Crystallography and dynamics in superspace

The discovery in materials of long range order without lattice periodicity has generated a very broad field of research with the introduction of totally new concepts. The common feature of this new state of matter is the translational symmetry recovered by adding supplementary dimensions to the physical space, defining a crystallographic superspace. Along these supplementary directions, specific low excitations, so-called phasons, are predicted and observed. These quasi-periodic crystals may present phase transitions as function of external parameters, which have then to be described as symmetry breakings within these crystallographic superspaces. Three different families of quasi-periodic crystals are usually considered: incommensurate modulated crystals, aperiodic composites and quasicrystals. Their crystallography and dynamics are discussed