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

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

Table 3.1.2.2 

H. Burzlaffa and H. Zimmermannb

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

Bravais type of latticeLattice parametersMetric tensorProjections
ConventionalPrimitiveConventionalPrimitive/transf.Relations of the components
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}]   [Scheme scheme116]
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}]   [Scheme scheme117]
β, α = γ = 90° β, α = γ = 90°
mC a1 = a2, c P(C) [g'_{11} = {\textstyle{1 \over 4}}(g_{11} + g_{22})] [Scheme scheme118]
(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}]   [Scheme scheme119]
α = β = γ = 90° α = β = γ = 90°  
oC a1 = a2, c P(C) [g'_{11} = {\textstyle{1 \over 4}}(g_{11} + g_{22})] [Scheme scheme120]
(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})] [Scheme scheme121]
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}] [Scheme scheme122]
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}]   [Scheme scheme123]
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})] [Scheme scheme124]
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})] [Scheme scheme125]
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}]   [Scheme scheme126]
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}]   [Scheme scheme127]
α = β = γ = 90° α = β = γ = 90°
cI a1 = a2 = a3 P(I) [g'_{11} = {\textstyle{3 \over 4}}g_{11}] [Scheme scheme128]
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}}]
cF a1 = a2 = a3 P(F) [g'_{11} = {\textstyle{1 \over 2}}g_{11}] [Scheme scheme129]
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[link]). Symbols in parentheses are standard symbols, see Table 2.1.1.1[link] .
[{\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).]