Wigner–Seitz cell
The Wigner–Seitz cell, named after
The unique property of a crystal is that its atoms are arranged in a regular three-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.
A Wigner–Seitz cell is an example of a
A Wigner–Seitz cell, like any primitive cell, is a
Overview
Background
The concept of Voronoi decomposition was investigated by Peter Gustav Lejeune Dirichlet, leading to the name Dirichlet domain. Further contributions were made from Evgraf Fedorov, (Fedorov parallelohedron), Georgy Voronoy (Voronoi polyhedron),[1][2] and Paul Niggli (Wirkungsbereich).[3]
The application to condensed matter physics was first proposed by Eugene Wigner and Frederick Seitz in a 1933 paper, where it was used to solve the Schrödinger equation for free electrons in elemental sodium.[4] They approximated the shape of the Wigner–Seitz cell in sodium, which is a truncated octahedron, as a sphere of equal volume, and solved the Schrödinger equation exactly using periodic boundary conditions, which require at the surface of the sphere. A similar calculation which also accounted for the non-spherical nature of the Wigner–Seitz cell was performed later by John C. Slater.[5]
There are only five topologically distinct polyhedra which tile three-dimensional space, ℝ3. These are referred to as the parallelohedra. They are the subject of mathematical interest, such as in higher dimensions.[6] These five parallelohedra can be used to classify the three dimensional lattices using the concept of a projective plane, as suggested by John Horton Conway and Neil Sloane.[7] However, while a topological classification considers any affine transformation to lead to an identical class, a more specific classification leads to 24 distinct classes of voronoi polyhedra with parallel edges which tile space.[3] For example, the rectangular cuboid, right square prism, and cube belong to the same topological class, but are distinguished by different ratios of their sides. This classification of the 24 types of voronoi polyhedra for Bravais lattices was first laid out by Boris Delaunay.[8]
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.[9]
It can be shown mathematically that a Wigner–Seitz cell is a
Constructing the cell
The general mathematical concept embodied in a Wigner–Seitz cell is more commonly called a
The cell may be chosen by first picking a
For a 3-dimensional lattice, the steps are analogous, but in step 2 instead of drawing perpendicular lines, perpendicular planes are drawn at the midpoint of the lines between the lattice points.
As in the case of all primitive cells, all area or space within the lattice can be filled by Wigner–Seitz cells and there will be no gaps.
Nearby lattice points are continually examined until the area or volume enclosed is the correct area or volume for a
Topological class (the affine equivalent parallelohedron) | ||||||
---|---|---|---|---|---|---|
Truncated octahedron | Elongated dodecahedron | Rhombic dodecahedron | Hexagonal prism | Cube | ||
Bravais lattice | Primitive cubic | Any | ||||
Face-centered cubic | Any | |||||
Body-centered cubic | Any | |||||
Primitive hexagonal | Any | |||||
Primitive rhombohedral | ||||||
Primitive tetragonal | Any | |||||
Body-centered tetragonal | ||||||
Primitive orthorhombic | Any | |||||
Base-centered orthorhombic | Any | |||||
Face-centered orthorhombic | Any | |||||
Body-centered orthorhombic | ||||||
Primitive monoclinic | Any | |||||
Base-centered monoclinic | , | , | ||||
, | ||||||
Primitive triclinic | where |
one time |
where |
Composite lattices
For
Symmetry
The Wigner–Seitz cell always has the same
Brillouin zone
In practice, the Wigner–Seitz cell itself is actually rarely used as a description of
See also
References
- S2CID 118441072.
- S2CID 199547003.
- ^ ISSN 0232-1300.
- .
- ISSN 0031-899X.
- S2CID 13277804.
- ^ Austin, Dave (2011). "Fedorov's Five Parallelohedra". American Mathematical Society. Archived from the original on 2019-01-03.
- S2CID 120358504.
- ^ ISBN 978-0030839931.
- ISSN 2399-6528.
- ISBN 978-981-02-2059-4.
- ISBN 978-0123044600.
- ISBN 978-1568812205.