Coordinate triplets and symmetry operations

Fischer, W.; Koch, E.

doi:10.1107/97809553602060000522

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. 11.1, p. 810

Section 11.1.1. Coordinate triplets and symmetry operations

W. Fischer^a and E. Koch^a ^*

^a Institut für Mineralogie, Petrologie und Kristallographie, Philipps-Universität, D-35032 Marburg, Germany
Correspondence e-mail: kochelke@mailer.uni-marburg.de

11.1.1. Coordinate triplets and symmetry operations

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The coordinate triplets of a general position , as given in the space-group tables, can be taken as a shorthand notation for the symmetry operations of the space group. Each coordinate triplet $[\tilde{x},\tilde{y},\tilde{z}]$ corresponds to the symmetry operation that maps a point with coordinates x, y, z onto a point with coordinates $[\tilde{x},\tilde{y},\tilde{z}]$ . The mapping of x, y, z onto $[\tilde{x},\tilde{y},\tilde{z}]$ is given by the equations: $[\eqalign{\tilde{x} &= W_{11}x + W_{12}y + W_{13}z + w_{1}\cr \tilde{y} &= W_{21}x + W_{22}y + W_{23}z + w_{2}\cr \tilde{z} &= W_{31}x + W_{32}y + W_{33}z + w_{3}.}]$ If, as usual, the symmetry operation is represented by a matrix pair, consisting of a $[(3 \times 3)]$ matrix W and a $[(3 \times 1)]$ column matrix w, the equations read $[\tilde{{\bi x}} = ({\bi W}, {\bi w}){\bi x} = {\bi W}{\bi x} + {\bi w}]$ with $[\eqalign{{\bi x} &= \pmatrix{x\cr y\cr z\cr},\quad \quad \tilde{{\bi x}}{\hbox to1pt{}} = \pmatrix{\tilde{x}\cr \tilde{y}\cr \tilde{z}\cr},\cr {\bi w} &= \pmatrix{w_{1}\cr w_{2}\cr w_{3}\cr},\quad {\bi W} = \pmatrix{W_{11} &W_{12} &W_{13}\cr W_{21} &W_{22} &W_{23}\cr W_{31} &W_{32} &W_{33}\cr}.}]$ W is called the rotation part and $[{\bi w} = {\bi w}_{g} + {\bi w}_{l}]$ the translation part; w is the sum of the intrinsic translation part $[{\bi w}_{g}]$ (glide part or screw part ) and the location part $[{\bi w}_{l}]$ (due to the location of the symmetry element) of the symmetry operation.

Example

The coordinate triplet $[- x + y, y, - z + {1 \over 2}]$ stands for the symmetry operation with rotation part $[{\bi W} = \pmatrix{\bar{1} &1 &0\cr 0 &1 &0\cr 0 &0 &\bar{1}\cr}]$ and with translation part $[{\bi w} = \pmatrix{0\cr 0\cr {1 \over 2}\cr}.]$ Matrix multiplication yields $[\pmatrix{\tilde{x}\cr \tilde{y}\cr \tilde{z}\cr} = \pmatrix{\bar{1} &1 &0\cr 0 &1 &0\cr 0 &0 &\bar{1}\cr} \pmatrix{x\cr y\cr z\cr} + \pmatrix{0\cr 0\cr {1 \over 2}\cr} = \pmatrix{- x + y\cr y\cr - z + {1 \over 2}\cr}.]$

Using the above relation, the assignment of coordinate triplets to symmetry operations given as pairs (W, w) is straightforward.

International Tables for Crystallography (2006). Vol. A. ch. 11.1, p. 810