# How do the notions of Crystal structure, unit cell and supercell fit together?

I'm trying to fully understand the notion of crystal structures with some basic implications. I understand that it is conveniently described by a lattice which is the infinite repetition of points in three-dimensional euclidean space given by the integer linear combinations of some lattice vectors. I'm taking the example of Silicon (Si) for which we should have:

x = (5.44, 0, 0)

y = (0, 5.44, 0)

z = (0, 0, 5.44)

as the three lattice vectors. Additionally I read about the concept of unit cell which is given by linear combinations of fractional coefficients (not integers, this time) given by ratios of the lattice vectors x_i as well as the notion of supercell , which in my understanding is just a stack of multiple unit cells ...?

I just can't seem to fit well all these different notions together and have a clear view of the underlying object. I tried to visualize the silicon structure and this resulted in even more confusion as I cannot understand what it does represent. For Si, I expect to have a single atom of silicon, while here probably the visualization refers to a supercell? ### Crystal structure:

Crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric patterns that repeat along the principal directions of three-dimensional space in matter .

### Bravais (conventional) Lattice

The Bravais lattice is the basic building block from which all crystals can be constructed. The concept originated as a topological problem of finding the number of different ways to arrange points in space where each point would have an identical “atmosphere”. That is each point would be surrounded by an identical set of points as any other point, so that all points would be indistinguishable from each other. Mathematician Auguste Bravais discovered that there were 14 different collections of the groups of points, which are known as Bravais lattices. These lattices fall into seven different "crystal systems”, as differentiated by the relationship between the angles between sides of the “unit cell” and the distance between points in the unit cell. The unit cell is the smallest group of atoms, ions or molecules that, when repeated at regular intervals in three dimensions, will produce the lattice of a crystal system. .

### Primitive cell

A primitive cell is a unit cell that contains exactly one lattice point. For unit cells generally, lattice points that are shared by n cells are counted as 1/n of the lattice points contained in each of those cells; so for example a primitive unit cell in three dimensions which has lattice points only at its eight vertices is considered to contain 1/8 of each of them. An alternative conceptualization is to consistently pick only one of the n lattice points to belong to the given unit cell (so the other 1-n lattice points belong to adjacent unit cells) .

### Wigner–Seitz cell

A Wigner–Seitz cell is an example of a primitive cell, which is a unit cell containing exactly one lattice point. For any given lattice, there are an infinite number of possible primitive cells. However there is only one Wigner–Seitz cell for any given lattice. It is the locus of points in space that are closer to that lattice point than to any of the other lattice points.

A Wigner–Seitz cell, like any primitive cell, is a fundamental domain for the discrete translation symmetry of the lattice. The primitive cell of the reciprocal lattice in momentum space is called the Brillouin zone .

### Super cell

In solid-state physics and crystallography, a crystal structure is described by a unit cell. There are an infinite number of unit cells with different shapes and sizes which can describe the same crystal. Say that a crystal structure is described by a unit cell U. The supercell S of unit cell U is a cell which describes the same crystal, but has larger volume than cell U. Many methods which use a supercell perturbate it somehow to determine properties which cannot be determined by the initial cell .

For example: Disordered compounds are crucially important for fundamental science and industrial applications. Yet most available methods to explore solid-state material properties require ideal periodicity, which, strictly speaking, does not exist in this type of materials. The supercell approximation is a way to imply periodicity to disordered systems while preserving “disordered” properties at the local level .