Table 9.1.7.2

Burzlaff, H.; Zimmermann, H.

doi:10.1107/97809553602060000517

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

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International Tables for Crystallography (2006). Vol. A. ch. 9.1, pp. 746-747

Table 9.1.7.2

H. Burzlaff^a and H. Zimmermann^b ^*

^a Universität 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

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

Bravais lattice^†	Lattice parameters		Metric tensor			Projections
Bravais lattice^†	Conventional	Primitive	Conventional	Primitive/transf.^‡	Relations of the components	Projections
aP	$[\matrix{a, b, c\hfill\cr \alpha, \beta, \gamma\hfill\cr}]$	$[\matrix{a, b, c\hfill\cr \alpha, \beta, \gamma\hfill\cr}]$	$[\matrix{g_{11} &g_{12} &g_{13}\hfill\cr &g_{22} &g_{23}\hfill\cr & &g_{33}\hfill\cr}]$	$[\matrix{g_{11} &g_{12} &g_{13}\hfill\cr &g_{22} &g_{23}\hfill\cr & &g_{33}\hfill\cr}]$
mP	$[\matrix{a, b, c\hfill\cr \beta, \alpha = \gamma = 90^{\circ}\hfill\cr}]$	$[\matrix{a, b, c\hfill\cr \beta, \alpha = \gamma = 90^{\circ}\hfill\cr}]$	$[\matrix{g_{11} &0 &g_{13}\hfill\cr &g_{22} &0\hfill\cr & &g_{33}\hfill\cr}]$	$[\matrix{g_{11} &0 &g_{13}\hfill\cr &g_{22} &0\hfill\cr & &g_{33}\hfill\cr}]$
$[\matrix{mC\hfill\cr (mS)\hfill\cr}]$		$[\matrix{a_{1} = a_{2}, c\hfill\cr \gamma, \alpha = \beta\hfill\cr}]$		$[{\bi P}(C)]$	$[\matrix{g'_{11} = {\textstyle{1 \over 4}}(g_{11} + g_{22})\hfill\cr g'_{12} = {\textstyle{1 \over 4}}(g_{11} - g_{22})\hfill\cr g'_{13} = {1 \over 2}g_{13}\hfill\cr \cr g_{11} = 2(g'_{11} + g'_{12})\hfill\cr g_{22} = 2(g'_{11} - g'_{12})\hfill\cr g_{13} = 2g'_{13}\hfill\cr}]$
$[\matrix{mC\hfill\cr (mS)\hfill\cr}]$				$[\matrix{g'_{11} &g'_{12} &g'_{13}\hfill\cr &g'_{11} &g'_{13}\hfill\cr &&g_{33}\hfill\cr}]$
oP	$[\matrix{a, b, c\hfill\cr \alpha = \beta = \gamma = 90^{\circ}\hfill\cr}]$	$[\matrix{a, b, c\hfill\cr \alpha = \beta = \gamma = 90^{\circ}\hfill\cr}]$	$[\matrix{g_{11} &0 &0\hfill\cr &g_{22} &0\hfill\cr & &g_{33}\hfill\cr}]$	$[\matrix{g_{11} &0 &0\hfill\cr &g_{22} &0\hfill\cr & &g_{33}\hfill\cr}]$
$[\matrix{oC\hfill\cr (oS)\hfill\cr}]$		$[\matrix{a_{1} = a_{2}, c\hfill\cr \gamma, \alpha = \beta = 90^{\circ}\hfill\cr}]$		$[{\bi P}(C)]$	$[\matrix{g'_{11} = {\textstyle{1 \over 4}}(g_{11} + g_{22})\hfill\cr g'_{12} = {\textstyle{1 \over 4}}(g_{11} - g_{22})\hfill\cr \cr g_{11} = 2(g'_{11} + g'_{12})\hfill\cr g_{22} = 2(g'_{11} - g'_{12})\hfill\cr}]$
$[\matrix{oC\hfill\cr (oS)\hfill\cr}]$				$[\matrix{g'_{11} &g'_{12} &0\hfill\cr &g'_{11} &0\hfill\cr & &g_{33}\hfill\cr}]$
oI		$[\matrix{a_{1} = a_{2} = a_{3}\hfill\cr \alpha, \beta, \gamma\hfill\cr \cos \alpha + \cos \beta\hfill\cr\quad + \cos \gamma = -1\hfill\cr}]$		$[{\bi P}(I)]$	$[\matrix{g'_{12} = {\textstyle{1 \over 4}}(-g_{11} - g_{22} + g_{33})\hfill\cr g'_{13} = {\textstyle{1 \over 4}}(-g_{11} + g_{22} - g_{33})\hfill\cr g'_{23} = {\textstyle{1 \over 4}}(g_{11} - g_{22} - g_{33})\hfill\cr \cr g_{11} = -2(g'_{12} + g'_{13})\hfill\cr g_{22} = -2(g'_{12} + g'_{23})\hfill\cr g_{33} = -2(g'_{13} + g'_{23})\hfill\cr}]$
oI				$[\displaylines{\matrix{-\tilde g &g'_{12} &g'_{13}\hfill\cr &-\tilde g &g'_{23}\hfill\cr & &-\tilde g\hfill\cr}\hfill\cr \tilde{g} = g'_{12} + g'_{13} + g'_{23}\hfill}]$
oF		$[\openup1pt\matrix{a, b, c\hfill\cr \alpha, \beta, \gamma\hfill\cr \cos \alpha = {\displaystyle{-a^{2} + b^{2} + c^{2} \over 2bc}}\hfill\cr \cos \beta = {\displaystyle{a^{2} + b^{2} + c^{2} \over 2ac}}\hfill\cr \cos \gamma = {\displaystyle{a^{2} + b^{2} - c^{2} \over 2ab}}\hfill\cr\cr}]$		$[{\bi P}(F)]$	$[\matrix{g'_{12} = {\textstyle{1 \over 4}}\;g_{33}\hfill\cr g'_{13} = {\textstyle{1 \over 4}}\;g_{22}\hfill\cr g'_{23} = {\textstyle{1 \over 4}}\;g_{11}\hfill\cr \cr g_{11} = 4g'_{23}\hfill\cr g_{22} = 4g'_{13}\hfill\cr g_{33} = 4g'_{12}\hfill\cr}]$
oF				$[\displaylines{\openup-4pt\matrix{\tilde{g}_{1} &g'_{12} &g'_{13}\hfill\cr &\tilde{g}_{2} &g'_{23}\hfill\cr & &\tilde{g}_{3}\hfill\cr}\hfill\cr \tilde{g}_{1} = g'_{12} + g'_{13}\hfill\cr\tilde{g}_{2} = g'_{12} + g'_{23}\hfill\cr \tilde{g}_{3} = g'_{13} + g'_{23}\hfill\cr}]$
tP	$[\matrix{a_{1} = a_{2}, c\hfill\cr \alpha = \beta = \gamma = 90^{\circ}\hfill\cr}]$	$[\matrix{a_{1} = a_{2}, c\hfill\cr \alpha = \beta = \gamma = 90^{\circ}\hfill\cr}]$	$[\matrix{g_{11} &0 &0\hfill\cr &g_{11} &0\hfill\cr & &g_{33}\hfill\cr}]$	$[\matrix{g_{11} &0 &0\hfill\cr &g_{11} &0\hfill\cr & &g_{33}\hfill\cr}]$
tI		$[\matrix{a_{1} = a_{2} = a_{3}\hfill\cr \gamma, \alpha = \beta\hfill\cr 2\cos \alpha + \cos \gamma = -1\hfill\cr}]$		$[{\bi P}(I)]$	$[\matrix{g'_{12} = {\textstyle{1 \over 4}}(-2g_{11} + g_{33})\hfill\cr g'_{13} = -{\textstyle{1 \over 4}}g_{33}\hfill\cr \cr g_{11} = 2(g'_{12} + g'_{13})\hfill\cr g_{33} = -4g'_{13}\hfill\cr}]$
tI				$[\eqalign{&\matrix{\bar{g} &g'_{12} &g'_{13}\hfill\cr &\bar{g} &g'_{13}\hfill\cr & &\bar{g}\hfill\cr}\cr &\bar{g} = -(g'_{12} + 2g'_{13})}]$
hR	$[\matrix{a_{1} = a_{2}, c\hfill\cr \alpha = \beta = 90^{\circ}\hfill\cr \gamma = 120^{\circ}\hfill\cr}]$	$[\matrix{a_{1} = a_{2} = a_{3}\hfill\cr \alpha = \beta = \gamma\hfill\cr}]$	$[\matrix{g_{11} &-{\textstyle{1 \over 2}}g_{11} &0\hfill\cr &}g_{11} &0\hfill\cr & &g_{33}\hfill\cr}]$	$[{\bi P}(R)]$	$[\matrix{g'_{11} = {\textstyle{1 \over 9}}(3g_{11} + g_{33})\hfill\cr g'_{12} = {\textstyle{1 \over 9}}(-{3 \over 2}g_{11} + g_{33})\hfill\cr \cr g_{11} = 2(g'_{11} - g'_{12})\hfill\cr g_{33} = 3(g'_{11} + 2g'_{12})\hfill\cr}]$
hR				$[\matrix{g'_{11} &g'_{12} &g'_{12}\hfill\cr &g'_{11} &g'_{12}\hfill\cr & &g'_{11}\hfill\cr}]$
hP		$[\matrix{a_{1} = a_{2}, c\hfill\cr \alpha = \beta = 90^{\circ}\hfill\cr \gamma = 120^{\circ}\hfill\cr}]$		$[\matrix{g_{11} &-{\textstyle{1 \over 2}}g_{11} &0\hfill\cr &g_{11} &0\hfill\cr & &g_{33}\hfill\cr}]$
cP	$[\matrix{a_{1} = a_{2} = a_{3}\hfill\cr \alpha = \beta = \gamma = 90^{\circ}\hfill\cr}]$	$[\matrix{a_{1} = a_{2} = a_{3}\hfill\cr \alpha = \beta = \gamma = 90^{\circ}\hfill\cr}]$	$[\matrix{g_{11} &0 &0\hfill\cr &g_{11} &0\hfill\cr & &g_{11}\hfill\cr}]$	$[\matrix{g_{11} &0 &0\hfill\cr &g_{11} &0\hfill\cr & &g_{11}\hfill\cr}]$
cI		$[\matrix{a_{1} = a_{2} = a_{3}\hfill\cr \alpha = \beta = \gamma = 109.5^{\circ}\hfill\cr \cos \alpha = -{\textstyle{1 \over 3}}\hfill\cr}]$		$[{\bi P}(I)]$	$[\matrix{g'_{11} = {\textstyle{3 \over 4}}g_{11}\hfill\cr g_{11} = {\textstyle{4 \over 3}}g'_{11}\hfill\cr}]$
cI				$[\matrix{g'_{11} &-{\textstyle{1 \over 3}}g'_{11} &-{\textstyle{1 \over 3}}g'_{11}\hfill\cr &}}g'_{11} &-{\textstyle{1 \over 3}}g'_{11}\hfill\cr& &}}g'_{11}\hfill\cr}]$
cF		$[\matrix{a_{1} = a_{2} = a_{3}\hfill\cr \alpha = \beta = \gamma = 60^{\circ}\cr}]$		$[{\bi P}(F)]$	$[\matrix{g'_{11} = {\textstyle{1 \over 2}}g_{11}\hfill\cr g_{11} = 2g'_{11}\hfill\cr}]$
cF				$[\matrix{g'_{11} &{\textstyle{1 \over 2}}g'_{11} &{\textstyle{1 \over 2}}g'_{11}\hfill\cr &}}g'_{11} &{\textstyle{1 \over 2}}g'_{11}\hfill\cr & &}}g'_{11}\hfill\cr}]$

^†The symbols for crystal families were adopted by the International Union of Crystallography in 1985; cf. de Wolff et al. (1985)

. Symbols in parentheses are standard symbols, see Table 2.1.2.1

.
^‡ $[{\bi P}(C) = \textstyle{1 \over 2}(110/\bar{1}10/002), {\bi P}(I) = \textstyle{1 \over 2}(\bar{1}\bar{1}1/1\bar{1}1/11\bar{1}), {\bi P}(F) = \textstyle{1 \over 2}(011/101/110), {\bi P}(R)=\textstyle{1 \over 3}(\bar{1}2\bar{1}/\bar{2}11/111).]$