Table 8.3.5.2

Wondratschek, H.

doi:10.1107/97809553602060000516

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. 8.3, p. 738

Table 8.3.5.2

H. Wondratschek^a ^*

^a Institut für Kristallographie, Universität, D-76128 Karlsruhe, Germany
Correspondence e-mail: hans.wondratschek@physik.uni-karlsruhe.de

Table 8.3.5.2 | top | pdf |
Example of a space-group generation $[{\cal G}: {P6_{1}22 \equiv D_{6}^{2}}]$ (No. 178)

	Coordinates	Symmetry operations
$[\hbox{\sf G}_{1}]$	(1) ;	Identity I
$[\!\matrix{\hbox{\sf G}_{2}\hfill\cr \hbox{\sf G}_{3}\hfill\cr \hbox{\sf G}_{4}\hfill\cr}]$	$[\!\left.\matrix{t(100)\hfill\cr t(010)\hfill\cr t(001)\hfill\cr}\right\}]$	$[\!\left\{\matrix{\hbox{These are the generating translations.}\hfill\cr {\cal G}_{4} \hbox{ is the group } {\cal T} \hbox{ of all translations}\hfill\cr \hbox{of } P6_{1}22\hfill\cr}\right.]$
$[ \hbox{\sf G}_{5}]$	$[(2)\ \bar{y},x - y, z + {1 \over 3};]$	Threefold screw rotation
$[ \hbox{\sf G}_{5}^{2}]$	$[(3)\ \bar{x} + y, \bar{x}, z + {2 \over 3};]$	Threefold screw rotation
$[ \hbox{\sf G}_{5}^{3} = t(001):]$	Now the space group $[{\cal G}_{5} \equiv P3_{1}]$ has been generated
$[ \hbox{\sf G}_{6}]$	$[(4)\ \bar{x},\bar{y}, z + {1 \over 2};]$	Twofold screw rotation
$[ \hbox{\sf G}_{6}* \hbox{\sf G}_{5}]$	$[(5)\ y,\bar{x} + y,z + {5 \over 6};]$	Sixfold screw rotation
$[ \hbox{\sf G}_{6}* \hbox{\sf G}_{5}^{2}]$	$[x - y,x,z + {7 \over 6} \sim (6)\ x - y,x, z + {1 \over 6};]$	Sixfold screw rotation
$[ \hbox{\sf G}_{6}^{2} = t(001):]$	Now the space group $[{\cal G}_{6} \equiv P6_{1}]$ has been generated
$[ \hbox{\sf G}_{7}]$	$[(7)\ y,x,\bar{z} + {1 \over 3};]$	Twofold rotation, direction of axis [110]
$[ \hbox{\sf G}_{7}* \hbox{\sf G}_{5}]$	$[(8)\ x - y,\bar{y},\bar{z};]$	Twofold rotation, axis [100]
$[ \hbox{\sf G}_{7}* \hbox{\sf G}_{5}^{2}]$	$[\bar{x},\bar{x} + y, \bar{z} - {1 \over 3} \sim (9)\ \bar{x},\bar{x} + y, \bar{z} + {2 \over 3};]$	Twofold rotation, axis [010]
$[ \hbox{\sf G}_{7}* \hbox{\sf G}_{6}]$	$[\bar{y},\bar{x},\bar{z} - {1 \over 6} \sim (10)\ \bar{y},\bar{x},\bar{z} + {5 \over 6};]$	Twofold rotation, axis $[[\bar{1}10]]$
$[ \hbox{\sf G}_{7}* \hbox{\sf G}_{6}* \hbox{\sf G}_{5}]$	$[\bar{x} + y,y, \bar{z} - {1 \over 2} \sim (11)\ \bar{x} + y, y,\bar{z} + {1 \over 2};]$	Twofold rotation, axis [120]
$[ \hbox{\sf G}_{7}* \hbox{\sf G}_{6}* \hbox{\sf G}_{5}^{2}]$	$[x,x - y, \bar{z} - {5 \over 6} \sim (12)\ x,x - y,\bar{z} + {1 \over 6};]$	Twofold rotation, axis [210]
$[ \hbox{\sf G}_{7}^{2} = \hbox{\sf I}]$	$[{\cal G}_{7} \sim P6_{1}22]$