Abstract
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.
Highlights
In the last part of the XXth century, the definition of the crystal changed
The lost of the translation symmetry has severe consequences on the determination of the properties of these so-called aperiodic crystals: where are the atoms?, what about the phase transitions in these crystals?, how to describe the collective vibrations in a crystal with no unit cell and so no Brillouin zone? The goal of this paper is to present the conceptual advances done in these fields up to now
Since the discovery of aperiodic crystal neutron diffraction coupled to X-ray and synchrotron diffraction has allowed the generalization of the standard crystallography towards superspace crystallography
Summary
In the last part of the XXth century, the definition of the crystal changed. The translation symmetry is no more a condition. The case of the three families of aperiodic crystals will be treated: incommensurate modulated crystals, aperiodic composites and quasicrystals
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