Translation-permutation operator algebra for the description of crystal structures. I. Ideal closest packing

Abstract

The foundation of an abstract algebra for the description of crystal structures is developed in terms of ideally closest-packed structures. All the spatial information of closest-packed structures can be derived in terms of (a) the geometry of the atoms (A positions) and the plane, triangular interstices (B and C positions) of a closest-packed p6mm monolayer, and (b) the permutations induced among these A, B, and C positions by translations from one monolayer midplane to another. The mathematical device used to correlate this information is a translation-permutation vector operator. Since this operator is closely related to the basic concept of a layer-by=layer crystal growth process, the algebra is readily extensible to crystal structures that are defective owing to many kinds of non-periodic elements of structure. Some examples are presented of the application of the algebra to closest-packed structures that are either ideal or exhibit one of two kinds of defect: classical stacking faults or point defects.