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

International Tables for Crystallography (2013). Vol. D, ch. 2.2, p. 314

Section 2.2.2. The lattice

K. Schwarza*

aInstitut für Materialchemie, Technische Universität Wien, Getreidemarkt 9/165-TC, A-1060 Vienna, Austria
Correspondence e-mail:

2.2.2. The lattice

| top | pdf | 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(]

  • (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(]

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[link] of International Tables for Crystallography, Volume A (2005)[link]]. 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[link] of International Tables for Crystallography, Volume A (2005)[link]]. The reciprocal lattice and the Brillouin zone

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Owing to the translational symmetry of a crystal, it is convenient to define a reciprocal lattice, which plays a dominating role in describing electrons in a solid. The three unit vectors of the reciprocal lattice [{\bf b}_{i}] are given according to the standard definition by [{\bf a}_{i}{\bf b}_{j}=2\pi\delta_{ij},\eqno(]where the factor [2\pi] is commonly used in solid-state physics in order to simplify many expressions. Strictly speaking (in terms of mathematics) this factor should not be included [see Section[link] of the present volume and Chapter 1.1[link] of International Tables for Crystallography, Volume B (2001)[link]], since the (complete) reciprocity is lost, i.e. the reciprocal lattice of the reciprocal lattice is no longer the direct lattice. [{\bf b}_{1}=2\pi{{{\bf a}_{2}\times{\bf a}_{3}}\over{{\bf a}_{1}\cdot{\bf a}_{2}\times{\bf a}_{3}}}\,\,\hbox{and cyclic permutations.}\eqno(]

In analogy to the direct lattice we define

  • (i) a vector of the reciprocal lattice (upper case) as[{\bf K}_{m}=m_{1}{\bf b}_{1}+m_{2}{\bf b}_{2}+m_{3}{\bf b} _{3}\,\,\hbox{with }m_{i}\hbox{ integer}\semi\eqno(]

  • (ii) a vector in the lattice (lower case) as[{\bf k}=k_{1}{\bf b}_{1}+k_{2}{\bf b}_{2}+k_{3}{\bf b} _{3}\,\, \hbox{with }k_{i}\hbox{ real}.\eqno(]From ([link] and ([link] it follows immediately that [{\bf T}_{n}{\bf K}_{m}=2\pi N\,\,\hbox{with }N\hbox{ an integer.}\eqno(]

A construction identical to the Wigner–Seitz cell delimits in reciprocal space a cell conventionally known as the first Brillouin zone (BZ), which is very important in the band theory of solids. There are 14 first Brillouin zones according to the 14 Bravais lattices.


International Tables for Crystallography (2001). Vol. B. Reciprocal space, edited by U. Shmueli, 2nd ed. Dordrecht: Kluwer Academic Publishers.
International Tables for Crystallography (2005). Vol. A. Space-group symmetry, edited by Th. Hahn, 5th ed. Heidelberg: Springer.

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