Easy and Uneasy Superspace Groups for Incommensurate Crystals

Abstract

The symmetry of an incommensurate crystal structure is usually expressed by its superspace group G. Alternatively one may examine the symmetry G’ of the n-fold superstructure which results when a free coefficient of the modulation vector is replaced by a rational number p/n. It turns out that G’ accounts for all symmetry elements of G only in certain cases: ‘easy’ superspace groups G. Even for them, this may depend on the parity of n. For many superspace groups, G’ never accounts for all symmetry elements, no matter whether n is odd or even: ‘uneasy’ superspace groups G. The symmetry groups which so far have been reliably shown to occur point to a strong and perhaps even exclusive preference for easy groups.