International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

International Tables for Crystallography (2006). Vol. A, ch. 9.1, p. 745

Section 9.1.7. Description of Bravais lattices

H. Burzlaffa and H. Zimmermannb*

aUniversität Erlangen–Nürnberg, Robert-Koch-Strasse 4a, D-91080 Uttenreuth, Germany, and bInstitut für Angewandte Physik, Lehrstuhl für Kristallographie und Strukturphysik, Universität Erlangen–Nürnberg, Bismarckstrasse 10, D-91054 Erlangen, Germany
Correspondence e-mail:  helmuth.zimmermann@knot.uni-erlangen.de

9.1.7. Description of Bravais lattices

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In Fig. 9.1.7.1[link], conventional cells for the 14 three-dimensional Bravais lattices are illustrated.

[Figure 9.1.7.1]

Figure 9.1.7.1 | top | pdf |

Conventional cells of the three-dimensional Bravais lattices (for symbols see Table 9.1.7.2[link]).

In Tables 9.1.7.1[link] and 9.1.7.2[link], the two- and three-dimensional Bravais lattices are described in detail. For each entry, the tables contain conditions that must be fulfilled by the lattice parameters and the metric tensor. These conditions are given with respect to two different basis systems, first the conventional basis related to symmetry, second a special primitive basis (see below). In columns 2 and 3, basis vectors not required by symmetry to be of the same length are designated by different letters. Columns 4 and 5 contain the metric tensors for the two related bases. Column 6 shows the relations between the components of the two tensors.

Table 9.1.7.1| top | pdf |
Two-dimensional Bravais lattices

Bravais latticeLattice parametersMetric tensorProjections
ConventionalPrimitive/transformation to primitive cellConventionalPrimitiveRelations of the components
mp [\matrix{a, b\hfill\cr \gamma\hfill\cr}] [\matrix{a, b\hfill\cr \gamma\hfill\cr}] [\matrix{g_{11} &g_{12}\hfill\cr &g_{22}\hfill\cr}] [\matrix{g_{11} &g_{12}\hfill\cr &g_{22}\hfill\cr}]   [Scheme scheme15]
op [\matrix{a, b\hfill\cr \gamma = 90^{\circ}\hfill\cr}] [\matrix{a, b\hfill\cr \gamma = 90^{\circ}\hfill\cr}] [\matrix{g_{11} &0\hfill\cr &g_{22}\hfill\cr}] [\matrix{g_{11} &0\hfill\cr &g_{22}\hfill\cr}]   [Scheme scheme16]
oc [\matrix{a_{1} = a_{2}\hfill, \gamma\hfill\cr {\bi P}(c)\hfill}] [\matrix{g'_{11} &g'_{12}\hfill\cr &g'_{22}\hfill\cr}] [\matrix{g'_{11} = {\textstyle{1 \over 4}}(g_{11} + g_{22})\hfill\cr g'_{12} = {\textstyle{1 \over 4}}(g_{11} - g_{22})\hfill\cr g_{11} = 2(g'_{11} + g'_{12})\hfill\cr g_{12} = 2(g'_{11} - g'_{12})\hfill\cr}] [Scheme scheme17]
tp [\matrix{a_{1} = a_{2}\hfill\cr \gamma = 90^{\circ}\hfill\cr}] [\matrix{a_{1} = a_{2}\hfill\cr \gamma = 90^{\circ}\hfill\cr}] [\matrix{g_{11} &0\hfill\cr &g_{11}\hfill\cr}] [\matrix{g_{11} &0\hfill\cr &g_{11}\hfill\cr}]   [Scheme scheme18]
hp [\matrix{a_{1} = a_{2}\hfill\cr \gamma = 120^{\circ}\hfill\cr}] [\matrix{a_{1} = a_{2}\hfill\cr \gamma = 120^{\circ}\hfill\cr}] [\matrix{g_{11} &-{\textstyle{1 \over 2}}\;g_{11}\hfill\cr &g_{11}\hfill\cr}] [\matrix{g_{11} &-{\textstyle{1 \over 2}}\;g_{11}\hfill\cr &g_{11}\hfill\cr}]   [Scheme scheme19]
The symbols for Bravais lattices were adopted by the International Union of Crystallography in 1985; cf. de Wolff et al. (1985)[link].
[{\bi P}(c) =\textstyle{1 \over 2}(11/\bar{1}1).]

Table 9.1.7.2| top | pdf |
Three-dimensional Bravais lattices

Bravais latticeLattice parametersMetric tensorProjections
ConventionalPrimitiveConventionalPrimitive/transf.Relations of the components
aP [\matrix{a, b, c\hfill\cr \alpha, \beta, \gamma\hfill\cr}] [\matrix{a, b, c\hfill\cr \alpha, \beta, \gamma\hfill\cr}] [\matrix{g_{11} &g_{12} &g_{13}\hfill\cr &g_{22} &g_{23}\hfill\cr & &g_{33}\hfill\cr}] [\matrix{g_{11} &g_{12} &g_{13}\hfill\cr &g_{22} &g_{23}\hfill\cr & &g_{33}\hfill\cr}]   [Scheme scheme1]
mP [\matrix{a, b, c\hfill\cr \beta, \alpha = \gamma = 90^{\circ}\hfill\cr}] [\matrix{a, b, c\hfill\cr \beta, \alpha = \gamma = 90^{\circ}\hfill\cr}] [\matrix{g_{11} &0 &g_{13}\hfill\cr &g_{22} &0\hfill\cr & &g_{33}\hfill\cr}] [\matrix{g_{11} &0 &g_{13}\hfill\cr &g_{22} &0\hfill\cr & &g_{33}\hfill\cr}]   [Scheme scheme2]
[\matrix{mC\hfill\cr (mS)\hfill\cr}] [\matrix{a_{1} = a_{2}, c\hfill\cr \gamma, \alpha = \beta\hfill\cr}] [{\bi P}(C)] [\matrix{g'_{11} = {\textstyle{1 \over 4}}(g_{11} + g_{22})\hfill\cr g'_{12} = {\textstyle{1 \over 4}}(g_{11} - g_{22})\hfill\cr g'_{13} = {1 \over 2}g_{13}\hfill\cr \cr g_{11} = 2(g'_{11} + g'_{12})\hfill\cr g_{22} = 2(g'_{11} - g'_{12})\hfill\cr g_{13} = 2g'_{13}\hfill\cr}] [Scheme scheme3]
[\matrix{g'_{11} &g'_{12} &g'_{13}\hfill\cr &g'_{11} &g'_{13}\hfill\cr &&g_{33}\hfill\cr}]
oP [\matrix{a, b, c\hfill\cr \alpha = \beta = \gamma = 90^{\circ}\hfill\cr}] [\matrix{a, b, c\hfill\cr \alpha = \beta = \gamma = 90^{\circ}\hfill\cr}] [\matrix{g_{11} &0 &0\hfill\cr &g_{22} &0\hfill\cr & &g_{33}\hfill\cr}] [\matrix{g_{11} &0 &0\hfill\cr &g_{22} &0\hfill\cr & &g_{33}\hfill\cr}]   [Scheme scheme4]
[\matrix{oC\hfill\cr (oS)\hfill\cr}] [\matrix{a_{1} = a_{2}, c\hfill\cr \gamma, \alpha = \beta = 90^{\circ}\hfill\cr}] [{\bi P}(C)] [\matrix{g'_{11} = {\textstyle{1 \over 4}}(g_{11} + g_{22})\hfill\cr g'_{12} = {\textstyle{1 \over 4}}(g_{11} - g_{22})\hfill\cr \cr g_{11} = 2(g'_{11} + g'_{12})\hfill\cr g_{22} = 2(g'_{11} - g'_{12})\hfill\cr}] [Scheme scheme5]
[\matrix{g'_{11} &g'_{12} &0\hfill\cr &g'_{11} &0\hfill\cr & &g_{33}\hfill\cr}]
oI [\matrix{a_{1} = a_{2} = a_{3}\hfill\cr \alpha, \beta, \gamma\hfill\cr \cos \alpha + \cos \beta\hfill\cr\quad + \cos \gamma = -1\hfill\cr}] [{\bi P}(I)] [\matrix{g'_{12} = {\textstyle{1 \over 4}}(-g_{11} - g_{22} + g_{33})\hfill\cr g'_{13} = {\textstyle{1 \over 4}}(-g_{11} + g_{22} - g_{33})\hfill\cr g'_{23} = {\textstyle{1 \over 4}}(g_{11} - g_{22} - g_{33})\hfill\cr \cr g_{11} = -2(g'_{12} + g'_{13})\hfill\cr g_{22} = -2(g'_{12} + g'_{23})\hfill\cr g_{33} = -2(g'_{13} + g'_{23})\hfill\cr}] [Scheme scheme6]
[\displaylines{\matrix{-\tilde g &g'_{12} &g'_{13}\hfill\cr &-\tilde g &g'_{23}\hfill\cr & &-\tilde g\hfill\cr}\hfill\cr \tilde{g} = g'_{12} + g'_{13} + g'_{23}\hfill}]
oF [\openup1pt\matrix{a, b, c\hfill\cr \alpha, \beta, \gamma\hfill\cr \cos \alpha = {\displaystyle{-a^{2} + b^{2} + c^{2} \over 2bc}}\hfill\cr \cos \beta = {\displaystyle{a^{2} + b^{2} + c^{2} \over 2ac}}\hfill\cr \cos \gamma = {\displaystyle{a^{2} + b^{2} - c^{2} \over 2ab}}\hfill\cr\cr}] [{\bi P}(F)] [\matrix{g'_{12} = {\textstyle{1 \over 4}}\;g_{33}\hfill\cr g'_{13} = {\textstyle{1 \over 4}}\;g_{22}\hfill\cr g'_{23} = {\textstyle{1 \over 4}}\;g_{11}\hfill\cr \cr g_{11} = 4g'_{23}\hfill\cr g_{22} = 4g'_{13}\hfill\cr g_{33} = 4g'_{12}\hfill\cr}] [Scheme scheme7]
[\displaylines{\openup-4pt\matrix{\tilde{g}_{1} &g'_{12} &g'_{13}\hfill\cr &\tilde{g}_{2} &g'_{23}\hfill\cr & &\tilde{g}_{3}\hfill\cr}\hfill\cr \tilde{g}_{1} = g'_{12} + g'_{13}\hfill\cr\tilde{g}_{2} = g'_{12} + g'_{23}\hfill\cr \tilde{g}_{3} = g'_{13} + g'_{23}\hfill\cr}]
tP [\matrix{a_{1} = a_{2}, c\hfill\cr \alpha = \beta = \gamma = 90^{\circ}\hfill\cr}] [\matrix{a_{1} = a_{2}, c\hfill\cr \alpha = \beta = \gamma = 90^{\circ}\hfill\cr}] [\matrix{g_{11} &0 &0\hfill\cr &g_{11} &0\hfill\cr & &g_{33}\hfill\cr}] [\matrix{g_{11} &0 &0\hfill\cr &g_{11} &0\hfill\cr & &g_{33}\hfill\cr}]   [Scheme scheme8]
tI [\matrix{a_{1} = a_{2} = a_{3}\hfill\cr \gamma, \alpha = \beta\hfill\cr 2\cos \alpha + \cos \gamma = -1\hfill\cr}] [{\bi P}(I)] [\matrix{g'_{12} = {\textstyle{1 \over 4}}(-2g_{11} + g_{33})\hfill\cr g'_{13} = -{\textstyle{1 \over 4}}g_{33}\hfill\cr \cr g_{11} = 2(g'_{12} + g'_{13})\hfill\cr g_{33} = -4g'_{13}\hfill\cr}] [Scheme scheme9]
[\eqalign{&\matrix{\bar{g} &g'_{12} &g'_{13}\hfill\cr &\bar{g} &g'_{13}\hfill\cr & &\bar{g}\hfill\cr}\cr &\bar{g} = -(g'_{12} + 2g'_{13})}]
hR [\matrix{a_{1} = a_{2}, c\hfill\cr \alpha = \beta = 90^{\circ}\hfill\cr \gamma = 120^{\circ}\hfill\cr}] [\matrix{a_{1} = a_{2} = a_{3}\hfill\cr \alpha = \beta = \gamma\hfill\cr}] [\matrix{g_{11} &-{\textstyle{1 \over 2}}g_{11} &0\hfill\cr &}g_{11} &0\hfill\cr & &g_{33}\hfill\cr}] [{\bi P}(R)] [\matrix{g'_{11} = {\textstyle{1 \over 9}}(3g_{11} + g_{33})\hfill\cr g'_{12} = {\textstyle{1 \over 9}}(-{3 \over 2}g_{11} + g_{33})\hfill\cr \cr g_{11} = 2(g'_{11} - g'_{12})\hfill\cr g_{33} = 3(g'_{11} + 2g'_{12})\hfill\cr}] [Scheme scheme10]
[\matrix{g'_{11} &g'_{12} &g'_{12}\hfill\cr &g'_{11} &g'_{12}\hfill\cr & &g'_{11}\hfill\cr}]
hP [\matrix{a_{1} = a_{2}, c\hfill\cr \alpha = \beta = 90^{\circ}\hfill\cr \gamma = 120^{\circ}\hfill\cr}] [\matrix{g_{11} &-{\textstyle{1 \over 2}}g_{11} &0\hfill\cr &g_{11} &0\hfill\cr & &g_{33}\hfill\cr}]   [Scheme scheme11]
cP [\matrix{a_{1} = a_{2} = a_{3}\hfill\cr \alpha = \beta = \gamma = 90^{\circ}\hfill\cr}] [\matrix{a_{1} = a_{2} = a_{3}\hfill\cr \alpha = \beta = \gamma = 90^{\circ}\hfill\cr}] [\matrix{g_{11} &0 &0\hfill\cr &g_{11} &0\hfill\cr & &g_{11}\hfill\cr}] [\matrix{g_{11} &0 &0\hfill\cr &g_{11} &0\hfill\cr & &g_{11}\hfill\cr}]   [Scheme scheme12]
cI [\matrix{a_{1} = a_{2} = a_{3}\hfill\cr \alpha = \beta = \gamma = 109.5^{\circ}\hfill\cr \cos \alpha = -{\textstyle{1 \over 3}}\hfill\cr}] [{\bi P}(I)] [\matrix{g'_{11} = {\textstyle{3 \over 4}}g_{11}\hfill\cr g_{11} = {\textstyle{4 \over 3}}g'_{11}\hfill\cr}] [Scheme scheme13]
[\matrix{g'_{11} &-{\textstyle{1 \over 3}}g'_{11} &-{\textstyle{1 \over 3}}g'_{11}\hfill\cr &}}g'_{11} &-{\textstyle{1 \over 3}}g'_{11}\hfill\cr& &}}g'_{11}\hfill\cr}]
cF [\matrix{a_{1} = a_{2} = a_{3}\hfill\cr \alpha = \beta = \gamma = 60^{\circ}\cr}] [{\bi P}(F)] [\matrix{g'_{11} = {\textstyle{1 \over 2}}g_{11}\hfill\cr g_{11} = 2g'_{11}\hfill\cr}] [Scheme scheme14]
[\matrix{g'_{11} &{\textstyle{1 \over 2}}g'_{11} &{\textstyle{1 \over 2}}g'_{11}\hfill\cr &}}g'_{11} &{\textstyle{1 \over 2}}g'_{11}\hfill\cr & &}}g'_{11}\hfill\cr}]
The symbols for crystal families were adopted by the International Union of Crystallography in 1985; cf. de Wolff et al. (1985)[link]. Symbols in parentheses are standard symbols, see Table 2.1.2.1[link] .
[{\bi P}(C) = \textstyle{1 \over 2}(110/\bar{1}10/002), {\bi P}(I) = \textstyle{1 \over 2}(\bar{1}\bar{1}1/1\bar{1}1/11\bar{1}), {\bi P}(F) = \textstyle{1 \over 2}(011/101/110), {\bi P}(R)=\textstyle{1 \over 3}(\bar{1}2\bar{1}/\bar{2}11/111).]

The last columns of Tables 9.1.7.1[link] and 9.1.7.2[link] show parallel projections of the appropriate conventional unit cells. Among the different possible choices of the primitive basis, as discussed in Sections 9.1.1[link]–9.1.5[link], the special primitive basis mentioned above is obtained according to the following rules:

  • (i) For each type of centring, only one transformation matrix [\bi P] is used to obtain the primitive cell as given in Tables 9.1.7.1[link] and 9.1.7.2[link]. The transformation obeys equation (9.1.1.1)[link].

  • (ii) Among the different possible transformations, those are preferred which result in a metric tensor with simple relations among its components, as defined in Tables 9.1.7.1[link] and 9.1.7.2[link].

If a primitive basis is chosen according to these rules, basis vectors of the conventional cell have parallel face-diagonal or body-diagonal orientation with respect to the basis vectors of the primitive cell. For cubic and rhombohedral lattices, the primitive basis vectors are selected such that they are symmetrically equivalent with respect to a threefold axis. In all cases, a face of the `domain of influence' is perpendicular to each basis vector of these primitive cells.








































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