Symmetry constraints for the dynamical matrix

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. 274-281

Section 2.1.3.1. Symmetry constraints for the dynamical matrix

G. Eckold^a ^*

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

2.1.3.1. Symmetry constraints for the dynamical matrix

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The elements of the $[3N\times 3N]$ dynamical matrix as introduced in Section 2.1.2.3 are given by $[D_{\kappa \kappa '}^{\alpha \beta }({\bf q}) = {1 \over {\sqrt {m_\kappa m_{\kappa '}}}}\sum\limits_{l'}{V_{\alpha \beta }(\kappa l,\kappa 'l')\exp[{i{\bf q}({\bf r}_{l'} - {\bf r}_l)}}]. \eqno (2.1.3.1) ]$ Using the matrix notation for the $[3\times3]$ submatrices introduced in (2.1.2.22a), this equation reads $[{\bf D}_{\kappa \kappa '}^{}({\bf q}) = {1 \over {\sqrt {m_\kappa m_{\kappa '}}}}\sum\limits_{l'}{{\bf V}(\kappa l,\kappa 'l')\exp[{i{\bf q}({\bf r}_{l'} - {\bf r}_l)}}]. \eqno (2.1.3.1a) ]$ Since the vector $[{\bf r}_{l'}-{\bf r}_{l}]$ corresponds to a vector of the direct lattice, the right-hand side of equation (2.1.3.1) is invariant with respect to changes of the wavevector q by an arbitrary reciprocal lattice vector g. Hence, the elements of the dynamical matrix represent periodic functions within the reciprocal space: $[{\bf D}({\bf q}+ {\bf g}) = {\bf D}({\bf q}). \eqno (2.1.3.2)]$ The same periodicity can also be assumed for the eigenvalues , or eigenfrequencies, and for the eigenvectors :² $[\eqalignno{ \omega _{{\bf q}+ {\bf g},j} &= \omega _{{\bf q},j}, &\cr {\bf e}({{\bf q}+ {\bf g},j}) &= {\bf e}({{\bf q},j}).&(2.1.3.3)} ]$ Consequently, we can restrict our discussion to wavevectors within the first Brillouin zone.

Owing to the symmetry of the atomic structure, not all of the force constants $[V_{\alpha \beta}(\kappa l, \kappa'l')]$ reflecting the interaction between atoms $[(\kappa l)]$ and $[(\kappa'l')]$ are independent. Rather, there are constraints to the elements of the dynamical matrix according to the space group of the crystal. In the following, these constraints will be considered in some detail. Suppose the space group contains a symmetry operation $[\{{\bf S}| {\bf v}({\bf S}) + {\bf x}(m)\}]$ .³ When applied to the crystal, this symmetry operation sends atom $[(\kappa l)]$ into another atom [({K}L)] and simultaneously atom $[(\kappa'l')]$ into [({K}'L')] . At the same time, the wavevector of a phonon is rotated from $[{\bf q}]$ into $[{\bf S}\,{\bf q}]$ . Hence, the elements of the dynamical matrix that describes the dynamics of the crystal after application of the symmetry operation may be written as $[\eqalignno{&D_{{K}{K}'}^{\alpha \beta }({\bf S}\, {\bf q}) &\cr&\quad= {1 \over {\sqrt {m_{K}m_{{K}'}}}}\sum\limits_{L'}{V_{\alpha \beta }({K}L,{K}'L')\exp[{i({\bf S}\, {\bf q})({\bf r}_{L'} - {\bf r}_L)}]}&\cr &&(2.1.3.4)}]$ or in submatrix notation $[{\bf D}_{{K}{K}'}^{}({\bf S}\, {\bf q}) = {1 \over {\sqrt {m_{K}m_{{K}'}}}}\sum\limits_{L'}{{\bf V}({K}L,{K}'L')\exp[{i({\bf S}\, {\bf q})({\bf r}_{L'} - {\bf r}_L)}]}. \eqno (2.1.3.4a) ]$ This submatrix can be related to the corresponding matrix $[{\bf D}_{\kappa \kappa'}({\bf q}) ]$ that describes the same dynamical behaviour, but in the unrotated crystal. To this end, we first consider the transformation of the force-constant matrices under the symmetry operation. Obviously, the interaction between atoms $[(\kappa l)]$ and $[(\kappa' l')]$ has to be of the same type as the interaction between [({K}L)] and [({K}'L')] .

Since the potential energy is invariant with respect to symmetry operations, the force constants are related via $[\eqalignno{&\textstyle\sum\limits_{\alpha \beta }{\textstyle\sum\limits_{\kappa l}{\textstyle\sum\limits_{\kappa 'l'}{V_{\alpha \beta }(\kappa l,\kappa 'l')\,u_{\kappa l}^\alpha \,u_{\kappa 'l'}^\beta }}} &\cr&\quad= \textstyle\sum\limits_{\alpha \beta }{\textstyle\sum\limits_{{K}L}{\textstyle\sum\limits_{{K}'L'}{V_{\alpha \beta }({K}L,{K}'L')\,u_{{K}L}^\alpha \,u_{{K}'L'}^\beta }}}&(2.1.3.5)} ]$ or in matrix notation $[\textstyle\sum\limits_{\kappa l}{\textstyle\sum\limits_{\kappa 'l'}{{\bf u}_{\kappa l}^{}\, {\bf V}(\kappa l,\kappa 'l') \,{\bf u}_{\kappa 'l'}^{}}} = \textstyle\sum\limits_{{K}L}{\textstyle\sum\limits_{{K}'L'}{{\bf u}_{{K}L}^{}\, {\bf V}({K}L,{K}'L') \,{\bf u}_{{K}'L'}^{}}}. \eqno (2.1.3.5a) ]$

Owing to the symmetry operation, the displacements of atoms $[(\kappa l)]$ and $[(\kappa' l')]$ are rotated and transferred to atoms [({K}L)] and [({K}'L')] , respectively (see Fig. 2.1.3.1). Thus, (2.1.3.5) can be rewritten as $[\eqalignno{&\textstyle\sum\limits_{\alpha \beta }{\textstyle\sum\limits_{\kappa l}{\textstyle\sum\limits_{\kappa 'l'}{V_{\alpha \beta }(\kappa l,\kappa 'l')\,u_\alpha (\kappa l)\,u_\beta (\kappa 'l')}}} &\cr&\quad= \textstyle\sum\limits_{\alpha \beta }{\textstyle\sum\limits_{\kappa l}{\textstyle\sum\limits_{\kappa 'l'}{V_{\alpha \beta }({K}L,{K}'L')\textstyle\sum\limits_\mu {S_{\alpha \mu }}u_\mu (\kappa l)\,\sum\limits_\nu {S_{\beta \nu }}u_\nu (\kappa 'l')}}}.&\cr&&(2.1.3.6)} ]$ Moreover, this relation is valid for arbitrary displacements and, hence, the matrices of force constants transform according to $[V_{\mu \nu }(\kappa l,\kappa 'l') = \textstyle\sum\limits_{\alpha \beta }{V_{\alpha \beta }({K}L,{K}'L') S_{\alpha \mu }S_{\beta \nu }} \eqno (2.1.3.7) ]$ or $[{\bf V}(\kappa l,\kappa 'l') = {\bf S}^{T}\, {\bf V}({K}L,{K}'L')\,{\bf S}. \eqno (2.1.3.7a) ]$ Using the fact that the matrix of rotation S is unitary $[({\bf S}^{-1}={\bf S}^{T}) ]$ , the inverse relation is obtained: $[{\bf V}({K}L,{K}'L') = {\bf S}\,{\bf V}(\kappa l,\kappa 'l')\, {\bf S}^{T}. \eqno (2.1.3.7b) ]$ Hence, the force-constant submatrices transform like tensors do. One has to bear in mind, however, that the matrices in equation (2.1.3.7b) correspond to different pairs of atoms as illustrated by Fig. 2.1.3.2. Using this result in equation (2.1.3.4a) and remembering the fact that atoms related by a symmetry operation have the same mass, we obtain $[\eqalignno{&D_{{K}{K}'}^{\alpha \beta }({\bf S}\, {\bf q}) &\cr&\quad= {1 \over {\sqrt {m_\kappa m_{\kappa '}}}}\sum\limits_{\mu \nu }{S_{\alpha \mu }S_{\beta \nu }\sum\limits_{l'}{V_{\mu \nu }(\kappa l,\kappa 'l')\exp[{i({\bf S}\, {\bf q})({\bf r}_{L'} - {\bf r}_L)}}]}& \cr &\quad= {1 \over {\sqrt {m_\kappa m_{\kappa '}}}}\sum\limits_{\mu \nu }{S_{\alpha \mu }S_{\beta \nu }\sum\limits_{l'}{V_{\mu \nu }(\kappa l,\kappa 'l')\exp[{i{\bf q}\, {\bf S}^{- 1} ({\bf r}_{L'} - {\bf r}_L)}}]}.&\cr &&(2.1.3.8)} ]$ The phase factor on the right-hand side contains the indices L and [L'] of those primitive cells into which the atoms $[(\kappa l)]$ and $[(\kappa'l')]$ are sent by the symmetry operation $[\{{\bf S}| {\bf v}({\bf S}) + {{\bf x}(m)}\} ]$ . In general, the phase is not conserved during the transformation and, hence, the sum over [l'] cannot simply be replaced by the matrix elements $[D_{\kappa \kappa '}^{\mu \nu }({\bf q})]$ . Rather, we have to consider the phase factor in more detail in order to find the transformation law for the dynamical matrix .

Figure 2.1.3.1 | top | pdf |

Transformation of atomic displacements by a symmetry operation.

Figure 2.1.3.2 | top | pdf |

Relation between interaction of symmetry-related atoms.

The position vectors of particles $[(\kappa l)]$ and [({K}L)] are related via $[\eqalignno{ {\bf r}_{{K}L}^o = {\bf r}_{K}^o + {\bf r}_L &= \{ {{\bf S} | {{\bf v} ({\bf S} ) + {\bf x} (m )} } \} \, {\bf r}_{\kappa l}^o &\cr &= {\bf S}\,{\bf r}_{\kappa l}^o + {\bf v} ({\bf S} ) + {\bf x} (m ) &\cr &= {\bf S}\, ({\bf r}_\kappa ^o + {\bf r}_l) + {\bf v} ({\bf S} ) + {\bf x} (m ) & (2.1.3.9)} ]$ and $[\eqalignno{ {\bf r}_\kappa ^o + {\bf r}_l &= {\bf S}^{- 1}\, ({\bf r}_{K}^o + {\bf r}_L) - {\bf v} ({\bf S} ) - {\bf x} (m )& \cr &= \{{{\bf S} | {{\bf v} ({\bf S} ) + {\bf x} (m )} } \}^{- 1} \, {\bf r}_{K}^o + {\bf S}^{- 1} \, {\bf r}_L. &\cr &&(2.1.3.9a)} ]$ Consequently, the vector appearing in the phase factor of equation (2.1.3.8) can be expressed as $[{\bf S}^{- 1} \, {\bf r}_L = {\bf r}_\kappa ^o + {\bf r}_l - \{{{\bf S} | {{\bf v} ({\bf S} ) + {\bf x} (m )} } \}^{- 1} \, {\bf r}_{K}^o. \eqno (2.1.3.10) ]$ When inserted into equation (2.1.3.8), the required transformation law for the dynamical matrix is obtained: $[\eqalignno{ D_{{K}{K}'}^{\alpha \beta }({\bf S}\, {\bf q}) &= {1 \over {\sqrt {m_\kappa m_{\kappa '}}}} \sum\limits_{\mu \nu }{S_{\alpha \mu }S_{\beta \nu }\sum\limits_{l'}{V_{\mu \nu }(\kappa l,\kappa 'l')}} &\cr &\quad\times \exp [{i{\bf q}\, ({\bf r}_{l'} - {\bf r}_l)} ] &\cr&\quad\times\exp [{i{\bf q}\, ({ \{{{\bf S} | {{\bf v} ({\bf S} ) + {\bf x}(m)} } \}^{- 1}\, {\bf r}_{K}^o - {\bf r}_\kappa ^o } )} ] &\cr &\quad\times \exp [{- i{\bf q}\, ({ \{{{\bf S} | {{\bf v} ({\bf S} ) + {\bf x}(m)} } \}^{- 1} \, {\bf r}_{{K}'}^o - {\bf r}_{\kappa '}^o } )} ] &\cr&&(2.1.3.11)} ]$ or $[\eqalignno{ D_{{K}{K}'}^{\alpha \beta }({\bf S} \,{\bf q}) &= \textstyle\sum\limits_{\mu \nu }{S_{\alpha \mu }S_{\beta \nu }D_{\kappa \kappa '}^{\mu \nu }({\bf q})}& \cr& \times\exp [{i{\bf q}\, ({ \{{{\bf S} | {{\bf v} ({\bf S} ) + {\bf x}(m)} } \}^{- 1} \, {\bf r}_{K}^o - {\bf r}_\kappa ^o } )} ] & \cr &\times\exp [{- i{\bf q}\, ({ \{{{\bf S} | {{\bf v} ({\bf S} ) + {\bf x}(m)} } \}^{- 1} \, {\bf r}_{{K}'}^o - {\bf r}_{\kappa '}^o } )} ], & \cr&&(2.1.3.11a)} ]$ or in submatrix notation $[\eqalignno{ {\bf D}_{{K}{K}'} ({\bf S}\, {\bf q}) &= {\bf S}^{\rm T}\, {\bf D}_{\kappa \kappa '} ({\bf q})\, {\bf S}& \cr&\times \exp [{i{\bf q}\, ({ \{{{\bf S} | {{\bf v} ({\bf S} ) + {\bf x}(m)} } \}^{- 1}\, {\bf r}_{K}^o - {\bf r}_\kappa ^o } )} ] & \cr &\times \exp [{- i{\bf q}\, ({ \{{{\bf S} | {{\bf v} ({\bf S} ) + {\bf x}(m)} } \}^{- 1} \,{\bf r}_{{K}'}^o - {\bf r}_{\kappa '}^o } )} ]. &\cr&& (2.1.3.11b)} ]$ Again, these are relations between pairs of submatrices of the dynamical matrix. In contrast to the matrices of force constants, however, phase factors have to be considered here. This is because the symmetry operation $[\{{\bf S}| {\bf v}({\bf S}) + {{\bf x}(m)}\} ]$ may send different atoms $[\kappa]$ and $[\kappa']$ located within the same primitive cell (0) into atoms [({K}L)] and [({K}' L')] within different primitive cells L and [L'] as illustrated in Fig. 2.1.3.3. Therefore, the product of phase factors in equation (2.1.3.11) is in general different from unity.

Figure 2.1.3.3 | top | pdf |

Symmetry-related atoms in different primitive cells.

Irrespective of the particular primitive cells in which the atoms are located, however, the labels $[\kappa]$ and K of those atoms that are related by a symmetry operation are uniquely determined. Given the label $[\kappa]$ and a particular symmetry operation $[ \{{\bf S} | {\bf v}({\bf S}) + {\bf x}(m) \} ]$ , the label K may be represented by the function $[{K}=F_{o}(\kappa, \bf {S}), \eqno (2.1.3.12)]$ which represents the atom transformation table.⁴ With the definition of unitary transformation matrices $[\eqalignno{&\Gamma _{{K}\kappa }^{\alpha \mu }({\bf q}, \{{{\bf S} | {{\bf v} ({\bf S} )} + {\bf x} (m )} \}) &\cr&\quad= S_{\alpha \mu } \delta ({\kappa, F_o^{- 1}({K}, {\bf S})} ) \exp [{i{\bf q}\, ({ \{{{\bf S} | {{\bf v} ({\bf S} ) + {\bf x}(m)} } \}^{- 1} \, {\bf r}_{K}^o - {\bf r}_\kappa ^o } )} ],&\cr&&(2.1.3.13)} ]$ we are now able to formulate the transformation law for the dynamical matrix briefly as $[\eqalignno{&{\bf D} ({{\bf S}\, {\bf q}} ) &\cr&\quad= \boldGamma ({{\bf q}, \{{{\bf S} | {{\bf v} ({\bf S} ) + {\bf x} (m )} } \}} ) \, {\bf D} ({\bf q} ) \, \boldGamma ^ + ({{\bf q}, \{{{\bf S} | {{\bf v} ({\bf S} ) + {\bf x} (m )} } \}} ).&\cr&&(2.1.3.14)}]$ Obviously, with the help of equation (2.1.3.13), we can allocate a unitary matrix to each symmetry operation. These $[\boldGamma]$ matrices, however, do not form a representation of the crystal space group in the mathematical sense since the mapping $[ \{{{\bf S} | {{\bf v}({\bf S}) + {\bf x}(m)} } \}\rightarrow \boldGamma ({\bf q}, \{{{\bf S} | {{\bf v}({\bf S}) + {\bf x}(m)} } \}) ]$ is not a linear one. Rather, we obtain the following transformation matrix for the product of two symmetry operations: $[ \displaylines{\boldGamma ({\bf q},\{{{\bf S}_1 | {{\bf v}({{\bf S}_1 }) + {\bf x}({m_1 })}}\}\circ\{{{\bf S}_2 | {{\bf v}({{\bf S}_2 }) + {\bf x}({m_2 })}}\}) \hfill\cr\quad= \boldGamma ({{\bf S}_2 \, {\bf q},\{{{\bf S}_1 | {{\bf v}({{\bf S}_1 }) + {\bf x}({m_1 })}}\}}) \, \boldGamma ({{\bf q},\{{{\bf S}_2 | {{\bf v}({{\bf S}_2 }) + {\bf x}({m_2 })}}\}}). \hfill\cr\hfill (2.1.3.15)} ]$ The nonlinearity of the mapping is due to the fact that the first matrix on the right-hand side of this equation depends on the wavevector $[{\bf S}_{2}\,{\bf q} ]$ rather than on q. If we restrict our considerations to the symmetry operations of the space group G(q) of the wavevector q that leave the wavevector invariant modulo some reciprocal-lattice vector g(q, S), $[{\bf S}\, {\bf q}= {\bf q}- {\bf g}({\bf q},{\bf S}), \eqno (2.1.3.16) ]$ then equation (2.1.3.13) provides an ordinary (3N-dimensional) representation of this symmetry group.⁵ In the following, we denote the elements of the subgroup G(q) by $[ \{{{\bf R} | {{\bf v}({\bf R}) + {{\bf x}(m)} \}} } ]$ . The corresponding unitary and Hermitian transformation matrices can be reduced to $[\eqalignno{ &\Gamma _{{K}\kappa }^{\alpha \mu }({\bf q},\{{{\bf R}| {{\bf v}({\bf R})} + {\bf x}(m )}\}) &\cr &\quad= R_{\alpha \mu } \delta \left({\kappa, F_o^{- 1}({K}\semi{\bf R})}\right)& \cr &\quad \times \exp [{i{\bf q}\, ({{\bf R}^{- 1} \, {\bf r}_{K}^o - {\bf R}^{- 1} \, {\bf v}({\bf R}) - {\bf R}^{- 1} \, {\bf x}(m) - {\bf r}_\kappa ^o })}] &\cr &\quad= R_{\alpha \mu } \delta\left({\kappa, F_o^{- 1}({K}\semi{\bf R})}\right) &\cr &\quad \times \exp [{i{\bf R}\, {\bf q}\, ({{\bf r}_{K}^o - \{{{\bf R}| {{\bf v}({\bf R}) + {\bf x}(m)}}\}\, {\bf r}_\kappa ^o })}] &\cr &\quad= R_{\alpha \mu } \delta \left({\kappa, F_o^{- 1}({K}\semi{\bf R})}\right) &\cr &\quad \times \exp [{i{\bf q}\, ({{\bf r}_{K}^o -\{{{\bf R}| {{\bf v}({\bf R}) + {\bf x}(m)}}\}\, {\bf r}_\kappa ^o })}]. &\cr&&(2.1.3.17)} ]$ According to equation (2.1.3.14), they commute with the dynamical matrix: $[\boldGamma ({{\bf q},\{{{\bf R}| {{\bf v}({\bf R}) + {\bf x}(m )}}\}}) \, {\bf D}({\bf q}) \, \boldGamma ^{- 1}({{\bf q},\{{{\bf R}| {{\bf v}({\bf R}) + {\bf x}(m )}}\}}) = {\bf D}({\bf q}). \eqno (2.1.3.18) ]$ This relation contains the symmetry constraints for the dynamical matrix . The independent elements of D(q) may be obtained by application of equation (2.1.3.18) for every operation of the space group of the wavevector.

Another approach to the symmetry reduction of the dynamical matrix is based on group-theoretical considerations making use of the well known irreducible representations of symmetry groups. It is especially useful for the prediction of the form of eigenvectors and the investigation of degeneracies. Following the treatment of Maradudin & Vosko (1968), we consider the purely rotational elements of the space group G(q) that form the point group of the wavevector $[G_o({\bf q})=\{{\bf R}\}]$ . According to equation (2.1.3.17), we associate a matrix operator $[{\bf T}({{\bf q},{\bf R}}) = \exp [{i{\bf q}\, ({{\bf v}({\bf R}) + {\bf x}(m)})}] \, \boldGamma ({{\bf q},\{{{\bf R}| {{\bf v}({\bf R}) + {\bf x}(m )}}\}}) \eqno (2.1.3.19) ]$ to each of the elements of G_o(q). These matrix operators are uniquely determined by the rotations R and do not depend on the translational parts of the space-group operation $[\{{{\bf R}| {{\bf v}({\bf R}) + {{\bf x}(m)}\}}}]$ , as proven by inspection of the individual matrix elements: $[T_{{K}\kappa }^{\alpha \mu }({\bf q},{\bf R}) = R_{\alpha \mu } \delta \left({\kappa, F_o^{- 1}({K},{\bf R})}\right) \exp [{i{\bf q}\, ({{\bf r}_{K}^o - {\bf R}\, {\bf r}_\kappa ^o })}].\eqno (2.1.3.19a) ]$ These T matrices again commute with the dynamical matrix, $[{\bf T}({{\bf q},{\bf R}}) \, {\bf D}({\bf q}) \, {\bf T}^{- 1}({{\bf q},{\bf R}}) = {\bf D}({\bf q}), \eqno (2.1.3.20) ]$ but in contrast to the $[\boldGamma]$ matrices they do not provide an ordinary representation of the group G_o(q). For the multiplication of two symmetry elements R_i and R_j the following relation holds: $[\eqalignno{&{\bf T}({\bf q},{\bf R}_i) \, {\bf T}({\bf q},{\bf R}_j) &\cr&\quad= \exp [{i({{\bf q}- {\bf R}_i ^{- 1} \, {\bf q}})({{\bf v}({{\bf R}_j }) + {\bf x}({m_j })})}] \, {\bf T}({{\bf q},{\bf R}_i \circ {\bf R}_j }).&\cr&&(2.1.3.21)} ]$ According to equation (2.1.3.16), $[{\bf q}- {\bf R}_{\bf i}^{- {\bf 1}} \, {\bf q} ]$ is a reciprocal-lattice vector $[{\bf g}({{\bf q},{\bf R}_i^{- 1}}) ]$ and hence $[\eqalignno{{\bf T}({{\bf q},{\bf R}_i }) \, {\bf T}({{\bf q},{\bf R}_j }) &= \exp [{i{\bf g}({{\bf q},{\bf R}_i^{- 1}}) \, {\bf v}({{\bf R}_j })}] \, {\bf T}({{\bf q},{\bf R}_i \circ {\bf R}_j }) &\cr&= \varphi ({\bf q},{\bf R}_i, {\bf R}_j) \, {\bf T}({{\bf q},{\bf R}_i \circ {\bf R}_j }).&\cr&&(2.1.3.21a)} ]$ Thus, the T matrices provide not a normal but a multiplier representation of the group G_o(q). The phase factor on the right-hand side of equation (2.1.3.21a) is the complex multiplier characteristic for the (ordered) product of symmetry operations.

For wavevectors within the first Brillouin zone, the reciprocal-lattice vectors $[{\bf g}({{\bf q},{\bf R}_i^{- 1}})]$ are identically zero (see last footnote⁵) and the T representation is an ordinary one. The same is true if none of the symmetry elements of G_o(q) contains a fractional translation, i.e. for symmorphic space groups. Therefore, multipliers have to be taken into account only if nonsymmorphic space groups and wavevectors on the Brillouin-zone boundary are considered.

There are some other restrictions for the dynamical matrix arising from the fact that inverting the wavevector is equivalent to taking the complex conjugate dynamical matrix [c.f. equation (2.1.2.24)]: $[{\bf D}^*({\bf q}) = {\bf D}({- {\bf q}}). \eqno (2.1.3.22) ]$ Hence it is useful to extend our discussion to those symmetry operations that invert the phonon wavevector. Let us assume that the space group of the crystal contains an element $[\{{{\bf S}_ - | {{\bf v}({{\bf S}_ - })}}\} ]$ with $[{\bf S}_ - \, {\bf q}= - {\bf q}. \eqno (2.1.3.23) ]$ Using equation (2.1.3.14) we obtain $[{\bf D}({- {\bf q}}) = {\bf D}^*({\bf q}) = \boldGamma ({{\bf q},\{{{\bf S}_ - | {{\bf v}({{\bf S}_ - })}}\}}) \, {\bf D}({\bf q}) \, \boldGamma ^ + ({{\bf q},\{{{\bf S}_ - | {{\bf v}({{\bf S}_ - })}}\}}). \eqno (2.1.3.24) ]$ In order to provide a consistent description, we introduce an anti-unitary operator $[{\bf K}_{o}]$ which transforms an arbitrary vector $[{\boldPsi}]$ into its complex conjugate counterpart $[\boldPsi^* ]$ $[{\bf K} _o\, \boldPsi = \boldPsi^*.\eqno (2.1.3.25) ]$ Obviously, $[{\bf K}_o]$ does not commute with the dynamical matrix but exhibits the following transformation behaviour: $[{\bf K}_o \, {\bf D}({\bf q})\, {\bf K}_o = {\bf D}^*({\bf q}). \eqno (2.1.3.26) ]$ On the other hand, we infer from equation (2.1.3.24) that $[\eqalignno{ &{\bf K}_o \,\boldGamma ({{\bf q},\{{{\bf S}_ - | {{\bf v}({{\bf S}_ - })}}\}}) \, {\bf D}({\bf q}) \, \boldGamma ^ + ({{\bf q},\{{{\bf S}_ - | {{\bf v}({{\bf S}_ - })}}\}}){\bf K}_o &\cr&\quad= {\bf K}_o \,{\bf D}({- {\bf q}})\,{\bf K}_o & \cr &\quad= {\bf D}^*({- {\bf q}}) &\cr &\quad= {\bf D}({\bf q}), &\cr&& (2.1.3.27)} ]$ which provide the additional constraints for the dynamical matrix. In component form, this last relation can be written explicitly as $[ \eqalignno{&[{\exp({- i{\bf qr}_{K}}) \, D_{{K}{K}'}^{\mu \nu }({\bf q}) \, \exp({i{\bf qr}_{{K}'}}})]^* \cr&\quad= \textstyle\sum\limits_{\alpha \beta }{({S_ - })_{\mu \alpha }} \, [{\exp({- i{\bf qr}_\kappa }) \, D_{\kappa \kappa '}^{\alpha \beta }({\bf q}) \, \exp({i{\bf qr}_{\kappa '}}})] \, ({S_ - })_{\nu \beta } &\cr&& (2.1.3.28)} ]$ if particles ( $[\kappa l]$ ) and ( $[\kappa' l']$ ) are sent into ( [{K}L] ) and ( [{K}'L'] ) by the symmetry operation $[\{{{\bf S}_ - | {{\bf v}({{\bf S}_ - })}}\} ]$ , respectively.

If $[{\bf S}_{-}]$ represents the inversion $[\left(({S_ - })_{\alpha \beta } = - \delta _{\alpha \beta }\right) ]$ , in particular, then (2.1.3.28) reduces to $[{& [{\exp({- i{\bf qr}_{K}}) D_{{K}{K}'}^{\mu \nu }({\bf q}) \exp({i{\bf qr}_{{K}'}}})]^* = [{\exp({- i{\bf qr}_\kappa }) D_{\kappa \kappa '}^{\mu \nu }({\bf q}) \exp({i{\bf qr}_{\kappa '}}})]. \eqno(2.1.3.29)} ]$ Moreover, if every atom is itself a centre of inversion (e.g. the NaCl structure) ( $[{K}= \kappa]$ and $[{K}' = \kappa ' ]$ ), the matrix C(q) defined by $[C_{\kappa \kappa '}^{\alpha \beta }({\bf q}) = \exp({- i{\bf qr}_\kappa }) \, D_{\kappa \kappa '}^{\alpha \beta }({\bf q}) \,\exp({i{\bf qr}_{\kappa '}}) \eqno (2.1.3.30) ]$ is a real and symmetric matrix with real eigenvectors for arbitrary wavevectors q.

In terms of group theory we proceed as follows: We add to the space group of the wavevector G(q) the elements of the coset $[\{{{\bf S}_ - | {{\bf v}({{\bf S}_ - })}}\}\circ G({\bf q}) ]$ .⁶ This will result in a new space group which we call $[G({\bf q},-{\bf q})]$ . If instead of the matrix operator $[\boldGamma({\bf q},\{{\bf S}|{\bf v}({\bf S})+{\bf x}(m)\})]$ the anti-unitary operator $[{\bf K}_{o}\, {\boldGamma}({\bf q},\{ {\bf S}_{-}|{\bf v}({\bf S}_{-})+{\bf x}(m)\}) ]$ is assigned to those symmetry operations that invert the wavevector, then a representation of the whole group $[G({\bf q},-{\bf q}) ]$ is provided. Moreover, all these matrix operators commute with the dynamical matrix.

As before, let us restrict ourselves to the rotational parts of the symmetry operations. The point group of the wavevector $[G{ _o}({\bf q})]$ is enlarged by the coset $[{\bf S}_ - \circ G_o ({\bf q})]$ yielding the group $[G_{o}({\bf q}, -{\bf q})]$ . In analogy to equation (2.1.3.19), the elements of the left coset will be represented by the matrix operator $[\eqalignno{ {\bf T}({{\bf q},{\bf S}_ - \circ {\bf R}}) &= {\bf K}_{o} \, \exp [{- i{\bf q}\, ({{\bf v}({{\bf S}_ - \circ {\bf R}}) + {\bf x}(m)})}] &\cr&\quad\times \boldGamma ({{\bf q},\{{{\bf S}_ - \circ {\bf R}| {{\bf v}({{\bf S}_ - \circ {\bf R}}) + {\bf x}(m )}}\}})& \cr &= \exp [{i{\bf q}\, ({{\bf v}({{\bf S}_ - \circ {\bf R}}) + {\bf x}(m)})}] &\cr&\quad\times {\bf K}_{ o}\boldGamma ({{\bf q},\{{{\bf S}_ - \circ {\bf R}| {{\bf v}({{\bf S}_ - \circ {\bf R}}) + {\bf x}(m )}}\}}). &\cr&& (2.1.3.31)} ]$

The T matrix operators provide a multiplier corepresentation. The multipliers are not uniquely defined as in equation (2.1.3.21a). Rather, the definition depends on the type and the order of the symmetry operations involved. In order to distinguish between the different kinds of symmetry operations, we introduce the following notation:

$[\bar{\bf R}\in G_o ({{\bf q}, - {\bf q}}) ]$ is an arbitrary element of the point group.
$[{\bf R}\in G_o ({\bf q})]$ is an element of the point group of the wavevector $[G_{o}({\bf q})]$ which is a subgroup of $[G_{o}({{\bf q}, - {\bf q}})]$ . This element is represented by an unitary matrix operator.
$[{\bf A}\in {\bf S}_ - \circ G_o ({\bf q}) ]$ is an element of the coset $[{\bf S}_ - \circ G_o ({\bf q})]$ , represented by an anti-unitary operator.

The multiplication rule $[{\bf T}({\bf q},{\bar{\bf R}}_i ) \, {\bf T}({\bf q},{\bar{\bf R}}_j ) = \varphi ({\bf q},{\bar{\bf R}}_i, {\bar{\bf R}}_j ) \, {\bf T}({\bf q},{\bar {\bf R}}_i \circ {\bar{\bf R}}_j ) \eqno (2.1.3.32) ]$ is determined by the multipliers $[\eqalignno{\varphi ({{\bf q},{\bf R}_{i},{\bar{\bf R}}_{j}}) &= \exp[{i({\bf q}- {\bf R}_{i}^{- 1} \, {\bf q}) \, {\bf v}({\bar{\bf R}}_{j})}] & \cr\varphi ({{\bf q},{\bf A}_{i},{\bar{\bf R}}_{j}}) &= \exp[{- i({\bf q}+ {\bf A}_{i}^{- 1} \, {\bf q}) \, {\bf v}({\bar{\bf R}}_{j})}]. & (2.1.3.33)} ]$ Again, this representation reduces to an ordinary representation either for symmorphic space groups [all $[{\bf v}({\bar {\bf R}}_i) = {\bf 0}]$ ] or for wavevectors within the interior of the Brillouin zone.

All matrix operators of the T representation commute with the dynamical matrix. Hence, they may be used for the determination of independent elements of the dynamical matrix as well as for the determination of the form of eigenvectors compatible with the atomic structure.

2.1.3.1.1. Example

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As an example, we consider a crystal of tetragonal symmetry, space group [P4mm] , with lattice parameters a and c. The primitive cell spanned by the three mutually orthogonal vectors a, b and c contains ten atoms at the positions listed in Table 2.1.3.1 and shown in Fig. 2.1.3.4. Consequently, the dynamical matrix has $[30\times30]$ elements.

Table 2.1.3.1 | top | pdf |
Example structure in space group [P4mm]

Atom No.	x	y	z
1	0	0	0
2	0.5	0.5	0.6
3	0.2	0.1	0
4	0.8	0.9	0
5	0.9	0.8	0
6	0.1	0.8	0
7	0.2	0.9	0
8	0.8	0.1	0
9	0.9	0.2	0
10	0.1	0.2	0

Figure 2.1.3.4 | top | pdf |

Projection along the tetragonal z axis of the example structure given in Table 2.1.3.1.

The space group [P4mm] contains eight symmetry operations, namely

(1) the identity, denoted E;
(2) a 90° rotation around the z axis, denoted $[D_{90}^z]$ ;
(3) a 180° rotation around the z axis, denoted $[D^{z}_{180}]$ ;
(4) a 270° rotation around the z axis, denoted $[D^{z}_{270}]$ ;
(5) a mirror plane normal to the x axis, denoted $[{m}_{x}]$ ;
(6) a mirror plane normal to the y axis, denoted $[{m}_{y}]$ ;
(7) a mirror plane normal to the $[ [{\bar 110}] ]$ axis, denoted $[m_{ [{\bar 110}]}]$ ; and
(8) a mirror plane normal to the axis, denoted $[m_{ [{110}]}]$ .

Obviously, atoms No. 3 to 10 are chemically identical and have the same mass.

For the reduction of the dynamical matrix, we need the function $[F_{o}(\kappa,{\bf S}) ]$ , yielding the label of that atom into which κ is sent by the symmetry operation S. This function can be represented by the atom transformations shown in Table 2.1.3.2. This table displays the labels of atoms κ and K related by a particular symmetry operation and also the relative position $[{\bf r}_{l}-{\bf r}_{L}]$ of the primitive cells l and L where both atoms are located. This information is needed for the calculation of phase factors in the expression for the matrix operators T. Via the twofold axis, atom 6, for example, is transformed into atom 9 located within the cell which is shifted by the vector $[-{\bf a}-{\bf b}]$ .

Table 2.1.3.2 | top | pdf |
Atom transformation table

	Symmetry operation
Atom No.	E	$[D_{90}^z]$	$[D_{180}^z]$	$[D_{270}^z ]$	$[m_{x}]$	$[m_{y}]$	$[m_{ [{\bar 110}]}]$	$[m_{ [{110}]}]$
1	1
2	2	$[2-{\bf a} ]$	$[2-{\bf a}-{\bf b} ]$	$[2-{\bf b} ]$	$[2-{\bf a}]$	$[2-{\bf b} ]$	2	$[2-{\bf a}-{\bf b} ]$
3	3	$[9-{\bf a}]$	$[4-{\bf a}-{\bf b} ]$	$[6-{\bf b} ]$	$[8-{\bf a} ]$	$[7-{\bf b} ]$	10	$[5-{\bf a}-{\bf b} ]$
4	4	$[6-{\bf a} ]$	$[3-{\bf a}-{\bf b} ]$	$[9-{\bf b} ]$	$[7-{\bf a} ]$	$[8-{\bf b} ]$	5	$[10-{\bf a}-{\bf b} ]$
5	5	$[7-{\bf a} ]$	$[10-{\bf a}-{\bf b} ]$	$[8-{\bf b} ]$	$[6-{\bf a} ]$	$[9-{\bf b} ]$	4	$[3-{\bf a}-{\bf b} ]$
6	6	$[3-{\bf a} ]$	$[9-{\bf a}-{\bf b} ]$	$[4-{\bf b} ]$	$[5-{\bf a} ]$	$[10-{\bf b} ]$	8	$[7-{\bf a}-{\bf b} ]$
7	7	$[10-{\bf a} ]$	$[8-{\bf a}-{\bf b} ]$	$[5-{\bf b} ]$	$[4-{\bf a} ]$	$[3-{\bf b} ]$	9	$[6-{\bf a}-{\bf b} ]$
8	8	$[5-{\bf a} ]$	$[7-{\bf a}-{\bf b} ]$	$[10-{\bf b} ]$	$[3-{\bf a}]$	$[4-{\bf b} ]$	6	$[9-{\bf a}-{\bf b} ]$
9	9	$[4-{\bf a} ]$	$[6-{\bf a}-{\bf b} ]$	$[3-{\bf b}]$	$[10-{\bf a} ]$	$[5-{\bf b} ]$	7	$[8-{\bf a}-{\bf b} ]$
10	10	$[8-{\bf a} ]$	$[5-{\bf a}-{\bf b} ]$	$[7-{\bf b} ]$	$[9-{\bf a} ]$	$[6-{\bf b} ]$	3	$[4-{\bf a}-{\bf b} ]$

Let us first consider the case of phonons with infinite wavelengths and, hence, the symmetry reduction of the dynamical matrix at zero wavevector (the Γ point). Here, the point group of the wavevector is equivalent to the point group [4mm] of the lattice. According to equation (2.1.3.19a), we can immediately write down the transformation matrix for any of these symmetry operations. Using the notation $[\matrix{ {\bf E} = {\pmatrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \cr }}\hfill& {\bf D}_{90}^z = {\pmatrix{ 0 & {- 1}& 0 \cr 1 & 0 & 0 \cr 0 & 0 & 1 \cr }}\hfill\cr {\bf D}_{180}^z = {\pmatrix{ {- 1}& 0 & 0 \cr 0 & {- 1}& 0 \cr 0 & 0 & 1 \cr }}\hfill&{\bf D}_{270}^z = {\pmatrix{ 0 & 1 & 0 \cr {- 1}& 0 & 0 \cr 0 & 0 & 1 \cr}}\hfill \cr {\bf m}_x = {\pmatrix{ {- 1}& 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \cr}}\hfill& {\bf m}_y = {\pmatrix{ 1 & 0 & 0 \cr 0 & {- 1}& 0 \cr 0 & 0 & 1 \cr}}\hfill\cr {\bf m}_{ [{\bar 110}]} = {\pmatrix{ 0 & 1 & 0 \cr1 & 0 & 0 \cr 0 & 0 & 1 \cr }}\hfill& {\bf m}_{ [{110}]} = {\pmatrix{ 0 & {- 1}& 0 \cr {- 1}& 0 & 0 \cr 0 & 0 & 1 \cr }} \hfill\cr} ]$ for the three-dimensional vector representation of the symmetry elements, we obtain the T matrix operators $[{\bf T}({{\bf 0},{\bf E}}) = {\pmatrix{ {\bf E}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr 0 & {\bf E}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & {\bf E}& 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & {\bf E}& 0 & 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & {\bf E}& 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & {\bf E}& 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & {\bf E}& 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\bf E}& 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\bf E}& 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\bf E} \cr }}, ]$ $[{\bf T}({{\bf 0},{\bf D}_{90}^z }) = {\pmatrix{ {{\bf D}_{90}^z }& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr 0 & {{\bf D}_{90}^z }& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & {{\bf D}_{90}^z }& 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{\bf D}_{90}^z }& 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{\bf D}_{90}^z }& 0 & 0 \cr 0 & 0 & 0 & {{\bf D}_{90}^z }& 0 & 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & {{\bf D}_{90}^z }& 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{\bf D}_{90}^z } \cr 0 & 0 & {{\bf D}_{90}^z }& 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & {{\bf D}_{90}^z }& 0 & 0 & 0 \cr }}, ]$ $[{\bf T}({{\bf 0},{\bf D}_{180}^z }) = {\pmatrix{ {{\bf D}_{180}^z }& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr0 & {{\bf D}_{180}^z }& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & {{\bf D}_{180}^z }& 0 & 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & {{\bf D}_{180}^z }& 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{\bf D}_{180}^z } \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{\bf D}_{180}^z }& 0 \cr0 & 0 & 0 & 0 & 0 & 0 & 0 & {{\bf D}_{180}^z }& 0 & 0 \cr0 & 0 & 0 & 0 & 0 & 0 & {{\bf D}_{180}^z }& 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & {{\bf D}_{180}^z }& 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & {{\bf D}_{180}^z }& 0 & 0 & 0 & 0 & 0 \cr}}, ]$ $[{\bf T}({{\bf 0},{\bf D}_{270}^z }) = {\pmatrix{ {{\bf D}_{270}^z }& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr 0 & {{\bf D}_{270}^z }& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{\bf D}_{270}^z }& 0 \cr 0 & 0 & 0 & 0 & 0 & {{\bf D}_{270}^z }& 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & {{\bf D}_{270}^z }& 0 & 0 & 0 \cr0 & 0 & {{\bf D}_{270}^z }& 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{\bf D}_{270}^z } \cr 0 & 0 & 0 & 0 & {{\bf D}_{270}^z }& 0 & 0 & 0 & 0 & 0 \cr0 & 0 & 0 & {{\bf D}_{270}^z }& 0 & 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{\bf D}_{270}^z }& 0 & 0 \cr}}, ]$ $[{\bf T}({{\bf 0},{\bf m}_x }) = {\pmatrix{ {{\bf m}_x }& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr 0 & {{\bf m}_x }& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{\bf m}_x }& 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & {{\bf m}_x }& 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & {{\bf m}_x }& 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & {{\bf m}_x }& 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & {{\bf m}_x }& 0 & 0 & 0 & 0 & 0 & 0 \cr0 & 0 & {{\bf m}_x }& 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{\bf m}_x } \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{\bf m}_x }& 0 \cr }}, ]$ $[{\bf T}({{\bf 0},{\bf m}_y }) = {\pmatrix{{{\bf m}_y }& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr 0 & {{\bf m}_y }& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & {{\bf m}_y }& 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{\bf m}_y }& 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{\bf m}_y }& 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{\bf m}_y } \cr 0 & 0 & {{\bf m}_y }& 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & {{\bf m}_y }& 0 & 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & {{\bf m}_y }& 0 & 0 & 0 & 0 & 0 \cr0 & 0 & 0 & 0 & 0 & {{\bf m}_y }& 0 & 0 & 0 & 0 \cr}}, ]$ $[\displaylines{{\bf T}({{\bf 0},{\bf m}_{ [{\bar 110}]}}) =\hfill\cr \left({\matrix{{{\bf m}_{ [{\bar 110}]}}& 0 & 0 & 0 & 0 &0 & 0 & 0 & 0 & 0 \cr 0 & {{\bf m}_{ [{\bar 110}]}}& 0 & 0 & 0 &0 & 0 & 0 & 0 & 0 \cr0 & 0 & 0 & 0 & 0 &0 & 0 & 0 & 0 & {{\bf m}_{ [{\bar 110}]}} \cr 0 & 0 & 0 & 0 & {{\bf m}_{ [{\bar 110}]}}&0 & 0 & 0 & 0 & 0\cr 0 & 0 & 0 & {{\bf m}_{ [{\bar 110}]}}& 0 &0 & 0 & 0 & 0 & 0\cr0 & 0 & 0 & 0 & 0 &0 & 0 & {{\bf m}_{ [{\bar 110}]}}& 0 & {{\bf m}_y } \cr 0 & 0 & 0 & 0 & 0 &0 & 0 & 0 & {{\bf m}_{ [{\bar 110}]}}& 0\cr 0 & 0 & 0 & 0 & 0 &{{\bf m}_{ [{\bar 110}]}}& 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 &0 & {{\bf m}_{ [{\bar 110}]}}& 0 & 0 & 0 \cr 0 & 0 & {{\bf m}_{ [{\bar 110}]}}& 0 & 0 &0 & 0 & 0 & 0 & 0 \cr }}\right)}]$ and $[\displaylines{{\bf T}({{\bf 0},{\bf m}_{ [{110}]}}) =\hfill\cr \left({\matrix{ {{\bf m}_{ [{110}]}}& 0 & 0 & 0 & 0 &0 & 0 & 0 & 0 & 0\cr 0 & {{\bf m}_{ [{110}]}}& 0 & 0 & 0 &0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & {{\bf m}_{ [{110}]}}&0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 &0 & 0 & 0 & 0 & {{\bf m}_{ [{110}]}}\cr 0 & 0 & {{\bf m}_{ [{110}]}}& 0 & 0 &0 & 0 & 0 & 0 & 0 \cr0 & 0 & 0 & 0 & 0 &0 & {{\bf m}_{ [{110}]}}& 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 &{{\bf m}_{ [{110}]}}& 0 & 0 & 0 & 0\cr 0 & 0 & 0 & 0 & 0 &0 & 0 & 0 & {{\bf m}_{ [{110}]}}& 0 \cr 0 & 0 & 0 & 0 & 0 &0 & 0 & {{\bf m}_{ [{110}]}}& 0 & 0\cr 0 & 0 & 0 & {{\bf m}_{ [{110}]}}& 0 &0 & 0 & 0 & 0 & 0\cr }}\right).}]$ Since each of these matrices commutes with the dynamical matrix ( $[{\bf T}^{- 1} \, {\bf D}\, {\bf T}= {\bf D}]$ , with $[{\bf T}^{- 1} = {\bf T}^{T}]$ ), the following relations are obtained for the $[{\bf D}_{\kappa\kappa'}({\bf 0}) ]$ submatrices: $[\eqalign{ {\bf D}_{{11}}({\bf 0}) &= {\pmatrix{ {D_{11}^{11}}& 0 & 0 \cr 0 & {D_{11}^{11}}& 0 \cr 0 & 0 & {D_{11}^{33}} \cr }},\cr {\bf D}_{22}({\bf 0}) &= {\pmatrix{ {D_{22}^{11}}& 0 & 0 \cr 0 & {D_{22}^{11}}& 0 \cr 0 & 0 & {D_{22}^{33}} \cr }}, \cr {\bf D}_{13}({\bf 0}) &= {\bf D}_{{270}}^z \, {\bf D}_{16}({\bf 0}) \, {\bf D}_{90}^z = {\pmatrix{ {D_{16}^{22}}& {- D_{16}^{12}}& {- D_{16}^{23}} \cr {- D_{16}^{12}}& {D_{16}^{11}}& {D_{16}^{13}} \cr {- D_{16}^{23}}& {D_{16}^{13}}& {D_{16}^{33}} \cr }},\cr &= {\bf D}_{180}^z \, {\bf D}_{14}({\bf 0}) \, {\bf D}_{180}^z = {\pmatrix{ {D_{14}^{11}}& {D_{14}^{12}}& {- D_{14}^{13}} \cr {D_{14}^{12}}& {D_{14}^{22}}& {- D_{14}^{23}} \cr {- D_{14}^{13}}& {- D_{14}^{23}}& {D_{14}^{33}} \cr }}, \cr &= {\bf D}_{90}^z \, {\bf D}_{19}({\bf 0}) \, {\bf D}_{270}^z = {\pmatrix{ {D_{19}^{22}}& {- D_{19}^{12}}& {D_{19}^{23}} \cr {- D_{19}^{12}}& {D_{19}^{11}}& {- D_{19}^{13}} \cr {D_{19}^{23}}& {- D_{19}^{13}}& {D_{19}^{33}} \cr }}, \cr &= {\bf m}_x \, {\bf D}_{18}({\bf 0}) \, {\bf m}_x = {\pmatrix{ {D_{18}^{11}}& {- D_{18}^{12}}& {- D_{18}^{13}} \cr {- D_{18}^{12}}& {D_{18}^{22}}& {D_{18}^{23}} \cr {- D_{18}^{13}}& {D_{18}^{23}}& {D_{18}^{33}} \cr }}, \cr &= {\bf m}_y \, {\bf D}_{17}({\bf 0}) \, {\bf m}_y = {\pmatrix{ {D_{17}^{11}}& {- D_{17}^{12}}& {D_{17}^{13}} \cr {- D_{17}^{12}}& {D_{17}^{22}}& {- D_{17}^{23}} \cr {D_{17}^{13}}& {- D_{17}^{23}}& {D_{17}^{33}} \cr }}, \cr &= {\bf m}_{ [{\bar 110}]} \, {\bf D}_{1,10}({\bf 0}) \, {\bf m}_{ [{\bar 110}]} = {\pmatrix{ {D_{1,10}^{22}}& {D_{1,10}^{12}}& {D_{1,10}^{23}} \cr {D_{1,10}^{12}}& {D_{1,10}^{11}}& {D_{1,10}^{13}} \cr {D_{1,10}^{23}}& {D_{1,10}^{13}}& {D_{1,10}^{33}} \cr }}, \cr &= {\bf m}_{ [{110}]} \, {\bf D}_{15}({\bf 0}) \, {\bf m}_{ [{110}]} = {\pmatrix{ {D_{15}^{22}}& {D_{15}^{12}}& {- D_{15}^{23}} \cr {D_{15}^{12}}& {D_{15}^{11}}& {- D_{15}^{13}} \cr {- D_{15}^{23}}& {- D_{15}^{13}}& {D_{15}^{33}} \cr }}, \cr} ]$ and so on for the other submatrices.

For nonzero wavevectors q along $[{\bf a}^*]$ ( $[{\bf q}=h{\bf a}^*]$ ), the point group $[G_{o}({\bf q})]$ contains the identity and the mirror plane $[m_{y}]$ only. The respective T matrix operators are the same as for the Γ point: $[{\bf T}({\bf q},{\bf m}_{y}) = {\bf T}({\bf 0},{\bf m}_{y}).]$ There are, however, symmetry elements that invert the wavevector, namely $[D_{{180}}^z]$ and $[m_{x}]$ . Hence the enlarged group $[G_{o}({\bf q},-{\bf q}) ]$ consists of the elements E, $[m_{y}]$ , $[m_{x}]$ and $[D_{{180}}^z]$ . Inspection of the atom transformation table yields the remaining matrix operators: $[\displaylines{{\bf T}(h{\bf a}^*,{\bf D}_{180}^z ) =\hfill\cr \exp(- 2\pi ih)\left({\matrix{ {{\bf D}_{180}^z \exp(2\pi ih)}&\!\! 0 &\!\! 0 &\!\! 0 &\!\! 0 &\!\!0 &\!\! 0 &\!\! 0 &\!\! 0 &\!\! 0\cr 0 &\!\! {{\bf D}_{180}^z }&\!\! 0 &\!\! 0 &\!\! 0 &\!\!0 &\!\! 0 &\!\! 0 &\!\! 0 &\!\! 0\cr 0 &\!\! 0 &\!\! 0 &\!\! {{\bf D}_{180}^z }&\!\! 0 &\!\!0 &\!\! 0 &\!\! 0 &\!\! 0 &\!\! 0 \cr 0 &\!\! 0 &\!\! {{\bf D}_{180}^z }&\!\! 0 &\!\! 0 &\!\!0 &\!\! 0 &\!\! 0 &\!\! 0 &\!\! 0 \cr 0 &\!\! 0 &\!\! 0 &\!\! 0 &\!\! 0 &\!\!0 &\!\! 0 &\!\! 0 &\!\! 0 &\!\! {{\bf D}_{180}^z } \cr0 &\!\! 0 &\!\! 0 &\!\! 0 &\!\! 0 &\!\!0 &\!\! 0 &\!\! 0 &\!\! {{\bf D}_{180}^z }&\!\! 0\cr0 &\!\! 0 &\!\! 0 &\!\! 0 &\!\! 0 &\!\!0 &\!\! 0 &\!\! {{\bf D}_{180}^z }&\!\! 0 &\!\! 0 \cr0 &\!\! 0 &\!\! 0 &\!\! 0 &\! \!0 &\!\!0 &\! \!{{\bf D}_{180}^z }&\! \!0 &\!\! 0 &\!\! 0 \cr0 &\! \!0 &\!\! 0 &\! \!0 &\!\! 0 &\!\!{{\bf D}_{180}^z }&\!\! 0 &\!\! 0 &\!\! 0 &\!\! 0 \cr 0 &\! \!0 &\! \!0 &\!\! 0 &\!\! {{\bf D}_{180}^z }&\!\!0 &\!\! 0 &\!\! 0 &\!\! 0 &\!\! 0 \cr}}\right) {\bf K}_o }]$ and $[\displaylines{{\bf T}({h{\bf a}^*,{\bf m}_x }) =\hfill\cr \exp(- 2\pi ih)\left({\matrix{{{\bf m}_x \exp({2\pi ih})}& 0 & 0 & 0 & 0 &0 & 0 & 0 & 0 & 0 \cr 0 & {{\bf m}_x }& 0 & 0 & 0 &0 & 0 & 0 & 0 & 0\cr 0 & 0 & 0 & 0 & 0 &0 & 0 & {{\bf m}_x }& 0 & 0 \cr0 & 0 & 0 & 0 & 0 &0 & {{\bf m}_x }& 0 & 0 & 0 \cr0 & 0 & 0 & 0 & 0 &{{\bf m}_x }& 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & {{\bf m}_x }&0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & {{\bf m}_x }& 0 &0 & 0 & 0 & 0 & 0 \cr 0 & 0 & {{\bf m}_x }& 0 & 0 &0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 &0 & 0 & 0 & 0 & {{\bf m}_x } \cr0 & 0 & 0 & 0 & 0 &0 & 0 & 0 & {{\bf m}_x }& 0 \cr }}\right) {\bf K}_{o}. }]$ Being anti-unitary, the corresponding inverse operators are⁷ $[\eqalign{ {\bf T}^{- 1}(h{\bf a}^*,{\bf D}_{{180}}^z) &= {\bf K}_{o} \, {\bf T}^ + (h{\bf a}^*,{\bf D}_{{180}}^z), \cr {\bf T}^{-1}(h{\bf a}^*,{\bf m}_x) &= {\bf K}_{o} \, {\bf T}^ + (h{\bf a}^*,{\bf m}_x). \cr} ]$ The invariance of the dynamical matrix with respect to the similarity transformation ( $[{\bf T}^{-1} \, {\bf D}\, {\bf T}= {\bf D}]$ ) using any of these operators leads to the following relations for wavevectors along $[{\bf a}^*]$ :⁸ $[\eqalign{ & {\rm For}\,\,{\bf \bar R}= {\bf m}_{y}: \cr & {\pmatrix{ {{\bf D}_{11}}& {{\bf D}_{12}}& {{\bf D}_{13}}& {{\bf D}_{14}}& {{\bf D}_{15}}& {{\bf D}_{16}}& {{\bf D}_{17}}& {{\bf D}_{18}}& {{\bf D}_{19}}& {{\bf D}_{1,10}} \cr {}& {{\bf D}_{22}}& {{\bf D}_{23}}& {{\bf D}_{24}}& {{\bf D}_{25}}& {{\bf D}_{26}}& {{\bf D}_{27}}& {{\bf D}_{28}}& {{\bf D}_{29}}& {{\bf D}_{2,10}} \cr {}& {}& {{\bf D}_{33}}& {{\bf D}_{34}}& {{\bf D}_{35}}& {{\bf D}_{36}}& {{\bf D}_{37}}& {{\bf D}_{38}}& {{\bf D}_{39}}& {{\bf D}_{3,10}} \cr {}& {}& {}& {{\bf D}_{44}}& {{\bf D}_{45}}& {{\bf D}_{46}}& {{\bf D}_{47}}& {{\bf D}_{48}}& {{\bf D}_{49}}& {{\bf D}_{4,10}} \cr {}& {}& {}& {}& {{\bf D}_{55}}& {{\bf D}_{56}}& {{\bf D}_{57}}& {{\bf D}_{58}}& {{\bf D}_{59}}& {{\bf D}_{5,10}} \cr {}& {}& {}& {}& {}& {{\bf D}_{66}}& {{\bf D}_{67}}& {{\bf D}_{68}}& {{\bf D}_{69}}& {{\bf D}_{6,10}} \cr {}& {}& {}& {}& {}& {}& {{\bf D}_{77}}& {{\bf D}_{78}}& {{\bf D}_{79}}& {{\bf D}_{7,10}} \cr {}& {}& {}& {}& {}& {}& {}& {{\bf D}_{88}}& {{\bf D}_{89}}& {{\bf D}_{8,10}} \cr {}& {}& {}& {}& {}& {}& {}& {}& {{\bf D}_{99}}& {{\bf D}_{9,10}} \cr {}& {}& {}& {}& {}& {}& {}& {}& {}& {{\bf D}_{10,10}} \cr }} \cr= &{\pmatrix{ {\tilde{\bf D}_{{\bf 11}}}& {\tilde{\bf D}_{{\bf 12}}}& {\tilde{\bf D}_{{\bf 1}7}}& {\tilde{\bf D}_{{\bf 1}8}}& {\tilde{\bf D}_{{\bf 1}9}}& {\tilde{\bf D}_{{\bf 1},10}}& {\tilde{\bf D}_{{\bf 1}3}}& {\tilde{\bf D}_{{\bf 1}4}}& {\tilde{\bf D}_{{\bf 1}5}}& {\tilde{\bf D}_{{\bf 1}6}} \cr {}& {\tilde{\bf D}_{{\bf 22}}}& {\tilde{\bf D}_{{\bf 2}7}}& {\tilde{\bf D}_{{\bf 2}8}}& {\tilde{\bf D}_{{\bf 2}9}}& {\tilde{\bf D}_{{\bf 2},10}}& {\tilde{\bf D}_{{\bf 2}3}}& {\tilde{\bf D}_{{\bf 2}4}}& {\tilde{\bf D}_{{\bf 2}5}}& {\tilde{\bf D}_{{\bf 2}6}} \cr {}& {}& {\tilde{\bf D}_{77}}& {\tilde{\bf D}_{78}}& {\tilde{\bf D}_{79}}& {\tilde{\bf D}_{7,10}}& {\tilde{\bf D}_{73}}& {\tilde{\bf D}_{74}}& {\tilde{\bf D}_{75}}& {\tilde{\bf D}_{76}} \cr {}& {}& {}& {\tilde{\bf D}_{88}}& {\tilde{\bf D}_{89}}& {\tilde{\bf D}_{8,10}}& {\tilde{\bf D}_{83}}& {\tilde{\bf D}_{84}}& {\tilde{\bf D}_{85}}& {\tilde{\bf D}_{86}} \cr {}& {}& {}& {}& {\tilde{\bf D}_{99}}& {\tilde{\bf D}_{9,10}}& {\tilde{\bf D}_{93}}& {\tilde{\bf D}_{94}}& {\tilde{\bf D}_{95}}& {\tilde{\bf D}_{96}} \cr {}& {}& {}& {}& {}& {\tilde{\bf D}_{10,10}}& {\tilde{\bf D}_{10,3}}& {\tilde{\bf D}_{10,4}}& {\tilde{\bf D}_{10,5}}& {\tilde{\bf D}_{10,6}} \cr {}& {}& {}& {}& {}& {}& {\tilde{\bf D}_{33}}& {\tilde{\bf D}_{34}}& {\tilde{\bf D}_{35}}& {\tilde{\bf D}_{36}} \cr {}& {}& {}& {}& {}& {}& {}& {\tilde{\bf D}_{44}}& {\tilde{\bf D}_{45}}& {\tilde{\bf D}_{46}} \cr {}& {}& {}& {}& {}& {}& {}& {}& {\tilde{\bf D}_{55}}& {\tilde{\bf D}_{56}} \cr {}& {}& {}& {}& {}& {}& {}& {}& {}& {\tilde{\bf D}_{66}} \cr }} \cr &{\rm with}\,\,\tilde{\bf D}_{kl} = {\bf m}_{ y} \, {\bf D}_{kl} \, {\bf m}_{ y} = {\pmatrix{ {D_{kl}^{11}}& {- D_{kl}^{12}}& {D_{kl}^{13}} \cr {- D_{kl}^{21}}& {D_{kl}^{22}}& {- D_{kl}^{23}} \cr {D_{kl}^{31}}& {- D_{kl}^{32}}& {D_{kl}^{33}} \cr }} \cr}, ]$ $[ \eqalign{ & {\rm for}\,\,{\bf \bar R}= {\bf D}_{180}^z: \cr & {\pmatrix{ {{\bf D}_{11}}& {{\bf D}_{12}}& {{\bf D}_{13}}& {{\bf D}_{14}}& {{\bf D}_{15}}& {{\bf D}_{16}}& {{\bf D}_{17}}& {{\bf D}_{18}}& {{\bf D}_{19}}& {{\bf D}_{1,10}} \cr {}& {{\bf D}_{22}}& {{\bf D}_{23}}& {{\bf D}_{24}}& {{\bf D}_{25}}& {{\bf D}_{26}}& {{\bf D}_{27}}& {{\bf D}_{28}}& {{\bf D}_{29}}& {{\bf D}_{2,10}} \cr {}& {}& {{\bf D}_{33}}& {{\bf D}_{34}}& {{\bf D}_{35}}& {{\bf D}_{36}}& {{\bf D}_{37}}& {{\bf D}_{38}}& {{\bf D}_{39}}& {{\bf D}_{3,10}} \cr {}& {}& {}& {{\bf D}_{44}}& {{\bf D}_{45}}& {{\bf D}_{46}}& {{\bf D}_{47}}& {{\bf D}_{48}}& {{\bf D}_{49}}& {{\bf D}_{4,10}} \cr {}& {}& {}& {}& {{\bf D}_{55}}& {{\bf D}_{56}}& {{\bf D}_{57}}& {{\bf D}_{58}}& {{\bf D}_{59}}& {{\bf D}_{5,10}} \cr {}& {}& {}& {}& {}& {{\bf D}_{66}}& {{\bf D}_{67}}& {{\bf D}_{68}}& {{\bf D}_{69}}& {{\bf D}_{6,10}} \cr {}& {}& {}& {}& {}& {}& {{\bf D}_{77}}& {{\bf D}_{78}}& {{\bf D}_{79}}& {{\bf D}_{7,10}} \cr {}& {}& {}& {}& {}& {}& {}& {{\bf D}_{88}}& {{\bf D}_{89}}& {{\bf D}_{8,10}} \cr {}& {}& {}& {}& {}& {}& {}& {}& {{\bf D}_{99}}& {{\bf D}_{9,10}} \cr {}& {}& {}& {}& {}& {}& {}& {}& {}& {{\bf D}_{10,10}} \cr }}\cr = &{\pmatrix{ {\tilde{\bf D}_{11}^* }& {\tilde{\bf D}_{12}^* }& {\tilde{\bf D}_{14}^* } & {\tilde{\bf D}_{13}^* }& {\tilde{\bf D}_{1,10}^* }& {\tilde{\bf D}_{19}^* }& {\tilde{\bf D}_{18}^* }& {\tilde{\bf D}_{17}^* }& {\tilde{\bf D}_{16}^* }& {\tilde{\bf D}_{15}^* } \cr {}& {\tilde{\bf D}_{22}^* }& {\tilde{\bf D}_{24}^* }& {\tilde{\bf D}_{23}^* }& {\tilde{\bf D}_{2,10}^* }& {\tilde{\bf D}_{29}^* }& {\tilde{\bf D}_{28}^* }& {\tilde{\bf D}_{27}^* } & {\tilde{\bf D}_{26}^* }& {\tilde{\bf D}_{25}^* } \cr {}& {}& {\tilde{\bf D}_{44}^* }& {\tilde{\bf D}_{43}^* }& {\tilde{\bf D}_{4,10}^* }& {\tilde{\bf D}_{49}^* }& {\tilde{\bf D}_{48}^* }& {\tilde{\bf D}_{47}^* }& {\tilde{\bf D}_{46}^* }& {\tilde{\bf D}_{45}^* } \cr {}& {}& {}& {\tilde{\bf D}_{33}^* }& {\tilde{\bf D}_{3,10}^* }& {\tilde{\bf D}_{39}^* }& {\tilde{\bf D}_{38}^* }& {\tilde{\bf D}_{37}^* }& {\tilde{\bf D}_{36}^* }& {\tilde{\bf D}_{35}^* } \cr {}& {}& {}& {}& {\tilde{\bf D}_{10,10}^* }& {\tilde{\bf D}_{10,9}^* }& {\tilde{\bf D}_{10,8}^* }& {\tilde{\bf D}_{10,7}^* }& {\tilde{\bf D}_{10,6}^* }& {\tilde{\bf D}_{10,5}^* } \cr {}& {}& {}& {}& {}& {\tilde{\bf D}_{99}^* }& {\tilde{\bf D}_{98}^* } & {\tilde{\bf D}_{97}^* }& {\tilde{\bf D}_{96}^* }& {\tilde{\bf D}_{95}^* } \cr {}& {}& {}& {}& {}& {}& {\tilde{\bf D}_{88}^* }& {\tilde{\bf D}_{87}^* }& {\tilde{\bf D}_{86}^* }& {\tilde{\bf D}_{85}^* } \cr {}& {}& {}& {}& {}& {}& {}& {\tilde{\bf D}_{77}^* }& {\tilde{\bf D}_{76}^* }& {\tilde{\bf D}_{75}^* } \cr {}& {}& {}& {}& {}& {}& {}& {}& {\tilde{\bf D}_{66}^* }& {\tilde{\bf D}_{65}^* } \cr {}& {}& {}& {}& {}& {}& {}& {}& {}& {\tilde{\bf D}_{55}^* } \cr }}\cr & {\rm with}\,\, \tilde{\bf D}_{kl}^{} = {\bf D}_{180}^z \, {\bf D}_{kl}^{} \, {\bf D}_{180}^z = {\pmatrix{ {D_{kl}^{11}}& {D_{kl}^{12}}& {- D_{kl}^{13}} \cr {D_{kl}^{21}}& {D_{kl}^{22}}& {- D_{kl}^{23}} \cr {- D_{kl}^{31}}& {- D_{kl}^{32}}& {D_{kl}^{33}} \cr }} \cr} ]$ and $[\eqalign{ & {\rm for}\,\,{\bf \bar R}= {\bf m}_x: \cr & {\pmatrix{ {{\bf D}_{11}}& {{\bf D}_{12}}& {{\bf D}_{13}}& {{\bf D}_{14}}& {{\bf D}_{15}}& {{\bf D}_{16}}& {{\bf D}_{17}}& {{\bf D}_{18}}& {{\bf D}_{19}}& {{\bf D}_{1,10}} \cr {}& {{\bf D}_{22}}& {{\bf D}_{23}}& {{\bf D}_{24}}& {{\bf D}_{25}}& {{\bf D}_{26}}& {{\bf D}_{27}}& {{\bf D}_{28}}& {{\bf D}_{29}}& {{\bf D}_{2,10}} \cr {}& {}& {{\bf D}_{33}}& {{\bf D}_{34}}& {{\bf D}_{35}}& {{\bf D}_{36}}& {{\bf D}_{37}}& {{\bf D}_{38}}& {{\bf D}_{39}}& {{\bf D}_{3,10}} \cr {}& {}& {}& {{\bf D}_{44}}& {{\bf D}_{45}}& {{\bf D}_{46}}& {{\bf D}_{47}}& {{\bf D}_{48}}& {{\bf D}_{49}}& {{\bf D}_{4,10}} \cr {}& {}& {}& {}& {{\bf D}_{55}}& {{\bf D}_{56}}& {{\bf D}_{57}}& {{\bf D}_{58}}& {{\bf D}_{59}}& {{\bf D}_{5,10}} \cr {}& {}& {}& {}& {}& {{\bf D}_{66}}& {{\bf D}_{67}}& {{\bf D}_{68}}& {{\bf D}_{69}}& {{\bf D}_{6,10}} \cr {}& {}& {}& {}& {}& {}& {{\bf D}_{77}}& {{\bf D}_{78}}& {{\bf D}_{79}}& {{\bf D}_{7,10}} \cr {}& {}& {}& {}& {}& {}& {}& {{\bf D}_{88}}& {{\bf D}_{89}}& {{\bf D}_{8,10}} \cr {}& {}& {}& {}& {}& {}& {}& {}& {{\bf D}_{99}}& {{\bf D}_{9,10}} \cr {}& {}& {}& {}& {}& {}& {}& {}& {}& {{\bf D}_{10,10}} \cr }}\cr = &{\pmatrix{ {\tilde{\bf D}_{11}^* }& {\tilde{\bf D}_{12}^* }& {\tilde{\bf D}_{18}^* } & {\tilde{\bf D}_{17}^* }& {\tilde{\bf D}_{16}^* }& {\tilde{\bf D}_{15}^* }& {\tilde{\bf D}_{14}^* }& {\tilde{\bf D}_{13}^* }& {\tilde{\bf D}_{1,10}^* }& {\tilde{\bf D}_{19}^* } \cr {}& {\tilde{\bf D}_{22}^* }& {\tilde{\bf D}_{28}^* }& {\tilde{\bf D}_{27}^* }& {\tilde{\bf D}_{26}^* }& {\tilde{\bf D}_{25}^* }& {\tilde{\bf D}_{24}^* }& {\tilde{\bf D}_{23}^* }& {\tilde{\bf D}_{2,10}^* }& {\tilde{\bf D}_{29}^* } \cr {}& {}& {\tilde{\bf D}_{88}^* }& {\tilde{\bf D}_{87}^* }& {\tilde{\bf D}_{86}^* }& {\tilde{\bf D}_{85}^* }& {\tilde{\bf D}_{84}^* }& {\tilde{\bf D}_{83}^* }& {\tilde{\bf D}_{8,10}^* }& {\tilde{\bf D}_{89}^* } \cr {}& {}& {}& {\tilde{\bf D}_{77}^* }& {\tilde{\bf D}_{76}^* }& {\tilde{\bf D}_{75}^* }& {\tilde{\bf D}_{74}^* }& {\tilde{\bf D}_{73}^* }& {\tilde{\bf D}_{7,10}^* }& {\tilde{\bf D}_{79}^* } \cr {}& {}& {}& {}& {\tilde{\bf D}_{66}^* }& {\tilde{\bf D}_{65}^* }& {\tilde{\bf D}_{64}^* }& {\tilde{\bf D}_{63}^* }& {\tilde{\bf D}_{6,10}^* }& {\tilde{\bf D}_{69}^* } \cr {}& {}& {}& {}& {}& {\tilde{\bf D}_{55}^* }& {\tilde{\bf D}_{54}^* } & {\tilde{\bf D}_{53}^* }& {\tilde{\bf D}_{5,10}^* }& {\tilde{\bf D}_{59}^* } \cr {}& {}& {}& {}& {}& {}& {\tilde{\bf D}_{44}^* }& {\tilde{\bf D}_{43}^* }& {\tilde{\bf D}_{4,10}^* }& {\tilde{\bf D}_{49}^* } \cr {}& {}& {}& {}& {}& {}& {}& {\tilde{\bf D}_{33}^* }& {\tilde{\bf D}_{3,10}^* }& {\tilde{\bf D}_{39}^* } \cr {}& {}& {}& {}& {}& {}& {}& {}& {\tilde{\bf D}_{10,10}^* }& {\tilde{\bf D}_{10,9}^* } \cr {}& {}& {}& {}& {}& {}& {}& {}& {}& {\tilde{\bf D}_{99}^* } \cr }} \cr & {\rm with}\,\, \tilde{\bf D}_{kl}^{} = {\bf m}_x \, {\bf D}_{kl}^{} \, {\bf m}_x = {\pmatrix{ {D_{kl}^{11}}& {- D_{kl}^{12}}& {- D_{kl}^{13}} \cr {- D_{kl}^{21}}& {D_{kl}^{22}}& {D_{kl}^{23}} \cr {- D_{kl}^{31}}& {D_{kl}^{32}}& {D_{kl}^{33}} \cr }}. \cr} ]$ For the submatrix $[{\bf D}_{{11}}]$ (and similarly also for $[{\bf D}_{12}]$ and $[{\bf D}_{22}]$ ) we can combine the three relations and obtain $[\eqalign{{\bf D}_{{11}} &= {\bf m}_x \, {\bf D}_{11}^* \, {\bf m}_x = {\pmatrix{ {D_{11}^{11*} }& {- D_{11}^{12*} }& {- D_{11}^{13*} } \cr {- D_{11}^{12}}& {D_{11}^{22*} }& {D_{11}^{23*} } \cr {- D_{11}^{13}}& {D_{11}^{23}}& {D_{11}^{33*} } \cr }} \cr&= {\bf m}_y \, {\bf D}_{11}^{} \, {\bf m}_y = {\pmatrix{ {D_{11}^{11}}& {- D_{11}^{12}}& {D_{11}^{13}} \cr {- D_{11}^{12*} }& {D_{11}^{22}}& {- D_{11}^{23}} \cr {D_{11}^{13*} }& {- D_{11}^{23*} }& {D_{11}^{33}} \cr }} \cr&= {\bf D}_{180}^z \, {\bf D}_{11}^{*} \, {\bf D}_{180}^z = {\pmatrix{ {D_{11}^{11*} }& {D_{11}^{12*} }& {- D_{11}^{13*} } \cr {D_{11}^{12}}& {D_{11}^{22*} }& {- D_{11}^{23*} } \cr {- D_{11}^{13}}& {- D_{11}^{23}}& {D_{11}^{33*} } \cr }}.} ]$ Hence $[\eqalign{ & D_{11}^{11} = D_{11}^{11*} = \alpha \quad {\rm real} \cr & D_{11}^{12} = - D_{11}^{12} = 0 \cr & D_{11}^{13} = - D_{11}^{13*} = i \, \beta \quad {\rm imaginary} \cr & D_{11}^{22} = D_{11}^{22*} = \gamma \quad {\rm real} \cr & D_{11}^{23} = - D_{11}^{23} = 0 \cr & D_{11}^{33} = D_{11}^{33*} = \delta \quad {\rm real} \cr & \quad \cr} \Rightarrow {\bf D}_{{11}} = {\pmatrix{ \alpha & 0 & {i\beta } \cr 0 & \gamma & 0 \cr {- i\beta }& 0 & \delta \cr }.} ]$ Obviously, the symmetry considerations lead to a remarkable reduction of the independent elements of the dynamical matrix.