Anharmonic Debye–Waller Factor in the X-ray Absorption Fine Structure (XAFS) Spectroscopy of Silver: A Classical Statistical Approach

Abstract

The anharmonic Debye–Waller (DW) factor in the X-ray absorption fine structure (XAFS) spectroscopy of crystalline silver (Ag) is explicitly examined, accounting for thermal disorder. A framework combining classical statistical mechanics with the correlated Einstein model is employed to describe atomic interactions and lattice vibrations. The DW factor characterizes attenuation of the XAFS amplitude induced by thermal motion and is essential for extracting structural and thermodynamic information from XAFS spectra. The derived thermodynamic XAFS parameters incorporate atomic correlations and anharmonicity, accounting for nearest-neighbor effects on both the absorber and backscatterer. The analytical expressions for these parameters are derived in a simple, explicit, temperature-dependent form, making them convenient for XAFS analysis. The applicability of the classical approximation is quantitatively assessed through comparison with quantum-mechanical expressions for lattice vibrations. Adopting a conservative criterion of less than 10% deviation, the classical description is reliable for temperatures T $\gtrsim$ 148  K, and the quantitative analysis is restricted to this range. Numerical results for Ag show good agreement with predictions from the anharmonic correlated Debye model, fitting method, and experimental data within the validated temperature range.

Results

at lower temperatures illustrate the breakdown of the classical approximation and are not used for quantitative interpretation. Theoretical and experimental Fourier transform magnitudes |χ(R)| from k² weighted spectra match closely near the first-shell peak. Overall, the present approach provides an effective framework for anharmonic XAFS DW analysis in Ag at elevated temperatures and is extendable to other metals.

Graphical abstract

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