Dualistic interpretation of the symmetry of incommensurate structures

The symmetry of an incommensurately modulated structure may be described in terms of the basic structure (B) and the modulation pattern (M). This description contrasts with the superspace-group approach, in which the structure is defined in a space of dimension 3 + d, where d is the number of rationally independent modulation vectors. Space groups G_B and G_M are defined, consisting of symmetry operations of B and M which are simply interrelated. These groups together characterize the total symmetry; they lead to a classification which for d = 1 is equivalent to the superspace groups. With this dualistic approach, all symmetry operations can be based on symmetry elements in the space of the crystal, and the lattice types can be composed simply from those of G_B and G_M. .