Classification of space groups for quasicrystals with (2+1)-reducible point group

Abstract

Quasicrystalline structures can be viewed as a cut through a periodic structure in a higherdimensional space. In this paper, the crystallography of 5-dimensional periodic structures which can give rise to a quasi crystal in this way is developed. For the five 5d Bravais classes giving rise to quasicrystals periodic in one direction and with symmetry of order 5, 8, 10 or 12 in the plane perpendicular to that direction, a complete classification of point groups and space groups is given. We find a total of 150 distinct space groups, which can be attributed to 44 point groups. For each of these space groups, a set of generators of its point group and the associated non-primitive translations are tabulated, and for the non-symmorphic space groups the indices of the systematic extinctions are given. All the results are expressed in terms of a geometrically natural lattice basis.