Exploiting symmetries: Speeding up the computational study of solid solutions
Symmetry is a prevalent feature of nature at all scales. For example, our naked eyes can easily identify symmetries in the bodily shape of countless organisms. Symmetry is also very important in the fields of physics and chemistry, especially in the microscopic realm of atoms and molecules. Crystals, which are highly ordered materials, can even have multiple types of symmetry at the same time, such as rotational symmetry, inversion symmetry, and translational symmetry.
Lately, alongside rapid progress in computer science, researchers have developed computational methods that seek to predict the physical properties of crystals based on their electronic structure. In practice, however, pure and perfectly symmetric crystals are seldom used. This is because a crystal's properties can be tuned as desired by alloying them with other materials or randomly substituting certain atoms with other elements, i.e., doping.
Accordingly, materials scientists are seeking computationally efficient approaches to analyze such alloys and substituted crystals, also known as solid solutions. The "supercell method" is one such approach and is widely used to model crystal structures with random substitutions of different atoms. The symmetry of crystals, however, is actually a problem when using this technique. In crystals, there can be many substitution patterns that are physically equivalent to other substitutions if we simply translate or rotate them. Findings these symmetric substitution patterns is not very meaningful, and thus their calculation when using the supercell method is a waste of time.
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