Lattice reduction using a Euclidean algorithm

The need to reduce a periodic structure given in terms of a large supercell and associated lattice generators arises frequently in different fields of application of crystallography, in particular in the ab initio theoretical modelling of materials at the atomic scale. This paper considers the reduction of crystals and addresses the reduction associated with the existence of a commensurate translation that leaves the crystal invariant, providing a practical scheme for it. The reduction procedure hinges on a convenient integer factorization of the full period of the cycle (or grid) generated by the repeated applications of the invariant translation, and its iterative reduction into sub-cycles, each of which corresponds to a factor in the decomposition of the period. This is done in successive steps, each time solving a Diophantine linear equation by means of a Euclidean reduction algorithm in order to provide the generators of the reduced lattice.