A periodic crystal is a structure with, ideally, sharp diffraction peaks on the positions of a reciprocal lattice. The structure then is invariant under the translations of the direct lattice. Periodicity here means lattice periodicity. Any structure without this property is aperiodic. For example, an amorphous system is aperiodic. An aperiodic crystal is a structure with sharp diffraction peaks, but without lattice periodicity. Therefore, amorphous systems are not aperiodic crystals. The positions of the sharp diffraction peaks of an aperiodic crystal belong to a vector module of
The basis vectors \(a_i^*\) are supposed to be independent over the rational
There are four classes of aperiodic structures, but these classes have an overlap:
modulatedcrystal phases (See incommensurate modulated crystal phases),
- incommensurate composite structures (See incommensurate composites),
- quasicrystals (see quasicrystals),
- and incommensurate magnetic structures (See incommensurate magnetic structures).