Crystallography of Quasicrystals; Application to Icosahedral Symmetry

Crystallographic concepts are extended to quasicrystalline structures. The fact is used that any d-dimensional non-crystallographic point group has D-dimensional representations (D > d) which are compatible with periodicity, i.e. can be viewed as point groups of D-dimensional periodic structures. These periodic structures can be classified by conventional crystallographic methods: Bravais lattices, point groups and space groups can be calculated. This program is carried out for icosahedral symmetry, for which the minimal dimension D is 6. There are surprisingly few types of 6-dimensional crystals with icosahedral symmetry: 3 Bravais lattice types, 2 point groups and totally 11 space groups can occur. The non-symmorphic space groups lead to systematic extinctions of Bragg peaks, similar to the case of ordinary crystals. These extinctions, if present, can help to distinguish between quasiperiodic and competing glass models for icosahedral quasicrystals.