The direct lattice and the Wigner–Seitz cell

Schwarz, K.

doi:10.1107/97809553602060000639

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 2.2, p. 294

Section 2.2.2.1. The direct lattice and the Wigner–Seitz cell

K. Schwarz^a ^*

^a Institut für Materialchemie, Technische Universität Wien, Getreidemarkt 9/165-TC, A-1060 Vienna, Austria
Correspondence e-mail: kschwarz@theochem.tuwein.ac.at

2.2.2.1. The direct lattice and the Wigner–Seitz cell

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The three unit-cell vectors $[{\bf a}_{1}]$ , $[{\bf a}_{2}]$ and $[{\bf a}_{3}]$ define the parallelepiped of the unit cell. We define

(i) a translation vector of the lattice (upper case) as a primitive vector (integral linear combination) of all translations $[{\bf T}_{n}=n_{1}{\bf a}_{1}+n_{2}{\bf a}_{2}+n_{3}{\bf a}_{3}\,\,\hbox{with }n_{i}\hbox{ integer},\eqno(2.2.2.1)]$
(ii) but a vector in the lattice (lower case) as $[{\bf r}=x_{1}{\bf a}_{1}+x_{2}{\bf a}_{2}+x_{3}{\bf a} _{3}\,\,\hbox{with }x_{i}\hbox{ real}.\eqno(2.2.2.2)]$

From the seven possible crystal systems one arrives at the 14 possible space lattices, based on both primitive and non-primitive (body-centred, face-centred and base-centred) cells, called the Bravais lattices [see Chapter 9.1 of International Tables for Crystallography, Volume A (2005)]. Instead of describing these cells as parallelepipeds, we can find several types of polyhedra with which we can fill space by translation. A very important type of space filling is obtained by the Dirichlet construction. Each lattice point is connected to its nearest neighbours and the corresponding bisecting (perpendicular) planes will delimit a region of space which is called the Dirichlet region, the Wigner–Seitz cell or the Voronoi cell. This cell is uniquely defined and has additional symmetry properties.

When we add a basis to the lattice (i.e. the atomic positions in the unit cell) we arrive at the well known 230 space groups [see Part 3 of International Tables for Crystallography, Volume A (2005)].

References

International Tables for Crystallography (2005). Vol. A. Space-group symmetry, edited by Th. Hahn, 5th ed. Heidelberg: Springer.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 2.2, p. 294