World Library  
Flag as Inappropriate
Email this Article

Wigner-Seitz cell

Article Id: WHEBN0023072717
Reproduction Date:

Title: Wigner-Seitz cell  
Author: World Heritage Encyclopedia
Language: English
Subject: Voronoi diagram, Brillouin zone, Elongated dodecahedron
Collection:
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Wigner-Seitz cell

The Wigner–Seitz cell, named after Eugene Wigner and Frederick Seitz, is a type of Voronoi cell 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 material properties consequent to this symmetry. The Wigner–Seitz cell is a means to achieve this.

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 cell 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.

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 direct space without leaving any gaps or holes.

The Wigner–Seitz cell in the reciprocal space is known as the first Brillouin zone. It is made by drawing planes normal to the segments joining nearest lattice points to a particular lattice point, through the midpoints of such segments.

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, 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.fr:Maille (cristallographie) ko:원시세포 ms:Sel primitif nl:Eenheidscel ru:Ячейка Вигнера-Зейтца ru:Примитивная ячейка uk:Примітивна комірка zh:维格纳-赛兹原胞

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.