The Hexagonality Law in Real Materials IV: Oxides

Chemists judge oxide structures with two rules of thumb: Pauling's third rule (octahedra that share faces are unstable) and Goldschmidt's tolerance factor (a large A cation pushes a perovskite towards hexagonal stacking). DICTO proposed that three-dimensional stacking rules can replace such rules of thumb. We show that both rules are measured by one number, the hexagonality h of the oxygen stacking (#66). In a hexagonal perovskite ABO3 of any stacking, each B octahedron shares faces with 2h neighbours and corners with 6(1 − h). In a layered battery cathode LiMO2, the lithium–metal octahedra that share a face number 2h per lithium site on average. Measured data follow the law: the volumes of four BaRuO3 polytypes lie on one straight line in h (within 0.2% for the three precisely measured ones), and pressure walks the structure down h, 9R → 4H → 6H → 3C. In LixCoO2 on charge, the H1-3 phase puts all its face-sharing into the empty gaps and keeps lithium only where it shares no face, giving the bound x ≤ 1 − h for the stable phases; its layer spacing, 4.52 Å, lies on the law line between O3 and O1 (4.525 Å). The law has limits: in BaMnO3 the volume follows it but the energy does not. The Theorem and Proposition are proved in the paper; the counts were checked from explicit coordinates; all material data are published values (sources listed). The ideas and names are DICTO's. Computation and proof with Claude (Anthropic). Sources checked are listed in the paper. The study (scripts and results) is attached. Licence CC BY-SA 4.0. More at dictospheres.com.