Learning Interatomic Force Coefficients from X-ray Thermal Diffuse Scattering Data

Abstract

We present a fully automated framework for extracting interatomic force constants (IFCs) directly from X-ray thermal diffuse scattering (TDS) data. By formulating scattering intensity as a differentiable function of a symmetry-reduced IFC parameterization, we enable gradient-based optimization via direct, Cholesky-based sampling of correlated atomic displacements at thermal equilibrium. This approach bypasses the computational bottleneck of repeated Hessian matrix diagonalizations, significantly accelerating the inversion process. Benchmark tests demonstrate that the framework accurately recovers ground-truth IFCs and phonon dispersion relations, providing a robust, high-throughput pathway for studying lattice dynamics across diverse crystalline materials. This method bridges the gap between experimental observations and computational modeling, enabling the direct integration of TDS data into the refinement of high-fidelity inter-atomic potentials.

Figures (7)

• Figure 1: Schematic representation of thermal diffuse scattering. In the absence of atomic vibrations, elastic scattering leads to sharp Bragg peaks in the scattering pattern. Atomic vibrations introduce diffuse signal around the Bragg peaks due to inelastic scattering. The specific pattern of the diffuse scattering signal reflects the correlation of atomic vibrations governed by interatomic force, making TDS a direct probe of the interatomic force constants (IFCs) in materials.
• Figure 2: Schematic of the differentiable optimization loop. The forward pass (bold arrows) generates predicted TDS intensities through stochastic displacement sampling and FFT-based scattering calculations. During the backward pass (thin arrow), gradients of the loss function are propagated to update the parameters $\phi$ iteratively.
• Figure 3: Illustration of the IFC hierarchy for the first five neighbor shells. The 0-th shell ($a_0$) corresponds to the self-interaction term of the Hessian matrix. Coefficients $b_k, c_k, d_k, e_k,$ and $f_k$ correspond to the force constants for the 1st, 2nd, 3rd, 4th, and 5th coordination shells, respectively.
• Figure 4: a) Comparison of the computed TDS signals at 300 K on the plane of $q_z = 0$ (a.k.a. the HK(L=0) slice in the reciprocal space). The sections marked as "Target" correspond to the U3 EAM potential. The section marked as "Initial" corresponds to the MEAM potential. The section marked as "Fit" corresponds to the IFCs after the training. b) The error of the computed TDS signals based on the initial and fitted IFCs when compared with the target. c) The training loss (light purple) is shown together with the moving average (dark purple) and the final value (dashed). d) The phonon dispersion relation computed from the Target, Initial, and Fitted IFCs. The curves corresponding to the Target and Fitted IFCs are nearly indistinguishable here. The inset shows the first Brillouin zone with the paths connecting high symmetry points shown in red.
• Figure 5: The evolution of the fitted IFCs corresponding to different neighboring shells during training. The initial values of IFCs correspond to the MEAM potential, while the target values correspond to the U3 EAM potential.