Space groups of trigonal and hexagonal quasiperiodic crystals of rank 4

Abstract

As a pedagogical illustration of the Fourier-space approach to the crystallography of quasiperiodic crystals, a simple derivation is given of the space-group classification scheme for hexagonal and trigonal quasiperiodic crystals of rank 4. The categories, which can be directly inferred from the Fourier-space forms of the hexagonal and trigonal space groups for periodic crystals, describe general hexagonal or trigonal quasiperiodic crystals of rank 4, which include but are not limited to modulated crystals and intergrowth compounds. When these general categories are applied to the special case of modulated crystals, it is useful to present them in ways that emphasize each of the subsets of Bragg peaks that can serve as distinct lattices of main reflections. These different settings of the general rank-4 space groups correspond precisely to the superspace-group description of (3 + 1) modulated crystals given by de Wolff, Jannsen & Janner [Acta Cryst. (1981), A37, 625–636]. As a demonstration of the power of the Fourier-space approach, the space groups for hexagonal and trigonal quasiperiodic crystals of arbitrary finite rank are derived in a companion paper [Lifshitz & Mermin (1994). Acta Cryst. A50, 85–97].