Symmetry of incommensurate crystal phases. I. Commensurate basic structures

The superspace-group approach [Janner & Janssen (1977), Phys. Rev. B, 15, 643-658] is used to solve the symmetry problem of incommensurate crystal phases in the case of displacive- and occupation-wave modulation. Generalization is given to cover magnetic modulation as well. The symmetry conditions imposed by the superspace group on the crystal structure are derived and applied to the following incommensurate crystals, whose structures have been discussed in the literature independently from the present point of view: K₂SeO₄, 2H-TaSe₂, NaNO₂ and Cr. The superspace groups describing the symmetry of these compounds are indicated and the structural implications of the corresponding symmetry elements discussed.