Example

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. 277-281

Section 2.1.3.1.1. Example

G. Eckold^a ^*

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

2.1.3.1.1. Example

| top | pdf |

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.