Method of enumeration of crystal superstructures based on fcc, bcc, and hexagonal close packings

A new method of enumeration of crystal superstructures is described. It involves (a) constructing of superlattices using Hermite Normal Form, (b) generating of superstructures by labeling of points of the base lattice, with non-unique solutions sorted off, (c) reducing the primitive cell of a superlattice to the Niggli cell, (d) assigning a Bravais lattice type to superlattice and changing the supercell to a standard setting of Bravaise basis vectors, (e) calculating the space group symmetry of the superstructure using the CRYCOM program of crystal structure comparison. The geometrical and symmetrical characteristics has been presented for the superlatices derived from the fcc, bcc and hcp types of the base lattice. As an application of the present method, the critical compositions of Fe-Cr alloys observed in studies of pitting corrosion has been explained as being conformed with the conditions of ordered superstructure formation giving rise to characteristic crystallographic symmetries.