On the arithmetic classification of crystal structures

An arithmetic criterion for the classification of crystal structures with points in their unit cell (`-lattices') was described by Pitteri & Zanzotto [Acta Cryst. (1998), A54 359-373]. In this paper, a systematic analysis of monoatomic 2-lattices is given, showing that there exist 29 distinct arithmetic types of these structures, some of which share the same space groups. As all monoatomic 2-lattices are constituted by a single crystallographic orbit, these structures are also classified by the established criterion of Fischer & Koch [Koch & Fischer (1975). Acta Cryst. A31, 88-95; Fischer & Koch (1996). International Tables for Crystallography, Vol. A. Dordrecht: Kluwer] involving the lattice complexes. The two classifications are found to coincide in this simplest case. By examination of some examples taken from the allotropes of the elements, it is also shown how the arithmetic criterion can be used to classify more complex crystals, such as monoatomic 4-lattices. This gives a group-theoretical framework for distinguishing structures when the space-group classification fails to do so, and Fischer & Koch's criterion, as presented in the literature, may not be immediately applied.