Crystallographic symmetries of quasicrystals

Abstract

Abstract The crystallographic symmetries of atomic positions in a quasicrystal are considered within a direct-superspace approach involving acceptance regions which conceptually correspond to crystal forms in space. The symmetries of the embedded quasicrystal structure are elements of an n-dimensional multi-metrical space group. The corresponding rotational parts are either Euclidean rotations, or hyperbolic rotations or a product of both (multimetrical, thus) giving rise in space to combined rotational and scaling symmetries. For the example of the decagonal phase Al 78 Mn 22, structural relations are shown to exist in space which are due to multimetrical symmetries in superspace.