The Hexagonality Law in Real Materials III: Alloys and Hydrogen

Hydrogen is the enemy of titanium implants and of the zirconium tubes that hold nuclear fuel: it gathers in the metal and turns it brittle. We show that the places hydrogen can sit follow the Hexagonality Law (#66) exactly. In a close-packed metal of any stacking, per metal atom, the pairs of tetrahedral holes that share a face, and the pairs of octahedral holes that share a face, both number h, the share of hexagonal layers. Two hydrogen atoms cannot sit in one face-sharing tetrahedral pair, so at most 2 − h tetrahedral sites per atom can be filled, and a hydride MHx with hydrogen in tetrahedral sites needs the metal locally to have h ≤ 2 − x: the brittle dihydrides need fully cubic stacking. DICTO proposed to test the law on alloys, starting from Ti-6Al-4V and Ti-Nb-Zr. Across six alloy families the law sorts them on one scale: exact for the hole and hydrogen counts; the composition of the magnesium LPSO alloys is a straight line in h in the authors' own model; the energies of cobalt and iron stackings come close (second-neighbour share 0.15–0.17); the faults of the high-entropy alloy CrCoNi depart clearly. In the standard fault model of these steels and alloys, Olson and Cohen's interface energy σ is exactly twice the second-neighbour layer coupling, so it measures the distance from the law. The Theorem and Proposition are proved in the paper; 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.