Real Polytypes and the Hexagonality Law: Silicon Carbide, Silicon, Germanium and Cerium against the Straight Line

The Hexagonality Law (#66) says that in ideal stackings every kind of piece totals a straight line in the share h of hexagonal layers. DICTO proposed to test it against real crystals. In the language physicists use for stackings, the axial Ising (ANNNI) model, we prove that a property is a straight line in h exactly when its couplings between layers two and three apart vanish, J2 = J3 = 0, and that its stacking faults then stand intrinsic : extrinsic : twin = 2 : 2 : 1, the ratios of #67. We then read published data. The computed volumes of the silicon carbide polytypes lie on the line within the precision of the published numbers; the measured ones scatter between laboratories by more than the whole effect. Silicon and germanium bend slightly but clearly, by about 0.04% of the volume: their volume couplings J2 are of the size of J1, a relaxation that ideal geometry does not have. Germanium's stacking-fault energies come close to the ideal 2 : 2 : 1 (2.25 : 2.06 : 1), silicon's do not (4.1 : 2.4 : 1). Silicon carbide's band gap is far from straight. In cerium, the double-hexagonal form at h = 1/2 has 1.2% more volume than the cubic form. The straight line is the ideal part of every stacking; the departures measure what real crystals add. The proposition is proved in the paper; all crystal data are published values (sources listed), fitted with short scripts attached in the study. The ideas and names are DICTO's. Computation and proof with Claude (Anthropic). Sources checked are listed in the paper. Licence CC BY-SA 4.0. More at dictospheres.com.