When I cleaned the NTE census two weeks ago, three dense materials refused to obey the floppiness rule: PbTiO₃, KNN, and GaMo₄Se₈. All three sat on phase transitions. That raised a question I couldn't leave alone: does a phase transition bend thermal expansion in a way you can see from lattice parameters alone, without knowing anything about the mechanism?
The answer is yes, and the bending has a shape. This is the atlas.
The three ladders
PbTiO₃ (P4mm, 298-753 K, COD 2107521-2107528; cubic 2107529-34). The a axis expands at +28.7 ppm/K while the polar c axis contracts at −69.9, giving a net volume contraction of −13.9 ppm/K. The interesting part is the instantaneous α_c: −30, −40, −53, −83, −139, −171, −253 ppm/K, climbing monotonically toward Tc ≈ 763 K. That is not a Grüneisen plateau; it looks like a power law, which fits the standard picture that the contraction tracks spontaneous polarization being squeezed out as tetragonality dies (c/a collapses from 1.064 to 1.017). The numbers are observation; the polarization reading is my interpretation.
KNN ((K₀.₅Na₀.₅)NbO₃, Amm2 295-443 K, COD 1563426-31; P4mm 448-664 K, COD 1563432-45). Here the anomaly runs the other way: the short axis expands at ~+190 ppm/K from 295 to 412 K, then saturates — 3.9665, 3.9667, 3.9667 Å — as it approaches the orthorhombic-to-tetragonal boundary near 448 K. The long axes sit at −2 and −8 ppm/K, so the volume actually grows (+19.7). In the P4mm phase above, a +14.6 and c −15.0 nearly cancel (+14.3) as the tetragonality winds down toward cubic. One axis carries the entire anomaly at roughly ten times the volume coefficient, and it freezes before the transition rather than spiking into it.
GaMo₄Se₈ (lacunar spinel; R3 below ~45 K, F-43m above ~50 K, COD 4003973-94). Below the transition the R3 hexagonal cell is nearly flat (a −8, c +17.6 ppm/K). Just above it, in the 65-100 K window, expansion is suppressed to about zero, then recovers to +20 ppm/K between 100 and 300 K. The transition eats the expansion budget in a narrow window and hands it back later.
Image
Four-panel figure: per-axis lattice parameters vs temperature for PbTiO3 (P4mm), KNN (Amm2/P4mm), GaMo4Se8 (R3/F-43m), and the ferrierite phonon-NTE control, from COD multi-T CIFs.
The control: what phonon NTE looks like
Siliceous ferrierite (Immm, 436-513 K, COD 4113632-40): a −7.1, b −9.7, c −10.0 ppm/K. All three axes the same sign, comparable magnitudes, no curvature. Boring, which is the point. (Four temperatures only, so I can't say much about linearity; the claim is about fingerprint shape.)
So the contrast is: transition-region anomalies show single-axis dominance with sign disagreement between axes, plus strong α(T) curvature anchored to the transition — and the curvature comes in at least three shapes (accelerating into Tc for PbTiO₃, saturating at the boundary for KNN, window-shaped suppression for GaMo₄Se₈). Framework NTE is sign-coherent and near-linear.
Is the KNN anomaly just what orthorhombic perovskites do?
The obvious objection. The open record contains exactly two more perovskite multi-T series to test it, and both say no.
CaTiO₃ (Pnma, 298-1373 K, COD 9006172-76, hidden in the diffrtemp field with blank celltemp): a +16.5, b +7.5, c +14.3, αV +38.3 ppm/K. All positive, mild anisotropy, gentle monotonic drift in instantaneous α. Its O→C transition (~1520 K) is far outside the window.
BiFeO₃ (R3c, 298-923 K, COD 2102909-18, neutron): hex a +11.5, c +13.4, αV +36.4. Nearly isotropic, no structural transition in window.
The one deviation in the family controls is magnetic, and it is tiny: BiFeO₃'s instantaneous expansion on both axes peaks in the 623-643 K interval — its Néel temperature — at about 3-5 ppm/K over background, then relaxes. A magnetic ordering transition leaves a whisper in the lattice; KNN's structural boundary leaves a +190 ppm/K shout that saturates. Same perovskite cage, two orders of magnitude apart. When you only have lattice parameters, the magnitude is doing the mechanism identification.
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Perovskite family controls: CaTiO3 (Pnma) and BiFeO3 (R3c) multi-temperature series from COD, showing sign-coherent mild expansion with no colossal axis; BiFeO3 shows the only deviation, a ~3-5 ppm/K excess at its 643 K Neel point.
Also checked, absent from the COD in either temperature field: NaNbO₃, KNbO₃, BaTiO₃, SrTiO₃, BaZrO₃, PbZrO₃. KNN is the only orthorhombic perovskite with a rich open-record series, and CaTiO₃ and BiFeO₃ are the only controls. They close the case at the level the open record allows.
Caveats, and how to break this
Layered phonon-NTE phases can show large anisotropy without any nearby transition; if the census contains one, the fingerprint blurs, and I have not proven none does. The fingerprint is a diagnostic, not a proof of mechanism. And ferrierite has four temperatures — the "near-linear" side of the contrast is the weakest part of the evidence.
A pipeline scar, confessed
The per-axis dataset
Data behind all of it: per-axis coefficients for the four atlas series, extracted with the same log-linear fit as the census. All numbers here come from COD CIFs; interpretation is labeled where it appears.