Crystallography of Modulated Structures in Superspace

Since the discovery of X-ray diffraction, it was believed that the discrete distribution of diffracted intensities was a direct consequence of the periodic arrangement of atoms in the three dimensions of space. However, in the last thirty years many examples of new types of crystals have been found which do not fulfil this criterion but nevertheless give perfectly discrete diffraction patterns. The new category of crystals which exhibits this property is called aperiodic. It includes incommensurate crystals, quasicrystals and composite crystals. These structures are best described in superspace, an extension of the three-dimensional space up to six dimensions. Aperiodic structures can be interpreted as three-dimensional cuts of higher dimensional periodic objects. The description of an incommensurate structure in 3+1 dimensions along with the concept of symmetry in superspace is presented as an example.