Crystallography of quasicrystals ; application to icosahedral symmetry

Abstract

Crystallographic concepts are extended to quasicrystalline structures and applied to icosahedral quasicrystals. 2-dimensional N fold rotational symmetries are shown to be compatible with Bravais lattices in (at least) ϕ (N) dimensions, where ϕ (N) is the Euler number, while for 3-dimensional icosahedral symmetry the minimal dimension is 6. The case of icosahedral crystallography is worked out in detail. A complete classification of six-dimensional periodic structures with icosahedral symmetry is derived. There are surprisingly few types of 6-dimensional « crystallographic objects » with icosahedral symmetry, namely 3 Bravais lattice types, 2 point groups, and 11 inequivalent space groups. The problem of equivalence of icosahedral space groups is studied in detail. Similar to the case of ordinary 3-dimensional crystals, nonsymmorphic space group symmetries lead to extinction of Bragg peaks. These extinctions are calculated systematically