Aperiodic crystal

Aperiodic crystals lack three-dimensional translational symmetry but still exhibit three-dimensional long-range order.^[1] In other words, they are periodic crystals in higher dimensions. They are classified into three different categories: incommensurate modulated structures, incommensurate composite structures, and quasicrystals.^[2]

The diffraction patterns of aperiodic crystals contain two sets of peaks, which include "main reflections" and "satellite reflections".^[1] Main reflections are usually stronger in intensity and span a lattice defined by three-dimensional reciprocal lattice vectors${\displaystyle (a^{*},b^{*},c^{*})}$. Satellite reflections are weaker in intensity and are known as "lattice ghosts". These reflections do not correspond to any lattice points in physical space and cannot be indexed with the original three vectors. To understand aperiodic crystal structures, one must use the superspace approach.^[3] In materials science, "superspace" or higher-dimensional space refers to the concept of describing the structures and properties of materials in terms of dimensions beyond the three dimensions of physical space. This may involve using mathematical models to describe the behavior of atoms or molecules in a materials in four, five, or even higher dimensions.^[4]