The Hexagonality Law in Real Materials II: Entropy and Pressure

The Hexagonality Law (#66) says that in ideal stackings every kind of piece is a straight line in the share h of hexagonal layers; in the physicists' layer model this is a property with only a nearest term (#68). DICTO proposed to test the law's form on two quantities that decide which stacking a crystal takes: its entropy and its enthalpy under pressure. Entropy: Mau and Huse wrote the stacking entropy of hard spheres in the same layer model; at close packing their nearest term is 55 and the next 6 (in 10⁻⁵ kB per sphere), so the entropy is about 90% of the law's form, and the farther terms grow toward melting (37, 18 and 9). Pressure: we prove that if a crystal's volume follows the law's form, pressure shifts only the balance between cubic and hexagonal layers and never reorders the longer stackings among themselves, and that it keeps the fault ratios 2 : 2 : 1. From published volumes and energies, at first order, pressure strengthens cubic 3C silicon carbide against 6H, 4H and 2H, as experiments find (6H turns into 3C above about 4.5 GPa); in silicon and germanium, whose volumes bend away from the straight line, pressure tilts them toward hexagonal-type stackings, mainly through the farther term. The proposition is proved in the paper; all material data are published values (sources listed); the pressure estimate is first order. The ideas and names are DICTO's. Computation and proof with Claude (Anthropic). Sources checked are listed in the paper. The studies (scripts and results) are attached. Licence CC BY-SA 4.0. More at dictospheres.com.