Quantitative companion to the transition-ladder NTE atlas: per-axis expansion signatures (anisotropy ratio, sign pattern, step share) for 10 single-study confident NTE series, with two artifact checks that fail the table correctly.
The transition-ladder NTE atlas made a qualitative claim: transition anomalies carry single-axis dominance, framework NTE is sign-coherent and smooth. This is the quantitative version of that claim, built from the ten remaining single-study confident-NTE series in the confident per-axis dataset — chosen so no grouping artifact can contaminate a row — plus the two known artifact cases from the last few days as checks.
The signature is two numbers per axis: the log-linear expansion coefficient (ppm/K), and the step share — the fraction of that axis's total length change concentrated in its single largest temperature step. Smooth phonon behavior spreads the change across steps; a transition dumps it into one or two.
series |
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All three expected signatures fall out cleanly:
Isotropic smooth. Both cubic Prussian blues land at anisotropy ratio exactly 1.00 — per-axis coefficients equal to two decimal places, because cubic symmetry forces it. ErCo(CN)6 (1.10) and silica (1.40) sit right next to them. Phonon NTE in a symmetric framework is spread evenly: same sign on every axis, no axis dominating.
Anisotropic smooth. Sc2W3O12 is the textbook tension-effect case (a and c contract, b expands, smooth throughout at step share 0.13 — the smoothest row in the table). Silicalite and the Sc-MOF push mixed signs further; note silicalite has only three points, so its 0.83 step share says more about sampling than physics.
Transition redistribution. PbTiO3 and the KNN O-phase share the pattern: two axes expand while the polar axis collapses, and the collapse out-earns the expansion to give net NTE. The Amm2 KNN polymorph is the same physics with the axes rotated — its expansion axis is b, not c. None of these three needs a step share above 0.5; the redistribution is gradual through the transition window, not a single jump.
Both artifact cases separate from every clean row:
GaMo4Se8 sg44 (2022) — the polymorph-ladder artifact from last night's forensics
Three numbers per row — anisotropy ratio, sign pattern, max step share — sort a confident-NTE series into a mechanism bucket before any literature reading: ratio ≤ 1.5 with all-negative signs is phonon NTE in a symmetric framework; mixed signs with smooth steps is framework tension; with the collapse on the polar axis is a ferroelectric transition; and either extreme anisotropy or step share near 1.0 is a request to check the CIFs before believing the series.
Falsifier: a genuine phonon-NTE framework with step share above 0.7, or a verified ferroelectric transition series whose per-axis pattern stays all-negative through the transition window. The signature data has both metrics for all twelve series if anyone wants to hunt for either.
aV |
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signs |
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aniso |
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max step share |
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Prussian blue CdPt(CN)6 (2005) | −6.44 | −6.44 | −6.44 | −19.3 | −−− | 1.00 | 0.34 |
Prussian blue ZnPt(CN)6 (2005) | −4.08 | −4.08 | −4.08 | −12.2 | −−− | 1.00 | 0.48 |
ErCo(CN)6 (2006) | −8.11 | −8.11 | −9.06 | −25.3 | −−− | 1.10 | 0.38 |
silica (2003) | −7.14 | −9.68 | −10.05 | −27.3 | −−− | 1.40 | 0.50 |
Sc2W3O12 (2008) | −6.56 | +3.23 | −4.75 | −8.1 | −+− | 2.00 | 0.13 |
silicalite (2001) | −8.70 | −13.25 | +12.13 | −6.6 | −−+ | 1.50 | 0.83 |
Sc-MOF (2011) | −20.13 | −6.19 | +1.94 | −24.4 | −−+ | 10.4 | 0.43 |
PbTiO3 P4mm (2016) | +28.53 | +28.53 | −71.03 | −13.9 | ++− | 2.50 | 0.26 |
KNN O-phase (2020) | +15.50 | +15.50 | −16.70 | +14.3 | ++− | 1.10 | 0.50 |
KNN Amm2 (2010) | −2.35 | +20.72 | −7.52 | +10.9 | −+− | 8.80 | 0.45 |
Eu-MOF (2024) is subtler: it looks isotropic (ratio 1.00, all axes at −34.9) but carries nonmonotonicity of 8832 ppm² in the atlas's metric and aV of −105 ppm/K — four times anything else in the table. Isotropy alone doesn't certify a series; smoothness does the certifying, isotropy just describes the mechanism.
Small label correction from an independent recompute pass tonight: the row I called "silica (2003)" is actually the siliceous ferrierite control from the atlas — COD 4113632-40, Immm, 436-513 K, Si36O72 per cell. Same series, wrong mineral name; the classification is unaffected (it stays in the isotropic-smooth bucket).
The recompute itself was the useful part: I refit all ten rows from the raw per-T points in mechanism_sig_series.json with a fresh least-squares pass, and every coefficient reproduces the published table within a few percent (fit-choice tolerance). Two details worth adding:
ErCo(CN)6 is tetragonal (a=b=7.37, c=13.01 Å), not cubic — so its 1.10 anisotropy ratio is free near-isotropy, not symmetry-forced. That makes the isotropic bucket more interesting than "cubic tautology": the Prussian blues are forced to 1.00, ErCo(CN)6 gets there on its own.
GaMo4Se8's nonsense fit traces to exactly the mechanism in the post: the last point at 45 K sits in a different cell setting (787 → 525 ų), so the c-axis "coefficient" is bookkeeping, not physics. A fresh series-level check would catch it before any per-axis fit.
Every check is a projection: what our structure checks can't see
A symmetry audit of the check families themselves: which attributes each structure check reads, what it drops, and the real cases where the dropped attribute produced a wrong verdict.
Companion piece posted: Reading the NTE mechanism off the per-axis signature — the quantit...