Subperiodic crystal

Cristal sous-périodique (Fr). Subperiodische Kristall (Ge). Cristallo subperiodico (It). 亜周期結晶 (Ja). Cristal subperiódico (Sp).

A subperiodic crystal is a crystal showing periodicity in a subspace of the three-dimensional Euclidean space. Subperiodic crystals are modular crystal structures in which the orientation / position of successive modules results in missing periodicity in at least one direction.

Subperiodic crystals are classified in the following categories.

• Diperiodic crystals, showing periodicity along two of the three directions of three-dimensional Euclidean space; examples include polytypes built by layers whose sequence is non-periodic ("disordered"), thin films, graphene.
• Monoperiodic crystals, showing periodicity along one of the three directions of three-dimensional Euclidean space; examples include polytypes built by rods whose sequence is non-periodic ("disordered") in both directions.

The building modules can be of three types.

• Layers or sheets, whose symmetry is described by one of the 80 types of layer group; if the stacking of layers is periodic, a conventional, triperiodic crystal is obtained, although the period along the stacking direction may be much longer than the period along any direction within the modules; otherwise, the crystal is diperiodic and its diffraction pattern shows diffuse scattering along the direction of missing periodicity.
• Rods or chains, whose symmetry is described by one of the 75 types of rod group; if the stacking of rods is periodic in both directions a conventional, triperiodic crystal is obtained, although the period along the stacking directions may be much longer than the period along the rod; otherwise, the crystal is diperiodic or monoperiodic depending on whether the lack of periodicity occurs in one or both directions. Accordingly, its diffraction pattern shows diffuse scattering along the direction(s) of missing periodicity.
• Blocks or bricks, whose symmetry is described by one of the 32 types of crystallographic point group; if the stacking of modules is periodic in all the three directions, a conventional, triperiodic crystal is obtained; otherwise, the solid is diperiodic, monoperiodic or non-periodic depending on whether the lack of periodicity occurs in one, two or three directions. Accordingly, its diffraction pattern shows diffuse scattering along the direction(s) of missing periodicity. A non-periodic solid is no longer a crystal.