Table 10.2.1.2

Klapper, H.; Hahn, Th.

doi:10.1107/97809553602060000521

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. 10.2, p. 806

Table 10.2.1.2

H. Klapper^a ^‡ and Th. Hahn^a ^*

^a Institut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany
Correspondence e-mail: hahn@xtl.rwth-aachen.de

Table 10.2.1.2 | top | pdf |
Polar axes and nonpolar directions in the 21 noncentrosymmetric crystal classes

All directions normal to an evenfold rotation axis and along rotoinversion axes are nonpolar. All directions other than those in the column `Nonpolar directions' are polar. A symbol like [u0w] refers to the set of directions obtained for all possible values of u and w, in this case to all directions normal to the b axis, i.e. parallel to the plane (010). Symmetrically equivalent sets of nonpolar directions are placed between semicolons; the sequence of these sets follows the sequence of the symmetry directions in Table 2.2.4.1 .

System	Class	Polar (symmetry) axes	Nonpolar directions
Triclinic	1	None	None
Monoclinic	2	[010]	[u0w]
Unique axis b	m	None	[010]
Monoclinic	2	[001]	[uv0]
Unique axis c	m	None	[001]
Orthorhombic	222	None	[0vw]; [u0w]; [uv0]
Orthorhombic	mm 2	[001]	[uv0]
Tetragonal	4	[001]	[uv0]
	$[\bar{4}]$	None	[001]; [uv0]
	422	None	[uv0]; [0vw] [u0w];
			[uuw] [ $[u\bar{u}w]$ ]
	4mm	[001]	[uv0]
	$[\bar{4}2m]$	None	[uv0]; [0vw] [u0w]
	$[\bar{4}m2]$	None	[uv0]; [uuw] [ $[u\bar{u}w]$ ]
Trigonal (Hexagonal axes)	3	[001]	None
	321	[100], [010], $[\hbox{[}\bar{1}\bar{1}0\hbox{]}]$	[u2uw] $[\hbox{[}\overline{2u}\bar{u}w\hbox{]}\ \hbox{[}u\bar{u}w\hbox{]}]$
	312	$[\hbox{[}1\bar{1}0\hbox{]}]$ , [120], $[\hbox{[}\bar{2}\bar{1}0\hbox{]}]$	[uuw] $[\hbox{[}\bar{u}0w\hbox{]}\ \hbox{[}0\bar{v}w\hbox{]}]$
	3m1	[001]	[100] [010] $[\hbox{[}\bar{1}\bar{1}0\hbox{]}]$
	31m	[001]	$[\hbox{[}1\bar{1}0\hbox{]}]$ [120] $[\hbox{[}\bar{2}\bar{1}0\hbox{]}]$
Trigonal (Rhombohedral axes)	3	[111]	None
	32	$[\hbox{[}1\bar{1}0\hbox{]}]$ , $[\hbox{[}01\bar{1}\hbox{]}]$ , $[\hbox{[}\bar{1}01\hbox{]}]$	[uuw] [uvv] [uvu]
	3m	[111]	$[\hbox{[}1\bar{1}0\hbox{]}\ \hbox{[}01\bar{1}\hbox{]}\ \hbox{[}\bar{1}01\hbox{]}]$
Hexagonal	6	[001]	[uv0]
	$[\bar{6}]$	None	[001]
	622	None	[u2uw] [ $[\overline{2u}\bar{u}w\hbox {] [}u\bar{u}w]$ ]
			[uuw] [ $[\bar{u}0w\hbox{] [}0\bar{v}w]$ ]
	6mm	[001]	[uv0]
	$[\bar{6}m2]$	$[\hbox{[}1\bar{1}0\hbox{]}]$ , [120], $[\hbox{[}\bar{2}\bar{1}0\hbox{]}]$	[uuw] [ $[\bar{u}0w\hbox{] [}0\bar{v}w]$ ]
	$[\bar{6}2m]$	[100], [010], $[\hbox{[}\bar{1}\bar{1}0\hbox{]}]$	[u2uw] $[\hbox{[}\overline{2u}\bar{u}w\hbox{]}\ \hbox{[}u\bar{u}w\hbox{]}]$
Cubic	$[\!\left.\matrix{23\hfill\cr \bar{4}3m\hfill\cr}\right\}]$	$[\!\matrix{\hbox{Four threefold}\hfill\cr\quad \hbox{axes along }\langle 111\rangle\hfill\cr}]$	$[\!\matrix{\hbox{[}0vw\hbox{]}\ \hbox{[}u0w\hbox{]}\ \hbox{[}uv0\hbox{]}\cr \hbox{[}0vw\hbox{]}\ \hbox{[}u0w\hbox{]}\ \hbox{[}uv0\hbox{]}\cr}]$
Cubic	432	$[{\hbox{None}}{\hbox to 3.9pc{}}\left\{\matrix{\cr\cr\cr}\right.]$	$[\!\matrix{\hbox{[}0vw\hbox{]}\ \hbox{[}u0w\hbox{]}\ \hbox{[}uv0\hbox{]};\hfill\cr \hbox{[}uuw\hbox{]}\ \hbox{[}uvv\hbox{]}\ \hbox{[}uvu\hbox{]};\hfill\cr \hbox{[}u\bar{u}w\hbox{]}\ \hbox{[}uv\bar{v}\hbox{]}\ \hbox{[}\bar{u}vu\hbox{]}\hfill\cr}]$