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arxiv: 2605.31107 · v1 · pith:2PA77PW3new · submitted 2026-05-29 · ❄️ cond-mat.mtrl-sci · physics.data-an

Clustering in atom probe tomography data: coordination number metric, percolation-based parameter scaling, and size effects

Mykola Lazarev , John Banhart This is my paper

Pith reviewed 2026-06-28 22:01 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci physics.data-an
keywords atom probe tomographyclusteringcoordination numberpercolationprecipitatessize effectsreconstruction artifactsphase nucleation
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The pith

A composition-based coordination number metric combined with percolation scaling improves clustering of nanoscale features in atom probe tomography data and compensates for reconstruction artifacts better than solute-density approaches.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a clustering method for atom probe tomography that uses a coordination-number metric defined from local atomic composition rather than Euclidean distances or solute atoms alone. It scales clustering properties to percolation thresholds to produce a self-similar variable that stays nearly constant across metrics, parameters, and structural variations. Simulations incorporating detection efficiency, spatial resolution limits, and other typical artifacts show that this coordination approach handles heterogeneous dilations more effectively and yields better results than conventional solute-density methods, including for small precipitates where matrix interfaces broaden the clustering response.

Core claim

The central claim is that a coordination-number metric based on composition, when combined with scaling of clustering properties to the corresponding percolation thresholds, defines a self-similar variable that is nearly invariant with respect to the choice of metric, clustering parameters, and structural disorder; this variable enables formal transfer of optimal parameters between methods and, in simulations that include finite spatial resolution, detection efficiency, and reconstruction artifacts, the coordination-number approach compensates for heterogeneous dilations and outperforms solute-density-based methods in all tested scenarios.

What carries the argument

The coordination-number metric based on local composition, which incorporates solvent atoms and is scaled to percolation thresholds to yield an invariant self-similar clustering variable.

If this is right

  • Optimal clustering parameters identified with one metric can be transferred to another metric via the self-similar variable.
  • Small precipitates exhibit broadened clustering curves because the precipitate-matrix interface alters the local composition spectrum.
  • The method remains effective when typical APT artifacts such as limited detection efficiency and spatial resolution are present.
  • Clustering descriptions become largely independent of the specific distance or density metric once scaled to the percolation threshold.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same percolation-scaling step might reduce sensitivity to reconstruction algorithm choices across different atom probe instruments.
  • The approach could be tested on experimental datasets where precipitate sizes are independently verified by another microscopy technique to check consistency of the invariant variable.
  • Extending the coordination definition to include higher-order neighbor shells might further stabilize the scaled variable in very dilute alloys.

Load-bearing premise

Percolation thresholds can be reliably identified and used for scaling in mixed solvent-solute APT structures, and the composition-based coordination definition captures relevant spatial information without new biases from detection efficiency or reconstruction artifacts.

What would settle it

A direct comparison on simulated APT datasets with known precipitate locations and controlled heterogeneous dilation artifacts in which the coordination-number method shows no improvement or produces a scaled variable that varies strongly with metric choice.

Figures

Figures reproduced from arXiv: 2605.31107 by John Banhart, Mykola Lazarev.

Figure 1
Figure 1. Figure 1: FIG. 1. A two-dimensional example illustrating the definition of coordination numbers based on [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Pair correlation functions [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The correlations between the average coordination number [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Clustering curves in the bulk depending on the control parameters of the kMCN method. [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Fraction of clustered solutes [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. (a) Cluster size distributions obtained by the MCN method for three values of the pa [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Dependencies of the optimal control parameter on the composition and size of the [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Cluster size distributions obtained by the kMCN method at [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The dependence of the optimal self-similar variable [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Comparison of quality metrics ( [PITH_FULL_IMAGE:figures/full_fig_p025_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. The impact of heterogeneous strain on the results of cluster analysis obtained by kMCN [PITH_FULL_IMAGE:figures/full_fig_p026_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Dependences of clustering quality metrics on control parameters separately for each of [PITH_FULL_IMAGE:figures/full_fig_p028_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Comparison of the kMCN and DBCSAN methods applied to heterogeneously deformed [PITH_FULL_IMAGE:figures/full_fig_p030_13.png] view at source ↗
read the original abstract

The ability to identify nanometer-scale nuclei of new phases in atom probe tomography (APT) is often limited by the sensitivity of clustering algorithms to user-defined control parameters. Conventional approaches typically rely on the Euclidean distance metric and consider only solute atoms, thereby discarding the solvent atoms that contain most of the spatial information. Here, we introduce a coordination-number metric based on the composition and apply it to higher-order clustering. Using various metrics, we investigate percolation in typical APT structures. By scaling clustering properties to the corresponding percolation thresholds, we define a self-similar variable that is almost invariant with respect to metrics, clustering parameters, and structural disorder. This variable provides a relevant description of clustering and enables the formal transfer of optimal parameters between clustering methods. We also study the characteristic clustering behavior in small precipitates and quantify how the precipitate-matrix interface alters the composition spectrum and broadens the clustering curve. Finally, using simulations that incorporate finite spatial resolution, detection efficiency, and other APT reconstruction artifacts, we show that the approach based on coordination numbers effectively compensates for heterogeneous dilations and outperforms solute-density-based methods in all tested scenarios.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces a coordination-number metric derived from local composition for clustering in atom probe tomography (APT) data. It applies this metric within a higher-order clustering framework, identifies percolation thresholds in typical APT structures (solvent + solute), and scales clustering properties to those thresholds to obtain a self-similar variable claimed to be nearly invariant to metric choice, clustering parameters, and structural disorder. The work further examines characteristic clustering in small precipitates, the effect of the precipitate-matrix interface on the composition spectrum, and demonstrates via simulations that incorporate finite spatial resolution, detection efficiency, and reconstruction artifacts that the coordination-number approach compensates for heterogeneous dilations and outperforms conventional solute-density methods in all tested cases.

Significance. If the central claims hold, the self-similar variable and associated parameter-transfer procedure would constitute a useful methodological advance for reducing user-parameter sensitivity in APT clustering analysis. The explicit inclusion of detection efficiency and reconstruction artifacts in the validation simulations is a strength; reproducible demonstration that the percolation threshold remains extractable under realistic atom-loss conditions would strengthen the result.

major comments (2)
  1. [Simulations section (detection-efficiency results)] The central claim that the coordination-number metric “effectively compensates for heterogeneous dilations” in the presence of detection efficiency (20–60 % atom removal) rests on the untested premise that percolation thresholds remain identifiable and stable on the resulting incomplete graphs. The abstract states that simulations include detection efficiency, but without explicit quantification of threshold shift or loss of percolation signature near interfaces and in dilute regions, the invariance of the self-similar variable cannot be verified.
  2. [Percolation-threshold definition and self-similar variable] The self-similar variable is defined by scaling to percolation thresholds measured on the same data set to which the coordination metric is applied. If the threshold extraction itself depends on the metric and on the same incomplete point set, the claimed parameter-free invariance may be partly by construction; an independent validation (e.g., against known synthetic precipitates or cross-validation with an external length scale) is needed to establish that the scaling is not circular.
minor comments (2)
  1. [Abstract] The abstract is information-dense; splitting the final sentence or adding a short clause on the range of detection efficiencies tested would improve readability.
  2. [Methods / metric definition] Notation for the coordination number (e.g., whether it is a local average or a graph degree) should be defined at first use with an explicit equation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive review. The two major comments raise important points about the robustness of the percolation analysis under detection efficiency and the potential circularity in the self-similar variable. We address each below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: The central claim that the coordination-number metric “effectively compensates for heterogeneous dilations” in the presence of detection efficiency (20–60 % atom removal) rests on the untested premise that percolation thresholds remain identifiable and stable on the resulting incomplete graphs. The abstract states that simulations include detection efficiency, but without explicit quantification of threshold shift or loss of percolation signature near interfaces and in dilute regions, the invariance of the self-similar variable cannot be verified.

    Authors: We agree that explicit quantification of percolation threshold stability under atom removal would strengthen the claims. While the simulations section already incorporates detection efficiencies of 20–60 % and shows the coordination-number metric outperforming density-based methods, we did not provide a dedicated quantification of threshold shifts or signature loss near interfaces. In the revised manuscript we will add a new panel or supplementary figure reporting the percolation threshold as a function of detection efficiency, including separate curves for interface and dilute regions, to directly verify that the signature remains identifiable and the self-similar variable remains stable. revision: yes

  2. Referee: The self-similar variable is defined by scaling to percolation thresholds measured on the same data set to which the coordination metric is applied. If the threshold extraction itself depends on the metric and on the same incomplete point set, the claimed parameter-free invariance may be partly by construction; an independent validation (e.g., against known synthetic precipitates or cross-validation with an external length scale) is needed to establish that the scaling is not circular.

    Authors: We acknowledge the referee’s concern that scaling to a threshold extracted from the same incomplete point set could introduce some degree of circularity. The current manuscript demonstrates invariance across metrics and parameters on the same data, which supports transferability, but does not include an external check against known ground-truth structures. In the revision we will add a validation subsection using synthetic precipitates of known size and composition; we will show that the scaled variable recovers the expected clustering behavior when compared against these independent length scales, thereby confirming that the invariance is not solely an artifact of the threshold definition. revision: yes

Circularity Check

0 steps flagged

No significant circularity; scaling to measured percolation thresholds is a standard data-collapse technique validated empirically.

full rationale

The abstract describes defining a self-similar variable by scaling clustering properties to percolation thresholds identified on the same APT structures. This is a conventional normalization step in percolation analysis to test invariance across parameters, not a reduction by construction. No equations or self-citations are provided that would make the claimed invariance tautological; the paper instead reports empirical checks in simulations that include detection efficiency and reconstruction artifacts. The derivation chain remains independent of its inputs and does not match any enumerated circularity pattern.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the coordination-number definition and percolation scaling are presented as methodological innovations without listed assumptions or fitted constants.

pith-pipeline@v0.9.1-grok · 5733 in / 1010 out tokens · 22021 ms · 2026-06-28T22:01:47.801781+00:00 · methodology

discussion (0)

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