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Scientific tutorial · Computational Atomic-Resolution STEM: Concepts and Methods · Scientific Computing ·

When Can We Add Atomic Images Together?

An evidence-synchronized visual guide to when HAADF-STEM atomic responses can be added, how geometry changes non-additivity, and when higher image order is needed.

How to use this tutorial

This is one tutorial with three reading depths rather than three separate versions. Core: follow the main story, figures, findings, and limitations. Technical: add the Methods depth when you want the construction of matched subsets, simulation controls, validation, and numerical checks. Expert/reproducibility: add Technical depth, equations, provenance, and Sources for the formal decomposition and evidence boundaries. Start with the Core route and open deeper layers only when useful; no other tutorial is a prerequisite.

Learning objectives

By the end of the Core route, you should be able to explain why a HAADF-STEM atomic feature is an imaging response rather than a literal atom picture; distinguish one-body addition, pair corrections, and a residual beyond pair order; explain why transverse and beam-direction separations must be treated separately; distinguish full-image error from error normalized by a weak interaction residual; state what the tested Au2/Au3 evidence supports and does not support; and explain why the 0.75 Å applicability boundary is operational rather than a universal physical threshold.

Start with what the microscope measures

HAADF-STEM—high-angle annular dark-field scanning transmission electron microscopy—moves a focused electron probe across a specimen. At each probe position, a detector records electrons scattered into a high-angle annular range. Repeating that measurement across the scan builds a two-dimensional intensity map.

The bright features are therefore not literal photographs of spherical atoms. They are the microscope response produced by electron–specimen interaction under a particular beam, detector, geometry, sampling, and numerical configuration.

From electron scattering to image intensity

Schematic showing a focused electron probe scanning a specimen, high-angle scattering reaching an annular detector, and an image intensity map building pixel by pixel.
Figure 1 A focused probe is scanned across the specimen. High-angle scattered intensity is recorded at each scan position and becomes part of the HAADF-STEM image.

One atom gives a spatial response

Place one gold atom in the calculation while keeping the imaging conditions fixed. The result is a spatial intensity field. We can denote the response of atom A by I_{A}(r), where r labels position in the image.

The notation is useful, but the idea is simple: one selected atom, calculated on its own under the comparison conditions, produces one matched singleton image.

One atom, one microscope response

A single gold atom at a fixed specimen position linked to a smooth two-dimensional intensity response rather than a literal atom-shaped photograph.
Figure 2 The image of a singleton is a spatial microscope response. Its bright feature is not a literal drawing of the atom.

The question

If we know the matched image produced by each atom separately, can we reconstruct the image of the atoms together simply by adding those images? If addition fails, how much extra image structure requires pair-specific or higher-order terms?

Can two isolated responses predict the joint image?

Diagram showing atom A and atom B calculated separately, their images added to make an additive prediction, and that prediction compared with the joint two-atom calculation and residual.
Figure 3 The central test compares an additive prediction assembled from matched singleton calculations with the image calculated for the atoms jointly.

Why addition can fail

The additive construction is a hypothesis, not a law. When atoms occupy geometries that strongly couple image formation, the joint calculation can contain intensity structure absent from either isolated response. In particular, atoms separated along the beam can still lie close together in projection.

We describe their separation across the image plane by d_perp. Geometry along the beam also matters, so one distance alone does not completely describe the response.

Geometry can make responses non-independent

Two-panel schematic contrasting atoms separated in projection with atoms strongly overlapping along the beam, and defining transverse projected separation d perpendicular.
Figure 4 Projected separation is measured across the image plane. Strong projected overlap can produce a joint response that is poorly represented by isolated responses alone.

A worked example: pair residual

Use deliberately simple arithmetic before looking at full images. At one pixel, suppose the matched singleton calculations give intensities 4 and 5. Addition predicts 9. If the joint two-atom calculation gives 11, then the residual at that pixel is 2.

Another pixel could have a negative residual. The research does not judge the interaction from one pixel: it compares the complete images and residual fields.

Measure the part that addition cannot explain

Five-panel worked example showing singleton A, singleton B, their additive prediction, the joint pair image, and the signed residual obtained by subtraction.
Figure 5 Subtract the additive prediction from the joint calculation. The remaining signed field is the pair contribution at the image level.

What does matched mean?

The singleton images are not generic pictures of a gold atom pasted into the scene. For a matched subset, the selected atom or atoms retain their exact positions from the complete geometry while the other atoms are removed. The fixed comparison context is preserved.

This matters because the residual should change because atoms were present together—not because we silently moved them or changed the microscope configuration.

Remove atoms, not the geometry

Three-panel schematic showing a full three-atom geometry, a singleton retaining atom B at its exact original position, and a pair retaining atoms A and C at their exact original positions.
Figure 6 Matched subsets keep the selected atoms at their exact positions in the fixed comparison geometry. Only membership changes.
Methods: How matching is enforced

In the Au3 control implementation, indexed singleton and pair subsets are constructed directly from the complete fixed-cell geometry at the original coordinates. The same image grid and microscope configuration are then used for the decomposition. This makes the subtraction an operational comparison of atomic membership.

The one-body reconstruction

\[I^{(1)}_S(\mathbf r)=\sum_{i\in S} I_{\{i\}}(\mathbf r)\]
(1)

Add the matched singleton image for every atom in the structure. The superscript (1) indicates that only one-body terms have been included.

Start by adding every matched singleton

Diagram showing three matched singleton response images being summed into a one-body reconstruction for a three-atom structure.
Figure 7 The one-body reconstruction is built from the matched response of each atom calculated separately.

Now isolate what a pair adds

Calculate atoms i and j together, then subtract both matched singleton images. What remains is the pair-specific image contribution. A zero field would mean exact one-body additivity for that pair under the tested configuration.

Irreducible pair contribution

\[\Delta I_{ij}=I_{\{i,j\}}-I_{\{i\}}-I_{\{j\}}\]
(2)

The pair contribution is image content present in the joint pair calculation that cannot be represented by the two matched singleton responses.

Subtract what the single atoms already explain

Visual equation showing the joint pair image minus the two matched singleton images leaving a signed pair residual field.
Figure 8 The pair residual has a precise operational meaning because every term belongs to the same matched comparison.

What if every pair is still not enough?

For three atoms, first construct the one-body sum and then add all three exact matched pair contributions. Compare that pair-corrected reconstruction with the full three-atom calculation. Any field that remains cannot be represented by the matched one- and two-body terms used in that reconstruction.

Irreducible three-body image term

\[\Delta I_{ijk}=I_{\{i,j,k\}}-\sum_{a<b}\Delta I_{ab}-\sum_a I_{\{a\}}\]
(3)

This is an image-level inclusion–exclusion residual. It is not a direct count of microscopic three-particle scattering paths.

Build the image in layers

Layered diagram showing a three-atom full image, the singleton sum, all three pair corrections, the pair-corrected reconstruction, and the remaining three-body image residual.
Figure 9 One-body terms form the baseline; exact pair terms are added next; the remaining difference is the irreducible three-atom image-level residual.
Technical depth: General subset form
\[\Delta I_T=I_T-\sum_{U\subsetneq T}\Delta I_U\]

The same bookkeeping can be written recursively for any matched atomic subset T. The tutorial only needs the one-, two-, and three-atom cases above.

Now test the idea against the frozen study

The preceding figures were explanatory. From this point onward, the tutorial switches to Paper 1 evidence. The numerical plots below use frozen result records or retained publication figures, and their captions state the evidence boundary explicitly. Numerical values and qualifications were rechecked against the final frozen Paper 1 evidence state on 2026-09-24.

Pair non-additivity is strongly geometry dependent

Frozen three-panel Paper 1 pair-response map over transverse and beam-direction separation, showing relative pair residual magnitude, scale-adjusted residual, and optimal scale on the sampled Au2 production grid.
Figure 10 Frozen publication figure from the 0.50 Å production pair map. The panels show the exact sampled 6 × 5 geometry grid; they are discrete cells rather than an interpolated physical phase diagram.

Read one cross-section before reading the whole map

At fixed beam-direction separation d_parallel = 2 Å, the 0.50 Å production-map relative pair contribution R_12 = ||ΔI_12||_2 / ||I_12||_2 changes from 12.604515% at exact projected overlap to 9.319616% at d_perp = 0.25 Å, 0.294951% at 1 Å, and 0.008963% at 3 Å.

The complementary exact-overlap cross-section holds d_perp = 0 and varies d_parallel: 43.059447%, 33.012768%, 12.604515%, 34.843657%, and 39.098840% at 0.5, 1, 2, 4, and 6 Å. Together the two slices show both orders-of-magnitude transverse suppression and strongly non-monotonic axial dependence. The result should not be simplified to a universal one-distance decay law.

The same result as one simple cross-section

Four frozen Au2 production-map points at fixed beam-direction separation 2 angstrom show relative pair contribution falling from 12.604515 percent at projected overlap to 0.008963 percent at 3 angstrom transverse separation.
Figure 11 Frozen 0.50 Å production-map cross-section at d_parallel = 2 Å. The logarithmic vertical axis keeps the strong-overlap and tiny 3 Å residual visible on the same plot.

Projected overlap still depends strongly on beam-direction separation

Five frozen Au2 production-map points at exact projected overlap show strongly non-monotonic relative pair contribution as beam-direction separation changes from 0.5 to 6 angstrom.
Figure 12 Frozen 0.50 Å production-map cross-section at d_perp = 0. The sampled pair contribution falls to 12.60% at d_parallel = 2 Å and rises again above 34% at 4–6 Å; this is a discrete cross-section, not a continuous law.

A sophisticated correction can pass and still be unnecessary

Twelve independent non-column Au2 validation cases split across three transverse separations, comparing logarithmic full-pair relative L2 errors for the frozen rank-3 correction and pure matched singleton addition.
Figure 13 Frozen validation records. The rank-3 correction passed its preregistered absolute gates, while the later additive null had lower error in 10 of 12 cases. The head-to-head additive comparison was post-hoc, not a preregistered superiority test.

Passing a gate is not the same as earning extra complexity

Here rank 3 means that, after translating pair-residual images to a common projected midpoint, the compact representation retained three singular-value-decomposition (SVD) spatial modes. The rank, applicability boundary, and interpolation rule were fixed using development data before independent validation.

Across the 12 authorized non-column validation geometries, the frozen rank-3 model had median full-image error 0.1353% and maximum 0.8862%. Pure matched singleton addition had median 0.0992% and maximum 0.9127%, and was more accurate in 10 of the 12 cases.

The scientifically useful conclusion is not that the pair residual is unreal. It is that these complete non-column pair images were already approximately additive at the sub-percent level, so the learned correction was not supported as an overall predictive improvement in this tested regime.

Full-image accuracy and residual accuracy are different questions

Scatter plot for the same twelve validation predictions showing sub-percent rank-3 error when normalized by the complete pair image but roughly 97 to 197 percent error when normalized by the much weaker interaction residual.
Figure 14 Same 12 frozen validation cases, two denominators. Accurate reconstruction of the complete image does not imply high-fidelity reconstruction of a weak residual field.
Technical depth: Why relative L2 here?

The primary Paper 1 metric is not a human-perceptual similarity score. Compared images share the same physical scan grid, atomic coordinates, microscope configuration, and intensity convention, so each pixel represents the same physical probe position and an intensity mismatch is part of the reconstruction error.

Relative L2 measures the complete registered image-field discrepancy relative to the reference image. Residual normalization changes the scientific target from full-image fidelity to fidelity of the weaker non-additive component. Global scale adjustment diagnoses amplitude versus spatial mismatch, while cosine similarity checks whether residual shape persists under refinement.

SSIM and human visual judgments address perceptual similarity instead. That is a separate observer question, not the acceptance target here.

Why the denominator changes the conclusion

When ΔI_12 is tiny compared with I_12, an error that is negligible relative to the full image can be comparable to the entire residual itself. The validation therefore supports approximate additivity of the full image. It does not establish that the weak pair residual was reconstructed accurately relative to its own norm.

Under strong overlap, every exact pair can still be insufficient

Frozen Paper 1 publication panel comparing the full Au3 image, one-body reconstruction, exact pair-corrected reconstruction, and irreducible three-atom residual for beam-aligned, near-overlap, and asymmetric non-column geometries.
Figure 15 Retained frozen publication figure. Beam-aligned and near-overlap controls retain substantial residuals after every exact matched pair term; the asymmetric non-column control is nearly exhausted by pair order. The authoritative 0.50 Å source arrays are preserved in the durable Paper 1 research archive rather than duplicated into this portfolio repository.

The same decomposition gives three very different answers

Beam aligned: E1 = 28.860416%, E2 = 10.313025%, F2 = 64.265848%. Near overlap: E1 = 13.944548%, E2 = 4.849940%, F2 = 65.219811%. Asymmetric non-column: E1 = 0.522053%, E2 = 0.008114%, F2 = 98.445841%.

E1 is the singleton-only full-image error. E2 is the error after every exact matched pair term is included, and is also the full-image-normalized magnitude of the remaining three-atom residual. F2 asks what fraction of the original singleton discrepancy the pairs removed. Improvement and final adequacy are therefore separate questions.

The strong-overlap residuals survive the tested refinement

Two-panel numerical stability summary comparing 0.50 and 0.25 angstrom multislice thickness for the near-overlap and beam-aligned Au3 pair-truncation residuals.
Figure 16 Pre-specified slice-thickness refinement. Both strong-overlap controls passed the frozen stability gates. This supports persistence under the tested refinement; it is not a proof of universal numerical convergence.

A skeptical alternative was tested directly

For near overlap, reducing slice thickness from 0.50 to 0.25 Å changed E2 by only 0.000335 percentage points and the residual field by 0.009539% relative to the refined residual norm. For beam alignment, E2 changed by 0.006531 percentage points; the residual field changed by 0.054798% and had cosine similarity 0.9999999986.

After the original confirmatory protocol, the asymmetric non-column control was also subjected to a separately frozen reviewer-motivated 0.50-to-0.25 Å slice-thickness check. Its E1, E2, and F2 changes were 0.0001305, 0.00000145, and 0.000665 percentage points; the residual field changed by 0.202% with cosine similarity 0.999998. It passed the reused stability tolerances, but remains post-hoc robustness evidence.

A second post-review check expanded the beam-aligned lateral cell from 15×12 to 21×18 Å while preserving relative geometry and scan registration. E2 changed from 10.3130% to 10.2920%; the irreducible residual field changed by 0.783% and retained cosine similarity 0.99999947. This supports persistence under the tested padding, not universal cell-size convergence.

The 0.75 Å value came from a stopped, ordered search

Four preregistered projected-separation candidate domains shown in order: 0.25 and 0.50 angstrom fail the frozen development gates, 0.75 angstrom is the first passing domain with rank 3, and 1.00 angstrom is not inspected.
Figure 17 Development-only applicability selection. The first predeclared passing candidate was d_perp ≥ 0.75 Å. Because the search used discrete candidate boundaries and stopped at the first pass, 0.75 Å is a grid-resolved operational boundary, not a universal physical threshold.

What the boundary does—and does not—mean

The candidate domains were d_perp ≥ 0.25, 0.50, 0.75, and 1.00 Å, evaluated in that fixed order. The 0.25 and 0.50 Å domains failed cross-validation criteria even when direct reconstruction was accurate. The 0.75 Å domain was the first to pass every frozen gate, with rank 3 the smallest eligible representation, so the protocol stopped before inspecting 1.00 Å.

This is not a universal physical transition at 0.75 Å. It identifies the first tested development domain in which the unchanged compact representation generalized adequately according to study-specific decision criteria.

The scientific lesson

For the configurations tested here, effective image order is geometry dependent. Independently tested non-column Au2 images were already approximately additive at the sub-percent full-image level. Under strong projected overlap, pair non-additivity became much larger; in the tested beam-aligned and near-overlap Au3 controls, even every exact pair contribution left substantial higher-order image residuals. Use the simplest image representation that the geometry and evidence support.

Scope and limitations

The study uses controlled static-lattice multislice calculations for Au2 and Au3 under fixed imaging conditions. Frozen-phonon thermal diffuse scattering was not included, nor were post-simulation shot noise or detector-degradation models. Scan-spacing convergence was not established, and lateral-cell sensitivity was tested only in the later beam-aligned robustness check rather than as a systematic convergence study. The reported magnitudes and 0.75 Å development boundary therefore characterize this deterministic simulation study, not finite-temperature experimental detectability thresholds or universal material laws.

Concept glossary

Key terms used consistently throughout the tutorial.
TermMeaning in this tutorial
Matched subset Selected atoms keep their exact coordinates from the full structure while the other atoms are removed and the comparison configuration is kept fixed.
One-body reconstruction The sum of all matched singleton images.
Pair residual ΔI_ij The part of a matched two-atom image not explained by adding its two matched singleton images.
Rank-3 representation A compact development-stage description of aligned pair-residual images using three retained SVD spatial modes; the rank was fixed before independent validation.
Higher-order residual What remains after the included lower-order matched contributions are subtracted; an image-level inclusion–exclusion quantity, not a unique microscopic scattering path.
R_12 Pair-residual magnitude normalized by the complete two-atom image.
E1 Full-image error of the singleton-only reconstruction.
E2 Full-image error after every exact matched pair term is included for the Au3 controls.
F2 Fraction of the original one-body discrepancy removed by the exact pair terms.
Operational boundary A boundary selected by a declared numerical rule over the sampled development grid, not automatically a physical transition.
Multislice simulation A numerical electron-microscopy method that propagates the electron wave through successive slices of the specimen potential.
Static lattice A calculation with atoms fixed at specified positions; no frozen-phonon thermal-displacement ensemble is included here.
Preregistered / frozen gate An acceptance criterion fixed before the confirmatory validation result is inspected, so the threshold is not adjusted to fit that result.
Post-hoc comparison An analysis performed after the confirmatory decision was fixed; informative, but not presented as a preregistered superiority test.
Relative L2 error Magnitude of the registered pixelwise error field divided by the reference-image magnitude; used here for physical reconstruction fidelity rather than human-perceptual similarity.
Scale-adjusted residual Error remaining after fitting one global multiplicative intensity scale; used diagnostically to separate overall amplitude mismatch from spatial-field mismatch.
Cosine similarity Field-shape alignment of two residual vectors independent of overall magnitude; used here in numerical-robustness checks.

Check your understanding

1. Why is a bright HAADF-STEM feature not a direct picture of an atom?

2. If ΔI_12 = 0, what does that say about the matched two-atom image?

3. Why can geometries with similar three-dimensional atom–atom distance behave differently when d_perp and d_parallel differ?

4. The frozen rank-3 Au2 representation passed its preregistered gates, yet simple addition had lower error in 10 of 12 post-hoc comparisons. What is the defensible conclusion?

5. What does a substantial Au3 E2 show, and what does it not identify?

6. Why should the 0.75 Å value not be interpreted as a universal experimental threshold?

Answers: Check your understanding

1. The feature is an imaging-system response, not the physical outline of the atom.

2. The joint pair image is exactly reproduced by the two matched singleton images at image level for that tested configuration.

3. Projected overlap and propagation along the beam are different geometric controls; the frozen evidence shows strong transverse suppression and non-monotonic axial dependence.

4. The tested non-column full images are already approximately additive at sub-percent error, so the learned correction is not scientifically necessary there despite being a real fitted correction.

5. Substantial E2 means exact pair terms do not fully reconstruct the complete image; it does not identify a unique microscopic three-particle scattering pathway or many-body potential.

6. The value is the first sampled development boundary that passed the frozen numerical criteria, so it is operational and grid-resolved rather than a universal physical transition.

Sources and further reading

Geometry-Resolved Few-Body Decomposition of HAADF-STEM Image Formation. Submission-ready unpublished manuscript. The first nine explanatory schematics teach the method. Within the later evidence section, two figures reuse retained frozen publication figures from the research repository; the remaining evidence figures visualize frozen production-map cross-sections, validation, normalization, refinement, and boundary-selection records. The underlying research arrays remain governed by the durable Paper 1 archive rather than being duplicated into the portfolio.

Explore a connected question

Optional reading; each tutorial supplies its own background.
TutorialWhat it adds
Building Fast, Trustworthy Models of Electron Microscope Images Explore how accurate atomic rendering and independent tests guide a fast approximation.
What Can an Electron Microscope Image Tell Us About Atomic Structure? Explore why a trustworthy image model still needs uncertainty and qualification before structural inference.