The Wigner-Seitz cell (named after E. P. Wigner and Frederick Seitz) is a geometrical construction which helps in the study of crystalline material in solid-state physics. The unique property of a crystal is that its atoms are arranged in a regular, 3-dimensional array, which is called a lattice. All the properties attributed to crystalline materials stem from this highly ordered structure. Such a structure exhibits discrete translational symmetry. In order to model and study such a periodic system, one needs a mathematical "handle" to describe the symmetry and hence draw conclusions about the consequences of this symmetry. The Wigner-Seitz cell is a means to achieve this.
The Wigner-Seitz cell around a lattice point is defined as the locus of points in space which are closer to that lattice point than to any of the other lattice points.
It can be shown mathematically that a Wigner-Seitz Cell is a primitive cell spanning the entire Bravais lattice without leaving any gaps or holes.
The Wigner-Seitz cell in the reciprocal lattice is known as the first Brillouin zone.
Constructing the cell
The cell may be chosen by first picking a lattice point. Then, lines are drawn to all nearby (closest) lattice points. At the midpoint of each line, another line is drawn normal to each of the first set of lines.
In the case of a three-dimensional lattice, a perpendicular plane is drawn at the midpoint of the lines between the lattice points. By using this method, the smallest area (or volume) is enclosed in this way and is called the Wigner-Seitz primitive cell. All area (or space) within the lattice will be filled by this type of primitive cell and will leave no gaps.
General mathematical concept
The general mathematical concept embodied in a Wigner-Seitz cell is more commonly called a Voronoi cell. Though the Wigner-Seitz cell in itself is not of paramount importance in direct space, it is extremely important in the reciprocal space. The Wigner-Seitz cell in the reciprocal space is called the Brillouin zone, which contains the information about whether a material will be a conductor, semi-conductor or an insulator.