Derivation of symbols and coordinate triplets

Fischer, W.; Koch, E.

doi:10.1107/97809553602060000523

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. 11.2, pp. 812-816
https://doi.org/10.1107/97809553602060000523

Chapter 11.2. Derivation of symbols and coordinate triplets¹

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

Chapter 11.2 describes a method for deriving the symbol of a symmetry operation (i.e. its type, the location of its symmetry element, and possibly its glide or screw vector) from its coordinate triplet as given in the `general position' of a space group. The reverse procedure is also explained. A table shows the coordinate triplets and the 3 × 3 matrices for all symmetry operations of crystallographic point groups and for all orientations of the corresponding symmetry elements.

Keywords: coordinate triplets; symbols; symmetry operations; symmetry elements; matrices for point-group symmetry operations.

11.2.1. Derivation of symbols for symmetry operations from coordinate triplets or matrix pairs

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In the space-group tables, all symmetry operations with $[0 \leq w \lt 1]$ are listed explicitly. As a consequence, the number of entries under the heading Symmetry operations equals the multiplicity of the general position. For space groups with centred unit cells, $[w \geq 1]$ may result if the centring translations are applied to the explicitly listed coordinate triplets. In those cases, all w values have been reduced modulo 1 for the derivation of the corresponding symmetry operations (see Section 2.2.9 ). In addition to the tabulated symmetry operations, each space group contains an infinite number of further operations obtained by application of integral lattice translations. In many cases, it is not trivial to obtain the additional symmetry operations (cf. Part 4 ) from the ones listed. Therefore, a general procedure is described below by which symbols for symmetry operations as described in Section 11.1.2 may be derived from coordinate triplets or, more specifically, from the corresponding matrix pairs (W, w). [For a similar treatment of this topic, see Wondratschek & Neubüser (1967).] This procedure may also be applied to cases where space groups are given in descriptions not contained in International Tables. In practice, two cases may be distinguished:

(i) The matrix W is the unit matrix : $[{\bi W} = {\bi I} = \pmatrix{1 &0 &0\cr 0 &1 &0\cr 0 &0 &1\cr}.]$ In this case, the symmetry operation is a translation with translation vector w.

Example

$[\eqalignno{&{\hbox to -8pt{}}x + {\textstyle{1 \over 2}}, y + {\textstyle{1 \over 2}}, z\;\;\Rightarrow \;\;{\bi W} = \pmatrix{1 &0 &0\cr 0 &1 &0\cr 0 &0 &1\cr} = {\bi I},\quad {\bi w} = \pmatrix{{1 \over 2}\cr {1 \over 2}\cr0\cr}\cr &{\hbox to -8pt{}}\!\!\Rightarrow\;\; \hbox{translation with translation vector }{\bf w} = {\textstyle{1 \over 2}{\bf a}} + {\textstyle{1 \over 2}}{\bf b}\cr &\quad(C \hbox{ centring}).}]$

(ii) The matrix W is not the unit matrix: $[{\bi W} \neq {\bi I.}]$ In this case, one calculates the trace, $[\hbox{tr}({\bi W}) = W_{11} + W_{22} + W_{33}]$ , and the determinant, $[\det ({\bi W})]$ , and identifies the type of the rotation part of the symmetry operation from Table 11.2.1.1 .

Table 11.2.1.1 | top | pdf |
Identification of the type of the rotation part of the symmetry operation

$[\det({\bi W})]$	$[\hbox{tr}({\bi W})]$
$[\det({\bi W})]$	−3	−2	−1	0	1	2	3
1			2	3	4	6	1
−1	$[\bar{1}]$	$[\bar{6}]$	$[\bar{4}]$	$[\bar{3}]$	m

One has to distinguish three subcases:

(a) W corresponds to a rotoinversion. The inversion point X is obtained by solving the equation $[({\bi W},{\bi w}){\bi x} = {\bi x}]$ . For a rotoinversion other than $[\bar{1}]$ , the location of the axis follows from the equation $[({\bi W},{\bi w})^{2}{\bi x} = ({\bi W}^{2}, {\bi W}{\bi w} + {\bi w}){\bi x} = {\bi x}]$ . The rotation sense may be found either by geometrical inspection of a pair of points related by the symmetry operation or by the algebraic procedure described below.

Example

$[\eqalign{&z,\! -\! y + {\textstyle{1 \over 2}},\! -\! x + {\textstyle{1 \over 2}} \;\;\Rightarrow \;\;{\bi W} = \pmatrix{0 &0 &1\cr 0 &\bar{1} &0\cr \bar{1} &0 &0\cr},\quad {\bi w} = \pmatrix{0\cr {1 \over 2}\cr {1 \over 2}\cr}\cr &\Rightarrow \hbox{tr}({\bi W}) = - 1,\quad \det ({\bi W}) = - 1\cr &\Rightarrow \hbox{fourfold rotoinversion;}\cr &({\bi W}, {\bi w}){\bi x} = {\bi x} \;\;\Rightarrow \;\;\pmatrix{0 &0 &1\cr 0 &\bar{1} &0\cr \bar{1} &0 &0\cr} \pmatrix{x\cr y\cr z\cr} + \pmatrix{0\cr {1 \over 2}\cr {1 \over 2}\cr} = \pmatrix{x\cr y\cr z\cr}\cr &\Rightarrow x = y = z = {\textstyle{1 \over 4}}\cr &\Rightarrow \hbox{inversion point at}\ {\textstyle{1 \over 4} {1 \over 4} {1 \over 4}};\cr &({\bi W}^{2}, {\bi W}{\bi w} + {\bi w}){\bi x} = {\bi x}\cr &\Rightarrow \pmatrix{\bar{1} &0 &0\cr 0 &1 &0\cr 0 &0 &\bar{1}\cr} \pmatrix{x\cr y\cr z\cr} + \pmatrix{{1 \over 2}\cr 0\cr {1 \over 2}\cr} = \pmatrix{x\cr y\cr z\cr}\cr &\Rightarrow x = z = {\textstyle{1 \over 4}},\ y \hbox{ undetermined}\cr &\Rightarrow \hbox{rotoinversion axis at } {\textstyle{1 \over 4}}\; y\; {\textstyle{1 \over 4}};\cr &({\bi W}, {\bi w}) \pmatrix{{1 \over 4}\cr {1 \over 4}\cr z\cr} = \pmatrix{0 &0 &1\cr 0 &\bar{1} &0\cr \bar{1} &0 &0\cr} \pmatrix{{1 \over 4}\cr {1 \over 4}\cr z\cr} + \pmatrix{0\cr {1 \over 2}\cr {1 \over 2}\cr} = \pmatrix{z\cr {1 \over 4}\cr {1 \over 4}\cr}\cr &\Rightarrow \hbox{the rotation sense is negative as verified by}\cr &{\hbox to 14pt{}} \hbox{geometrical inspection.}\cr &\Rightarrow \bar{4}^{-}\ {\textstyle{1 \over 4}}, y, {\textstyle{1 \over 4}};\ {\textstyle{1 \over 4}}, {\textstyle{1 \over 4}}, {\textstyle{1 \over 4}}.}\hfill]$
(b) W corresponds to an n-fold rotation. (W, w) is thus either a rotation or a screw rotation . To distinguish between these alternatives, $[({\bi W}, {\bi w})^{n} = ({\bi I}, {\bi t})]$ has to be calculated. For $[{\bi t} = {\bi o}]$ , $[({\bi W}, {\bi w})]$ describes a pure rotation, the rotation axis of which is found by solving $[({\bi W}, {\bi w}){\bi x} = {\bi x}]$ . For $[{\bi t} \neq {\bi o}]$ , (W, w) describes a screw rotation with screw part $[{\bi w}_{g} = (1/n){\bi t}]$ . The location of the screw axis is found as the set of fixed points for the corresponding pure rotation $[({\bi W}, {\bi w}_{l})]$ with $[{\bi w}_{l} = {\bi w} - {\bi w}_{g}]$ , i.e. by solving $[({\bi W}, {\bi w}_{l}){\bi x} = {\bi x}]$ . The sense of the rotation may be found either by geometrical inspection or by the algebraic procedure described below.

Example

$[{\hbox to -3pt{}}\eqalign{&-z, -x + {\textstyle{1 \over 2}}, y \;\;\Rightarrow \;\;{\bi W} = \pmatrix{0 &0 &\bar{1}\cr \bar{1} &0 &0\cr 0 &1 &0\cr},\quad {\bi w} = \pmatrix{0\cr {1 \over 2}\cr 0\cr}\cr &\Rightarrow \hbox{tr}({\bi W}) = 0,\quad \det ({\bi W}) = 1\cr &\Rightarrow \hbox{threefold rotation or screw rotation};\cr &({\bi W}, {\bi w})^{3} = ({\bi W}^{3}, {\bi W}^{2}{\bi w} + {\bi W}{\bi w} + {\bi w})\cr &\hbox{and}\cr &{\bi W}^{2}{\bi w} + {\bi W}{\bi w} + {\bi w} = {\bi t} = \pmatrix{-{1 \over 2}\cr\hfill {1 \over 2}\cr \hfill {1 \over 2}\cr}\cr &\Rightarrow \hbox{threefold screw rotation with screw part}\cr &\quad\;\; {\bi w}_{g} = {\textstyle{1 \over 3}}\;{\bi t} = \pmatrix{-{1 \over 6}\cr \hfill {1 \over 6}\cr \hfill {1 \over 6}\cr};\cr &({\bi W},{\bi w}_{l}){\bi x} = {\bi x}\cr &\Rightarrow \pmatrix{0 &0 &\bar{1}\cr \bar{1} &0 &0\cr 0 &1 &0\cr} \pmatrix{x\cr y\cr z\cr} + \pmatrix{\hfill {1 \over 6}\cr \hfill {1 \over 3}\cr -{1 \over 6}\cr} = \pmatrix{x\cr y\cr z\cr}\cr &\Rightarrow y = {\textstyle{1 \over 3}} - x;\quad z = {\textstyle{1 \over 6}} - x;\quad x \hbox{ undetermined}\cr &\Rightarrow \hbox{screw axis at } x, {\textstyle{1 \over 3}} - x,{\textstyle{1 \over 6}} - x\cr &\qquad\!\! \hbox{(for the sense of rotation see example below)}.}\hfill]$
(c) W corresponds to a (glide) reflection. The glide character is now found by means of the equation $[({\bi W}, {\bi w})^{2} = ({\bi I}, {\bi W}{\bi w} + {\bi w}) = ({\bi I}, {\bi t})]$ . For $[{\bi t} = {\bi o}]$ , (W, w) describes a pure reflection and the location of the mirror plane follows from $[({\bi W}, {\bi w}){\bi x} = {\bi x}]$ . For $[{\bi t} \neq {\bi o}]$ , (W, w) corresponds to a glide reflection with glide part $[{\bi w}_{g} = {\textstyle{1 \over 2}}\;{\bi t}]$ . The location of the glide plane is the set of fixed points for the corresponding pure reflection $[({\bi W},{\bi w}_{l}) = ({\bi W},{\bi w} - {\textstyle{1 \over 2}}\;{\bi t})]$ and is thus calculated by solving $[({\bi W}, {\bi w}_{l}){\bi x} = {\bi x}]$ .

Example

$[\eqalign{&-\!y + {\textstyle{1 \over 2}}, -\! x, z + {\textstyle{3 \over 4}} \;\;\Rightarrow\;\; {\bi W} = \pmatrix{0 &\bar{1} &0\cr \bar{1} &0 &0\cr 0 &0 &1\cr},\quad {\bi w} = \pmatrix{{1 \over 2}\cr 0\cr {3 \over 4}\cr}\cr &\Rightarrow \hbox{tr}({\bi W}) = 1,\quad\det ({\bi W}) = -1\cr &\Rightarrow \hbox{reflection or glide reflection};\cr &({\bi W}, {\bi w})^{2} = ({\bi W}^{2}, {\bi W}{\bi w} + {\bi w}),\quad{\bi W}{\bi w} + {\bi w} = {\bi t} = \pmatrix{\hfill {1 \over 2}\cr - {1 \over 2}\cr \hfill {3 \over 2}\cr}\cr &\Rightarrow \hbox{glide reflection with glide part}\cr& {\bi w}_{g} = {\textstyle{1 \over 2}}\ {\bi t} = \pmatrix{\hfill {1 \over 4}\cr - {1 \over 4}\cr \hfill {3 \over 4}\cr};\cr &({\bi W}, {\bi w}_{l}){\bi x} = {\bi x}\;\; \Rightarrow\; \;\pmatrix{0 &\bar{1} &0\cr \bar{1} &0 &0\cr 0 &0 &1\cr} \pmatrix{x\cr y\cr z\cr} + \pmatrix{{1 \over 4}\cr {1 \over 4}\cr 0\cr} = \pmatrix{x\cr y\cr z\cr}\cr &\Rightarrow y = - x + {\textstyle{1 \over 4}};\ x, z \hbox{ undetermined}\cr &\Rightarrow \hbox{glide plane } d \hbox{ at } x,\! -\! x + {\textstyle{1 \over 4}}, z\cr &\Rightarrow d({\textstyle{1 \over 4}},\! -\! {\textstyle{1 \over 4}}, {\textstyle{3 \over 4}})\quad x,\! -\! x + {\textstyle{1 \over 4}}, z.}\hfill]$

The sense of a pure or screw rotation or of a rotoinversion may be calculated as follows: One takes two arbitrary points $[P_{0}]$ and $[P_{1}]$ on the rotation axis, $[P_{0}]$ having the lower value for the free parameter of the axis. One takes a point $[P_{2}]$ not lying on the axis and generates $[P_{3}]$ from $[P_{2}]$ by the symmetry operation under consideration. One calculates the determinant d of the matrix $[({\bi v}_{1}, {\bi v}_{2}, {\bi v}_{3})]$ composed of the components of vectors $[{\bi v}_1 = \overrightarrow{P_{0}P_{1}}, {\bi v}_{2} = \overrightarrow{P_{0}P_{2}}]$ and $[{\bi v}_{3} = \overrightarrow{P_{0}P_{3}}]$ . For rotations or screw rotations, the sense is positive for $[d \gt 0]$ and negative for $[d \lt 0]$ . For rotoinversions, the sense is positive for $[{d \lt 0}]$ and negative for $[d \gt 0]$ .

Example

According to the example in (b) above, the triplet $[-z,\! -\!x + {\textstyle{1 \over 2}}, y]$ represents a threefold screw rotation with screw part $[\openup3pt\pmatrix{- {1 \over 6}\cr \hfill {1 \over 6}\cr \hfill {1 \over 6}\cr}]$ and screw axis at $[x,{\textstyle{1 \over 3}} -x,{\textstyle{1 \over 6}} -x]$ . To obtain the sense of the rotation, the points $[0{\textstyle{1 \over 3}}\;{\textstyle{1 \over 6}}]$ and $[{\textstyle{1 \over 6}}\;{\textstyle{1 \over 6}}0]$ are used as $[P_{0}]$ and $[P_{1}]$ on the axis and the points 000 and $[0{\textstyle{1 \over 2}}0]$ as $[P_{2}]$ and $[P_{3}]$ outside the axis. The resulting vectors are $[\eqalign{&{\bi v}_{1} = \pmatrix{\hfill {1 \over 6}\cr - {1 \over 6}\cr - {1 \over 6}\cr},\quad {\bi v}_{2} = \pmatrix{\hfill 0\cr - {1 \over 3}\cr - {1 \over 6}\cr},\quad {\bi v}_{3} = \pmatrix{\hfill 0\cr \hfill {1 \over 6}\cr - {1 \over 6}\cr}\cr &\Rightarrow d = \left|\matrix{\hfill {1 \over 6} &\hfill 0 &\hfill 0\cr - {1 \over 6} &- {1 \over 3} &\hfill {1 \over 6}\cr - {1 \over 6} &- {1 \over 6} &- {1 \over 6}\cr}\right| = {1 \over 72} \;\gt\; 0\cr &\Rightarrow \hbox{the sense of the rotation is positive}\cr &\Rightarrow 3^{+} (- {\textstyle{1 \over 6}},{\textstyle{1 \over 6}},{\textstyle{1 \over 6}})\quad x,{\textstyle{1 \over 3}} - x,{\textstyle{1 \over 6}} - x.}]$

11.2.2. Derivation of coordinate triplets from symbols for symmetry operations

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A particular symmetry operation is uniquely described by its symbol, as introduced in Section 11.1.2 , and the coordinate system to which it refers. In the examples of the previous section, the symbols have been derived from the coordinate triplets representing the respective symmetry operations. Inversely, the pair (W, w) of the symmetry operation and the coordinate triplet of the image point can be deduced from the symbol.

(i) For all symmetry operations of space groups, the rotation parts W referring to conventional coordinate systems are listed in Tables 11.2.2.1 and 11.2.2.2 as matrices for point-group symmetry operations. For rotoinversions, the position of the inversion point at 0, 0, 0 is not explicitly given.

Table 11.2.2.1 | top | pdf |
Matrices for point-group symmetry operations and orientation of corresponding symmetry elements, referred to a cubic, tetragonal, orthorhombic, monoclinic, triclinic or rhombohedral coordinate system (cf. Table 2.1.2.1 )

Symbol of symmetry operation and orientation of symmetry element	Transformed coordinates $[\tilde{x},\tilde{y},\tilde{z}]$	Matrix W	Symbol of symmetry operation and orientation of symmetry element	Transformed coordinates $[\tilde{x},\tilde{y},\tilde{z}]$	Matrix W	Symbol of symmetry operation and orientation of symmetry element	Transformed coordinates $[\tilde{x},\tilde{y},\tilde{z}]$	Matrix W	Symbol of symmetry operation and orientation of symmetry element	Transformed coordinates $[\tilde{x},\tilde{y},\tilde{z}]$	Matrix W
1		$[\pmatrix{1 &0 &0\cr 0 &1 &0\cr 0 &0 &1\cr}]$	$[\matrix{2 {\hbox to 12pt{}}0,0,z\cr \cr {\hbox to 13pt{}}[001]\cr}]$	$[\bar{x},\bar{y},z]$	$[\pmatrix{\bar{1} &0 &0\cr 0 &\bar{1} &0\cr 0 &0 &1\cr}]$	$[\matrix{2 {\hbox to 12pt{}}0,y,0\cr \cr {\hbox to 13pt{}}[010]\cr}]$	$[\bar{x},y,\bar{z}]$	$[\pmatrix{\bar{1} &0 &0\cr 0 &1 &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{2 {\hbox to 12pt{}}x,0,0\cr \cr {\hbox to 12pt{}}[100]\cr}]$	$[x,\bar{y},\bar{z}]$	$[\pmatrix{1 &0 &0\cr 0 &\bar{1} &0\cr 0 &0 &\bar{1}\cr}]$
$[\matrix{3^{+} &x,x,x\cr \cr &[111]\cr}]$		$[\pmatrix{0 &0 &1\cr 1 &0 &0\cr 0 &1 &0\cr}]$	$[\matrix{3^{+} {\hbox to 6pt{}}x,\bar{x},\bar{x}\cr \cr{\hbox to 14pt{}}[1\bar{1}\bar{1}]\cr}]$	$[\bar{z},\bar{x},y]$	$[\pmatrix{0 &0 &\bar{1}\cr \bar{1} &0 &0\cr 0 &1 &0\cr}]$	$[\matrix{3^{+} {\hbox to 6pt{}}\bar{x},x,\bar{x}\cr \cr{\hbox to 14pt{}}[\bar{1}1\bar{1}]\cr}]$	$[z,\bar{x},\bar{y}]$	$[\pmatrix{0 &0 &1\cr \bar{1} &0 &0\cr 0 &\bar{1} &0\cr}]$	$[\matrix{3^{+} {\hbox to 6pt{}}\bar{x},\bar{x},x\cr \cr{\hbox to 13pt{}}[\bar{1}\bar{1}1]\cr}]$	$[\bar{z},x,\bar{y}]$	$[\pmatrix{0 &0 &\bar{1}\cr 1 &0 &0\cr 0 &\bar{1} &0\cr}]$
$[\matrix{3^{-} &x,x,x\cr \cr &[111]\cr}]$		$[\pmatrix{0 &1 &0\cr 0 &0 &1\cr 1 &0 &0\cr}]$	$[\matrix{3^{-} {\hbox to 6pt{}}x,\bar{x},\bar{x}\cr \cr {\hbox to 14pt{}}[1\bar{1}\bar{1}]\cr}]$	$[\bar{y},z,\bar{x}]$	$[\pmatrix{0 &\bar{1} &0\cr 0 &0 &1\cr \bar{1} &0 &0\cr}]$	$[\matrix{3^{-} {\hbox to 6pt{}}\bar{x},x,\bar{x}\cr \cr {\hbox to 14pt{}}[\bar{1}1\bar{1}]\cr}]$	$[\bar{y},\bar{z},x]$	$[\pmatrix{0 &\bar{1} &0\cr 0 &0 &\bar{1}\cr 1 &0 &0\cr}]$	$[\matrix{3^{-} {\hbox to 6pt{}}\bar{x},\bar{x},x\cr \cr{\hbox to 13pt{}}[\bar{1}\bar{1}1]\cr}]$	$[y,\bar{z},\bar{x}]$	$[\pmatrix{0 &1 &0\cr 0 &0 &\bar{1}\cr \bar{1} &0 &0\cr}]$
			$[\matrix{2 {\hbox to 12pt{}}x,x,0\cr \cr{\hbox to 13pt{}}[110]\cr}]$	$[y,x,\bar{z}]$	$[\pmatrix{0 &1 &0\cr 1 &0 &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{2 {\hbox to 12pt{}}x,0,x\cr \cr{\hbox to 13.5pt{}}[101]\cr}]$	$[z,\bar{y},x]$	$[\pmatrix{0 &0 &1\cr 0 &\bar{1} &0\cr 1 &0 &0\cr}]$	$[\matrix{2 {\hbox to 11pt{}}0,y,y\cr \cr{\hbox to 14pt{}}[011]\cr}]$	$[\bar{x},z,y]$	$[\pmatrix{\bar{1} &0 &0\cr 0 &0 &1\cr0 &1 &0\cr}]$
			$[\matrix{2 {\hbox to 12pt{}}x,\bar{x},0\cr \cr{\hbox to 13pt{}}[1\bar{1}0]\cr}]$	$[\bar{y},\bar{x},\bar{z}]$	$[\pmatrix{0 &\bar{1} &0\cr \bar{1} &0 &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{2 {\hbox to 12pt{}}\bar{x},0,x\cr \cr{\hbox to 13.5pt{}}[\bar{1}01]\cr}]$	$[\bar{z},\bar{y},\bar{x}]$	$[\pmatrix{0 &0 &\bar{1}\cr 0 &\bar{1} &0\cr \bar{1} &0 &0\cr}]$	$[\matrix{2 {\hbox to 11pt{}}0,y,\bar{y}\cr \cr{\hbox to 13.5pt{}}[01\bar{1}]\cr}]$	$[\bar{x},\bar{z},\bar{y}]$	$[\pmatrix{\bar{1} &0 &0\cr 0 &0 &\bar{1}\cr 0 &\bar{1} &0\cr}]$
			$[\matrix{4^{+} {\hbox to 5pt{}}0,0,z\cr \cr{\hbox to 14.5pt{}}[001]\cr}]$	$[\bar{y},x,z]$	$[\pmatrix{0 &\bar{1} &0\cr 1 &0 &0\cr 0 &0 &1\cr}]$	$[\matrix{4^{+} {\hbox to 6pt{}}0,y,0\cr \cr{\hbox to 13pt{}}[010]\cr}]$	$[z,y,\bar{x}]$	$[\pmatrix{0 &0 &1\cr 0 &1 &0\cr\bar{1} &0 &0\cr}]$	$[\matrix{4^{+} {\hbox to 6pt{}}x,0,0\cr \cr{\hbox to 12pt{}}[100]\cr}]$	$[x,\bar{z},y]$	$[\pmatrix{1 &0 &0\cr 0 &0 &\bar{1}\cr 0 &1 &0\cr}]$
			$[\matrix{4^{-} {\hbox to 5pt{}}0,0,z\cr \cr{\hbox to 14.5pt{}}[001]\cr}]$	$[y,\bar{x},z]$	$[\pmatrix{0 &1 &0\cr \bar{1} &0 &0\cr 0 &0 &1\cr}]$	$[\matrix{4^{-} {\hbox to 6pt{}}0,y,0\cr \cr{\hbox to 13pt{}}[010]\cr}]$	$[\bar{z},y,x]$	$[\pmatrix{0 &0 &\bar{1}\cr 0 &1 &0\cr 1 &0 &0\cr}]$	$[\matrix{4^{-} {\hbox to 6pt{}}x,0,0\cr \cr{\hbox to 12pt{}}[100]\cr}]$	$[x,z,\bar{y}]$	$[\pmatrix{1 &0 &0\cr 0 &0 &1\cr 0 &\bar{1} &0\cr}]$
$[\matrix{\bar{1} &0,0,0\cr}]$	$[\bar{x},\bar{y},\bar{z}]$	$[\pmatrix{\bar{1} &0 &0\cr 0 &\bar{1} &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{m {\hbox to 10pt{}}x,y,0\cr \cr{\hbox to 13pt{}}[001]\cr}]$	$[x,y,\bar{z}]$	$[\pmatrix{1 &0 &0\cr 0 &1 &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{m {\hbox to 10pt{}}x,0,z\cr \cr{\hbox to 14pt{}}[010]\cr}]$	$[x,\bar{y},z]$	$[\pmatrix{1 &0 &0\cr 0 &\bar{1} &0\cr 0 &0 &1\cr}]$	$[\matrix{m {\hbox to 9pt{}}0,y,z\cr \cr{\hbox to 14pt{}}[100]\cr}]$	$[\bar{x},y,z]$	$[\pmatrix{\bar{1} &0 &0\cr 0 &1 &0\cr 0 &0 &1\cr}]$
$[\matrix{\bar{3}^{+} &x,x,x\cr \cr&[111]\cr}]$	$[\bar{z},\bar{x},\bar{y}]$	$[\pmatrix{0 &0 &\bar{1}\cr \bar{1} &0 &0\cr 0 &\bar{1} &0\cr}]$	$[\matrix{\bar{3}^{+} {\hbox to 6pt{}}x,\bar{x},\bar{x}\cr \cr{\hbox to 14pt{}}[1\bar{1}\bar{1}]\cr}]$	$[z,x,\bar{y}]$	$[\pmatrix{0 &0 &1\cr 1 &0 &0\cr 0 &\bar{1} &0\cr}]$	$[\matrix{\bar{3}^{+} {\hbox to 6pt{}}\bar{x},x,\bar{x}\cr \cr{\hbox to 14pt{}}[\bar{1}1\bar{1}]\cr}]$	$[\bar{z},x,y]$	$[\pmatrix{0 &0 &\bar{1}\cr 1 &0 &0\cr 0 &1 &0\cr}]$	$[\matrix{\bar{3}^{+} {\hbox to 6pt{}}\bar{x},\bar{x},x\cr \cr{\hbox to 13pt{}}[\bar{1}\bar{1}1]\cr}]$	$[z,\bar{x},y]$	$[\pmatrix{0 &0 &1\cr \bar{1} &0 &0\cr 0 &1 &0\cr}]$
$[\matrix{\bar{3}^{-} &x,x,x\cr \cr&[111]\cr}]$	$[\bar{y},\bar{z},\bar{x}]$	$[\pmatrix{0 &\bar{1} &0\cr 0 &0 &\bar{1}\cr \bar{1} &0 &0\cr}]$	$[\matrix{\bar{3}^{-} {\hbox to 6pt{}}x,\bar{x},\bar{x}\cr \cr{\hbox to 14pt{}}[1\bar{1}\bar{1}]\cr}]$	$[y,\bar{z},x]$	$[\pmatrix{0 &1 &0\cr 0 &0 &\bar{1}\cr 1 &0 &0\cr}]$	$[\matrix{\bar{3}^{-} {\hbox to 6pt{}}\bar{x},x,\bar{x}\cr \cr{\hbox to 14pt{}}[\bar{1}1\bar{1}]\cr}]$	$[y,z,\bar{x}]$	$[\pmatrix{0 &1 &0\cr 0 &0 &1\cr \bar{1} &0 &0\cr}]$	$[\matrix{\bar{3}^{-} {\hbox to 6pt{}}\bar{x},\bar{x},x\cr \cr{\hbox to 13pt{}}[\bar{1}\bar{1}1]\cr}]$	$[\bar{y},z,x]$	$[\pmatrix{0 &\bar{1} &0\cr 0 &0 &1\cr 1 &0 &0\cr}]$
			$[\matrix{m {\hbox to 10pt{}}x,\bar{x},z\cr \cr{\hbox to 14pt{}}[110]\cr}]$	$[\bar{y},\bar{x},z]$	$[\pmatrix{0 &\bar{1} &0\cr \bar{1} &0 &0\cr 0 &0 &1\cr}]$	$[\matrix{m {\hbox to 10pt{}}\bar{x},y,x\cr \cr{\hbox to 14pt{}}[101]\cr}]$	$[\bar{z},y,\bar{x}]$	$[\pmatrix{0 &0 &\bar{1}\cr 0 &1 &0\cr \bar{1} &0 &0\cr}]$	$[\matrix{m {\hbox to 10pt{}}x,y,\bar{y}\cr \cr{\hbox to 13pt{}}[011]\cr}]$	$[x,\bar{z},\bar{y}]$	$[\pmatrix{1 &0 &0\cr 0 &0 &\bar{1}\cr 0 &\bar{1} &0\cr}]$
			$[\matrix{m {\hbox to 10pt{}}x,x,z\cr \cr{\hbox to 14pt{}}[1\bar{1}0]\cr}]$		$[\pmatrix{0 &1 &0\cr 1 &0 &0\cr 0 &0 &1\cr}]$	$[\matrix{m {\hbox to 10pt{}}x,y,x\cr \cr{\hbox to 14pt{}}[\bar{1}01]\cr}]$		$[\pmatrix{0 &0 &1\cr 0 &1 &0\cr 1 &0 &0\cr}]$	$[\matrix{m {\hbox to 10pt{}}x,y,y\cr \cr{\hbox to 13pt{}}[01\bar{1}]\cr}]$		$[\pmatrix{1 &0 &0\cr 0 &0 &1\cr 0 &1 &0\cr}]$
			$[\matrix{\bar{4}^{+} {\hbox to 5pt{}}0,0,z\cr \cr{\hbox to 14.5pt{}}[001]\cr}]$	$[y,\bar{x},\bar{z}]$	$[\pmatrix{0 &1 &0\cr \bar{1} &0 &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{\bar{4}^{+} {\hbox to 6pt{}}0,y,0\cr \cr{\hbox to 13.5pt{}}[010]\cr}]$	$[\bar{z},\bar{y},x]$	$[\pmatrix{0 &0 &\bar{1}\cr 0 &\bar{1} &0\cr 1 &0 &0\cr}]$	$[\matrix{\bar{4}^{+} {\hbox to 6pt{}}x,0,0\cr \cr{\hbox to 12pt{}}[100]\cr}]$	$[\bar{x},z,\bar{y}]$	$[\pmatrix{\bar{1} &0 &0\cr 0 &0 &1\cr 0 &\bar{1} &0\cr}]$
			$[\matrix{\bar{4}^{-} {\hbox to 5pt{}}0,0,z\cr \cr{\hbox to 14.5pt{}}[001]\cr}]$	$[\bar{y},x,\bar{z}]$	$[\pmatrix{0 &\bar{1} &0\cr 1 &0 &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{\bar{4}^{-} {\hbox to 6pt{}}0,y,0\cr \cr{\hbox to 13pt{}}[010]\cr}]$	$[z,\bar{y},\bar{x}]$	$[\pmatrix{0 &0 &1\cr 0 &\bar{1} &0\cr \bar{1} &0 &0\cr}]$	$[\matrix{\bar{4}^{-} {\hbox to 6pt{}}x,0,0\cr \cr{\hbox to 12pt{}}[100]\cr}]$	$[\bar{x},\bar{z},y]$	$[\pmatrix{\bar{1} &0 &0\cr 0 &0 &\bar{1}\cr 0 &1 &0\cr}]$

Table 11.2.2.2 | top | pdf |
Matrices for point-group symmetry operations and orientation of corresponding symmetry elements, referred to a hexagonal coordinate system (cf. Table 2.1.2.1 )

Symbol of symmetry operation and orientation of symmetry element	Transformed coordinates $[\tilde{x},\tilde{y},\tilde{z}]$	Matrix W	Symbol of symmetry operation and orientation of symmetry element	Transformed coordinates $[\tilde{x},\tilde{y},\tilde{z}]$	Matrix W	Symbol of symmetry operation and orientation of symmetry element	Transformed coordinates $[\tilde{x},\tilde{y},\tilde{z}]$	Matrix W
1		$[\pmatrix{1 &0 &0\cr 0 &1 &0\cr 0 &0 &1\cr}]$	$[\matrix{3^{+} {\hbox to 8pt{}}0,0,z\cr \cr{\hbox to 16pt{}}[001]\cr}]$	$[\bar{y},x - y,z]$	$[\pmatrix{0 &\bar{1} &0\cr 1 &\bar{1} &0\cr 0 &0 &1\cr}]$	$[\matrix{3^{-} {\hbox to 8pt{}}0,0,z\cr \cr{\hbox to 15.5pt{}}[001]\cr}]$	$[y - x,\bar{x},z]$	$[\pmatrix{\bar{1} &1 &0\cr \bar{1} &0 &0\cr 0 &0 &1\cr}]$
$[\matrix{2 {\hbox to 12pt{}}0,0,z\cr \cr{\hbox to 14pt{}}[001]\cr}]$	$[\bar{x},\bar{y},z]$	$[\pmatrix{\bar{1} &0 &0\cr 0 &\bar{1} &0\cr 0 &0 &1\cr}]$	$[\matrix{6^{+} {\hbox to 8pt{}}0,0,z\cr \cr{\hbox to 16pt{}}[001]\cr}]$		$[\pmatrix{1 &\bar{1} &0\cr 1 &0 &0\cr 0 &0 &1\cr}]$	$[\matrix{6^{-} {\hbox to 8pt{}}0,0,z\cr \cr{\hbox to 15.5pt{}}[001]\cr}]$		$[\pmatrix{0 &1 &0\cr \bar{1} &1 &0\cr 0 &0 &1\cr}]$
$[\matrix{2 {\hbox to 12pt{}}x,x,0\cr \cr{\hbox to 14pt{}}[110]\cr}]$	$[y,x,\bar{z}]$	$[\pmatrix{0 &1 &0\cr 1 &0 &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{2 {\hbox to 14.5pt{}}x,0,0\cr \cr{\hbox to 15pt{}}[100]\cr}]$	$[x - y,\bar{y},\bar{z}]$	$[\pmatrix{1 &\bar{1} &0\cr 0 &\bar{1} &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{2 {\hbox to 14pt{}}0,y,0\cr \cr{\hbox to 15pt{}}[010]\cr}]$	$[\bar{x},y - x,\bar{z}]$	$[\pmatrix{\bar{1} &0 &0\cr \bar{1} &1 &0\cr 0 &0 &\bar{1}\cr}]$
$[\matrix{2 {\hbox to 12pt{}}x,\bar{x},0\cr \cr{\hbox to 14pt{}}[1\bar{1}0]\cr}]$	$[\bar{y},\bar{x},\bar{z}]$	$[\pmatrix{0 &\bar{1} &0\cr \bar{1} &0 &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{2 {\hbox to 14.5pt{}}x,2x,0\cr \cr{\hbox to 10.5pt{}}[120]\cr}]$	$[y - x,y,\bar{z}]$	$[\pmatrix{\bar{1} &1 &0\cr 0 &1 &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{2 {\hbox to 14pt{}}2x,x,0\cr \cr{\hbox to 10.5pt{}}[210]\cr}]$	$[x,x - y,\bar{z}]$	$[\pmatrix{1 &0 &0\cr 1 &\bar{1} &0\cr 0 &0 &\bar{1}\cr}]$
$[\matrix{\bar{1} {\hbox to 12pt{}}0,0,0\cr}]$	$[\bar{x},\bar{y},\bar{z}]$	$[\pmatrix{\bar{1} &0 &0\cr 0 &\bar{1} &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{\bar{3}^{+} {\hbox to 8pt{}}0,0,z\cr \cr{\hbox to 16pt{}}[001]\cr}]$	$[y,y - x,\bar{z}]$	$[\pmatrix{0 &1 &0\cr \bar{1} &1 &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{\bar{3}^{-} {\hbox to 8pt{}}0,0,z\cr \cr{\hbox to 16pt{}}[001]\cr}]$	$[x - y,x,\bar{z}]$	$[\pmatrix{1 &\bar{1} &0\cr 1 &0 &0\cr 0 &0 &\bar{1}\cr}]$
$[\matrix{m {\hbox to 10pt{}}x,y,0\cr \cr{\hbox to 13.5pt{}}[001]\cr}]$	$[x,y,\bar{z}]$	$[\pmatrix{1 &0 &0\cr 0 &1 &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{\bar{6}^{+} {\hbox to 8pt{}}0,0,z\cr \cr{\hbox to 16pt{}}[001]\cr}]$	$[y - x,\bar{x},\bar{z}]$	$[\pmatrix{\bar{1} &1 &0\cr \bar{1} &0 &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{\bar{6}^{-} {\hbox to 8pt{}}0,0,z\cr \cr{\hbox to 15.5pt{}}[001]\cr}]$	$[\bar{y},x - y,\bar{z}]$	$[\pmatrix{0 &\bar{1} &0\cr 1 &\bar{1} &0\cr 0 &0 &\bar{1}\cr}]$
$[\matrix{m {\hbox to 10pt{}}x,\bar{x},z\cr \cr{\hbox to 14.5pt{}}[110]\cr}]$	$[\bar{y},\bar{x},z]$	$[\pmatrix{0 &\bar{1} &0\cr \bar{1} &0 &0\cr 0 &0 &1\cr}]$	$[\matrix{m {\hbox to 12pt{}}x,2x,z\cr \cr{\hbox to 12pt{}}[100]\cr}]$		$[\pmatrix{\bar{1} &1 &0\cr 0 &1 &0\cr 0 &0 &1\cr}]$	$[\matrix{m {\hbox to 12pt{}}2x,x,z\cr \cr{\hbox to 11.5pt{}}[010]\cr}]$		$[\pmatrix{1 &0 &0\cr 1 &\bar{1} &0\cr 0 &0 &1\cr}]$
$[\matrix{m {\hbox to 10pt{}}x,x,z\cr \cr{\hbox to 14.5pt{}}[1\bar{1}0]\cr}]$		$[\pmatrix{0 &1 &0\cr 1 &0 &0\cr 0 &0 &1\cr}]$	$[\matrix{m {\hbox to 12pt{}}x,0,z\cr \cr{\hbox to 16.5pt{}}[120]\cr}]$	$[x - y,\bar{y},z]$	$[\pmatrix{1 &\bar{1} &0\cr 0 &\bar{1} &0\cr 0 &0 &1\cr}]$	$[\matrix{m {\hbox to 12pt{}}0,y,z\cr \cr{\hbox to 16pt{}}[210]\cr}]$	$[\bar{x},y - x,z]$	$[\pmatrix{\bar{1} &0 &0\cr \bar{1} &1 &0\cr 0 &0 &1\cr}]$

(ii) The location part $[{\bi w}_{l}]$ of w may easily be derived from $[({\bi W}, {\bi w}_{l}) \pmatrix{x_{0}\cr y_{0}\cr z_{0}\cr} = \pmatrix{x_{0}\cr y_{0}\cr z_{0}\cr},\quad i.e. \ {\bi w}_{l} = ({\bi I} - {\bi W}) \pmatrix{x_{0}\cr y_{0}\cr z_{0}\cr}]$ with $[x_{0},y_{0},z_{0}]$ being the coordinate triplet of the inversion point of a rotoinversion or the coordinate triplet of an arbitrary fixed point of any other symmetry operation. The intrinsic translation part $[{\bi w}_{g}]$ of w is given explicitly in the symbol of the symmetry operation, so that the translation part w is obtained as $[{\bi w} = {\bi w}_{g} + {\bi w}_{l} = \pmatrix{w_{1}\cr w_{2}\cr w_{3}\cr}.]$
(iii) The coordinate triplet $[\tilde{x},\tilde{y},\tilde{z}]$ corresponding to the symmetry operation is now given by $[\eqalign{\tilde{x} &= {W}_{11} x + {W}_{12} y + {W}_{13} z + {w}_{1}\cr \tilde{y} &= {W}_{21} x + {W}_{22} y + {W}_{23} z + {w}_{2} \cr \tilde{z} &= {W}_{31} x + {W}_{32} y + {W}_{33} z + {w}_{3}.\cr}]$

Example

$[4^{-}(0,0,{\textstyle{3 \over 4}})\ {\textstyle{1 \over 4}},\! - {\textstyle{1 \over 4}}, z]$ tetragonal system $[4^{-}\ 0,0,z \; \Rightarrow\; {\bi W} = \pmatrix{0 &1 &0\cr \bar{1} &0 &0\cr 0 &0 &1\cr}\hbox{ from Table 11.2.2.1}.\hfill]$ $[x_{0} = {\textstyle{1 \over 4}}, y_{0} = - {\textstyle{1 \over 4}}, z_{0} = 0]$ is a fixed point of $[4^{-}\ {\textstyle{1 \over 4}},\! -\! {\textstyle{1 \over 4}}, z]$ , i.e. a point on the screw axis. $[\eqalign{&{\bi w}_{l} = ({\bi I} - {\bi W}) \pmatrix{x_{0}\cr y_{0}\cr z_{0}\cr}\cr &\Rightarrow {\bi w}_{l} = \pmatrix{1 &\bar{1} &0\cr 1 &1 &0\cr 0 &0 &0\cr} \pmatrix{\hfill {1 \over 4}\cr - {1 \over 4}\cr \hfill0\cr} = \pmatrix{{1 \over 2}\cr 0\cr 0\cr};\cr &{\bi w} = {\bi w}_{g} + {\bi w}_{l}\cr &\Rightarrow {\bi w} = \pmatrix{0\cr 0\cr {3 \over 4}\cr} + \pmatrix{{1 \over 2}\cr 0\cr 0\cr} = \pmatrix{{1 \over 2}\cr 0\cr {3 \over 4}\cr}\cr &\Rightarrow \tilde{x} = y + {\textstyle{1 \over 2}},\quad \tilde{y} = - x,\quad \tilde{z} = z + {\textstyle{3 \over 4}}.}\hfill]$

References

Wondratschek, H. & Neubüser, J. (1967). Determination of the symmetry elements of a space group from the `general positions' listed in International Tables for X-ray Crystallography, Vol. I. Acta Cryst. 23, 349–352.Google Scholar

International Tables for Crystallography (2006). Vol. A. ch. 11.2, pp. 812-816
https://doi.org/10.1107/97809553602060000523

Chapter 11.2. Derivation of symbols and coordinate triplets1

11.2.1. Derivation of symbols for symmetry operations from coordinate triplets or matrix pairs

Example

Example

Example

Example

Example

11.2.2. Derivation of coordinate triplets from symbols for symmetry operations

Example

References

Chapter 11.2. Derivation of symbols and coordinate triplets¹