# Aperiodic crystal

Cristal apériodique (Fr). Cristallo aperiodico (It). 非周期性結晶 (Ja).

## Definition

The definition of aperiodic crystal was included in the definition of crystal proposed by the IUCr Commission on Aperiodic Structures (International Union of Crystallography, 1992): by crystal we mean any solid having an essentially discrete diffraction diagram and aperiodic crystal we mean any crystal in which three-dimensional lattice periodicity can be considered to be absent. As an extension, the latter term will also include those crystals in which three-dimensional periodicity is too weak to describe significant correlations in the atomic configuration, but which can be properly described by crystallographic methods developed for actual aperiodic crystals.

For practical purposes, however, many scientists currently working in the field use a narrower definition of aperiodic crystal, namely

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

$\textbf{k}=\sum_{i=1}^n h_i \textbf{a}_i^* ( \textrm{integer}\ h_i).$

The basis vectors $\textbf{a}_i^*$ 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: