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

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

Section 5.2.1. Transformations

H. Arnolda

aInstitut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany

5.2.1. Transformations

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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 xyz 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[link] and 11.2.2.2[link] .

Again, we can introduce the augmented [ (4 \times 4) ] matrix (cf. Chapter 8.1[link] )[\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 xyz, 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.








































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