Crystal topologies – the achievable and inevitable symmetries

Abstract

The link between the crystal topology and symmetry is examined, focusing on the conditions under which a structure with a given topology can exhibit a certain symmetry. By defining embeddings for quotient graphs (finite representations of crystal topologies) and the corresponding nets (the graph-theoretical equivalents of structures), a strong relationship between the automorphisms of the quotient graphs and the symmetry of the embedded net is established. This allows one to constrain the relative node positions under the premise that an embedding of a net has a certain symmetry, and allows one to assign nodes to equivalents of Wyckoff positions. Two-dimensional examples as well as known crystal structures are used to illustrate the findings. A comparison with a related publication and a discussion on whether constraints on distances between atoms and on bond angles result in restrictions on symmetry without causing confusion conclude the work.