Scientific tutorial · Computational Atomic-Resolution STEM: Concepts and Methods · Scientific Computing ·
Building Fast, Trustworthy Models of Electron Microscope Images
How an accurate one-body atlas, a continuous interaction measure, independent gate testing, and abstention combine into an interpretable fast surrogate.
Choose your reading depth
Core explanation: read the visible story, example, findings, and limitations for the main argument. Open Methods for how the comparison works, or Technical depth for equations and assumptions. These are choices about detail, not labels for readers. No other tutorial is required.
Start with the image
HAADF-STEM (high-angle annular dark-field scanning transmission electron microscopy) scans a focused electron probe across a specimen and records electrons scattered to large angles. The resulting image is a two-dimensional intensity map of a three-dimensional object. Bright features can carry information about atoms, but the image also depends on the beam, detector, atomic arrangement, and measurement conditions.
A surrogate is a cheaper approximation to an expensive calculation. Here the reference is multislice simulation, which propagates an electron wave through successive slices of a specimen. A useful fast model must reproduce isolated-atom images accurately before we ask whether it also needs interactions between atoms.
An atlas is a library of simulated atomic images indexed by relevant conditions. A gate chooses a modelling action from geometry and validation evidence. Abstention means sending an uncertain case for further assessment instead of pretending the fast approximation is qualified.
The design problem
How can a fast HAADF-STEM surrogate stay interpretable while spending complexity only where atomic geometry indicates that interactions may matter?
Methods: Separate rendering from interaction physics
The surrogate study treats accurate one-body rendering, continuous interaction strength, and discrete model selection as different problems. This prevents a poor atomic renderer from being mistaken for physical non-additivity.
A worked example
Illustrative error budget, not an additional experiment: suppose a simple atomic spot model misses the reference image by 7%, while the interaction you hope to capture changes it by 0.1%. Adding an interaction correction before fixing the spot model obscures the question. First compare the atlas with exact single-atom sums; then compare those exact sums with joint simulations.
These comparisons separate representation error from physical non-additivity. Even after detecting an interaction, test the proposed correction on independent cases: a correction that improves one geometry may worsen another.
Technical depth: Environment-conditioned one-body atlas
The empirical atlas A may depend on species, depth, sampling phase, and microscope configuration.
Why the renderer matters
Table 1
| Representation | Full-image relative L2 error | Consequence |
|---|---|---|
| Calibrated isotropic Gaussian | Median about 6.86% on Au2 development data | Too inaccurate for a sub-percent interaction question. |
| Depth- and sampling-phase-aware atlas | Frozen 18-case test: median 0.0715%; mean 0.0621%; maximum 0.0918% | One-body rendering error was reduced by roughly two orders of magnitude. |
Technical depth: Continuous interaction strength
The continuous score is measured before assigning a discrete modelling action.
Operational decision regions
Table 2
| Transverse separation | Action region | Evidence status |
|---|---|---|
| d_perp <= 0.75 Å | Interaction-resolvable | Development-supported; not independently confirmed. |
| 0.75 Å < d_perp < 2.0 Å | Uncertain | Do not force either additive or corrected treatment. |
| d_perp >= 2.0 Å | Additive-safe | Independent evidence at 2.5 Å; not validation of the entire region. |
Use an independent set to challenge the gate
On 12 already-computed independent few-body non-column geometries that were not used to develop the atlas or gate, the atlas reproduced exact additive images with 0.0561–0.0710% error. At d_perp = 2.5 Å, physical non-additivity was only 0.0172–0.0291%, corroborating additive-safe behaviour.
A useful negative result
The frozen rank-1 pair correction improved two of four external cases and worsened two. Detectable non-additivity therefore did not justify a universal low-rank correction. A trustworthy surrogate must be able to abstain.
Technical depth: Geometry-gated routing
These are operational decisions over a continuous interaction landscape, not physical discontinuities. The small-separation route does not designate a validated correction.
What remains conditional
The numerical boundaries belong to the tested Au system and fixed imaging configuration. The interaction-resolvable boundary remains development-supported, and the rejected rank-1 correction must not be presented as a validated interaction model.
Sources and further reading
Geometry-Gated Interpretable Surrogates for HAADF-STEM Image Formation. Manuscript in preparation. This explanation distinguishes illustrative examples from the frozen results; its numerical findings apply to the configurations described above.