Wigner-Seitz cell

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<Font color="blue"> Maille de Wigner-Seitz </Font>(''Fr''). <Font color="red"> Wigner-Seitz Zell </Font>(''Ge''). <Font color="green"> Celda de Wigner-Seitz</Font> (''Sp'').<Font color="black"> Cella di Wigner-Seitz </Font>(''It'').
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Synonym: Voronoi domain
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<font color="blue">Maille de Wigner-Seitz</font> (''Fr''). <font color="red">Wigner-Seitz-Zelle, Wirkungsbereich</font> (''Ge''). <font color="black">Cella di Wigner-Seitz</font> (''It''). <font color="green">Celda de Wigner-Seitz</font> (''Sp'').
== Definition ==
== Definition ==

Latest revision as of 14:50, 20 November 2017

Synonym: Voronoi domain

Maille de Wigner-Seitz (Fr). Wigner-Seitz-Zelle, Wirkungsbereich (Ge). Cella di Wigner-Seitz (It). Celda de Wigner-Seitz (Sp).

Definition

W-S-1.gif

The Wigner-Seitz cell is a polyhedron obtained by connecting a lattice point P to all other lattice points and drawing the planes perpendicular to these connecting lines and passing through their midpoints (Fig. 1). The polyhedron enclosed by these planes is the Wigner-Seitz cell. This construction is called the Dirichlet construction. The cell thus obtained is a primitive cell and it is possible to fill up the whole space by translation of that cell.


The Wigner-Seitz cell of a body-centred cubic lattice I is a cuboctahedron (Fig. 2) and the Wigner-Seitz cell of a face-centred cubic lattice F is a rhomb-dodecahedron (Fig. 3). In reciprocal space this cell is the first Brillouin zone. Since the reciprocal lattice of a body-centred lattice is a face-centred lattice and reciprocally, the first Brillouin zone of a body-centred cubic lattice is a rhomb-dodecahedron and that of a face-centred cubic lattice is a cuboctahedron.

W-S-2.gifW-S-3.gif

The inside of the Wigner-Seitz cell has been called the domain of influence by Delaunay (1933). It is also called the Dirichlet domain or Voronoi domain. The domain of influence of lattice point P thus consists of all points Q in space that are closer to this lattice point than to any other lattice point or at most equidistant to it (such that OP ≤ |t - OP| for any vector tL).

See also

  • Chapter 3.1.1.4 of International Tables for Crystallography, Volume A, 6th edition
  • Chapter 1.5 of International Tables for Crystallography, Volume B
  • Chapters 1.2 and 2.2 of International Tables for Crystallography, Volume D