Bravais type of lattice† | Lattice parameters | Metric tensor | Projections |
Conventional | Primitive | Conventional | Primitive/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}]](/teximages/acch3o1/acch3o1fd46.gif) |
![[\matrix{g_{11} &g_{12} &g_{13}\hfill\cr &g_{22} &g_{23}\hfill\cr & &g_{33}\hfill\cr}]](/teximages/acch3o1/acch3o1fd46.gif) |
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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}]](/teximages/acch3o1/acch3o1fd48.gif) |
![[\matrix{g_{11} &0 &g_{13}\hfill\cr &g_{22} &0\hfill\cr & &g_{33}\hfill\cr}]](/teximages/acch3o1/acch3o1fd48.gif) |
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β, α = γ = 90° |
β, α = γ = 90° |
mC |
a1 = a2, c |
P(C) |
![[g'_{11} = {\textstyle{1 \over 4}}(g_{11} + g_{22})]](/teximages/acch3o1/acch3o1fi143.gif) |
|
(mS) |
γ, α = β |
![[\matrix{g'_{11} &g'_{12} &g'_{13}\hfill\cr &g'_{11} &g'_{13}\hfill\cr &&g_{33}\hfill\cr}]](/teximages/acch3o1/acch3o1fd50.gif) |
![[g'_{12} = {\textstyle{1 \over 4}}(g_{11} - g_{22})]](/teximages/acch3o1/acch3o1fi144.gif) |
![[g'_{13} = {1 \over 2}g_{13}]](/teximages/acch3o1/acch3o1fi150.gif) |
|
![[g_{11} = 2(g'_{11} + g'_{12})]](/teximages/acch3o1/acch3o1fi145.gif) |
![[g_{22} = 2(g'_{11} - g'_{12})]](/teximages/acch3o1/acch3o1fi152.gif) |
![[g_{13} = 2g'_{13}]](/teximages/acch3o1/acch3o1fi153.gif) |
oP |
a, b, c |
a, b, c |
![[\matrix{g_{11} &0 &0\hfill\cr &g_{22} &0\hfill\cr & &g_{33}\hfill\cr}]](/teximages/acch3o1/acch3o1fd51.gif) |
![[\matrix{g_{11} &0 &0\hfill\cr &g_{22} &0\hfill\cr & &g_{33}\hfill\cr}]](/teximages/acch3o1/acch3o1fd51.gif) |
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α = β = γ = 90° |
α = β = γ = 90° |
|
oC |
a1 = a2, c |
P(C) |
![[g'_{11} = {\textstyle{1 \over 4}}(g_{11} + g_{22})]](/teximages/acch3o1/acch3o1fi143.gif) |
|
(oS) |
γ, α = β = 90° |
![[\matrix{g'_{11} &g'_{12} &0\hfill\cr &g'_{11} &0\hfill\cr & &g_{33}\hfill\cr}]](/teximages/acch3o1/acch3o1fd53.gif) |
![[g'_{12} = {\textstyle{1 \over 4}}(g_{11} - g_{22})]](/teximages/acch3o1/acch3o1fi144.gif) |
|
![[g_{11} = 2(g'_{11} + g'_{12})]](/teximages/acch3o1/acch3o1fi145.gif) |
![[g_{22} = 2(g'_{11} - g'_{12})]](/teximages/acch3o1/acch3o1fi152.gif) |
oI |
a1 = a2 = a3 |
P(I) |
![[g'_{12} = {\textstyle{1 \over 4}}(-g_{11} - g_{22} + g_{33})]](/teximages/acch3o1/acch3o1fi158.gif) |
|
α, β, γ |
![[\matrix{-\tilde g &g'_{12} &g'_{13}\hfill\cr &-\tilde g &g'_{23}\hfill\cr & &-\tilde g\hfill\cr}]](/teximages/acch3o1/acch3o1fd54.gif) |
![[g'_{13} = {\textstyle{1 \over 4}}(-g_{11} + g_{22} - g_{33})]](/teximages/acch3o1/acch3o1fi159.gif) |
cos α + cos β + cos γ = −1 |
![[g'_{23} = {\textstyle{1 \over 4}}(g_{11} - g_{22} - g_{33})]](/teximages/acch3o1/acch3o1fi160.gif) |
|
![[\tilde{g} = g'_{12} + g'_{13} + g'_{23}]](/teximages/acch3o1/acch3o1fi161.gif) |
![[g_{11} = -2(g'_{12} + g'_{13})]](/teximages/acch3o1/acch3o1fi162.gif) |
![[g_{22} = -2(g'_{12} + g'_{23})]](/teximages/acch3o1/acch3o1fi163.gif) |
![[g_{33} = -2(g'_{13} + g'_{23})]](/teximages/acch3o1/acch3o1fi164.gif) |
oF |
a, b, c |
![[{\bi P}(F)]](/teximages/acch3o1/acch3o1fi165.gif) |
![[g'_{12} = {\textstyle{1 \over 4}}\;g_{33}]](/teximages/acch3o1/acch3o1fi166.gif) |
|
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}]](/teximages/acch3o1/acch3o1fd55.gif) |
![[g'_{13} = {\textstyle{1 \over 4}}\;g_{22}]](/teximages/acch3o1/acch3o1fi167.gif) |
![[\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}}}]](/teximages/acch3o1/acch3o1fd56.gif) |
![[g'_{23} = {\textstyle{1 \over 4}}\;g_{11}]](/teximages/acch3o1/acch3o1fi168.gif) |
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![[\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}]](/teximages/acch3o1/acch3o1fd57.gif) |
![[g_{11} = 4g'_{23}]](/teximages/acch3o1/acch3o1fi169.gif) |
![[g_{22} = 4g'_{13}]](/teximages/acch3o1/acch3o1fi170.gif) |
![[g_{33} = 4g'_{12}]](/teximages/acch3o1/acch3o1fi171.gif) |
tP |
a1 = a2, c |
a1 = a2, c |
![[\matrix{g_{11} &0 &0\hfill\cr &g_{11} &0\hfill\cr & &g_{33}\hfill\cr}]](/teximages/acch3o1/acch3o1fd58.gif) |
![[\matrix{g_{11} &0 &0\hfill\cr &g_{11} &0\hfill\cr & &g_{33}\hfill\cr}]](/teximages/acch3o1/acch3o1fd58.gif) |
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|
α = β = γ = 90° |
α = β = γ = 90° |
tI |
a1 = a2 = a3 |
![[{\bi P}(I)]](/teximages/acch3o1/acch3o1fi172.gif) |
![[g'_{12} = {\textstyle{1 \over 4}}(-2g_{11} + g_{33})]](/teximages/acch3o1/acch3o1fi173.gif) |
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γ, α = β |
![[\matrix{\bar{g} &g'_{12} &g'_{13}\hfill\cr &\bar{g} &g'_{13}\hfill\cr & &\bar{g}\hfill\cr}]](/teximages/acch3o1/acch3o1fd60.gif) |
![[g'_{13} = -{\textstyle{1 \over 4}}g_{33}]](/teximages/acch3o1/acch3o1fi174.gif) |
2 cos α + cos γ = −1 |
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|
![[\bar{g} = -(g'_{12} + 2g'_{13})]](/teximages/acch3o1/acch3o1fi175.gif) |
![[g_{11} = 2(g'_{12} + g'_{13})]](/teximages/acch3o1/acch3o1fi176.gif) |
|
![[g_{33} = -4g'_{13}]](/teximages/acch3o1/acch3o1fi177.gif) |
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}]](/teximages/acch3o1/acch3o1fd61.gif) |
![[{\bi P}(R)]](/teximages/acch3o1/acch3o1fi178.gif) |
![[g'_{11} = {\textstyle{1 \over 9}}(3g_{11} + g_{33})]](/teximages/acch3o1/acch3o1fi179.gif) |
|
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}]](/teximages/acch3o1/acch3o1fd62.gif) |
![[g'_{12} = {\textstyle{1 \over 9}}(-{3 \over 2}g_{11} + g_{33})]](/teximages/acch3o1/acch3o1fi180.gif) |
γ = 120° |
|
![[g_{11} = 2(g'_{11} - g'_{12})]](/teximages/acch3o1/acch3o1fi181.gif) |
![[g_{33} = 3(g'_{11} + 2g'_{12})]](/teximages/acch3o1/acch3o1fi182.gif) |
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}]](/teximages/acch3o1/acch3o1fd63.gif) |
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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}]](/teximages/acch3o1/acch3o1fd64.gif) |
![[\matrix{g_{11} &0 &0\hfill\cr &g_{11} &0\hfill\cr & &g_{11}\hfill\cr}]](/teximages/acch3o1/acch3o1fd64.gif) |
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|
α = β = γ = 90° |
α = β = γ = 90° |
cI |
a1 = a2 = a3 |
P(I) |
![[g'_{11} = {\textstyle{3 \over 4}}g_{11}]](/teximages/acch3o1/acch3o1fi183.gif) |
|
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}]](/teximages/acch3o1/acch3o1fd66.gif) |
![[g_{11} = {\textstyle{4 \over 3}}g'_{11}]](/teximages/acch3o1/acch3o1fi184.gif) |
![[\cos \alpha = -{\textstyle{1 \over 3}}]](/teximages/acch3o1/acch3o1fi185.gif) |
cF |
a1 = a2 = a3 |
P(F) |
![[g'_{11} = {\textstyle{1 \over 2}}g_{11}]](/teximages/acch3o1/acch3o1fi186.gif) |
|
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}]](/teximages/acch3o1/acch3o1fd67.gif) |
![[g_{11} = 2g'_{11}]](/teximages/acch3o1/acch3o1fi187.gif) |
†The symbols for Bravais types of lattices were adopted by the International Union of Crystallography in 1985; cf. de Wolff et al. (1985 ![[link]](/graphics/greenarr.gif) ). Symbols in parentheses are standard symbols, see Table 2.1.1.1 ![[link]](/graphics/greenarr.gif)
.
‡
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