Symmetry coordinates

Eckold, G.

doi:10.1107/97809553602060000638

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

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. D. ch. 2.1, pp. 283-286

Section 2.1.3.4. Symmetry coordinates

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. Symmetry coordinates

| top | pdf |

So far, we have used the 3N Cartesian coordinates of all atoms within a primitive cell in order to describe the dynamics of the crystal lattice. Within this coordinate system, the elements of the dynamical matrix can be calculated on the basis of specific models for interatomic interactions. The corresponding eigenvectors or normal coordinates are some linear combinations of the Cartesian components. With respect to these normal coordinates, which are specific to each particular crystal, the dynamical matrix has diagonal form and contains the squares of the eigenfrequencies reflecting the interatomic forces.

As shown in Sections 2.1.3.1 and 2.1.3.3, there are constraints for the dynamical matrix due to the symmetry of the crystal lattice and, hence, eigenvectors must obey certain transformation laws. Not all arbitrary linear combinations of the Cartesian coordinates can form an eigenvector. Rather, there are symmetry-adapted coordinates or simply symmetry coordinates compatible with a given structure that can be used to predict the general form of eigenvectors without the need for any particular model of interatomic interactions. These symmetry coordinates can be determined on the basis of the irreducible multiplier representations introduced in the previous section.

From the T-matrix operators and the representation matrices $[\boldtau^{(s)}({\bf q},{\bf R})]$ of a particular irreducible multiplier representation we may define another matrix operator $[{\bf P}^{(s)}({\bf q}) ]$ with the elements $[{\bf P}_{\lambda \lambda' }^{(s)}({\bf q}) = ({{f_s }/{| G |}})\textstyle\sum\limits_{\bf R}{\tau _{\lambda \lambda' }^{(s)*}({\bf q},{\bf R})}\,{\bf T}({\bf q},{\bf R}). \eqno (2.1.3.51) ]$ When applied to an arbitrary 3N-dimensional vector Ψ built from the Cartesian coordinates of the individual atoms, this operator yields the particular component of Ψ that transforms according to the irreducible representation s. Hence it acts as a projection operator. Defining a set of $[f_{s}]$ vectors by $[{\bf E}({\bf q},s\lambda) = {\bf P}_{\lambda \lambda '}^{(s)}({\bf q}) \, \boldPsi, \quad\lambda = 1, \ldots, f_s, \eqno (2.1.3.52) ]$ we obtain $[{{\bf T}({\bf q},{\bf R}') \, {\bf E}({\bf q},s\lambda) = ({{f_s }/{| G |}})\textstyle\sum\limits_{\bf R}{\tau _{\lambda \lambda'}^{(s)*} ({\bf q},{\bf R})\,{\bf T}({\bf q},{\bf R}') \, {\bf T}({\bf q},{\bf R}) \, \boldPsi}.\eqno(2.1.3.53)} ]$ Using the multiplication rule (2.1.3.21a), it can be shown that the right-hand side of this equation reduces to $[({{f_s }/{| G |}})\textstyle\sum\limits_{\bf R}{\sum\limits_{\lambda '' = 1}^{f_s }{\tau _{\lambda ''\lambda }^{(s)}({\bf q},{\bf R}')\,\tau _{\lambda ''\lambda' }^{(s)*}({\bf q},{\bf R}' \circ {\bf R})}}\,{\bf T}({\bf q},{\bf R}' \circ {\bf R}) \, \boldPsi, ]$ which can also be written in the form $[\textstyle\sum\limits_{\lambda '' = 1}^{f_s }{\tau _{\lambda ''\lambda }^{(s)}({\bf q},{\bf R}') \, ({{f_s }/{| G |}}})\textstyle\sum\limits_{\bf R}{\tau _{\lambda ''\lambda' }^{(s)*}({\bf q},{\bf R})}\,{\bf T}({\bf q},{\bf R}) \, \boldPsi, ]$ since if R runs over all symmetry operations of the group $[G_{o}({\bf q})]$ the same is true for the product $[{\bf R}'\circ {\bf R} ]$ . Comparing this expression with the definitions (2.1.3.51) and (2.1.3.52) we obtain $[{\bf T}({\bf q},{\bf R}') \, {\bf E}({\bf q},{s\lambda)}=\textstyle\sum\limits_{\lambda '' = 1}^{f_s }{\tau _{\lambda ''\lambda }^{(s)}({\bf q},{\bf R}')\,{\bf E}({\bf q},{s\lambda '')}}. \eqno (2.1.3.54) ]$ Hence the set of vectors $[{\bf E}({\bf q},s\lambda)]$ span an irreducible vector space and transform into each other in just the same way as the eigenvectors $[{\bf e}({\bf q}, sa\lambda)]$ of the dynamical matrix do.

If the corresponding irreducible representation s appears only once in the decomposition of the 3N-dimensional T representation, then the vector space provided by the $[{\bf E}({\bf q},s\lambda) ]$ , $[\lambda=1,\ldots,f_{s}]$ , is uniquely determined. Consequently, these basis vectors themselves may be regarded as eigenvectors of the dynamical matrix. In this case, symmetry considerations alone determine the polarization of lattice vibrations irrespective of the particular interatomic interactions.

If, on the other hand, the multiplicity $[c_{s}]$ of the representation s is larger than 1, the most that can be inferred is that each of the vectors $[{\bf E}({\bf q},s\lambda)]$ is some linear combination of the $[c_{s}]$ eigenvectors $[{\bf e}({\bf q}, sa\lambda) ]$ , $[a=1,\ldots,c_s]$ . By an appropriate choice of the different generating vectors $[\boldPsi_{a}]$ in (2.1.3.52), it is, however, always possible to find a set of $[c_{s}]$ pairwise orthogonal vectors $[{\bf E}({\bf q},sa\lambda)]$ that span the same vector space as the eigenvectors $[{\bf e}({\bf q}, sa\lambda) ]$ . If we repeat this procedure for every irreducible representation s contributing to the T representation, we obtain 3N linearly independent vectors, the symmetry coordinates, that generate a new coordinate system within the 3N-dimensional space of atomic displacements. With respect to this coordinate system the dynamical matrix is reduced to a symmetry-adapted block-diagonal form.

In order to show this, let us denote the matrix elements of the transformed dynamical matrix by $[\bar D_{sa\lambda }^{s'a'\lambda '}({\bf q})]$ ( $[\lambda=1,\ldots,f_{s} ]$ , $[a=1,\ldots,c_{s}]$ ) and the components of the symmetry coordinates by $[E_\kappa ^\alpha ({\bf q},sa\lambda)]$ ( $[\kappa=1,\ldots,N ]$ , $[\alpha=1,2,3]$ ). Then the following equation holds, since the dynamical matrix D(q) commutes with the T-matrix operators and since the symmetry coordinates transform according to (2.1.3.54): $[\eqalignno{& \overline D _{sa\lambda }^{s'a'\lambda '}({\bf q}) &\cr&\quad= \textstyle\sum\limits_{\kappa \alpha }\textstyle\sum\limits_{\kappa ' \beta }E_\kappa ^{\alpha *}({\bf q},sa\lambda )\, D_{\kappa \kappa ' }^{\alpha \beta }({\bf q})\, E_{\kappa ' }^\beta ({\bf q},s'a'\lambda ') &\cr &\quad = \textstyle\sum\limits_{\kappa \alpha }\textstyle\sum\limits_{\kappa' \beta }E_\kappa ^{\alpha *}({\bf q},sa\lambda )&\cr&\quad\quad\times\textstyle\sum\limits_{\kappa _1 \alpha _1 }\textstyle\sum\limits_{\kappa _2 \alpha _2 }\{(T^{- 1}({\bf q},{\bf R}))_{\kappa \kappa _1 }^{\alpha \alpha _1 }\, D_{\kappa _1 \kappa _2 }^{\alpha _1 \alpha _2 }({\bf q})\, (T({\bf q},{\bf R}))_{\kappa _2 \kappa ' }^{\alpha _2 \beta }\}&\cr&\quad \quad \times E_{\kappa ' }^\beta ({\bf q},s'a'\lambda ') &\cr &\quad = \textstyle\sum\limits_{\kappa _1 \alpha _1 }\textstyle\sum\limits_{\kappa _2 \alpha _2 }\textstyle\sum\limits_{\mu = 1}^{f_s }\textstyle\sum\limits_{\mu ' = 1}^{f_s }\tau _{\mu \lambda }^{(s)*}({\bf q},{\bf R})\, E_{\kappa _1 }^{\alpha _1^* }({\bf q},sa\mu )&\cr&\quad \quad \times D_{\kappa _1 \kappa _2 }^{\alpha _1 \alpha _2 }({\bf q})\,\tau _{\mu' \lambda' }^{(s')}({\bf q},{\bf R})\, E_{\kappa _2 }^{\alpha _2 }({\bf q},s'a'\lambda ') &\cr &\quad = \textstyle\sum\limits_{\mu = 1}^{f_s }\textstyle\sum\limits_{\mu ' = 1}^{f_s }\tau _{\mu \lambda }^{(s)*}({\bf q},{\bf R})\,\tau _{\mu ' \lambda '}^{(s')}({\bf q},{\bf R})\,\overline D _{sa\mu }^{s'a'\lambda '}({\bf q}).&\cr&& (2.1.3.55)} ]$ Owing to the orthogonality of the irreducible representation, we obtain after summation over all symmetry elements R and division by the order of the group $[\eqalignno{ \overline D _{sa\lambda }^{s' a' \lambda ' }({\bf q}) &= (1/{| G |})\textstyle\sum\limits_{\bf R}{\textstyle\sum\limits_{\mu = 1}^{f_s }{\textstyle\sum\limits_{\mu ' = 1}^{f_s }{\tau _{\mu \lambda }^{(s)*}({{\bf q},{\bf R}})}}\,\tau _{\mu' \lambda' }^{(s')}({{\bf q},{\bf R}})\,\overline D _{sa\mu }^{s' a' \mu '}({\bf q})} &\cr &= (1/ {f_s })\textstyle\sum\limits_{\mu = 1}^{f_s }{\textstyle\sum\limits_{\mu ' = 1}^{f_s }{\delta _{\mu \mu' }\delta _{\lambda \lambda'}\delta _{ss' }\,\overline D _{sa\mu }^{s' a' \mu ' }({\bf q})}} &\cr &= (1/{f_s })\delta _{\lambda \lambda '}\delta _{ss'}\textstyle\sum\limits_{\mu = 1}^{f_s }{\overline D _{sa\mu }^{sa' \mu }({\bf q})}.& (2.1.3.56)} ]$ This equation proves the block-diagonal form of the transformed dynamical matrix $[\bar{\bf D}]$ . Hence, with respect to the symmetry coordinates, the dynamical matrix can be represented by submatrices $[{\bar {\bf D}}^{(s)}({\bf q})]$ of dimension $[c_{s}\times c_s]$ that are determined by the individual irreducible representations (s): $[\overline {\bf D}({\bf q}) = {\pmatrix{ {\overline {\bf D}^{(1)}({\bf q})}& {{\bf 0}}& \,s & {{\bf 0}} \cr {{\bf 0}}& {\overline {\bf D}^{(2)}({\bf q})}& \,s & {{\bf 0}} \cr \vdots & \vdots & \ddots & \vdots \cr {{\bf 0}}& {{\bf 0}}& \,s & \ddots \cr }}. \eqno (2.1.3.57) ]$ The elements of the submatrices are given by $[\overline D _{aa' }^{(s)}({\bf q}) = \textstyle\sum\limits_{\kappa \alpha }{\textstyle\sum\limits_{\kappa \prime \beta }{E_\kappa ^{\alpha *}({{\bf q},sa\lambda })\, D_{\kappa \kappa ' }^{\alpha \beta }({\bf q})\, E_{\kappa '}^\beta ({{\bf q},sa' \lambda })}}, \eqno (2.1.3.58) ]$ and must be independent of λ. Obviously, a submatrix $[\overline {\bf D}^{(s)}({\bf q}) ]$ may appear once, twice or three times on the diagonal, according to the dimensionality $[f_{s}]$ of the respective irreducible representation.

The eigenvectors and eigenvalues of the block-diagonalized dynamical matrix can be collected from the eigenvectors and eigenvalues of the individual submatrices. Hence, the eigenvectors of $[\overline {\bf D}^{(s)}({\bf q})]$ correspond to the $[c_{s}]$ non-degenerate phonons of symmetry s.

2.1.3.4.1. Example

| top | pdf |

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. 283-286