A computational comparison of three models used to simulate transmission electron microscopy (TEM) diffraction found notably larger low-angle gaps in strontium titanate (STO), a heavy, anisotropic crystal. The absolute differences there ranged from 10^-1.5 I0 to 10^-2 I0, corresponding to errors of the order of units of percent—larger than the gaps seen in diamond.
In diamond, the correlated-motion frozen-phonon model (MD-FPM) showed the largest differences from the other approaches. Its central-Bragg discrepancy was about 10^-3 I0 and remained comparable near HOLZ positions at around 110 mrad.
A comparison of three computational views
The study examined when complex absorptive potentials, or CAP, agree or disagree with frozen-phonon approaches in TEM diffraction simulations. It focused on elastic-channel intensity—the part separated from the thermal diffuse scattering signal—and on the intensity of Bragg spots.
The researchers compared CAP, represented by parametrized complex form factors, with correlated atomic-motion FPM and independent harmonic Einstein FPM. All three approaches were tested through multislice propagation in diamond and STO.
The frozen-phonon averages used 100 correlated-motion snapshots for each material and 50 Einstein-FPM snapshots. The paper states that the Einstein version converged faster.
Patterns that look alike still hide differences
At 50 nm diamond thickness, the two frozen-phonon variants produced qualitatively similar thermal diffuse scattering patterns, but MD-FPM spread intensity more broadly around Bragg spots. Their total-intensity patterns also showed Kikuchi lines and HOLZ reflections near 100 mrad.
To compare the models across scattering angles, the researchers combined intensity around rings of constant angle. They then calculated pairwise absolute differences and relative differences at Bragg reflections.
The detector changes the comparison
The detector’s collection area changed the apparent agreement in the diamond case. With a 110 mrad circular aperture, the maximum inter-model discrepancy fell to approximately 10^-4.5 I0, which indicated better agreement with a larger collection area under the tested conditions.
That improvement did not remove every difference. Diamond still showed the largest discrepancies for MD-FPM at high-angle HOLZ positions, while STO retained notable high-angle differences, although some fell below the study’s convergence threshold.
Thickness and convergence matter
Thickness added another separation between the models. The diffuse thermal-scattering background in frozen-phonon maps increased as the simulated specimen became thicker, while it was absent from CAP maps. In STO, MD-FPM differences from the other models increased at low-to-mid scattering angles around 50 mrad as thickness grew.
The maximum-thickness profile comparisons used diamond at 50 nm and STO at 30 nm. Those values define the main thickness range examined in the reported profile analysis.
The researchers assessed convergence by calculating the standard deviation over the last 50 snapshot additions. The reported values were 5.65 × 10^-5 I0 for diamond MD-FPM and 1.65 × 10^-5 I0 for diamond Einstein FPM; for STO, they were 6.73 × 10^-5 and 2.71 × 10^-5 I0, respectively. Differences below these thresholds were treated as inconclusive.
A conditional conclusion
The authors regard correlated MD-FPM as the most physically complete of the three models. Their practical conclusion is conditional: CAP may be adequate for light-element systems, moderate specimen thicknesses and large detector collection angles, while larger errors are expected for heavy, anisotropic systems and at high scattering angles.
The evidence is bounded by the design. The study is a computational series of multislice calculations involving diamond and STO at 300 kV, with no reported randomization procedure or experimental control group.
The manuscript is an arXiv version 1 preprint dated 25 Aug 2026, with no journal or DOI reported in the supplied metadata. Funding, conflicts of interest, data availability and code availability were also not reported.
Paper data and sources
Original title: Thermal diffuse scattering in TEM: complex absorptive potentials compared to the frozen phonon model
Authors: Martin Hájek, Ján Rusz
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text