Wigner–Seitz cell
Encyclopedia
The Wigner–Seitz cell, named after Eugene Wigner and Frederick Seitz
, is a type of Voronoi cell used in the study of crystal
line material in solid-state physics
.
The unique property of a crystal
is that its atom
s 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 material properties consequent to this symmetry. The Wigner–Seitz cell is a means to achieve this.
. 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 parallelepiped. 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
.
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.
. 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.
. 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.
Frederick Seitz
Frederick Seitz was an American physicist and a pioneer of solid state physics. Seitz was president of Rockefeller University, and president of the United States National Academy of Sciences 1962–1969. He was the recipient of the National Medal of Science, NASA's Distinguished Public Service...
, is a type of Voronoi cell used in the study of crystal
Crystal
A crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is known as crystallography...
line material in solid-state physics
Solid-state physics
Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the large-scale properties of solid materials result from...
.
The unique property of a crystal
Crystal
A crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is known as crystallography...
is that its atom
Atom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...
s are arranged in a regular 3-dimensional array called a lattice
Lattice (group)
In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...
. All the properties attributed to crystalline materials stem from this highly ordered structure. Such a structure exhibits discrete
Discrete mathematics
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not...
translational symmetry
Translational symmetry
In geometry, a translation "slides" an object by a a: Ta = p + a.In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation...
. 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 material properties consequent to this symmetry. The Wigner–Seitz cell is a means to achieve this.
Wigner–Seitz cell
A Wigner–Seitz cell is an example of another kind of Primitive cellPrimitive 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...
. 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 parallelepiped. 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
Fundamental domain
In geometry, the fundamental domain of a symmetry group of an object is a part or pattern, as small or irredundant as possible, which determines the whole object based on the symmetry. More rigorously, given a topological space and a group acting on it, the images of a single point under the group...
or the Voronoi cell. The primitive cell of the reciprocal lattice
Reciprocal lattice
In physics, the reciprocal lattice of a lattice is the lattice in which the Fourier transform of the spatial function of the original lattice is represented. This space is also known as momentum space or less commonly k-space, due to the relationship between the Pontryagin duals momentum and...
in momentum space is called the Brillouin zone
Brillouin zone
In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. The boundaries of this cell are given by planes related to points on the reciprocal lattice. It is found by the same method as for the Wigner–Seitz cell in the Bravais lattice...
.
Definition
The Wigner–Seitz cell around a lattice point is defined as the locusLocus (mathematics)
In geometry, a locus is a collection of points which share a property. For example a circle may be defined as the locus of points in a plane at a fixed distance from a given point....
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
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...
spanning the entire Bravais lattice without leaving any gaps or holes.
The Wigner–Seitz cell in the reciprocal lattice
Reciprocal lattice
In physics, the reciprocal lattice of a lattice is the lattice in which the Fourier transform of the spatial function of the original lattice is represented. This space is also known as momentum space or less commonly k-space, due to the relationship between the Pontryagin duals momentum and...
is known as the first Brillouin zone
Brillouin zone
In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. The boundaries of this cell are given by planes related to points on the reciprocal lattice. It is found by the same method as for the Wigner–Seitz cell in the Bravais lattice...
. It is made by drawing planes normal to the lines joining nearest lattice points to a particular lattice point.
Constructing the cell
The cell may be chosen by first picking a lattice pointLattice (group)
In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...
. Then, lines are drawn to all nearby (closest) lattice points. At the midpoint of each line, another line is drawn normal
Surface normal
A surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a...
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 diagramVoronoi diagram
In mathematics, a Voronoi diagram is a special kind of decomposition of a given space, e.g., a metric space, determined by distances to a specified family of objects in the space...
. Though the Wigner–Seitz cell in itself is not of paramount importance in the direct space, it is extremely important in the reciprocal space
Reciprocal lattice
In physics, the reciprocal lattice of a lattice is the lattice in which the Fourier transform of the spatial function of the original lattice is represented. This space is also known as momentum space or less commonly k-space, due to the relationship between the Pontryagin duals momentum and...
. The Wigner–Seitz cell in the reciprocal space is called the Brillouin zone
Brillouin zone
In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. The boundaries of this cell are given by planes related to points on the reciprocal lattice. It is found by the same method as for the Wigner–Seitz cell in the Bravais lattice...
, which contains the information about whether a material will be a conductor
Electrical conductor
In physics and electrical engineering, a conductor is a material which contains movable electric charges. In metallic conductors such as copper or aluminum, the movable charged particles are electrons...
, semiconductor
Semiconductor
A semiconductor is a material with electrical conductivity due to electron flow intermediate in magnitude between that of a conductor and an insulator. This means a conductivity roughly in the range of 103 to 10−8 siemens per centimeter...
or an insulator.