Experimental Techniques

Synchrotron Sources and Beamlines

Synchrotron radiation sources are essential for X-ray absorption fine structure (XAFS) experiments due to their ability to produce highly tunable, intense X-ray beams with exceptional properties not achievable with conventional laboratory sources. Third-generation synchrotrons, such as the European Synchrotron Radiation Facility (ESRF), the Advanced Photon Source (APS), and SPring-8, operate by accelerating electrons to relativistic speeds in storage rings, generating synchrotron radiation through bending magnets, undulators, and wigglers. These facilities deliver X-ray fluxes exceeding 10^{12} photons per second in the hard X-ray regime, enabling high signal-to-noise ratios in absorption measurements even for weakly absorbing samples.^[34]^[35]^[36] XAFS beamlines at these synchrotrons incorporate specialized optical components to select and condition the X-ray beam for precise energy tuning and minimal distortion. A key element is the double-crystal monochromator, typically using silicon (Si) crystals in the (111) or (220) reflection, which provides energy resolution better than 0.1 eV by diffracting X-rays according to Bragg's law while scanning the energy across absorption edges. Focusing mirrors, often coated with rhodium or platinum and operated near grazing incidence, collimate and concentrate the beam to micrometer spot sizes, enhancing spatial resolution for micro-XAFS studies. Harmonic rejection is achieved through detuning the monochromator crystals or using detuned mirrors to suppress higher-order harmonics, ensuring a clean monochromatic beam with harmonic content below 10^{-4} relative to the fundamental.^[37]^[38]^[39] The operational energy range of XAFS beamlines spans typically 2 keV to 40 keV, covering K-edges from sodium (Na, ~1.07 keV) to uranium (U, K-edge ~115.6 keV, though typically up to L-edges ~17 keV) and L-edges of heavier elements. This broad tunability allows probing of a wide array of elements across the periodic table, from light transition metals to actinides, by selecting the appropriate absorption edge for element-specific structural and electronic information.^[40]^[41] Compared to laboratory X-ray tubes, which produce bremsstrahlung and characteristic lines with limited tunability and flux, synchrotrons offer 10^6 to 10^9 times higher brightness, defined as photons per unit area per unit solid angle per unit bandwidth per second. This superior brightness enables XAFS measurements on dilute samples with concentrations below 1 at.%, such as trace elements in biological or environmental materials, where lab sources would require impractically long acquisition times or fail to detect signals.^[42]^[43] As of 2025, upgrades to fourth-generation, diffraction-limited sources like the ESRF Extremely Brilliant Source (EBS), completed in 2022, and the APS Upgrade (APS-U), operational since 2025, have further enhanced XAFS capabilities by reducing emittance to 110 pm·rad, achieving up to 100 times higher coherence and stability. These advancements support time-resolved operando studies of dynamic processes, such as catalytic reactions, with sub-second temporal resolution and microfocus beams for in situ imaging.^[44]^[45]^[46]

Sample Preparation and Detection Methods

Sample preparation for X-ray absorption fine structure (XAFS) measurements requires samples that are homogeneous on a scale of approximately 1 μm to ensure uniform absorption across the beam.^[41] Stability under vacuum conditions is essential, as many experiments occur in ultrahigh vacuum or helium atmospheres to minimize scattering and oxidation effects.^[41] Common sample forms include fine powders, thin films, and liquid solutions; for instance, solid powders are typically ground to particle sizes below 1 μm and mixed with inert matrices such as boron nitride (BN) or alumina (Al₂O₃) to achieve optimal thickness, often around 1 mm with 70-90% absorption at the edge.^[41] Cryogenic cooling, such as liquid nitrogen or helium temperatures, is frequently employed to reduce beam-induced damage, particularly for radiation-sensitive biological or aqueous samples.^[41] Detection methods in XAFS primarily involve measuring the absorption of X-rays by the sample, with transmission and fluorescence modes being the most widely used.^[47] In transmission mode, ionization chambers detect the incident beam intensity (I₀) upstream and the transmitted intensity (I_t) downstream, making it suitable for concentrated samples where the total absorption is typically 1-2 absorption lengths to balance signal-to-noise ratio.^[41] This mode requires precise sample thickness control and is ideal for homogeneous, bulk materials like pellets pressed from powdered samples.^[41] Fluorescence mode detects X-ray emission from the sample following absorption, which is advantageous for dilute systems or thick, bulk samples where transmission is impractical.^[41] Energy-dispersive detectors, such as multi-element germanium (Ge) arrays, are commonly used, offering energy resolution around 250 eV FWHM and handling count rates up to approximately 280,000 counts per second per element for concentrations as low as parts per million.^[41] However, self-absorption effects in thick samples distort the signal, necessitating corrections based on sample geometry and composition.^[41] For surface-sensitive studies, electron yield detection modes are employed, particularly in ultrahigh vacuum environments.^[47] Partial electron yield measures photoelectrons from specific depths, providing surface selectivity, while total electron yield integrates all emitted electrons for a probing depth of a few nanometers, suitable for thin films or adsorbates on surfaces.^[47] Time-resolved XAFS is enabled by quick-EXAFS techniques using rapid-scanning monochromators, achieving resolutions down to 1 ms or better for capturing dynamic structural changes in operando conditions.^[48] These setups often employ channel-cut crystal monochromators oscillated at high frequencies, paired with fast-response detectors like gridded ionization chambers.^[49]

Data Analysis

Pre-processing and Normalization

Pre-processing of X-ray absorption fine structure (XAFS) data begins with the conversion of raw intensity measurements into absorption coefficients, typically expressed as μ(E), where μ(E) = -ln(I/I₀) for transmission mode or proportional to fluorescence yield for detection mode, followed by corrections for instrumental artifacts to ensure data quality.^[6] Common artifacts include glitches arising from multiple-beam diffraction in the monochromator crystals, which manifest as sharp discontinuities in the spectrum and require identification and interpolation or exclusion of affected energy regions.^[50] Detector dead-time effects, particularly in fluorescence measurements with high count rates, lead to non-linear response and are corrected using standard formulas that account for pulse pile-up, often implemented in software like PyMca or Larch.^[51] Edge energy determination is crucial for aligning spectra and isolating the absorption feature, commonly defined as the inflection point of the absorption edge—identified as the maximum of the first derivative of μ(E)—or alternatively as the energy at half the height of the total edge jump from pre- to post-edge absorption.^[18] This E₀ value serves as the threshold for subsequent transformations and is refined iteratively during analysis to minimize distortions in the fine structure signal. Normalization involves subtracting a smooth pre-edge background, often fitted using a Victoreen polynomial or low-order polynomial to approximate atomic absorption below the edge, followed by division of the raw μ(E) by the edge jump Δμ(E₀) to yield the normalized spectrum μ(E)/μ₀(E), which scales the data between 0 (pre-edge) and 1 (far post-edge).^[6] An advanced method, implemented in the MBACK algorithm, normalizes the entire spectrum to tabulated mass absorption coefficients by fitting a background function excluding the edge region, reducing operator bias and achieving reproducibility within 0.2% error.^[52] This step isolates the oscillatory fine structure from smooth atomic contributions. To analyze the extended fine structure, the normalized post-edge data is converted to photoelectron wave number space using k = \sqrt{\frac{2m(E - E_0)}{\hbar^2}} where m is the electron mass, E is the photon energy, E₀ is the edge energy, and ℏ is the reduced Planck's constant; windowing functions (e.g., Hanning) are applied to select the post-edge region and suppress edge and pre-edge contributions.^[51] The EXAFS function χ(k) is then extracted via spline background subtraction, where a piecewise cubic spline is fitted to the post-edge μ(E) or χ(E) with knots spaced to capture low-frequency atomic background without overlapping the oscillatory fine structure, typically using algorithms like Autobk in modern software such as Larch that optimize the spline based on a minimum knot distance parameter Rbkg ≈ 1 Å.^[53]^[54] This ensures χ(k) represents the pure fine structure modulation, ready for Fourier transform and modeling, while avoiding subtraction of true structural signals.^[6]

Modeling, Fitting, and Interpretation

Theoretical modeling of X-ray absorption fine structure (XAFS) spectra relies on ab initio calculations to generate theoretical phase and amplitude functions that simulate the experimental data. The FEFF code, a widely used real-space multiple-scattering program, performs these calculations by solving the Schrödinger equation within a muffin-tin potential approximation, enabling the computation of XAFS spectra including both single- and multiple-scattering contributions up to high orders.^[55] For more accurate simulations, particularly in systems with non-spherical potentials, the MXAN code employs a full-potential approach based on the muffin-tin approximation extended to include non-muffin-tin corrections, allowing quantitative fitting of X-ray absorption near-edge structure (XANES) spectra through iterative structural parameter variations.^[56] The fitting process involves nonlinear least-squares minimization to refine structural parameters by comparing the experimental fine-structure function, χ(k), to its theoretical counterpart generated from codes like FEFF or MXAN. This is achieved using algorithms such as Levenberg-Marquardt, which minimize the reduced chi-squared statistic, χ², defined as the sum of squared residuals weighted by the number of independent data points, typically over a range of photoelectron wavevectors k. Key refinable parameters include the interatomic distance R, coordination number N, and mean-square disorder σ², which account for bond lengths, atomic multiplicities, and thermal/structural disorder, respectively; these are adjusted iteratively while fixing or constraining others like the energy shift E₀ to avoid correlations. The EXAFS equation, as detailed in the spectral regions section, provides the basis for this theoretical χ(k). Error analysis in XAFS fitting assesses the reliability of refined parameters through the correlation matrix, derived from the Hessian of the χ² surface, which quantifies parameter interdependencies and yields statistical uncertainties at the 68% confidence level (1σ). Validation often involves Fourier transforms of the fitted χ(k), where peaks in the radial distribution function confirm the structural model by matching interatomic distances after phase correction, with residuals highlighting unmodeled features or systematic errors. Since the late 2010s, advanced machine learning methods have emerged as surrogates to accelerate XAFS inversion, using neural networks trained on large datasets of FEFF-generated spectra to predict structural parameters directly from experimental data, reducing computation times from hours to seconds while maintaining accuracy comparable to traditional fitting. These approaches, such as Bayesian neural networks and interpretable multimodal models combining XANES with atomic pair distribution functions, also provide uncertainty quantification by propagating input noise through the model; as of 2025, they enable robust analysis of complex spectra.^[57]^[58]^[59] For interpretation, multi-shell fitting extends the single-scattering approximation to include multiple coordination shells in complex environments, refining parameters shell-by-shell while accounting for correlations via shared disorder terms. Combining XANES and EXAFS analyses yields a comprehensive local model, where XANES simulations from MXAN inform short-range geometry and electronic structure, and EXAFS from FEFF quantifies longer-range disorder and coordination. Modern software packages like Larch facilitate these workflows, integrating pre-processing, modeling, and fitting tools.^[56]^[60]

Applications

Local Atomic Structure Determination

X-ray absorption fine structure (XAFS), particularly the extended X-ray absorption fine structure (EXAFS) region, enables the determination of local atomic geometry, including bond lengths, coordination numbers, and disorder parameters, in materials lacking long-range crystalline order. This element-specific technique probes the immediate environment (typically 3–6 Å) around an absorbing atom by analyzing oscillations in the X-ray absorption coefficient due to backscattering from neighboring atoms. Unlike X-ray diffraction (XRD), which requires periodic structures for phase analysis, XAFS excels in disordered systems, providing interatomic distances with a precision of approximately 0.02 Å.^[10] In amorphous solids and glasses, XAFS derives radial distribution functions to quantify short-range order, such as Si–O bond distances in silicate networks. For instance, EXAFS studies of sodium silicate glasses reveal Si–O distances around 1.62 Å with coordination numbers of 4, reflecting tetrahedral coordination despite the lack of crystallinity. Seminal work by Greaves demonstrated this approach for mineral glasses, establishing XAFS as a tool for probing network connectivity in oxide glasses. Similarly, in silica-based systems, pressure-induced changes in Ge coordination from 4 to 5 in GeO₂–SiO₂ glasses have been tracked, yielding bond lengths accurate to ±0.01 Å.^[61]^[62]^[63] For nanomaterials, XAFS assesses size-dependent structural variations, such as reduced coordination numbers in smaller nanoparticles due to higher surface-to-volume ratios. In gold (Au) clusters, the first-shell Au–Au coordination number (N₁) decreases from ~12 in bulk to ~7–9 for ~1–2 nm particles, allowing estimation of mean particle size via empirical models. This method, applied to thiol-stabilized Au nanoparticles, also detects subtle bond contractions (e.g., 0.02–0.05 Å shorter Au–Au distances in smaller clusters), linking atomic structure to size effects.^[64]^[65] In catalytic systems, in-situ XAFS monitors dynamic changes at active sites under operational conditions, such as Pt–O bond formation in fuel cell electrocatalysts. For Pt/C cathodes in proton exchange membrane fuel cells, EXAFS reveals Pt–O coordination numbers rising from ~0.9 to 1.6 during cycling, correlating with oxidation and activity loss, while PtCo/C shows greater stability. These measurements, performed at potentials up to 1.2 V, track reversible Pt–O bond lengths (~2.0 Å) and inform catalyst durability.^[66] Biological applications leverage XAFS for element-selective probing of metal centers in proteins, where disorder precludes high-resolution crystallography. In hemoglobin, EXAFS determines the Fe–N_porphyrin distance as 2.06 ± 0.01 Å in deoxy form, with the Fe atom displaced approximately 0.20 Å above the plane according to EXAFS, though crystallographic data indicate 0.38 Å, highlighting differences in the methods' sensitivities. This approach has elucidated metal–ligand geometries in heme proteins, aiding understanding of oxygen binding without requiring ordered crystals.^[67]

Electronic and Chemical State Analysis

X-ray absorption near-edge structure (XANES) spectroscopy within XAFS provides critical insights into the electronic and chemical states of absorbing atoms by analyzing shifts in the absorption edge position, which correlate with oxidation state changes due to variations in effective nuclear charge. Higher oxidation states typically shift the edge to higher energies, as the increased binding energy of core electrons reflects greater electrostatic attraction from the nucleus. For instance, in copper complexes, the K-edge of Cu(II) species appears approximately 2-4 eV higher than that of Cu(I), enabling unambiguous differentiation of valence states in coordination compounds without requiring crystalline samples.^[68]^[69] Beyond edge positions, the shape of XANES spectra serves as a fingerprint for coordination chemistry, distinguishing geometries through multiple scattering effects and orbital hybridization patterns. In transition metals, tetrahedral coordination often produces more intense pre-edge features compared to octahedral due to greater d-p mixing, allowing identification of local symmetry in amorphous or disordered systems. For example, in 3d metals like iron and cobalt in silicate glasses, XANES analysis reveals shifts from octahedral to tetrahedral environments under varying conditions, informing bonding and reactivity.^[70]^[71] Pre-edge features in K-edge XANES of 3d transition metals arise from electric dipole-allowed transitions enabled by 3d-4p orbital mixing, providing a probe of electronic structure and site symmetry. The intensity and position of these pre-edges quantify d-p hybridization, which is stronger in lower symmetry sites, while the white line—a sharp peak at the edge—reflects the density of unoccupied d-states, with higher intensities indicating greater electron deficiency. In L-edge spectra of noble metals like platinum, white line variations directly map unoccupied 5d density of states, linking spectral features to electronic configuration.^[72]^[73] Operando XAFS studies leverage time-resolved XANES to track dynamic valence changes in working devices, such as lithium-ion battery cathodes, where manganese oxidation state evolves during charge-discharge cycles. In Li-rich layered oxides like Li1.2Mn0.54Co0.13Ni0.13O2, XANES reveals Mn valence shifts from +4 to lower states, correlating with capacity fade and structural stability under electrochemical operation.^[74]^[75] Recent advancements integrate XAFS with density functional theory (DFT) simulations to predict and interpret electronic states in quantum materials, enhancing accuracy for complex systems like halide perovskites. In lead/bismuth iodide perovskites, combined experimental XANES and DFT modeling elucidates how alloying affects unoccupied p-states and band gaps, guiding design of optoelectronic devices with tailored chemical environments.^[76]

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