The reciprocal lattice and the Brillouin zone

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 2.2.2.2. The reciprocal lattice and the Brillouin zone

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

2.2.2.2. 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 $[{\bf a}_{i}{\bf b}_{j}=2\pi\delta_{ij},\eqno(2.2.2.3)]$ 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 1.1.2.4 of the present volume and Chapter 1.1 of International Tables for Crystallography, Volume B (2001)], since the (complete) reciprocity is lost, i.e. the reciprocal lattice of the reciprocal lattice is no longer the direct lattice. $[{\bf b}_{1}=2\pi{{{\bf a}_{2}\times{\bf a}_{3}}\over{{\bf a}_{1}\cdot{\bf a}_{2}\times{\bf a}_{3}}}\,\,\hbox{and cyclic permutations.}\eqno(2.2.2.4)]$

In analogy to the direct lattice we define

(i) a vector of the reciprocal lattice (upper case) as $[{\bf K}_{m}=m_{1}{\bf b}_{1}+m_{2}{\bf b}_{2}+m_{3}{\bf b} _{3}\,\,\hbox{with }m_{i}\hbox{ integer}\semi\eqno(2.2.2.5)]$
(ii) a vector in the lattice (lower case) as $[{\bf k}=k_{1}{\bf b}_{1}+k_{2}{\bf b}_{2}+k_{3}{\bf b} _{3}\,\, \hbox{with }k_{i}\hbox{ real}.\eqno(2.2.2.6)]$ From (2.2.2.5) and (2.2.2.1) it follows immediately that $[{\bf T}_{n}{\bf K}_{m}=2\pi N\,\,\hbox{with }N\hbox{ an integer.}\eqno(2.2.2.7)]$

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.

References

International Tables for Crystallography (2001). Vol. B. Reciprocal space, edited by U. Shmueli, 2nd ed. Dordrecht: Kluwer Academic Publishers.Google Scholar

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