Bravais classes for the simplest incommensurate crystal phases

Through the reformulation of crystallography that treats periodic and quasiperiodic structures on an equal footing in three-dimensional Fourier space, a novel computation is given of the Bravais classes for the simplest kinds of incommensurately modulated crystals: (3 + 3) Bravais classes in the cubic system and (3 + 1) Bravais classes in any of the other six crystal systems. The contents of a Bravais class are taken to be sets of ordinary three-dimensional wave vectors inferred from a diffraction pattern. Because no finer distinctions are made based on the intensities of the associated Bragg peaks, a significantly simpler set of Bravais classes is found than Janner, Janssen & de Wolff [Acta Cryst. (1983). A39, 658-666] find by defining their Bravais classes in higher-dimensional superspace. In our scheme, the Janner, Janssen & de Wolff categories appear as different ways to describe identical sets of three-dimensional wave vectors when those sets contain crystallographic (3 + 0) sublattices belonging to more than a single crystallographic Bravais class. While such further discriminations are important to make when the diffraction pattern is well described by a strong lattice of main reflections and weaker satellite peaks, by not making them at the fundamental level of the Bravais class, the crystallographic description of all quasiperiodic materials is placed on a single unified foundation.