NMR and NQR study of the thermodynamically stable quasicrystals
In this chapter we present the hyperspace model of quasicrystals, and discuss the importance of the pseudo-Brillouin zone in determining the electronic properties and transport properties of quasicrystals. We also discuss structural properties of AlCuFe, AlCuRu, and AlPdMn, and review experiments designed to detect the fine structure in the pseudogaps of various quasicrystals. A. Quasilattices and Hyperspace Lattice periodicity is not necessary for a solid to have long range positional order.
... is fact was highlighted in the early 1930's when incommensurate crystals were discovered. These systems can be described as the result of two interpenetrating crystalline lattices whose lattice constants are inconmiensurate with each other. Therefore, the spatial periods of the two sublattices are related by an irrational number, and the overall lattice is not periodic. Since the two sublattices are periodic, however, one finds that these systems exhibit sharp diffraction peaks, though the diffi-action peaks are not equally spaced in reciprocal space as they are for a crystalline system . What distinguishes quasicrystals fi'om crystals, incommensurate crystals in particular, is that quasicrystals have non-crystallographic point symmetries. There are exactly 14 types of three dimensional crystalline lattices, the Bravais lattices, whose symmetries therefore comprise all the symmetries seen in ordinary crystals  . In theses crystalline lattices, there are no 5-, 8-, or 12-fold axes, and for this reason such symmetries are termed "noncrystallographic symmetries." Quasicrystals, by definition, have long-range order but noncrystallographic orientational symmetries, the most famous of which is icosahedral symmetry, as is found in the perfect quasicrystals of AlCuFe, AlCuRu, and AlPdMn. Because quasicrystals are not periodic-they do not have a unit cell-the mass density function in space is not a periodic function. However, mathematically it is possible to