Example

Eckold, G.

doi:10.1107/97809553602060000638

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 2.1, pp. 284-286

Section 2.1.3.4.1. Example

G. Eckold^a ^*

^a Institut für Physikalische Chemie, Universität Göttingen, Tammannstrasse 6, D-37077 Göttingen, Germany
Correspondence e-mail: geckold@gwdg.de

2.1.3.4.1. Example

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Let us try to find the symmetry coordinates corresponding to our sample structure introduced in Section 2.1.3.1.1 for $[{\bf q}={\bf 0}]$ . Using the irreducible representations displayed in Section 2.1.3.3.1, we write down the projection operator for representation $[\tau^{(1^ \pm)}]$ according to equation (2.1.3.51): $[\displaylines{{\bf P}_{11}^{(1^ \pm)}({\bf 0}) =\hfill\cr {1 \over 8}\left({\matrix{ {\boldSigma _1^ \pm }&\! 0 &\! 0 &\! 0 &\! 0 &\!0 &\! 0 &\! 0 &\! 0 &\! 0 \cr 0 &\! {\boldSigma _1^ \pm }&\! 0 &\! 0 &\! 0 &\!0 &\! 0 &\! 0 &\! 0 &\! 0 \cr 0 &\! 0 &\! {\bf E}&\! {{\bf D}_{180}^z }&\! {\pm {\bf m}_{ [{110}]}} &\!{{\bf D}_{90}^z }&\! {\pm {\bf m}_y }&\! {\pm {\bf m}_x }&\! {{\bf D}_{270}^z }&\! {\pm {\bf m}_{ [{\bar 110}]}}\cr 0 &\! 0 &\! {{\bf D}_{180}^z }&\! {\bf E}&\! {\pm {\bf m}_{ [{\bar 110}]}}&\!{{\bf D}_{270}^z }&\! {\pm {\bf m}_x }&\! {\pm {\bf m}_y }&\! {{\bf D}_{90}^z }&\! {\pm {\bf m}_{ [{110}]}}\cr 0 &\! 0 &\! {\pm {\bf m}_{ [{110}]}}&\! {\pm {\bf m}_{ [{\bar 110}]}}&\! {\bf E} &\!{\pm {\bf m}_x }&\! {{\bf D}_{270}^z }&\! {{\bf D}_{90}^z }&\! {\pm {\bf m}_y }&\! {{\bf D}_{180}^z } \cr 0 &\! 0 &\! {{\bf D}_{270}^z }&\! {{\bf D}_{90}^z }&\! {\pm {\bf m}_x } &\!{\bf E}&\! {\pm {\bf m}_{ [{110}]}}&\! {\pm {\bf m}_{ [{\bar 110}]}}&\! {{\bf D}_{180}^z }&\! {\pm {\bf m}_y }\cr 0 &\! 0 &\! {\pm {\bf m}_y }&\! {\pm {\bf m}_x }&\! {{\bf D}_{90}^z }&\!{\pm {\bf m}_{ [{110}]}}&\! {\bf E}&\! {{\bf D}_{180}^z }&\! {\pm {\bf m}_{ [{\bar 110}]}}&\! {{\bf D}_{270}^z } \cr 0 &\! 0 &\! {\pm {\bf m}_x }&\! {\pm {\bf m}_y }&\! {{\bf D}_{270}^z }&\!{\pm {\bf m}_{ [{\bar 110}]}}&\! {{\bf D}_{180}^z }&\! {\bf E}&\! {\pm {\bf m}_{ [{110}]}}&\! {{\bf D}_{90}^z } \cr 0 &\! 0 &\! {{\bf D}_{90}^z }&\! {{\bf D}_{270}^z }&\! {\pm {\bf m}_y }&\! {{\bf D}_{180}^z }&\! {\pm {\bf m}_{ [{\bar 110}]}}&\! {\pm {\bf m}_{ [{110}]}}&\! {\bf E}& {\pm {\bf m}_x }\cr 0 &\! 0 &\! {\pm {\bf m}_{ [{\bar 110}]}}&\! {\pm {\bf m}_{ [{110}]}}&\! {{\bf D}_{180}^z }&\!{\pm {\bf m}_y }&\! {{\bf D}_{90}^z }&\! {{\bf D}_{270}^z }&\! {\pm {\bf m}_x } &\! {\bf E} \cr }}\right)}]$ with the abbreviations $[\eqalign{ & \boldSigma _1^+ = {\bf E}+ {\bf D}_{90}^z + {\bf D}_{180}^z + {\bf D}_{270}^z + {\bf m}_x + {\bf m}_y + {\bf m}_{ [{110}]} + {\bf m}_{ [{\bar 110}]} \cr&= {\pmatrix{ 0 & 0 & 0 \cr 0 & 0 & 0 \cr 0 & 0 & 8 \cr }}, \cr & \boldSigma _1^ - = {\bf E}+ {\bf D}_{90}^z + {\bf D}_{180}^z + {\bf D}_{270}^z - {\bf m}_x - {\bf m}_y - {\bf m}_{ [{110}]} - {\bf m}_{ [{\bar 110}]} \cr&= {\pmatrix{ 0 & 0 & 0 \cr 0 & 0 & 0 \cr 0 & 0 & 0 \cr }} = {{\bf 0}}.\cr} ]$

From the results in Section 2.1.3.4, we expect to have five symmetry coordinates corresponding to representation $[\tau _{}^{(1^ +)} ]$ and three for $[\tau _{}^{(1^ -)}]$ according to the respective multiplicities. Let $[{\bf x}_{1}, {\bf y}_{1}, {\bf z}_{1}, {\bf x}_{2}, {\bf y}_{2}, {\bf z}_{2}, \ldots, {\bf x}_{10}, {\bf y}_{10}, {\bf z}_{10} ]$ denote the basis of the 30-dimensional space generated by the displacements of the ten atoms in the x, y and z directions, respectively. If we apply the projection operator $[{\bf P}_{11}^{1^ + }({\bf 0}) ]$ to the basis vector $[{\bf z}_{1}]$ , we obtain the first symmetry coordinate according to equation (2.1.3.52): $[{\bf E}({\bf 0},1^ + 11) = {\bf P}_{{11}}^{({1}^ +)}({\bf 0}) \,{\pmatrix{ 0 \cr 0 \cr 1 \cr 0 \cr 0 \cr 0 \cr \vdots \cr 0 \cr 0 \cr 0 \cr }} = {\bf P}_{{11}}^{({1}^ +)}({\bf 0}) \, {\bf z}_{1} = {\bf z}_{1}. ]$ In a similar way we may use the basis vectors $[{\bf z}_{2}]$ , $[{\bf x}_{3}]$ , $[{\bf z}_{3}]$ and $[{\bf x}_{5}]$ in order to generate the other symmetry coordinates: $[\eqalign{ {\bf E}({{\bf 0}},{1}^ + 21) &= {\bf P}_{{11}}^{({1}^ +)}({{\bf 0}})\, {\bf z}_2 = {\bf z}_2 \cr {\bf E}({\bf 0},{1}^ + 31) &= {\bf P}_{{11}}^{({1}^ +)}({\bf 0})\, {\bf x}_3 \cr&= {\textstyle{1 \over 8}} [{{\bf x}_3 - {\bf x}_4 - {\bf y}_5 - {\bf y}_6 + {\bf x}_7 - {\bf x}_8 + {\bf y}_9 + {\bf y}_{10}}] \cr {\bf E}({\bf 0},{1}^ + 41) &= {\bf P}_{{11}}^{({1}^ +)}({\bf 0}) \,{\bf z}_3 \cr&= {\textstyle{1 \over 8}} [{{\bf z}_3 + {\bf z}_4 + {\bf z}_5 + {\bf z}_6 + {\bf z}_7 + {\bf z}_8 + {\bf z}_9 + {\bf z}_{10}}] \cr {\bf E}({\bf 0},{1}^ + 51) &= {\bf P}_{{11}}^{({ 1}^ +)}({\bf 0})\,{\bf x}_5 \cr&= {\textstyle{1 \over 8}} [{- {\bf y}_3 + {\bf y}_4 + {\bf x}_5 - {\bf x}_6 + {\bf y}_7 - {\bf y}_8 + {\bf x}_9 - {\bf x}_{10}}]. \cr} ]$ (It can easily be shown that all the other basis vectors would lead to linearly dependent symmetry coordinates.)

Any eigenvector of the dynamical matrix corresponding to the irreducible representation $[\tau _{}^{(1^ +)}]$ is necessarily some linear combination of these five symmetry coordinates. Hence it may be concluded that for all lattice vibrations of this symmetry, the displacements of atoms 1 and 2 can only be along the tetragonal axis. Moreover, the displacements of atoms 3 to 10 have to be identical along z, and pairs of atoms vibrate in opposite directions within the xy plane.

For the representation $[\tau^{(1^ -)}]$ we obtain the following symmetry coordinates when $[{\bf P}_{11}^{(1^ -)}({\bf 0})]$ is applied to $[{\bf x}_{3} ]$ , $[{\bf z}_{3}]$ and $[{\bf x}_{5}]$ : $[\eqalign{{\bf E}({{\bf 0}},{1}^ - 11) &= {\bf P}_{{11}}^{({1}^ -)}({{\bf 0}}) \, {\bf x}_3 \cr&= {\textstyle{1 \over 8}} [{{\bf x}_3 - {\bf x}_4 + {\bf y}_5 - {\bf y}_6 - {\bf x}_7 + {\bf x}_8 + {\bf y}_9 - {\bf y}_{10}}] \cr {\bf E}({{\bf 0}},{1}^ - 21) &= {\bf P}_{{11}}^{({1}^ -)}({{\bf 0}}) \, {\bf z}_3 \cr&= {\textstyle{1 \over 8}} [{{\bf z}_3 + {\bf z}_4 - {\bf z}_5 + {\bf z}_6 - {\bf z}_7 - {\bf z}_8 + {\bf z}_9 - {\bf z}_{10}}]\cr {\bf E}({{\bf 0}},{1}^ - 31) &= {\bf P}_{{ 11}}^{({1}^ -)}({{\bf 0}}) \, {\bf x}_5 \cr&= {\textstyle{1 \over 8}} [{{\bf y}_3 - {\bf y}_4 + {\bf x}_5 + {\bf x}_6 + {\bf y}_7 - {\bf y}_8 - {\bf x}_9 - {\bf x}_{10}}]. \cr }]$

Obviously, none of the corresponding phonons exhibits any displacement of atoms 1 and 2. There is an antiphase motion of pairs of atoms not only within the tetragonal plane but also along the tetragonal z axis.

For the representations $[\tau ^{(3^ \pm)}]$ we obtain the following projection operators: $[\displaylines{{\bf P}_{11}^{(3^ \pm)}({{\bf 0}}) =\hfill\cr {1 \over 8}\left({\matrix{ {\boldSigma _3^ \pm }&\! 0 &\! 0 &\! 0 &\! 0 &\!0 &\! 0 &\! 0 &\! 0 &\! 0\cr 0 &\! {\boldSigma _3^ \pm }&\! 0 &\! 0 &\! 0 &\! 0 &\! 0 &\! 0 &\! 0 &\! 0 \cr 0 &\! 0 &\! {\bf E}&\! {{\bf D}_{180}^z }&\! {\mp {\bf m}_{ [{110}]}}&\!{- {\bf D}_{90}^z }&\! {\pm {\bf m}_y }&\! {\pm {\bf m}_x }&\! {- {\bf D}_{270}^z }&\! {\mp {\bf m}_{ [{\bar 110}]}} \cr 0 &\! 0 &\! {{\bf D}_{180}^z }&\! {\bf E}&\! {\mp {\bf m}_{ [{\bar 110}]}}&\!{- {\bf D}_{270}^z }&\! {\pm {\bf m}_x }&\! {\pm {\bf m}_y }&\! {- {\bf D}_{90}^z }&\! {\mp {\bf m}_{ [{110}]}}\cr 0 &\! 0 &\! {\mp {\bf m}_{ [{110}]}}&\! {\mp {\bf m}_{ [{\bar 110}]}}&\! {\bf E}&\!{\pm {\bf m}_x }&\! {- {\bf D}_{270}^z }&\! {- {\bf D}_{90}^z }&\! {\pm {\bf m}_y }&\! {{\bf D}_{180}^z } \cr 0 &\! 0 &\! {- {\bf D}_{270}^z }&\! {- {\bf D}_{90}^z }&\! {\pm {\bf m}_x } &\!{\bf E}&\! {\mp {\bf m}_{ [{110}]}}&\! {\mp {\bf m}_{ [{\bar 110}]}}&\! {{\bf D}_{180}^z }&\! {\pm {\bf m}_y }\cr 0 &\! 0 &\! {\pm {\bf m}_y }&\! {\pm {\bf m}_x }&\! {- {\bf D}_{90}^z }&\!{\mp {\bf m}_{ [{110}]}}&\! {\bf E}& {{\bf D}_{180}^z }&\! {\mp {\bf m}_{ [{\bar 110}]}}&\! {- {\bf D}_{270}^z } \cr 0 &\! 0 &\! {\pm {\bf m}_x }&\! {\pm {\bf m}_y }&\! {- {\bf D}_{270}^z } &\!{\mp {\bf m}_{ [{\bar 110}]}}&\! {{\bf D}_{180}^z }&\! {\bf E}&\! {\mp {\bf m}_{ [{110}]}}&\! {- {\bf D}_{90}^z } \cr 0 &\! 0 &\! {- {\bf D}_{90}^z }&\! {- {\bf D}_{270}^z }&\! {\pm {\bf m}_y }&\!{{\bf D}_{180}^z }&\! {\mp {\bf m}_{ [{\bar 110}]}}&\! {\mp {\bf m}_{ [{110}]}}&\! {\bf E}& \!{\pm {\bf m}_x } \cr 0 &\! 0 &\! {\mp {\bf m}_{ [{\bar 110}]}}&\! {\mp {\bf m}_{ [{110}]}}&\! {{\bf D}_{180}^z }&\!{\pm {\bf m}_y }&\! {- {\bf D}_{90}^z }&\! {- {\bf D}_{270}^z }&\! {\pm {\bf m}_x }&\! {\bf E} \cr }}\right)}]$ with $[\boldSigma _3^ \pm = {\bf E}- {\bf D}_{90}^z + {\bf D}_{180}^z - {\bf D}_{270}^z \pm {\bf m}_x \pm {\bf m}_y \mp {\bf m}_{ [{110}]} \mp {\bf m}_{ [{\bar 110}]} = {{\bf 0}}. ]$

Both representations appear three times in the decomposition of the T representation. Hence, we expect three phonons of each symmetry and also three linearly independent symmetry coordinates. These are generated if the projection operators are applied to the basis vectors $[{\bf x}_{3} ]$ , $[{\bf z}_{3}]$ and $[{\bf x}_{5}]$ : $[\eqalign{E({{\bf 0}},3^ \pm 11) &= {\bf P}_{{11}}^{(3^ +)}({{\bf 0}}) \, {\bf x}_3 \cr&= {\textstyle{1 \over 8}} [{{\bf x}_3 - {\bf x}_4 \pm {\bf y}_5 + {\bf y}_6 \pm {\bf x}_7 \mp {\bf x}_8 - {\bf y}_9 \mp {\bf y}_{10}}] \cr E({{\bf 0}},3^ \pm 21) &= {\bf P}_{{11}}^{(3^ +)}({{\bf 0}}) \, {\bf z}_3 \cr&= {\textstyle{1 \over 8}} [{{\bf z}_3 + {\bf z}_4 \mp {\bf z}_5 - {\bf z}_6 \pm {\bf z}_7 \pm {\bf z}_8 - {\bf z}_9 \mp {\bf z}_{10}}] \cr E({{\bf 0}},3^ \pm 11) &= {\bf P}_{{11}}^{(3^ +)}({{\bf 0}}) \, {\bf x}_5 \cr&= {\textstyle{1 \over 8}} [{\pm {\bf y}_3 \mp {\bf y}_4 + {\bf x}_5 \mp {\bf x}_6 - {\bf y}_7 + {\bf y}_8 \pm {\bf x}_9 - {\bf x}_{10}}]. \cr} ]$ Just as for representation $[\tau ^{(1^ -)}]$ , the symmetry coordinates corresponding to representations $[\tau ^{(3^ \pm)}]$ do not contain any component of atoms 1 and 2. Consequently, all lattice modes of these symmetries leave the atoms on the fourfold axis at their equilibrium positions at rest.

Representation $[\boldtau^{(2)}]$ is two-dimensional and appears eight times in the decomposition of the T representation. Hence, there are 16 doubly degenerate phonons of this symmetry. According to (2.1.3.51), four projection operators $[{\bf P}_{11}^{(2)}({\bf 0})]$ , $[{\bf P}_{21}^{(2)}({\bf 0})]$ , $[{\bf P}_{12}^{(2)}({\bf 0}) ]$ and $[{\bf P}_{22}^{(2)}({\bf 0})]$ can in principle be constructed, the latter two being, however, equivalent to the former ones: $[\displaylines{{\bf P}_{\scriptstyle 11 \hfill \atop \scriptstyle 22 \hfill}^{(2)}({\bf 0}) =\hfill\cr {1 \over 4}\left({\matrix{ {\boldSigma _2^ \pm }&\! 0 &\! 0 &\! 0 &\! 0 &\!0 &\! 0 &\! 0 &\! 0 &\! 0 \cr 0 &\! {\boldSigma _2^ \pm }&\! 0 &\! 0 &\! 0 &\!0 &\! 0 &\! 0 &\! 0 &\! 0 \cr 0 &\! 0 &\! {\bf E}&\! {- {\bf D}_{180}^z }&\! 0 &\!{\pm i{\bf D}_{90}^z }&\! 0 &\! 0 &\! {\mp i{\bf D}_{270}^z }&\! 0\cr 0 &\! 0 &\! {- {\bf D}_{180}^z }&\! {\bf E}&\! 0 &\!{\mp i{\bf D}_{270}^z }&\! 0 &\! 0 &\! {\pm i{\bf D}_{90}^z }&\! 0 \cr 0 &\! 0 &\! 0 &\! 0 &\! {\bf E}&\!0 &\! {\mp i{\bf D}_{270}^z }&\! {\pm i{\bf D}_{90}^z }&\! 0 &\! {- {\bf D}_{180}^z } \cr 0 &\! 0 &\! {\mp i{\bf D}_{270}^z }&\! {\pm i{\bf D}_{90}^z }&\! 0 &\!{\bf E}&\! 0 &\! 0 &\! {- {\bf D}_{180}^z }&\! 0 \cr 0 &\! 0 &\! 0 &\! 0 &\! {\pm i{\bf D}_{90}^z }&\!0 &\! {\bf E}&\! {- {\bf D}_{180}^z }&\! 0 &\! {\mp i{\bf D}_{270}^z } \cr 0 &\! 0 &\! 0 &\! 0 &\! {\mp i{\bf D}_{270}^z }&\!0 &\! {- {\bf D}_{180}^z }&\! {\bf E}&\! 0 &\! {\pm i{\bf D}_{90}^z } \cr 0 &\! 0 &\! {\pm i{\bf D}_{90}^z }&\! {\mp i{\bf D}_{270}^z }&\! 0 &\!{- {\bf D}_{180}^z }&\! 0 &\! 0 &\! {\bf E}&\! 0 \cr 0 &\! 0 &\! 0 &\! 0 &\! {- {\bf D}_{180}^z }&\!0 &\! {\pm i{\bf D}_{90}^z }&\! {\mp i{\bf D}_{270}^z }&\! 0 &\! {\bf E}\cr }}\right)}]$ and $[\displaylines{{\bf P}_{\scriptstyle 12 \hfill \atop \scriptstyle 21 \hfill}^{(2)}({\bf 0}) =\hfill\cr {1 \over 4}\left({\matrix{ {\boldDelta _2^ \pm }&\!\! 0 &\!\! 0 &\!\! 0 &\!\! 0 &\!\!0 &\!\! 0 &\!\! 0 &\!\! 0 &\!\! 0 \cr 0 &\!\! {\boldDelta _2^ \pm }&\!\! 0 &\!\! 0 &\!\! 0 &\!\!0 &\!\! 0 &\!\! 0 &\!\! 0 &\!\! 0 \cr 0 &\!\! 0 &\!\! 0 &\!\! 0 &\!\! {\pm i{\bf m}_{ [{110}]}}&\!\!0 &\!\! {- {\bf m}_y }&\!\! {{\bf m}_x }&\!\! 0 &\!\! {\mp i{\bf m}_{ [{\bar 110}]}} \cr 0 &\!\! 0 &\!\! 0 &\!\! 0 &\!\! {\mp i{\bf m}_{ [{\bar 110}]}} &\!\!0 &\!\! {{\bf m}_x }&\!\! {- {\bf m}_y }&\!\! 0 &\!\! {\pm i{\bf m}_{ [{110}]}} \cr 0 &\!\! 0 &\!\! {\pm i{\bf m}_{ [{110}]}}&\!\! {\mp i{\bf m}_{ [{\bar 110}]}}&\!\! 0 &\!\!{{\bf m}_x }&\!\! 0 &\!\! 0 &\!\! {- {\bf m}_y }&\!\! 0 \cr 0 &\!\! 0 &\!\! 0 &\!\! 0 &\!\! {{\bf m}_x }&\!\!0 &\!\! {\pm i{\bf m}_{ [{110}]}}&\!\! {\mp i{\bf m}_{ [{\bar 110}]}}&\!\! 0 &\!\! {- {\bf m}_y } \cr 0 &\!\! 0 &\!\! {- {\bf m}_y }&\!\! {{\bf m}_x }&\!\! 0 &\!\!{\pm i{\bf m}_{ [{110}]}}&\!\! 0 &\!\! 0 &\!\! {\mp i{\bf m}_{ [{\bar 110}]}}&\!\! 0 \cr 0 &\!\! 0 &\!\! {{\bf m}_x }&\!\! {- {\bf m}_y }&\!\! 0 &\!\!{\mp i{\bf m}_{ [{\bar 110}]}}&\!\! 0 &\!\! 0 &\!\! {\pm i{\bf m}_{ [{110}]}}&\!\! 0 \cr 0 &\!\! 0 &\!\! 0 &\!\! 0 &\!\! {- {\bf m}_y } &\!\!0 &\!\! {\mp i{\bf m}_{ [{\bar 110}]}}&\!\! {\pm i{\bf m}_{ [{110}]}}&\!\! 0 &\!\! {{\bf m}_x } \cr 0 &\!\! 0 &\!\! {\mp i{\bf m}_{ [{\bar 110}]}}&\!\! {\pm i{\bf m}_{ [{110}]}}&\!\! 0 &\!\!{- {\bf m}_y }&\!\! 0 &\!\! 0 &\!\! {{\bf m}_x }&\!\! 0 \cr }}\right)}]$ with $[\eqalign{ & \boldSigma _2^ \pm = {\bf E}\pm i{\bf D}_{90}^z - {\bf D}_{180}^z \mp i{\bf D}_{270}^z = {\pmatrix{ 2 & {\mp 2i}& 0 \cr {\pm 2i}& 2 & 0 \cr 0 & 0 & 0 \cr }}, \cr & \boldDelta _2^ \pm = {\bf m}_x - {\bf m}_y \pm i{\bf m}_{ [{110}]} \mp i{\bf m}_{ [{\bar 110}]} = {\pmatrix{ {- 2}& {\mp 2i}& 0 \cr {\mp 2i}& 2 & 0 \cr 0 & 0 & 0 \cr }}. \cr} ]$

The projection operator $[{\bf P}_{11}^{(2)}({\bf 0})]$ applied to the basis vectors $[{\bf x}_{1}]$ , $[{\bf x}_{2}]$ , $[{\bf x}_{3} ]$ , $[{\bf x}_{5}]$ , $[{\bf x}_{6}]$ , $[{\bf x}_{7}]$ , $[{\bf z}_{3}]$ and $[{\bf z}_{5}]$ yields eight symmetry coordinates for eight phonon modes with different eigenfrequencies. Owing to the degeneracy, each of these phonons has a counterpart with the same frequency but with a different linearly independent eigenvector. These new eigenvectors are built from another set of symmetry coordinates, which is generated if the other operator $[{\bf P}_{21}^{(2)}({\bf 0})]$ is applied to the same vectors $[{\bf x}_{1} ]$ , $[{\bf x}_{2}]$ , $[{\bf x}_{3}]$ , $[{\bf x}_{5}]$ , $[{\bf x}_{6}]$ , $[{\bf x}_{7}]$ , $[{\bf z}_{3}]$ and $[{\bf z}_{5} ]$ .The two sets of symmetry coordinates are $[\eqalign{ {\bf E}({\bf 0},211) &= {\bf P}_{11}^{(2)}({\bf 0}) \, {\bf x}_1 = {\textstyle{1 \over 2}} [{{\bf x}_1 + i{\bf y}_1 }] \cr {\bf E}({\bf 0},221) &= {\bf P}_{11}^{(2)}({\bf 0}) \, {\bf x}_2 = {\textstyle{1 \over 2}} [{{\bf x}_2 + i{\bf y}_2 }] \cr {\bf E}({\bf 0},231) &= {\bf P}_{11}^{(2)}({\bf 0}) \, {\bf x}_3 = {\textstyle{1 \over 4}} [{{\bf x}_3 + {\bf x}_4 + i{\bf y}_6 + i{\bf y}_9 }] \cr {\bf E}({\bf 0},241) &= {\bf P}_{11}^{(2)}({\bf 0}) \, {\bf x}_5 = {\textstyle{1 \over 4}} [{{\bf x}_5 + {\bf x}_{10} + i{\bf y}_7 + i{\bf y}_8 }] \cr {\bf E}({\bf 0},251) &= {\bf P}_{11}^{(2)}({\bf 0}) \, {\bf x}_6 = {\textstyle{1 \over 4}} [{{\bf x}_6 + {\bf x}_9 + i{\bf y}_3 + i{\bf y}_4 }] \cr {\bf E}({\bf 0},261) &= {\bf P}_{11}^{(2)}({\bf 0}) \, {\bf x}_7 = {\textstyle{1 \over 4}} [{{\bf x}_7 + {\bf x}_8 + i{\bf y}_5 + i{\bf y}_{10}}] \cr {\bf E}({\bf 0},271) &= {\bf P}_{11}^{(2)}({\bf 0}) \, {\bf z}_3 = {\textstyle{1 \over 4}} [{{\bf z}_3 - {\bf z}_4 - i{\bf z}_6 + i{\bf z}_9 }] \cr {\bf E}({\bf 0},281) &= {\bf P}_{11}^{(2)}({\bf 0}) \, {\bf z}_5 = {\textstyle{1 \over 4}} [{{\bf z}_5 - {\bf z}_{10} + i{\bf z}_7 - i{\bf z}_8 }] \cr &\cr {\bf E}({\bf 0},212) &= {\bf P}_{21}^{(2)}({\bf 0}) \, {\bf x}_1 = {\textstyle{1 \over 2}} [{- {\bf x}_1 + i{\bf y}_1 }] \cr {\bf E}({\bf 0},222) &= {\bf P}_{21}^{(2)}({\bf 0}) \, {\bf x}_2 = {\textstyle{1 \over 2}} [{- {\bf x}_2 + i{\bf y}_2 }] \cr {\bf E}({\bf 0},232) &= {\bf P}_{21}^{(2)}({\bf 0}) \, {\bf x}_3 = {\textstyle{1 \over 4}} [{- {\bf x}_7 - {\bf x}_8 + i{\bf y}_5 + i{\bf y}_{10}}] \cr {\bf E}({\bf 0},242) &= {\bf P}_{21}^{(2)}({\bf 0}) \, {\bf x}_5 = {\textstyle{1 \over 4}} [{- {\bf x}_6 - {\bf x}_9 + i{\bf y}_3 + i{\bf y}_4 }] \cr {\bf E}({\bf 0},252) &= {\bf P}_{21}^{(2)}({\bf 0}) \, {\bf x}_6 = {\textstyle{1 \over 4}} [{- {\bf x}_5 - {\bf x}_{10} + i{\bf y}_7 + i{\bf y}_8 }] \cr {\bf E}({\bf 0},262) &= {\bf P}_{21}^{(2)}({\bf 0}) \, {\bf x}_7 = {\textstyle{1 \over 4}} [{- {\bf x}_3 - {\bf x}_4 + i{\bf y}_6 + i{\bf y}_9 }] \cr {\bf E}({\bf 0},272) &= {\bf P}_{21}^{(2)}({\bf 0}) \, {\bf z}_3 = {\textstyle{1 \over 4}} [{- {\bf z}_7 + {\bf z}_8 - i{\bf z}_5 + i{\bf z}_{10}}] \cr {\bf E}({\bf 0},282) &= {\bf P}_{21}^{(2)}({\bf 0}) \, {\bf z}_5 = {\textstyle{1 \over 4}} [{{\bf z}_6 - {\bf z}_9 - i{\bf z}_3 + i{\bf z}_4 }] .\cr} ]$ Looking carefully at these sets of symmetry coordinates, one recognises that both vector spaces are spanned by mutually complex conjugate symmetry coordinates.

Collecting all symmetry coordinates as column vectors within a $[30\times 30]$ matrix we finally obtain the matrix shown in Fig. 2.1.3.7. For simplicity, only nonzero elements are displayed. This matrix can be used for the block-diagonalization of any dynamical matrix that describes the dynamical behaviour of our model crystal.

Figure 2.1.3.7 | top | pdf |

Matrix of symmetry coordinates at $[{\bf q}={\bf 0}]$ for the example structure given in Fig. 2.1.3.4 and Table 2.1.3.1.

References

International Tables for Crystallography (2006). Vol. D. ch. 2.1, pp. 284-286