Transformations

Arnold, H.

doi:10.1107/97809553602060000511

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. 5.2, p. 86

Section 5.2.1. Transformations

H. Arnold^a ^‡

^a Institut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany

5.2.1. Transformations

| top | pdf |

Symmetry operations are transformations in which the coordinate system, i.e. the basis vectors a, b, c and the origin O, are considered to be at rest, whereas the object is mapped onto itself. This can be visualized as a `motion' of an object in such a way that the object before and after the `motion' cannot be distinguished.

A symmetry operation $[ {\sf W} ]$ transforms every point X with the coordinates x, y, z to a symmetrically equivalent point $[ \tilde{X} ]$ with the coordinates $[\tilde{x}]$ , $[\tilde{y}]$ , $[\tilde{z}]$ . In matrix notation, this transformation is performed by $[ \eqalignno{\pmatrix{\tilde{x}\cr \tilde{y}\cr \tilde{z}\cr} &= \pmatrix{W_{11} &W_{12} &W_{13}\cr W_{21} &W_{22} &W_{23}\cr W_{31} &W_{32} &W_{33}\cr} \pmatrix{x\cr y\cr z\cr} + \pmatrix{w_{1}\cr w_{2}\cr w_{3}\cr}\cr &= \pmatrix{W_{11}x + W_{12}y + W_{13}z + w_{1}\cr W_{21}x + W_{22}y + W_{23}z + w_{2}\cr W_{31}x + W_{32}y + W_{33}z + w_{3}\cr}.} ]$ The $[ (3 \times 3) ]$ matrix W is the rotation part and the $[ (3 \times 1) ]$ column matrix w the translation part of the symmetry operation $[ {\sf W} ]$ . The pair (W, w) characterizes the operation uniquely. Matrices W for point-group operations are given in Tables 11.2.2.1 and 11.2.2.2 .

Again, we can introduce the augmented $[ (4 \times 4) ]$ matrix (cf. Chapter 8.1 ) $[\specialfonts{\bbsf W} = \pmatrix{{\bi W} &{\bi w}\cr {\bi o} &1\cr} = \pmatrix{W_{11} &W_{12} &W_{13} &w_{1}\cr W_{21} &W_{22} &W_{23} &w_{2}\cr W_{31} &W_{32} &W_{33} &w_{3}\cr 0 &0 &0 &1\cr}. ]$ The coordinates $[ \tilde{x}]$ , $[\tilde{y}]$ , $[\tilde{z}]$ of the point $[ \tilde{X} ]$ , symmetrically equivalent to X with the coordinates x, y, z, are obtained by $[ \eqalignno{\pmatrix{\tilde{x}\cr \tilde{y}\cr \tilde{z}\cr 1\cr} &= \pmatrix{W_{11} &W_{12} &W_{13} &w_{1}\cr W_{21} &W_{22} &W_{23} &w_{2}\cr W_{31} &W_{32} &W_{33} &w_{3}\cr 0 &0 &0 &1\cr} \pmatrix{x\cr y\cr z\cr 1\cr}\cr &= \pmatrix{W_{11}x + W_{12}y + W_{13}z + w_{1}\cr W_{21}x + W_{22}y + W_{23}z + w_{2}\cr W_{31}x + W_{32}y + W_{33}z + w_{3}\cr 1\cr},} ]$ or, in short notation, $[\specialfonts \tilde{\bbsf x} = {\bbsf W}{\bbsf x}. ]$ A sequence of symmetry operations can be obtained as a product of $[ (4 \times 4) ]$ matrices $[\specialfonts {\bbsf W} ]$ .

An affine transformation of the coordinate system transforms the coordinates $[\specialfonts{\bbsf x}]$ of the starting point $[\specialfonts {\bbsf x}' = {\bbsf Q}{\bbsf x} ]$ as well as the coordinates $[\specialfonts \tilde{\bbsf x}]$ of a symmetrically equivalent point $[\specialfonts \eqalignno{\tilde{\bbsf x}' &= {\bbsf Q} \tilde{\bbsf x}\cr &= {\bbsf Q} {\bbsf W}{\bbsf x}\cr &= {\bbsf Q} {\bbsf W}{\bbsf P} {\bbsf Q} {\bbsf x}\qquad (\hbox{with } {\bbsf P} = {\bbsf Q}^{-1})\cr &= {\bbsf Q} {\bbsf W}{\bbsf P} {\bbsf x}'.} ]$ Thus, the affine transformation transforms also the symmetry-operation matrix $[\specialfonts {\bbsf W} ]$ and the new matrix $[\specialfonts {\bbsf W}' ]$ is obtained by $[\specialfonts {\bbsf W}' = {\bbsf Q} {\bbsf W}{\bbsf P}. ]$

Example

Space group [ P4/n ] (85) is listed in the space-group tables with two origins; origin choice 1 with $[\bar{4}]$ , origin choice 2 with $[ \bar{1} ]$ as point symmetry of the origin. How does the matrix $[\specialfonts {\bbsf W} ]$ of the symmetry operation $[ \bar{4}^{+} ]$ 0, 0, z; 0, 0, 0 of origin choice 1 transform to the matrix $[\specialfonts {\bbsf W}' ]$ of symmetry operation $[ \bar{4}^{+} {1 \over 4}]$ , $[-{1 \over 4}]$ , z; $[ {1 \over 4}]$ , $[ -{1 \over 4} ]$ , 0 of origin choice 2?

In the space-group tables, origin choice 1, the transformed coordinates $[ \tilde{x}, \tilde{y},\tilde{z} = y, \bar{x}, \bar{z} ]$ are listed. The translation part is zero, i.e. $[ {\bi w} = (0/0/0) ]$ . In Table 11.2.2.1, the matrix W can be found. Thus, the $[ (4 \times 4) ]$ matrix $[\specialfonts {\bbsf W} ]$ is obtained: $[\specialfonts {\bbsf W} = \let\normalbaselines\relax\openup3pt\pmatrix{{\bi W} &{\bi w}\cr {\bi o} &{\bf 1}\cr} = \pmatrix{0 &1 &0 &0\cr \bar{1} &0 &0 &0\cr 0 &0 &\bar{1} &0\cr 0 &0 &0 &1\cr}. ]$

The transformation to origin choice 2 is accomplished by a shift vector p with components $[ {1 \over 4}]$ , $[- {1 \over 4}]$ , 0. Since this is a pure shift, the matrices P and Q are the unit matrix I. Now the shift vector q is derived: $[ {\bi q} = - {\bi P}^{-1} {\bi p} = - {\bi I}{\bi p} = - {\bi p} ]$ . Thus, the matrices $[\specialfonts {\bbsf P} ]$ and $[\specialfonts {\bbsf Q} ]$ are $[\specialfonts{\bbsf P} = \let\normalbaselines\relax\openup3pt\pmatrix{1 &0 &0 &{1 \over 4}\cr 0 &1 &0 &{\bar{1} \over 4}\cr 0 &0 &1 &0\cr 0 &0 &0 &1\cr}, \quad{\bbsf Q}{\rm = \openup1pt\pmatrix{ 1 &0 &0 &{\bar{1} \over 4}\cr \lower2pt\hbox{0} &\lower2pt\hbox{1} &\lower2pt\hbox{0} &\lower2pt\hbox{$1 \over 4$}\cr 0 &0 &1 &0\cr \raise2pt\hbox{0} &\raise2pt\hbox{0} &\raise2pt\hbox{0} &\raise2pt\hbox{1}\cr}}.]$ By matrix multiplication, the new matrix $[\specialfonts {\bbsf W}' ]$ is obtained: $[\specialfonts {\bbsf W}' = {\bbsf QWP} = \let\normalbaselines\relax\openup3pt\pmatrix{0 &1 &0 &{\bar{1} \over 2}\cr \bar{1} &0 &0 &0\cr 0 &0 &\bar{1} &0\cr 0 &0 &0 &1\cr}. ]$ If the matrix $[\specialfonts {\bbsf W}' ]$ is applied to x′, y′, z′, the coordinates of the starting point in the new coordinate system, we obtain the transformed coordinates $[ \tilde{x}']$ , $[\tilde{y}']$ , $[\tilde{z}']$ , $[ \pmatrix{\tilde{x}'\cr \tilde{y}'\cr \tilde{z}'\cr 1\cr} = \pmatrix{0 &1 &0 &{\bar{1} \over 2}\cr \bar{1} &0 &0 &0\cr 0 &0 &\bar{1} &0\cr 0 &0 &0 &1\cr} \pmatrix{x'\cr y'\cr z'\cr 1\cr} = \pmatrix{y' - {1 \over 2}\cr \bar{x}'\cr \bar{z}'\cr 1\cr}. ]$ By adding a lattice translation a, the transformed coordinates $[ {y + {1 \over 2},\bar{x},\bar{z}} ]$ are obtained as listed in the space-group tables for origin choice 2.

References

International Tables for Crystallography (2006). Vol. A. ch. 5.2, p. 86