Wigner-Seitz cell

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== See also ==
== See also ==
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*Section 3.1.1.4 of ''International Tables of Crystallography, Volume A'', 6<sup>th</sup> edition
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*Section 1.5 of ''International Tables of Crystallography, Volume B''
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*Sections 1.2 and 2.2 of ''International Tables of Crystallography, Volume D''
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Section 9.1 of ''International Tables of Crystallography, Volume A''<br>
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[[Category:Fundamental crystallography]]
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Section 1.5 of ''International Tables of Crystallography, Volume B''<br>
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Sections 1.2 and 2.2 of ''International Tables of Crystallography, Volume D''<br>
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[[Category:Fundamental crystallography]]<br>
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Revision as of 17:02, 11 April 2017

Maille de Wigner-Seitz (Fr). Wigner-Seitz Zell (Ge). Celda de Wigner-Seitz (Sp). Cella di Wigner-Seitz (It)

Definition

W-S-1.gif

The Wigner-Seitz cell is a 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 (Figure 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 (Figure 2) and the Wigner-Seitz cell of a face-centred cubic lattice F is a rhomb-dodecahedron (Figure 3). In reciprocal space this cell is the first Brillouin zone. Since the reciprocal lattice of 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 domain of influence by Delaunay (1933). It is also called 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

  • Section 3.1.1.4 of International Tables of Crystallography, Volume A, 6th edition
  • Section 1.5 of International Tables of Crystallography, Volume B
  • Sections 1.2 and 2.2 of International Tables of Crystallography, Volume D