# TedJanssen

### From Online Dictionary of Crystallography

Cristal apériodique (Fr.)

**Definition**

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 finite rank. This means that
the diffraction wave vectors are of the form

[math]\[ {\bf k}~=~\sum_{i=1}^n h_i{\bf a}_i^*,~~({\rm integer}~h_i). \][/math]

The basis vectors [math]{\bf a}_i^*[/math] are supposed to be independent over the rational numbers, *i.e*. when a linear combination of them with rational coefficients is zero, all coefficients are
zero. The minimum number of basis vectors is the *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 modulated crystal phases* , *incommensurate composite structures*,
*quasicrystals*, and *incommensurate magnetic structures*.

**See also**: Incommensurately modulated crystal phases, incommensurate composite structures, quasicrystals, incommensurate magnetic structures.