The concept of unit cells in the theory of quasicrystals

To describe the structure of icosahedral quasicrystals, the concept of (quasi-)unit cells is proposed as an alternative to the higher-dimensional approach. The problem of describing the structure of icosahedral quasicrystals splits into two stages: filling the cells with atoms and filling the space with cells. The only difference from ordinary periodic crystals is that four unit cells should be used instead of one, and an iterative inflation/deflation algorithm should be used instead of translations to fill the entire space with cells. Icosahedral packings are described as lists of cells, each of which is given its type, position, and orientation. Based on the developed algorithm, representative fragments of all three types of the Socolar–Steinhardt zonohedral packing were generated, which clearly illustrate the main structural features and hierarchical motifs of icosahedral quasicrystals. The theoretical possibility of calculating the intensities of X-ray reflections in the structural analysis of quasicrystals without using the higher-dimensional crystallography approaches is discussed. To do this, one should first calculate the partial structure factors for each type of unit cell, and then average them over the volume of the quasicrystal by using the derived substitution rules.