1.5: 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 finite rank. This means that the diffraction wave vectors are of the form
\[\mathbf{k}=\sum_{i-1}^{n}h_i\mathbf{a}_i^*, (integer\,h_i) \nonumber \]
The basis vectors \(a_i^*\) 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.