The Hexagonality Law in Real Materials V: Ice, Diamond and Nanowires

Ice, diamond and the semiconductor nanowires of new lasers and solar cells are all built from the same piece: a tetrahedron of bonds, in close-packed layers that stack cubic or hexagonal. We test the Hexagonality Law (#66) on them. The geometry is exact: in any stacking, the eclipsed bonds (whose neighbours face each other) number h/2 per atom, where h is the share of hexagonal layers. For six III-V semiconductors in four stackings each, computed by Belabbes et al., the conduction-band offset lies on the law's straight line within 4% of its full change and the crystal-field splitting within 7%. The polytype energies of the tetrahedral elements rank them by their distance from the law: tin 0.02, germanium 0.08, diamond 0.15, silicon 0.32, and silicon carbide 1.03; silicon carbide, the best-known material with many stacking forms, is the one furthest from the law. For ice, Tonauer et al. measured −37.7 J/mol for pure cubic ice turning hexagonal; the law predicts that the stacking-disordered ice of clouds releases 37.7 J/mol times its cubicity, and that its excess vapour pressure scales the same way. This closes the series on real materials. The Theorem is proved in the paper and checked from explicit coordinates; all material data are published values (sources listed); the ice result is a stated prediction. 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.