The arithmetic symmetry of colored crystals: classification of 2-color 2-lattices

A framework for the detailed classification of general crystal structures, based on an arithmetic criterion, has been proposed in recent years. In this paper it is shown how this method can also be applied to enumerate colored crystals. To illustrate this approach, the systematic classification in the simplest case, i.e. of `2-color 2-lattices', in two and three dimensions (two- and three-dimensional crystals with two differently colored atoms per unit translational cell) is presented. 51 distinct types of 2-color 2-lattices are found in three dimensions (ten types in two dimensions); this gives a complete catalog of the simplest crystal structures that are theoretically possible for two-element compounds. Among the 51 2-lattices, all those which already have a Strukturberichte denomination are retrieved, as well as the 22 `black-and-white lattices' considered in the theory of magnetic crystals. The symmetry hierarchies and symmetry-breaking possibilities for 2-color 2-lattices are also determined in two and three dimensions.