Example

Eckold, G.

doi:10.1107/97809553602060000638

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

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International Tables for Crystallography (2006). Vol. D. ch. 2.1, pp. 288-289

Section 2.1.3.5.3. Example

G. Eckold^a ^*

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

2.1.3.5.3. Example

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Let us consider the space group $[P\bar 6]$ . For wavevectors along the hexagonal axis, the point group $[G{_o}({\bf q})]$ consists of the three symmetry operations E, $[{D}_{120}^z]$ and $[{D}_{240}^z]$ . Being a cyclic group, its irreducible representations are one-dimensional (see Table 2.1.3.4). The mirror plane $[{m}_z]$ inverts the wavevector and the two threefold rotations are self-conjugated with respect to $[{m}_z]$ : $[\eqalign{ {\bf m}_z \, {\bf D}_{120}^z \, {\bf m}_z &= {\bf D}_{120}^z \cr {\bf m}_z \, {\bf D}_{240}^z \, {\bf m}_z &= {\bf D}_{240}^z. \cr} ]$ If we remember that for symmorphic space groups all multipliers are unity, we obtain the following conjugate representations according to (2.1.3.62): $[\eqalign{ \bar \tau ^{(1)} &= \tau ^{(1)*} = \tau ^{(1)} \cr \bar \tau ^{(2)} &= \tau ^{(2)*} = \tau ^{(3)} \cr \bar \tau ^{(3)} &= \tau ^{(3)*} = \tau ^{(2)}. \cr} ]$ Obviously, $[\bar \tau ^{(2)}]$ and $[\tau ^{(2)}]$ are inequivalent and, hence, pairs of phonons corresponding to representations $[\tau ^{(2)} ]$ and $[\tau ^{(3)}]$ , respectively, are degenerate. The two transverse acoustic phonon branches tin particular not only leave the Γ point with the same slope as determined by the elastic stiffness $[c_{44}=c_{2323}]$ (cf. Section 2.1.2.4 and Chapter 1.3 ) but are strictly identical throughout the whole Brillouin zone.

Table 2.1.3.4 | top | pdf |
Irreducible representations of the point group [3]

3	E	$[D_{120}^z ]$	$[D_{240}^z ]$
$[\tau ^{(1)}]$	1	1	1
$[\tau ^{(2)}]$	1	$[\exp({i2\pi /3})]$	$[\exp({- i2\pi /3})]$
$[\tau ^{(3)}]$	1	$[\exp({- i2\pi /3})]$	$[\exp({i2\pi /3})]$

Another example may illustrate the degeneracy of phonons at special wavevectors where the elements of the dynamical matrix are real quantities. Let us consider the nonsymmorphic space group [P6 _3] . For the Γ point ( $[{\bf q}={\bf 0}]$ ), the one-dimensional representations of this cyclic group are collected in Table 2.1.3.5. Obviously, $[\tau ^{(2)}({\bf 0})]$ and $[\tau ^{(6)}({\bf 0})]$ form a pair of complex-conjugated representations as well as $[\tau ^{(3)}({\bf 0})]$ and $[\tau ^{(5)}({\bf 0}) ]$ . Therefore, always two lattice vibrations of these symmetries exhibit the same frequencies. The eigenvectors for representation $[\tau ^{(6)}({\bf 0}) ]$ or $[\tau ^{(5)}({\bf 0})]$ can be combined with the eigenvectors of corresponding modes of representations $[\tau ^{(2)}({\bf 0})]$ or $[\tau ^{(3)}({\bf 0}) ]$ , respectively, to yield real quantities.

Table 2.1.3.5 | top | pdf |
Irreducible representations of the space group [P6_3] for $[{\bf q}={\bf 0}]$ (the $[\Gamma]$ point)

	E	$[D_{60}^z ]$	$[D_{120}^z ]$	$[D_{180}^z ]$	$[D_{240}^z ]$	$[D_{300}^z ]$
$[\tau^{(1)}({\bf 0}) ]$	1	1	1	1	1	1
$[\tau^{(2)}({\bf 0}) ]$	1	$[\exp({i\pi /3}) ]$	$[\exp({i2\pi /3}) ]$		$[\exp({i4\pi /3}) = \exp({- i2\pi /3}) ]$	$[\exp({i5\pi /3}) = \exp({- i\pi /3}) ]$
$[\tau^{(3)}({\bf 0}) ]$	1	$[\exp({i2\pi /3}) ]$	$[\exp({i4\pi /3}) =\exp(-i2\pi/3)]$	1	$[\exp({i2\pi /3}) ]$	$[\exp({i4\pi /3}) =\exp(-i2\pi/3)]$
$[\tau^{(4)}({\bf 0}) ]$	1		1		1
$[\tau^{(5)}({\bf 0}) ]$	1	$[\exp({i4\pi /3}) =\exp(-i2\pi/3)]$	$[\exp({i2\pi /3}) ]$	1	$[\exp({i4\pi /3}) =\exp(-i2\pi/3)]$	$[\exp({i2\pi /3}) ]$
$[\tau^{(6)}({\bf 0}) ]$	1	$[\exp({i5\pi /3}) =\exp(-i\pi/3)]$	$[\exp({i4\pi /3}) =\exp(-i2\pi/3)]$		$[\exp({i2\pi /3}) ]$	$[\exp({i\pi /3}) ]$

For wavevectors within the Brillouin zone along the hexagonal axis, the irreducible representations are the same as for the Γ point. However, the elements of the dynamical matrix are complex and symmetry does not yield any degeneracies. Hence phonons can be distinguished according to the six different representations.

At the Brillouin-zone boundary along the hexagonal axis ( $[{\bf q}={\bf c}^*/2 ]$ , the A point), one has to take into account multipliers of the form $[\exp[i{\bf q}\,{\bf v }({\bf R})]]$ since the space group is non-symmorphic. For symmetry operations without fractional translation ( [{E}] , $[{D}_{120}^z]$ , $[{D}_{240}^z]$ ) this factor is unity, whereas it equals the complex unit i for the other elements of the point group ( $[{D}_{60}^z]$ , $[{D}_{180}^z ]$ , $[{D}_{300}^z]$ ). Hence the six irreducible multiplier representations are as shown in Table 2.1.3.6. Now we have three pairs of complex-conjugate representations, namely: $[\tau ^{(1)}({\bf c}^*/2) ]$ and $[\tau ^{(4)}({\bf c}^*/2)]$ ; $[\tau ^{(2)}({\bf c}^*/2) ]$ and $[\tau ^{(3)}({\bf c}^*/2)]$ ; and $[\tau ^{(5)}({\bf c}^*/2) ]$ and $[\tau ^{(6)}({\bf c}^*/2)]$ . Again, pairs of phonons of corresponding representations are degenerate. As a consequence, the phonon dispersion curves need not approach the Brillouin-zone boundary with a horizontal slope but meet another branch with the opposite slope.

Table 2.1.3.6 | top | pdf |
Irreducible representations of the space group [P6 _3] for $[{\bf q}={\bf c}^*/2]$ (the A point)

	E	$[D_{60}^z ]$	$[D_{120}^z ]$	$[D_{240}^z]$	$[D_{300}^z ]$
$[\tau ^{(1)}({{\bf c}^*/2}) ]$	1	i	1	1	i
$[\tau ^{(2)}({{\bf c}^*/2}) ]$	1	$[i\exp({i\pi /3}) = \exp({- i\pi /6}) ]$	$[\exp({i2\pi /3}) ]$	$[\exp({i4\pi /3}) = \exp({- i2\pi /3}) ]$	$[i\exp({-i\pi /3}) = \exp({ i\pi /6}) ]$
$[\tau ^{(3)}({{\bf c}^*/2}) ]$	1	$[i\exp({i2\pi /3})=\exp(i\pi/6) ]$	$[\exp({i4\pi /3})=\exp(-i2\pi/3) ]$	$[\exp({i2\pi /3}) ]$	$[i\exp({-i2\pi /3})=\exp(-i\pi/6) ]$
$[\tau ^{(4)}({{\bf c}^*/2}) ]$	1		1	1
$[\tau ^{(5)}({{\bf c}^*/2}) ]$	1	$[i\exp({-i2\pi /3})=\exp(-i\pi/6) ]$	$[\exp({i2\pi /3}) ]$	$[\exp({i4\pi /3})=\exp(-i2\pi/3) ]$	$[i\exp({i2\pi /3}) = \exp({i\pi /6}) ]$
$[\tau ^{(6)}({{\bf c}^*/2}) ]$	1	$[i\exp(-i\pi/3)=\exp({i\pi /6}) ]$	$[\exp({i4\pi /3})=\exp(-i2\pi/3) ]$	$[\exp({i2\pi /3}) ]$	$[i\exp({i\pi /3})=\exp(-i\pi/6) ]$

In conclusion, group-theoretical considerations for wavevectors along the hexagonal axis yield at the centre (Γ point) as well as at the boundary (A point) of the first Brillouin zone pairs of degenerate phonon modes. Both modes belong to complex-conjugate representations. This result can be used in order to display the dispersion curves very clearly in an extended zone scheme plotting the phonon branches of different symmetries alternately from Γ to A and from A back to Γ as illustrated in Fig. 2.1.3.9. Here, the phonon dispersion for the room-temperature phase of KLiSO₄ is shown as an example. Note that irreducible representations are frequently denoted by the letters A, B, E, T instead of our notation $[\tau ^{(i)}]$ . T and E are reserved for representations that (at the Γ point) are triply and doubly degenerate, respectively. An index $[\pm]$ or g/u is often used to distinguish representations that are symmetric (gerade) and antisymmetric (ungerade) with respect to a prominent symmetry operation, e.g. a centre of inversion or, in the case of [P6_3] , the twofold axis. The total symmetric representation is always denoted by A. Hence in the preceding example all the representations $[\tau ^{(2)}]$ , $[\tau ^{(3)}]$ , $[\tau ^{(5)} ]$ and $[\tau ^{(6)}]$ are E-type representations since they are doubly degenerate at the zone centre due to time-reversal degeneracy . Moreover, $[\tau^ {(3)}]$ and $[\tau^ {(5)}]$ are symmetric with respect to $[{D}_{180}^z]$ . Therefore, the irreducible representations of Fig. 2.1.3.9 can be identified as $[{\rm A}=\tau^ {(1)}]$ , $[{\rm B}=\tau^ {(2)} ]$ , $[{\rm E}_1^ -=\tau ^{(2)}]$ , $[E_1^ +=\tau^ {(6)}]$ , $[E_1^ +=\tau ^{(5)}]$ and $[E_1^ +=\tau ^{(3)}]$ .

Figure 2.1.3.9 | top | pdf |

Low-frequency part of the phonon dispersion of KLiSO₄ at room temperature (space group P6₃). The phonons are arranged in an extended zone scheme according to the different irreducible representations [after Eckold & Hahn (1987)]. The symbols represent experimental data and the lines represent the results of model calculations.

It can be seen that all phonon branches cross the zone boundary continuously while changing their symmetry. This behaviour is a direct consequence of the time-reversal degeneracy.

References

International Tables for Crystallography (2006). Vol. D. ch. 2.1, pp. 288-289