SPACE-GROUPS OF TRIGONAL AND HEXAGONAL QUASI-PERIODIC CRYSTALS OF RANK-4

As a pedagogical illustration of the Fourier-space approach to the cry stallography of quasiperiodic crystals, a simple derivation is given o f the space-group classification scheme for hexagonal and trigonal qua siperiodic crystals of rank 4. The categories, which can be directly i nferred from the Fourier-space forms of the hexagonal and trigonal spa ce groups for periodic crystals, describe general hexagonal or trigona l quasiperiodic crystals of rank 4, which include but are not limited to modulated crystals and intergrowth compounds. When these general ca tegories are applied to the special case of modulated crystals, it is useful to present them in ways that emphasize each of the subsets of B ragg peaks that can serve as distinct lattices of main reflections. Th ese different settings of the general rank-4 space groups correspond p recisely to the superspace-group description of (3 + 1) modulated crys tals 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 & Mer min (1994). Acta Cryst. A50, 85-97].