Wigner–Seitz cell

The Wigner–Seitz cell, named after Eugene Wigner and Frederick Seitz, is a geometrical construction used 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 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.

Definition

The Wigner–Seitz cell around a lattice point is defined as the locus of points in space that 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. It is made by drawing planes normal to the lines joining nearest lattice points to a particular lattice point.

Constructing the cell

Construction of a Wigner–Seitz primitive 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, and the partition of the plane into these cells for a given set of point sites is known as a Voronoi diagram. Though the Wigner–Seitz cell in itself is not of paramount importance in the 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, semiconductor or an [[Insulator (electrical) |insulator]].

Primitive cell

Used predominantly in geometry, solid state physics, and mineralogy, particularly in describing crystal structure, a primitive cell is a minimum cell corresponding to a single lattice point of a structure with translational symmetry in 2 dimensions, 3 dimensions, or other dimensions. A lattice can be characterized by the geometry of its primitive cell.

The primitive cell is a fundamental domain with respect to translational symmetry only. In the case of additional symmetries a fundamental domain is smaller.

A crystal can be categorized by its lattice and the atoms that lie in a primitive cell (the basis). A cell will fill all the lattice space without leaving gaps by repetition of crystal translation operations.

Primitive translation vectors are used to define a crystal translation vector, , and also gives a lattice cell of smallest volume for a particular lattice. The lattice and translation vectors , , and are primitive if the atoms look the same from any lattice points using integers u₁, u₂, and u₃.

The primitive cell is defined by the primitive axes (vectors) , , and . The volume, V_c, of the primitive cell is given by the parallelepiped from the above axes as,

A Wigner–Seitz cell is an example of another kind of primitive cell. The primitive unit cell (or simply primitive cell) is a special case of unit cells which has only one lattice point combined and shared by eight other primitive cells. It is the most "primitive" cell one can construct, and it is a parrallelepiped. The general unit cell has an integral number of lattice points. The simple cubic lattice is the only primitive unit cell conventionally. The body centered cubic (BCC) and face centered cubic (FCC) lattices are simply unit cells, not primitive.

The general mathematical concept behind the primitive cell is termed the fundamental domain or the Voronoi cell. The primitive cell of the reciprocal lattice in momentum space is called the Brillouin zone.