Incommensurate, long-period and quasicrystalline structures. Static and dynamic aspects

Abstract

FROM INCOMMENSURATE TO QUASICRYSTALS.

It is by now a well established fact that even the ideal thermodynamic ground state of a solid state system is not necessarily periodic. In fact crystallography has always been forced to generalize its definitions and its symmetry notions. First from point group symmetry to space group symmetry, then to magnetic and colour symmetry and now one has to abandon the idea of periodicity. A periodic system may as well be characterized in direct space (its density function is invariant under a three- dimensional lattice translation group) as in reciprocal space (its Fourier components are restricted to the vectors of a reciprocal lattice). For the new class(es) of quasiperiodic systems it is simpler to characterize them in reciprocal space. One calls a system quasiperiodic if it has only Fourier components belonging to vectors that are linear integral combinations of a finite number of basis vectors. So these vectors are of the form

When n<4 the structure is periodic, otherwise it is only quasiperiodic.