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. .
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) .
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 .
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 .