# Aperiodic crystal

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

\[\mathbf{k}=\sum_{i-1}^{n}h_i\mathbf{a}_i^*, (integer\,h_i)\]

The basis vectors \(a_i^*\) are supposed to be independent over the rational *rank* of the vector module. If the rank *n* is larger than the space dimension, the structure is not periodic, but aperiodic.

### Applications

There are four classes of aperiodic structures, but these classes have an overlap:

*incommensurately*(See incommensurate modulated crystal phases),modulated crystal phases*incommensurate composite structures*(See incommensurate composites),*quasicrystals*(see quasicrystals),- and
*incommensurate magnetic structures*(See incommensurate magnetic structures).