Superspace Transformation Properties of Incommensurate Irreducible Distortions

Abstract

The so-called superspace groups have recently been introduced to characterize the symmetry properties of incommensurate structures (1–5), allowing a unified approach to selection rules and other regularities observed in this kind of phases. Essentially the theory defines an imaginary (3+d)-dimensional structure called a supercrystal which is determined by the real three dimensional structure. In this way a translational lattice symmetry is “recovered”. The space group associated with this “supercrystal” is the so-called superspace group of the corresponding incommensurate structure. The supercrystal is therefore defined in a space which apart from the three dimensional one includes a d-dimensional “internal” space, d being the number of independent incommensurate modulation wave vectors in the structural distortion.