R. Lifshitz et Nd. Mermin, BRAVAIS CLASSES AND SPACE-GROUPS FOR TRIGONAL AND HEXAGONAL QUASI-PERIODIC CRYSTALS OF ARBITRARY FINITE RANK, Acta crystallographica. Section A, Foundations of crystallography, 50, 1994, pp. 85-97
To demonstrate the power of the Fourier-space approach to crystallogra
phy, the Bravais classes and space groups of hexagonal and trigonal qu
asiperiodic crystals are derived for lattices of arbitrary finite rank
. The specification of the space groups for each Bravais class is give
n by an elementary extension of the rank-4 case. The conventional clas
sification of incommensurately modulated hexagonal and trigonal crysta
ls, previously derived using the superspace approach for Bravais class
es up to rank (3 + 3) [Janner, Janssen & de Wolff (1993). Acta Cryst.
A39, 658-666] and for superspace groups of rank (3 + 1) [de Wolff, Jan
ssen & Janner (1981). Acta Cryst. A37, 625-636], is easily extracted f
rom the general classification for modulations of any finite rank.