(International Tables for Crystallography) Crystal lattices

International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

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International Tables for Crystallography (2016). Vol. A, ch. 3.1, pp. 702-703

Table 3.1.2.2

H. Burzlaff^a and H. Zimmermann^b

Table 3.1.2.2| top | pdf |
Three-dimensional Bravais types of lattices

Bravais type of lattice^†	Lattice parameters		Metric tensor			Projections
Bravais type of lattice^†	Conventional	Primitive	Conventional	Primitive/transf.^‡	Relations of the components	Projections
aP	a, b, c	a, b, c	$[\matrix{g_{11} &g_{12} &g_{13}\hfill\cr &g_{22} &g_{23}\hfill\cr & &g_{33}\hfill\cr}]$	$[\matrix{g_{11} &g_{12} &g_{13}\hfill\cr &g_{22} &g_{23}\hfill\cr & &g_{33}\hfill\cr}]$
aP	α, β, γ	α, β, γ
mP	a, b, c	a, b, c	$[\matrix{g_{11} &0 &g_{13}\hfill\cr &g_{22} &0\hfill\cr & &g_{33}\hfill\cr}]$	$[\matrix{g_{11} &0 &g_{13}\hfill\cr &g_{22} &0\hfill\cr & &g_{33}\hfill\cr}]$
mP	β, α = γ = 90°	β, α = γ = 90°
mC		a₁ = a₂, c		P(C)	$[g'_{11} = {\textstyle{1 \over 4}}(g_{11} + g_{22})]$
(mS)		γ, α = β		$[\matrix{g'_{11} &g'_{12} &g'_{13}\hfill\cr &g'_{11} &g'_{13}\hfill\cr &&g_{33}\hfill\cr}]$	$[g'_{12} = {\textstyle{1 \over 4}}(g_{11} - g_{22})]$
					$[g'_{13} = {1 \over 2}g_{13}]$

					$[g_{11} = 2(g'_{11} + g'_{12})]$
					$[g_{22} = 2(g'_{11} - g'_{12})]$
					$[g_{13} = 2g'_{13}]$
oP	a, b, c	a, b, c	$[\matrix{g_{11} &0 &0\hfill\cr &g_{22} &0\hfill\cr & &g_{33}\hfill\cr}]$	$[\matrix{g_{11} &0 &0\hfill\cr &g_{22} &0\hfill\cr & &g_{33}\hfill\cr}]$
oP	α = β = γ = 90°	α = β = γ = 90°
oC		a₁ = a₂, c		P(C)	$[g'_{11} = {\textstyle{1 \over 4}}(g_{11} + g_{22})]$
(oS)		γ, α = β = 90°		$[\matrix{g'_{11} &g'_{12} &0\hfill\cr &g'_{11} &0\hfill\cr & &g_{33}\hfill\cr}]$	$[g'_{12} = {\textstyle{1 \over 4}}(g_{11} - g_{22})]$

					$[g_{11} = 2(g'_{11} + g'_{12})]$
					$[g_{22} = 2(g'_{11} - g'_{12})]$
oI		a₁ = a₂ = a₃		P(I)	$[g'_{12} = {\textstyle{1 \over 4}}(-g_{11} - g_{22} + g_{33})]$
		α, β, γ		$[\matrix{-\tilde g &g'_{12} &g'_{13}\hfill\cr &-\tilde g &g'_{23}\hfill\cr & &-\tilde g\hfill\cr}]$	$[g'_{13} = {\textstyle{1 \over 4}}(-g_{11} + g_{22} - g_{33})]$
		cos α + cos β + cos γ = −1			$[g'_{23} = {\textstyle{1 \over 4}}(g_{11} - g_{22} - g_{33})]$

				$[\tilde{g} = g'_{12} + g'_{13} + g'_{23}]$	$[g_{11} = -2(g'_{12} + g'_{13})]$
					$[g_{22} = -2(g'_{12} + g'_{23})]$
					$[g_{33} = -2(g'_{13} + g'_{23})]$
oF		a, b, c		$[{\bi P}(F)]$	$[g'_{12} = {\textstyle{1 \over 4}}\;g_{33}]$
		α, β, γ		$[\matrix{\tilde{g}_{1} &g'_{12} &g'_{13}\hfill\cr &\tilde{g}_{2} &g'_{23}\hfill\cr & &\tilde{g}_{3}\hfill\cr}]$	$[g'_{13} = {\textstyle{1 \over 4}}\;g_{22}]$
		$[\eqalign{&\cos \alpha =\cr&\quad {\displaystyle{-a^{2} + b^{2} + c^{2} \over 2bc}}\cr &\cos \beta =\cr&\quad {\displaystyle{a^{2} + b^{2} + c^{2} \over 2ac}}\cr &\cos \gamma =\cr&\quad {\displaystyle{a^{2} + b^{2} - c^{2} \over 2ab}}}]$			$[g'_{23} = {\textstyle{1 \over 4}}\;g_{11}]$

				$[\displaylines{\openup-4pt\hfill\cr \tilde{g}_{1} = g'_{12} + g'_{13}\hfill\cr\tilde{g}_{2} = g'_{12} + g'_{23}\hfill\cr \tilde{g}_{3} = g'_{13} + g'_{23}\hfill\cr}]$	$[g_{11} = 4g'_{23}]$
					$[g_{22} = 4g'_{13}]$
					$[g_{33} = 4g'_{12}]$
tP	a₁ = a₂, c	a₁ = a₂, c	$[\matrix{g_{11} &0 &0\hfill\cr &g_{11} &0\hfill\cr & &g_{33}\hfill\cr}]$	$[\matrix{g_{11} &0 &0\hfill\cr &g_{11} &0\hfill\cr & &g_{33}\hfill\cr}]$
tP	α = β = γ = 90°	α = β = γ = 90°
tI		a₁ = a₂ = a₃		$[{\bi P}(I)]$	$[g'_{12} = {\textstyle{1 \over 4}}(-2g_{11} + g_{33})]$
		γ, α = β		$[\matrix{\bar{g} &g'_{12} &g'_{13}\hfill\cr &\bar{g} &g'_{13}\hfill\cr & &\bar{g}\hfill\cr}]$	$[g'_{13} = -{\textstyle{1 \over 4}}g_{33}]$
		2 cos α + cos γ = −1
				$[\bar{g} = -(g'_{12} + 2g'_{13})]$	$[g_{11} = 2(g'_{12} + g'_{13})]$
					$[g_{33} = -4g'_{13}]$
hR	a₁ = a₂, c	a₁ = a₂ = a₃	$[\matrix{g_{11} &-{\textstyle{1 \over 2}}g_{11} &0\hfill\cr &\phantom{\textstyle{1 \over 2}}g_{11} &0\hfill\cr & &g_{33}\hfill\cr}]$	$[{\bi P}(R)]$	$[g'_{11} = {\textstyle{1 \over 9}}(3g_{11} + g_{33})]$
	α = β = 90°	α = β = γ		$[\matrix{g'_{11} &g'_{12} &g'_{12}\hfill\cr &g'_{11} &g'_{12}\hfill\cr & &g'_{11}\hfill\cr}]$	$[g'_{12} = {\textstyle{1 \over 9}}(-{3 \over 2}g_{11} + g_{33})]$
	γ = 120°
					$[g_{11} = 2(g'_{11} - g'_{12})]$
					$[g_{33} = 3(g'_{11} + 2g'_{12})]$
hP		a₁ = a₂, c		$[\matrix{g_{11} &-{\textstyle{1 \over 2}}g_{11} &0\hfill\cr &\phantom{ab\ }g_{11} &0\hfill\cr & &g_{33}\hfill\cr}]$
		α = β = 90°
		γ = 120°
cP	a₁ = a₂ = a₃	a₁ = a₂ = a₃	$[\matrix{g_{11} &0 &0\hfill\cr &g_{11} &0\hfill\cr & &g_{11}\hfill\cr}]$	$[\matrix{g_{11} &0 &0\hfill\cr &g_{11} &0\hfill\cr & &g_{11}\hfill\cr}]$
cP	α = β = γ = 90°	α = β = γ = 90°
cI		a₁ = a₂ = a₃		P(I)	$[g'_{11} = {\textstyle{3 \over 4}}g_{11}]$
		α = β = γ = 109.5°		$[\matrix{g'_{11} &-{\textstyle{1 \over 3}}g'_{11} &-{\textstyle{1 \over 3}}g'_{11}\hfill\cr &\phantom{{\textstyle{1 \over 3}}}g'_{11} &-{\textstyle{1 \over 3}}g'_{11}\hfill\cr& &\phantom{-{\textstyle{1 \over 3}}}g'_{11}\hfill\cr}]$	$[g_{11} = {\textstyle{4 \over 3}}g'_{11}]$
		$[\cos \alpha = -{\textstyle{1 \over 3}}]$			$[g_{11} = {\textstyle{4 \over 3}}g'_{11}]$
cF		a₁ = a₂ = a₃		P(F)	$[g'_{11} = {\textstyle{1 \over 2}}g_{11}]$
cF		α = β = γ = 60°		$[\matrix{g'_{11} &{\textstyle{1 \over 2}}g'_{11} &{\textstyle{1 \over 2}}g'_{11}\hfill\cr &\phantom{{\textstyle{1 \over 3}}}g'_{11} &{\textstyle{1 \over 2}}g'_{11}\hfill\cr & &\phantom{{\textstyle{1 \over 3}}}g'_{11}\hfill\cr}]$	$[g_{11} = 2g'_{11}]$

^†The symbols for Bravais types of lattices were adopted by the International Union of Crystallography in 1985; cf. de Wolff et al. (1985

). Symbols in parentheses are standard symbols, see Table 2.1.1.1

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