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

International Tables for Crystallography (2016). Vol. A, ch. 2.1, p. 168

Table 2.1.3.9 

Th. Hahna and A. Looijenga-Vosb

Table 2.1.3.9| top | pdf |
Cell parameters a′, b′, γ′ of the two-dimensional cell in terms of cell parameters a, b, c, α, β, γ of the three-dimensional cell for the projections listed in the space-group tables of Chapter 2.3[link]

Projection directionTriclinicMonoclinicOrthorhombic
Unique axis bUnique axis c
[001] [a' = a \sin \beta] [a' = a \sin \beta] [a' = a] [a' = a]
[b' = b \sin \alpha] [b' = b] [b' = b] [b' = b]
[\gamma' = 180^{\circ} - \gamma^{*}] [\gamma' = 90^{\circ}] [\gamma' = \gamma] [\gamma' = 90^{\circ}]
[100] [a' = b \sin \gamma] [a' = b] [a' = b \sin \gamma] [a' = b]
[b' = c \sin \beta] [b' = c \sin \beta] [b' = c] [b' = c]
[\gamma' = 180^{\circ} - \alpha^{*}] [\gamma' = 90^{\circ}] [\gamma' = 90^{\circ}] [\gamma' = 90^{\circ}]
[010] [a' = c \sin \alpha] [a' = c] [a' = c] [a' = c]
[b' = \alpha \sin \gamma] [b' = a] [b' = a \sin \gamma] [b' = a]
[\gamma' = 180^{\circ} - \beta^{*}] [\gamma' = \beta] [\gamma' = 90^{\circ}] [\gamma' = 90^{\circ}]

Projection directionTetragonal
[001] [a' = a]
[b' = a]
[\gamma' = 90^{\circ}]
[100] [a' = a]
[b' = c]
[\gamma' = 90^{\circ}]
[110] [a' = (a/2) \sqrt{2}]
[b' = c]
[\gamma' = 90^{\circ}]

Projection directionHexagonal
[001] [a' = a]
[b' = a]
[\gamma' = 120^{\circ}]
[100] [a' = (a/2) \sqrt{3}]
[b' = c]
[\gamma' = 90^{\circ}]
[210] [a' = a/2]
[b' = c]
[\gamma' = 90^{\circ}]

Projection directionRhombohedral
[111] [a' = {\displaystyle{2 \over \sqrt{3}}}\; a \sin (\alpha/2)]
[b' = {\displaystyle{2 \over \sqrt{3}}} \;a \sin (\alpha/2)]
[\gamma' = 120^{\circ}]
[[1\bar{1}0]] [a' = a \cos (\alpha/2)]
[b' = a]
[\gamma' = \delta]§
[[\bar{2}11]] [a' = {\displaystyle{1 \over \sqrt{3}}}\; a\sqrt{1 + 2 \cos \alpha}]
[b' = a \sin (\alpha/2)]
[\gamma' = 90^{\circ}]

Projection directionCubic
[001] [a' = a]
[b' = a]
[\gamma' = 90^{\circ}]
[111] [a' = a\sqrt{2/3}]
[b' = a\sqrt{2/3}]
[\gamma' = 120^{\circ}]
[110] [a' = (a/2)\sqrt{2}]
[b' = a]
[\gamma' = 90^{\circ}]
[\cos \alpha^{*} = {\displaystyle{\cos \beta \cos \gamma - \cos \alpha \over \sin \beta \sin \gamma}}; \;\cos \beta^{*} = {\displaystyle{\cos \gamma \cos \alpha - \cos \beta \over \sin \gamma \sin \alpha}};\; \cos \gamma^{*} = {\displaystyle{\cos \alpha \cos \beta - \cos \gamma \over \sin \alpha \sin \beta}}.]
The entry `Rhombohedral' refers to the primitive rhombohedral cell with [a = b = c], [\alpha = \beta = \gamma] (cf. Table 2.1.1.1[link]).
§[\cos \delta = {\displaystyle{\cos \alpha \over \cos \alpha/2\;}}].