Table 2.2.13.2

Hahn, Th.; Looijenga-Vos, A.

doi:10.1107/97809553602060000505

International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. A. ch. 2.2, pp. 30-31

Table 2.2.13.2

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 2.2.13.2 | top | pdf |
Zonal and serial reflection conditions for glide planes and screw axes (cf. Chapter 1.3 )

(a) Glide planes

Type of reflections	Reflection condition	Glide plane			Crystallographic coordinate system to which condition applies
Type of reflections	Reflection condition	Orientation of plane	Glide vector	Symbol
0kl		(100)	$[{\bf b}/2]$	b	$[\!\left.\matrix{\cr\cr\cr\cr\cr}\!\right\}\!\matrix{\hbox{Monoclinic } (a\hbox{ unique)},\hfill\cr\quad\hbox{Tetragonal}\hfill\cr}]$	$[\!\left.\matrix{\cr\cr\cr\cr\cr\cr\cr}\right\}\!\matrix{\hbox{Orthorhombic,}\hfill\cr\quad\hbox{Cubic}\hfill\cr}]$
			$[{\bf c}/2]$	c
			$[{\bf b}/2 + {\bf c}/2]$	n
	$[\!\matrix{k + l = 4n\hfill\cr \quad (k,l = 2n)\hfill\cr}]$ ^†		$[{\bf b}/4 \pm {\bf c}/4]$	d
h0l		(010)	$[{\bf c}/2]$	c	$[\!\left.\matrix{\cr\cr\cr\cr\cr}\!\right\}\!\matrix{\hbox{Monoclinic } (b\hbox{ unique)},\hfill\cr\quad\hbox{Tetragonal}\hfill\cr}]$	$[\!\left.\matrix{\cr\cr\cr\cr\cr\cr\cr}\right\}\!\matrix{\hbox{Orthorhombic,}\hfill\cr\quad\hbox{Cubic}\hfill\cr}]$
			$[{\bf a}/2]$	a
			$[{\bf c}/2 + {\bf a}/2]$	n
	$[\!\matrix{l + h = 4n\hfill\cr \quad (l,h = 2n)\hfill\cr}]$ ^†		$[{\bf c}/4 \pm {\bf a}/4]$	d
hk0		(001)	$[{\bf a}/2]$	a	$[\!\left.\matrix{\cr\cr\cr\cr\cr}\!\right\}\!\matrix{\hbox{Monoclinic (}c\ \hbox{unique)},\hfill\cr\quad\hbox{Tetragonal}\hfill\cr}]$	$[\!\left.\matrix{\cr\cr\cr\cr\cr\cr\cr}\right\}\!\matrix{\hbox{Orthorhombic,}\hfill\cr\quad\hbox{Cubic}\hfill\cr}]$
			$[{\bf b}/2]$	b
			$[{\bf a}/2 + {\bf b}/2]$	n
	$[\!\matrix{h + k = 4n\hfill\cr \quad (h,k = 2n)\hfill\cr}]$ ^†		$[{\bf a}/4 \pm {\bf b}/4]$	d
$[\matrix{h\bar{h}0l\hfill\cr 0k\bar{k}l\hfill\cr \bar{h}0hl\hfill\cr}]$		$[\left.\!\matrix{(11\bar{2}0)\hfill\cr (\bar{2}110)\hfill\cr (1\bar{2}10)\hfill\cr}\right\} \{11\bar{2}0\}]$	$[{\bf c}/2]$	c	$[\left.\vphantom{\matrix{(11\bar{2}0)\hfill\cr (\bar{2}110)\hfill\cr (1\bar{2}10)\hfill\cr}}\right\}\hbox{Hexagonal}]$
$[\matrix{hh.\overline{2h}.l\hfill\cr \overline{2h}.hhl\hfill\cr h.\overline{2h}.hl\hfill\cr}]$		$[\left.\!\matrix{(1\bar{1}00)\hfill\cr (01\bar{1}0)\hfill\cr (\bar{1}010)\hfill\cr}\right\} \{1\bar{1}00\}]$	$[{\bf c}/2]$	c	$[\left.\vphantom{\matrix{(11\bar{2}0)\hfill\cr (\bar{2}110)\hfill\cr (1\bar{2}10)\hfill\cr}}\right\}\hbox{Hexagonal}]$
$[\matrix{hhl\hfill\cr hkk\hfill\cr hkh\hfill\cr}]$	$[\matrix{l = 2n\hfill\cr h = 2n\hfill\cr k = 2n\hfill\cr}]$	$[\left.\!\matrix{(1\bar{1}0)\hfill\cr (01\bar{1})\hfill\cr (\bar{1}01)\hfill\cr}\right\} \{1\bar{1}0\}]$	$[\matrix{{\bf c}/2\hfill\cr {\bf a}/2\hfill\cr {\bf b}/2\hfill\cr}]$	$[\matrix{c,n\hfill\cr a,n\hfill\cr b,n\hfill\cr}]$	$[\left.\vphantom{\matrix{(11\bar{2}0)\hfill\cr (\bar{2}110)\hfill\cr (1\bar{2}10)\hfill\cr}}\right\}\hbox{Rhombohedral}]$ ^‡
$[hhl, h\bar{h}l]$		$[(1\bar{1}0), (110)]$	$[{\bf c}/2]$	c, n	$[\!\left.\matrix{\cr\cr\cr}\right\}\hbox{Tetragonal}]$ ^§	$[\!\left.\matrix{\cr\cr\cr\cr\cr\cr\cr\cr\cr\cr}\right\}\hbox{Cubic}]$ ^¶
$[hhl, h\bar{h}l]$		$[(1\bar{1}0), (110)]$	$[{\bf a}/4 \pm {\bf b}/4 \pm {\bf c}/4]$	d
$[hkk, hk\bar{k}]$		$[(01\bar{1}), (011)]$	$[{\bf a}/2]$	a, n
$[hkk, hk\bar{k}]$		$[(01\bar{1}), (011)]$	$[\pm {\bf a}/4 + {\bf b}/4 \pm {\bf c}/4]$	d
$[hkh, \bar{h}kh]$		$[(\bar{1}01), (101)]$	$[{\bf b}/2]$	b, n
$[hkh, \bar{h}kh]$		$[(\bar{1}01), (101)]$	$[\pm {\bf a}/4 \pm {\bf b}/4 + {\bf c}/4]$	d

(b) Screw axes

Type of reflections	Reflection conditions	Screw axis			Crystallographic coordinate system to which condition applies
Type of reflections	Reflection conditions	Direction of axis	Screw vector	Symbol
h00		[100]	$[{\bf a}/2]$	$[2_{1}]$	$[\left\{\matrix{\hbox{Monoclinic }(a\hbox{ unique}),\hfill\cr\quad\hbox{Orthorhombic, Tetragonal}\cr}\right.]$	$[\left.\matrix{\noalign{\vskip56pt}\cr}\right\}\hbox{Cubic}]$
			$[{\bf a}/2]$	$[4_{2}]$
			$[{\bf a}/4]$	$[4_{1},4_{3}]$
0k0		[010]	$[{\bf b}/2]$	$[2_{1}]$	$[\left\{\matrix{\hbox{Monoclinic }(b\hbox{ unique}),\hfill\cr\quad\hbox{Orthorhombic, Tetragonal}\cr}\right.]$	$[\left.\matrix{\noalign{\vskip56pt}\cr}\right\}\hbox{Cubic}]$
			$[{\bf b}/2]$	$[4_{2}]$
			$[{\bf b}/4]$	$[4_{1},4_{3}]$
00l		[001]	$[{\bf c}/2]$	$[2_{1}]$	$[\matrix{\left\{\matrix{\hbox{Monoclinic } (c\hbox{ unique}),\hfill\cr\quad\hbox{ Orthorhombic}\hfill\cr}\right.\cr\left.\matrix{\noalign{\vskip31pt}\cr}\right\}\hbox{Tetragonal}\hfill}]$	$[\left.\matrix{\noalign{\vskip56pt}\cr}\right\}\hbox{Cubic}]$
			$[{\bf c}/2]$	$[4_{2}]$
			$[{\bf c}/4]$	$[4_{1},4_{3}]$
000l		[001]	$[{\bf c}/2]$	$[6_{3}]$	$[\left.{\hbox to 6pt{}}\matrix{\noalign{\vskip54pt}}\right\}\hbox{Hexagonal}]$
			$[{\bf c}/3]$	$[3_{1},3_{2},6_{2},6_{4}]$
			$[{\bf c}/6]$	$[6_{1},6_{5}]$

^†Glide planes d with orientations (100), (010) and (001) occur only in orthorhombic and cubic F space groups. Combination of the integral reflection condition (hkl: all odd or all even) with the zonal conditions for the d glide planes leads to the further conditions given between parentheses.
^‡For rhombohedral space groups described with `rhombohedral axes' the three reflection conditions [(l = 2n, h = 2n, k = 2n)]

imply interleaving of c and n glides, a and n glides, b and n glides, respectively. In the Hermann–Mauguin space-group symbols, c is always used, as in R3c (161) and $[R\bar{3}c\ (167)]$ , because c glides occur also in the hexagonal description of these space groups.
^§For tetragonal P space groups, the two reflection conditions (hhl and $[h\bar{h}l]$ with [l = 2n]

) imply interleaving of c and n glides. In the Hermann–Mauguin space-group symbols, c is always used, irrespective of which glide planes contain the origin: cf. P4cc (103), $[P\bar{4}2c\ (112)]$ and $[P4/nnc\ (126)]$ .
^¶For cubic space groups, the three reflection conditions [(l = 2n, h = 2n, k = 2n)]

imply interleaving of c and n glides, a and n glides, and b and n glides, respectively. In the Hermann–Mauguin space-group symbols, either c or n is used, depending upon which glide plane contains the origin, cf. $[P\bar{4}3n\ (218)]$ , $[Pn\bar{3}n\ (222)]$ , $[Pm\bar{3}n\ (223)]$ vs $[F\bar{4}3c\ (219)]$ , $[Fm\bar{3}c\ (226)]$ , $[Fd\bar{3}c\ (228)]$ .