Symmorphic space groups of finite crystal lattices

Abstract

The symmorphic space group of a finite n-dimensional crystal lattice is studied and its factorization is presented. This lattice is determined by a direct product of cyclic groups and then its translation, point and space groups are defined as modular images of corresponding ones for a lattice of infinite extent. A factorization of holohedries (of an infinite lattice) is used in order to present a symmorphic finite space group as a direct product. As an example, and as a very special case, a space group in the case of the hypercubic lattice is studied and its irreducible representations are investigated. The results obtained suggest that for vectors lying on a surface of the first Brillouin zone (i.e. for which at least one coordinate is equal to 0 or 1/2) an additional index describing their symmetry properties should be introduced. This enables the authors to make a more detailed classification of states (energy levels). On the other hand, finite symmorphic space groups can be used within, e.g., the so-called finite-lattice approach.