The construction of orthogonal supercells of an arbitrary lattice

A lattice is called orthogonal if it possesses at least monoclinic symmetry. Necessary and sufficient conditions for the existence of orthogonal supercells of a given arbitrary lattice and procedures for calculating them are presented. In addition the conditions for the existence of supercells of higher symmetry, especially cubic and hexagonal, are discussed. Applications to the following crystallographic problems are suggested: geometrical twinning conditions, coincidence-site lattices, conventional supercells of primitive cells, criteria whether a given zonal net can belong to a tetragonal or hexagonal lattice, derivation of possible structural relationships of a lattice to such of higher symmetries from the geometry of the unit cell prior to the knowledge of the structure.