Mallard's law recast as a Diophantine system: fast and complete enumeration of possible twin laws by [reticular] [pseudo] merohedry

Mallard's law of observation states that known twins by reticular pseudomerohedry have low twin index m and low obliquity delta between a lattice row and a lattice plane. Crystals in the twin are related by exact reflection in the lattice plane, exact rotation by pi about the perpendicular to the lattice plane, or by exact rotation about the lattice row by 2kpi/q, with q = 2, 3, 4 or 6, and 1 less than or equal to k < q. In known cases, m is up to five or six, and &delta; is up to five or six degrees. Mallard's converse problem is then about finding all pairs of indices for rows and planes leading to twin indices not larger than m and to obliquities that are at most &delta;. Mallard's law is recast as the Diophantine pair constituted by the equality h &BULL; u = n and the inequality \h x u\ < n tan delta. If a primitive reference system is used, the integer n is either m, 2m or 3m. A direct general solution of this system for u given n, h, delta and lattice data is detailed. That straightforward solution involves a tiny two-dimensional grid for u, considerably reducing the number of permutations to be considered. If the primitive reference system is Buerger-reduced, then moduli for indices of solutions h cannot exceed 3m, thus establishing a simple way to produce complete solutions. A program called OBLIQUE was designed on such principles. An implementation is available for free execution at http://ylp.icpet.nrc.ca/oblique/. OBLIQUE is also an interactive tool in Toth Information Systems' Materials Toolkit (http://www.tothcanada.com/), an exploitation framework currently operating on CRYSTMET and ICSD crystal structure data. The example of quartz cell data is described with m(max) = 3 and delta(max) = 6degrees.