The aperiodic nature of incommensurately modulated structures
The discovery of aperiodic crystals is perhaps one of the most important event which has changed our vision on crystalline architectures since the discovery of diffraction 100 years ago. It was the merit of a Dutch crystallographer, P.M. de Wolff, to interpret their diffraction pattern as a three dimensional projection of a higher dimensional reciprocal lattice, idea which led directly to the generalization of the concept of crystal. Aperiodic crystals are currently described as periodic objects in higher-dimensional space, i.e. the superspace and their structures can be described in terms of 3-d cuts. Incommensurate structures, composite structures and quasicrystals all belong to aperiodic structures. Many interesting properties of superspace have been discovered which are also directly applicable to crystals in the conventional sense, i.e. crystals with 3-d periodicity. In particular the concept of structure type can be extended for a better understanding of structure relations. The notion of solid solution has also benefited from superspace considerations. Moreover, superspace is a very powerful tool for a better understanding of structure–property relations in materials science, e.g. luminescence properties could be directly associated to the description of structures in superspace.