Table 2.2.14.1

Hahn, Th.; Looijenga-Vos, A.

doi:10.1107/97809553602060000505

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. 2.2, p. 33

Table 2.2.14.1

Th. Hahn^a ^* and A. Looijenga-Vos^b ^‡

^a Institut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany, and ^bLaboratorium voor Chemische Fysica, Rijksuniversiteit Groningen, The Netherlands
Correspondence e-mail: hahn@xtl.rwth-aachen.de

Table 2.2.14.1 | 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 Part 7

Projection direction	Triclinic	Monoclinic		Orthorhombic	Projection direction	Tetragonal
Projection direction	Triclinic	Unique axis b	Unique axis c	Orthorhombic	Projection direction	Tetragonal
[001]	$[a' = a \sin \beta]$	$[a' = a \sin \beta]$			[001]
	$[b' = b \sin \alpha]$
	$[\gamma' = 180^{\circ} - \gamma^{*}]$ ^†	$[\gamma' = 90^{\circ}]$	$[\gamma' = \gamma]$	$[\gamma' = 90^{\circ}]$		$[\gamma' = 90^{\circ}]$
[100]	$[a' = b \sin \gamma]$		$[a' = b \sin \gamma]$		[100]
	$[b' = c \sin \beta]$	$[b' = c \sin \beta]$
	$[\gamma' = 180^{\circ} - \alpha^{*}]$ ^†	$[\gamma' = 90^{\circ}]$	$[\gamma' = 90^{\circ}]$	$[\gamma' = 90^{\circ}]$		$[\gamma' = 90^{\circ}]$
[010]	$[a' = c \sin \alpha]$				[110]	$[a' = (a/2) \sqrt{2}]$
	$[b' = \alpha \sin \gamma]$		$[b' = a \sin \gamma]$
	$[\gamma' = 180^{\circ} - \beta^{*}]$ ^†	$[\gamma' = \beta]$	$[\gamma' = 90^{\circ}]$	$[\gamma' = 90^{\circ}]$		$[\gamma' = 90^{\circ}]$

Projection direction	Hexagonal	Projection direction	Rhombohedral^‡	Projection direction	Cubic
[001]		[111]	$[a' = {\displaystyle{2 \over \sqrt{3}}}\; a \sin (\alpha/2)]$	[001]
			$[b' = {\displaystyle{2 \over \sqrt{3}}} \;a \sin (\alpha/2)]$
	$[\gamma' = 120^{\circ}]$		$[\gamma' = 120^{\circ}]$		$[\gamma' = 90^{\circ}]$
[100]	$[a' = (a/2) \sqrt{3}]$	$[[1\bar{1}0]]$	$[a' = a \cos (\alpha/2)]$	[111]	$[a' = a\sqrt{2/3}]$
					$[b' = a\sqrt{2/3}]$
	$[\gamma' = 90^{\circ}]$		$[\gamma' = \delta]$ ^§		$[\gamma' = 120^{\circ}]$
[210]		$[[\bar{2}11]]$	$[a' = {\displaystyle{1 \over \sqrt{3}}}\; a\sqrt{1 + 2 \cos \alpha}]$	[110]	$[a' = (a/2)\sqrt{2}]$
			$[b' = a \sin (\alpha/2)]$
	$[\gamma' = 90^{\circ}]$		$[\gamma' = 90^{\circ}]$		$[\gamma' = 90^{\circ}]$

^† $[\cos \alpha^{*} = {\displaystyle{\cos \beta \cos \gamma - \cos \alpha \over \sin \beta \sin \gamma}}; \;\cos \beta^{*} = {\displaystyle{\cos \gamma \cos \alpha - \cos \beta \over \sin \gamma \sin \alpha}};\; \cos \gamma^{*} = {\displaystyle{\cos \alpha \cos \beta - \cos \gamma \over \sin \alpha \sin \beta}}.]$
^‡The entry `Rhombohedral' refers to the primitive rhombohedral cell with $[a = b = c, \alpha = \beta = \gamma]$ (cf. Table 2.1.2.1

).
^§ $[\cos \delta = {\displaystyle{\cos \alpha \over \cos \alpha/2\;}}]$ .