The Crystallography of Quasicrystals
Abstract
A century ago, E. S. Fedorov, A. Schonflies, and W. Barlow, working independently, classified the 230 distinct symmetry groups of objects repeated periodically in three-dimensional space. The 230 space groups determine the symmetries of macroscopic properties and provide crystallographers with their most important tool in deducing structure; as such, they find use in biology, organic chemistry, and virology as well as in physics, geology, and materials science. With the discovery in 1984 of quasicrystals, well-ordered but aperiodic metallic alloys with crystallographically forbidden rotational symmetries, the need arose to develop a space-group theory for these new materials. Based on the work by Rokhsar, Wright, and Mermin^1 in two dimensions, Rabson, Mermin, Rokhsar, and Wright have classified all quasicrystal and crystal three -dimensional axial space groups.^2 Our classification, proceeding in reciprocal space, is elementary, does not rely on projecting higher-dimensional crystallographic space groups, and is valid for arbitrary rotational symmetry. As an additional illustration of quasicrystallographic space groups, I derive and demonstrate algorithms that produce two-dimensional tilings of rhombi with each of the possible plane-group symmetries.^3 While the analogous task for crystallographic plane groups is trivial, the lack of translational symmetry in a quasicrystal tiling makes these constructions interesting. Since the symmetry of a quasicrystal appears more naturally in reciprocal than in direct space, it is not surprising that some of the resulting tilings seem intricate, although in fact their symmetries are quite simple. ftn^1 Acta Cryst. A44, 197-211 (1988). ^2 "The Space Groups of Axial Crystals and Quasicrystals," preprint. ^3See also Rabson, Ho, and Mermin, Acta Cryst. A44, 678 (1988) and Acta Cryst. A45, 538 (1989).
- Publication:
-
Ph.D. Thesis
- Pub Date:
- 1991
- Bibcode:
- 1991PhDT........53R
- Keywords:
-
- SPACE GROUPS;
- Physics: Condensed Matter