EXAFS-Based Thermometry

• EXAFS-based thermometry is defined by determining temperature through analysis of the thermal damping of EXAFS oscillations, extracting MSRD (σ²) and higher cumulants.
• It combines lattice-dynamical models, such as the correlated Debye and Einstein approaches, with advanced methodologies like single-shot XFEL experiments to achieve femtosecond time resolution.
• Robust data reduction pipelines and careful shell isolation enable accurate calibration and mitigation of static disorder, ensuring reliable temperature reconstruction across different material classes.

EXAFS-based thermometry is the determination of sample temperature from the temperature dependence of the extended X-ray absorption fine structure, principally through the damping term $\exp(-2k^2\sigma^2)$, where $\sigma^2$ is the mean-square relative displacement (MSRD) of an absorber–scatterer pair. In this framework, temperature is inferred by fitting the EXAFS signal, extracting $\sigma^2$ or higher cumulants, and inverting a lattice-dynamical or pair-potential model. The method spans equilibrium synchrotron measurements, software-centered offline analysis, and single-shot XFEL implementations in which simultaneous measurement of $I_0(E)$ and $I_t(E)$ makes femtosecond-resolved EXAFS thermometry feasible by concatenating tens of shots into a several-hundred-eV post-edge range (Harmand et al., 2020).

1. Physical basis and formalism

The central observable in EXAFS thermometry is the temperature-dependent attenuation of EXAFS oscillations by thermal disorder. The photoelectron wave number is defined by

$k = \sqrt{\frac{2 m_e}{\hbar^2}(E-E_0)},$

and the Debye–Waller factor enters as

$\exp\!\big(-2k^2\sigma^2(T)\big).$

In a standard shell-resolved form, the EXAFS signal is written as

$$\chi(k)=\sum_j \frac{S_0^2 N_j F_j(k)}{kR_j^2} e^{-2k^2\sigma_j^2} e^{-2R_j/\lambda(k)} \sin\!\big(2kR_j+\phi_j(k)\big),$$

with $S_0^2$ the amplitude reduction factor, $N_j$ the path degeneracy, $\sigma^2$0 the half-path length, $\sigma^2$1 the photoelectron mean free path, and $\sigma^2$2 the total phase shift. In the multiple-scattering formulation used for monatomic metals, the signal is instead expressed as a sum over paths with FEFF-derived amplitudes and phases, each multiplied by $\sigma^2$3 (Kuzmin et al., 2024).

For thermometry, the inverse problem is to determine $\sigma^2$4 from the measured thermal disorder term. In harmonic treatments this means modeling $\sigma^2$5; in anharmonic systems it can also require the third and higher cumulants. The EDA package implements both a Gaussian model and a cumulant expansion up to $\sigma^2$6, with odd cumulants modifying phase and even cumulants modifying amplitude:

$\sigma^2$7

with $\sigma^2$8 corrected for energy-origin differences through $\sigma^2$9 (Kuzmin, 2021).

This formalism makes clear why EXAFS is sensitive to temperature yet not uniquely tied to it. $\sigma^2$0 may also contain static strain, defects, or configurational disorder, and phase transitions can alter both the structural model and the appropriate thermometric calibration. A plausible implication is that EXAFS-based thermometry is most robust when structural symmetry, path inventory, and static disorder are independently constrained.

2. Measurement architectures and XFEL single-shot implementation

At XFELs, single-shot EXAFS thermometry depends on simultaneous acquisition of incident and transmitted spectra. The demonstrated implementation uses two identical transmissive dispersive spectrometers, upstream and downstream of the sample, to record $\sigma^2$1 and $\sigma^2$2 simultaneously; the absorbance is then computed shot by shot after alignment of the two 2D spectrograms (Harmand et al., 2020). Each spectrometer uses a 10 $\sigma^2$3m-thick Si(220) membrane crystal analyzer, a YAG scintillator, optical microscope, and CCD camera. The spectral resolution is $\sigma^2$4 eV at 7.1 keV, below the Fe K-edge core-hole lifetime broadening of $\sigma^2$5 eV.

The stochastic spectral structure of hard-X-ray SASE is mitigated by two measures. First, careful matching of the two spectrometers minimizes spatial–spectral mismatch or “spatial chirp,” with a vertical scattering geometry selected for this purpose. Second, an “overcompressed” electron bunch mode broadens the single-shot bandwidth and suppresses regions of near-zero photons across the band so that normalization remains robust. In this mode at LCLS, the pulse duration is $\sigma^2$6 fs and the relative bandwidth is $\sigma^2$7 at $\sigma^2$8 keV, corresponding to a single-shot energy span of roughly 100 eV at the Fe K-edge (Harmand et al., 2020).

For Fe K-edge measurements on a 4 $\sigma^2$9m Fe foil in transmission geometry, a single-shot window of roughly 100 eV is achieved with a standard deviation $I_0(E)$0 of 0.02–0.03 in the no-sample absorbance metric, corresponding to $I_0(E)$1–3% noise. The pre-edge near 7115 eV and shoulder near 7125 eV are resolved in single-shot spectra. By retuning the central photon energy by $I_0(E)$2 eV between shots and concatenating tens of single-shot spectra, the usable spectral span extends into the EXAFS regime over several hundred eV. The reconstruction is explicitly shown for 1, 2, 4, 8, 16, 32, and 64 shots, and detailed EXAFS features are accessible with 10–20 shots (Harmand et al., 2020).

This architecture defines the practical frontier of femtosecond EXAFS thermometry. With $I_0(E)$3 above the edge of $I_0(E)$4–400 eV, the accessible Fe K-edge photoelectron range reaches approximately $I_0(E)$5–10 $I_0(E)$6, which is sufficient to fit nearest-neighbor MSRDs. Because the probe pulse is $I_0(E)$7 fs and each spectrum is self-normalized, the method is compatible with low repetition-rate or irreversible processes and can probe transient disorder at a defined pump–probe delay. The paper does not specify a pump–probe sequence in the EXAFS tests, but notes compatibility with femtosecond pump–probe thermometry at XPP (Harmand et al., 2020).

3. Data reduction, shell isolation, and fitting workflows

The reduction pipeline begins from simultaneous single-shot or conventional measurements of $I_0(E)$8 and $I_0(E)$9. For XFEL data, a 2D affine transformation function $I_t(E)$0 is derived from tens of reference shots without sample and is applied to register the downstream spectrogram to the upstream one. Transmittance and absorbance are then computed as

$I_t(E)$1

Pre-edge subtraction is followed by post-edge normalization through a smooth atomic-like background $I_t(E)$2, after which the EXAFS signal is extracted as

$I_t(E)$3

Windowing in $I_t(E)$4-space and Fourier transformation to $I_t(E)$5-space isolate the coordination shells used for fitting (Harmand et al., 2020).

The EDA workflow provides a software decomposition of these operations. EDAFORM converts beamline signals into $I_t(E)$6; EDAXANES determines $I_t(E)$7 and checks energy reproducibility; EDAEES models the pre-edge as $I_t(E)$8 and determines the atomic-like background in series,

$I_t(E)$9

with $k = \sqrt{\frac{2 m_e}{\hbar^2}(E-E_0)},$0, $k = \sqrt{\frac{2 m_e}{\hbar^2}(E-E_0)},$1, and $k = \sqrt{\frac{2 m_e}{\hbar^2}(E-E_0)},$2; EDAFT performs direct and back Fourier transforms with rectangular, Gaussian, Kaiser-Bessel, Hamming, or Norton-Beer F3 windows; EDAFEFF imports FEFF8/9 scattering tables; EDAFIT performs non-linear EXAFS fitting in $k = \sqrt{\frac{2 m_e}{\hbar^2}(E-E_0)},$3-space; EDARDF reconstructs $k = \sqrt{\frac{2 m_e}{\hbar^2}(E-E_0)},$4 by a regularization-like inversion; and EDACA computes configurationally averaged EXAFS from MD or MC configurations (Kuzmin, 2021).

In practical thermometry workflows, shell isolation is not a cosmetic step but a parameter-identifiability requirement. The Nyquist criterion in EDA is

$k = \sqrt{\frac{2 m_e}{\hbar^2}(E-E_0)},$5

so the chosen $k = \sqrt{\frac{2 m_e}{\hbar^2}(E-E_0)},$6- and $k = \sqrt{\frac{2 m_e}{\hbar^2}(E-E_0)},$7-ranges directly limit how many parameters can be fitted without overparameterization. EDA recommends using EDAFT to isolate shells, calibrating $k = \sqrt{\frac{2 m_e}{\hbar^2}(E-E_0)},$8 with a reference compound, and using FTEST to compare nested models such as Gaussian versus cumulant fits (Kuzmin, 2021).

Specific studies adopt these general principles in different ways. In lead-halide perovskites, data processing used Demeter 0.9.26 (Athena, Artemis), FEFF paths via IFEFFIT, multiple k-weights 1, 2, and 3, and a Hanning window. $k = \sqrt{\frac{2 m_e}{\hbar^2}(E-E_0)},$9 and $\exp\!\big(-2k^2\sigma^2(T)\big).$0 were determined at 20 K and then fixed for all temperatures, while the relative quantities $\exp\!\big(-2k^2\sigma^2(T)\big).$1, $\exp\!\big(-2k^2\sigma^2(T)\big).$2, and $\exp\!\big(-2k^2\sigma^2(T)\big).$3 were referenced to the lowest temperature to suppress sensitivity to $\exp\!\big(-2k^2\sigma^2(T)\big).$4 and $\exp\!\big(-2k^2\sigma^2(T)\big).$5 (Schuck et al., 2021). In the correlated Debye study on bcc and fcc metals, EXAFS $\exp\!\big(-2k^2\sigma^2(T)\big).$6 was analyzed from 2.5 to 17 $\exp\!\big(-2k^2\sigma^2(T)\big).$7 with a 10% Gaussian window, and the Fourier transforms were not phase-corrected (Kuzmin et al., 2024).

4. Temperature models and inversion strategies

The thermometric inversion is controlled by the chosen model for $\exp\!\big(-2k^2\sigma^2(T)\big).$8. For pair-relative vibrations, one commonly used expression is the Einstein model:

$\exp\!\big(-2k^2\sigma^2(T)\big).$9

where $$\chi(k)=\sum_j \frac{S_0^2 N_j F_j(k)}{kR_j^2} e^{-2k^2\sigma_j^2} e^{-2R_j/\lambda(k)} \sin\!\big(2kR_j+\phi_j(k)\big),$$0 is the Einstein frequency and $$\chi(k)=\sum_j \frac{S_0^2 N_j F_j(k)}{kR_j^2} e^{-2k^2\sigma_j^2} e^{-2R_j/\lambda(k)} \sin\!\big(2kR_j+\phi_j(k)\big),$$1 is the reduced mass. For crystalline solids with collective phonon modes, the correlated Debye model is often preferred:

$$\chi(k)=\sum_j \frac{S_0^2 N_j F_j(k)}{kR_j^2} e^{-2k^2\sigma_j^2} e^{-2R_j/\lambda(k)} \sin\!\big(2kR_j+\phi_j(k)\big),$$2

with $$\chi(k)=\sum_j \frac{S_0^2 N_j F_j(k)}{kR_j^2} e^{-2k^2\sigma_j^2} e^{-2R_j/\lambda(k)} \sin\!\big(2kR_j+\phi_j(k)\big),$$3 the correlated Debye temperature (Harmand et al., 2020).

In the dedicated study of monatomic metals, the correlated Debye formulation is expressed through the pair-projected vibrational density of states:

$$\chi(k)=\sum_j \frac{S_0^2 N_j F_j(k)}{kR_j^2} e^{-2k^2\sigma_j^2} e^{-2R_j/\lambda(k)} \sin\!\big(2kR_j+\phi_j(k)\big),$$4

and for an atom pair separated by $$\chi(k)=\sum_j \frac{S_0^2 N_j F_j(k)}{kR_j^2} e^{-2k^2\sigma_j^2} e^{-2R_j/\lambda(k)} \sin\!\big(2kR_j+\phi_j(k)\big),$$5,

$$\chi(k)=\sum_j \frac{S_0^2 N_j F_j(k)}{kR_j^2} e^{-2k^2\sigma_j^2} e^{-2R_j/\lambda(k)} \sin\!\big(2kR_j+\phi_j(k)\big),$$6

Only one materials parameter, $$\chi(k)=\sum_j \frac{S_0^2 N_j F_j(k)}{kR_j^2} e^{-2k^2\sigma_j^2} e^{-2R_j/\lambda(k)} \sin\!\big(2kR_j+\phi_j(k)\big),$$7, is then required to describe thermal disorder across shells, while lattice expansion is included through a constant offset $$\chi(k)=\sum_j \frac{S_0^2 N_j F_j(k)}{kR_j^2} e^{-2k^2\sigma_j^2} e^{-2R_j/\lambda(k)} \sin\!\big(2kR_j+\phi_j(k)\big),$$8 and a thermal expansion coefficient $$\chi(k)=\sum_j \frac{S_0^2 N_j F_j(k)}{kR_j^2} e^{-2k^2\sigma_j^2} e^{-2R_j/\lambda(k)} \sin\!\big(2kR_j+\phi_j(k)\big),$$9 (Kuzmin et al., 2024).

The EDA package does not implement built-in Einstein, Debye, or correlated Debye temperature models inside EDAFIT. Instead, it treats $S_0^2$0 as a per-shell fit parameter, after which $S_0^2$1 is modeled externally, or benchmarked with EDACA and EDARDF. This division of labor is explicit: EDA estimates structural parameters at each temperature, while the thermodynamic interpretation is carried out outside EDA (Kuzmin, 2021).

Anharmonic thermometry extends the inversion to higher cumulants. In lead-halide perovskites, the third cumulant obeys a $S_0^2$2-type classical approximation in the orthorhombic phase,

$S_0^2$3

where $S_0^2$4 is the harmonic force constant and $S_0^2$5 the cubic anharmonic force constant. The same work maps these constants onto a Morse potential,

$S_0^2$6

with

$S_0^2$7

where $S_0^2$8 and $S_0^2$9 are structure factors for the local octahedral environment (Schuck et al., 2021).

The choice among these models is system-dependent. The correlated Debye model is stated to be preferable for crystalline solids, especially monatomic bcc and fcc metals, while the Einstein model is often used for liquids or disordered states and for robust relative thermometry when limited $N_j$0-range is available (Harmand et al., 2020). This suggests that thermometric performance is governed as much by the validity of the lattice-dynamical model as by raw spectral quality.

5. Calibrated material classes and representative results

The strongest quantitative demonstrations of EXAFS-based thermometry currently come from two material classes: simple monatomic metals and lead-halide perovskites. In monatomic metals, the correlated Debye model was tested on bcc Cr, Mo, and W and fcc Cu and Ag, including contributions up to the 4th–7th coordination shell and multiple-scattering events up to the 4th order. The fitted Debye temperatures were 412 $N_j$1 16 K for Cr, 408 $N_j$2 5 K for Mo, 333 $N_j$3 4 K for W, 321 $N_j$4 3 K for Cu, and 222 $N_j$5 2 K for Ag. The method fails at low temperatures, where quantum effects dominate and MSRD values change only slightly, but becomes more accurate at higher temperatures, where the MSRD shows a near-linear dependence on temperature. At 293–300 K, typical deviations are within $N_j$6–20 K for Mo, W, Cu, and Ag, while chromium is notably worse; the average difference between experimental and obtained temperatures is reported as $N_j$7 K for all but Cr, where it is roughly four times larger (Kuzmin et al., 2024).

In lead-halide perovskites, the thermometric observable is the first-shell Pb–X path with $N_j$8, analyzed at the Pb $N_j$9 edge in transmission geometry. For MAPbI$\sigma^2$00, MAPbI$\sigma^2$01Cl$\sigma^2$02, and MAPbCl$\sigma^2$03, the orthorhombic-phase parallel MSRD follows an Einstein-model temperature dependence with fitted parameters: MAPbI$\sigma^2$04, $\sigma^2$05 K, $\sigma^2$06 THz, $\sigma^2$07 eV/\AA$\sigma^2$08; MAPbI$\sigma^2$09Cl$\sigma^2$10, $\sigma^2$11 K, $\sigma^2$12 THz, $\sigma^2$13 eV/\AA$\sigma^2$14; MAPbCl$\sigma^2$15, $\sigma^2$16 K, $\sigma^2$17 THz, $\sigma^2$18 eV/\AA$\sigma^2$19. In all three materials, the Einstein model derived in the orthorhombic phase also described $\sigma^2$20 reasonably well after the transition to tetragonal or cubic phases (Schuck et al., 2021).

The same perovskite study quantified anharmonicity through $\sigma^2$21. In the orthorhombic phase, $\sigma^2$22 was $\sigma^2$23 eV/\AA$\sigma^2$24 for MAPbI$\sigma^2$25, $\sigma^2$26 eV/\AA$\sigma^2$27 for MAPbI$\sigma^2$28Cl$\sigma^2$29, and $\sigma^2$30 eV/\AA$\sigma^2$31 for MAPbCl$\sigma^2$32. MAPbCl$\sigma^2$33 showed notably smaller $\sigma^2$34 in the orthorhombic phase, but increased anharmonicity after transition to the room-temperature phase; MAPbI$\sigma^2$35 showed decreased anharmonicity after transition; and the 2% Cl substitution produced only small differences from MAPbI$\sigma^2$36 in both phases (Schuck et al., 2021).

A concise comparison of representative calibrations is given below.

System class	Thermometric model	Representative calibration/result
bcc/fcc monatomic metals	Correlated Debye with multi-shell and multiple-scattering fitting	Cu: $\sigma^2$37 K; Ag: $\sigma^2$38 K; Mo: $\sigma^2$39 K; high-$\sigma^2$40 accuracy typically $\sigma^2$41–20 K
Lead-halide perovskites	First-shell Einstein calibration, optional $\sigma^2$42 refinement	MAPbI$\sigma^2$43: $\sigma^2$44 K, $\sigma^2$45 eV/\AA$\sigma^2$46
XFEL Fe K-edge implementation	MSRD fitting after stitched single-shot EXAFS	$\sigma^2$47 eV per shot; 10–20 shots yield several hundred eV and $\sigma^2$48–10 \AA$\sigma^2$49

These examples define two distinct operating modes. In simple metals, temperature is inferred from a many-shell, many-path inversion based on a single global $\sigma^2$50. In structurally softer and more anharmonic materials such as halide perovskites, temperature is inferred mainly from a first-shell calibration of $\sigma^2$51, with $\sigma^2$52 used as an anharmonic correction and as a phase-sensitive diagnostic.

6. Error sources, limitations, and methodological boundaries

A recurrent limitation is that EXAFS thermometry does not measure temperature directly; it measures local structural disorder and converts it into temperature under a model. This becomes problematic when the slope $\sigma^2$53 is small. In the correlated Debye study, the method fails at low temperatures because zero-point motion dominates and $\sigma^2$54 varies weakly with $\sigma^2$55. The authors identify a characteristic regime $\sigma^2$56 below which thermometry becomes insensitive and above which $\sigma^2$57 is nearly linear, stabilizing the inversion (Kuzmin et al., 2024).

Model inadequacy is a second limitation. The correlated Debye model assumes an isotropic Debye density of states and neglects anharmonicity; this is acceptable for distant shells in simple bcc and fcc metals but less accurate for nearest shells and for systems with complex phonon spectra or strong anisotropy. Conversely, the Gaussian approximation is adequate only for near-harmonic behavior; for strongly anharmonic systems, EDA recommends the cumulant model, while EDARDF reconstructs $\sigma^2$58 directly under positivity and smoothness constraints (Kuzmin, 2021).

Experimental systematics are especially important at XFELs. Shot-to-shot SASE jaggedness is mitigated by overcompressed operation and simultaneous $\sigma^2$59 normalization. Spatial chirp and spectrometer mismatch are reduced by vertical scattering geometry, careful alignment, and correction by $\sigma^2$60. Energy calibration drift is managed by deriving $\sigma^2$61 shortly before inserting the sample and by stitching spectra with overlapping energy windows. Bandwidth limitations mean that a single shot spans only $\sigma^2$62 eV, so a useful EXAFS $\sigma^2$63-range requires concatenation across central-energy retunes. Focused-beam operation introduces an additional bias: the XFEL probe can self-heat the sample and alter XANES features, so attenuation, larger spot size, or reduced exposure are recommended for thermometry (Harmand et al., 2020).

Material-specific complications also bound applicability. The transmission-based single-shot XFEL method does not support fluorescence-mode XAS, which limits use in dilute systems or many oxides. In the lead-halide perovskites, phase transitions near 150–165 K produce discontinuities in $\sigma^2$64, $\sigma^2$65, and anisotropy; phase misassignment therefore introduces systematic temperature error. The authors recommend treating phases separately and using XRD when possible (Schuck et al., 2021).

These constraints yield a consistent practical picture. EXAFS-based thermometry is most reliable when the structural model is fixed, the relevant shells are well isolated, $\sigma^2$66 and $\sigma^2$67 are constrained from baseline data, and the experiment operates in a regime where thermal disorder dominates over static disorder and zero-point motion. Under those conditions, the method can provide either equilibrium local thermometry or femtosecond-resolved transient thermometry through MSRD analysis, with the attainable precision determined jointly by $\sigma^2$68-range, percent-level noise, and the validity of the underlying temperature model.