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, p. 291

Section 2.1.3.7.1. Example

G. Eckold^a ^*

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

2.1.3.7.1. Example

| top | pdf |

As an example, let us once more consider the space group [P4mm ] . For $[{\bf q}={\bf 0}]$ , the character table shown in Table 2.1.3.8 summarizes all essential information about irreducible, vector and tensor representations. Obviously, the vector representation consists of the irreducible representations $[\tau ^{(1^ +)}]$ and $[\boldtau ^{(2)}]$ , the latter being two-dimensional. Γ-point phonons corresponding to these two representations are infrared active. All other lattice vibrations cannot be detected by absorption experiments.

Table 2.1.3.8 | top | pdf |
Character table of the space group [P4mm] for $[{\bf q}={\bf 0}]$ (the $[\Gamma]$ point)

	Symmetry operation
	E	$[D_{90}^z ]$	$[D_{180}^z]$	$[ D_{270}^z ]$			$[m_{ [{\bar 110}]}]$	$[m_{ [{110}]}]$
$[\chi _{\tau _{}^{(1^ +)}} ]$	1	1	1	1	1	1	1	1
$[\chi _{\tau _{}^{(1^ -)}} ]$	1	1	1	1
$[\chi _{\tau _{}^{(3^ +)}} ]$	1		1		1	1
$[\chi _{\tau^{(3^ -)} } ]$	1		1				1	1
$[\chi _{\boldtau ^{(2)}} ]$	2	0		0	0	0	0	0
$[\boldtau{_v}]$	$[\,\,\,{\pmatrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \cr }} ]$	$[\,\,\,{\pmatrix{ 0 & {- 1}& 0 \cr 1 & 0 & 0 \cr 0 & 0 & 1 \cr }}]$	$[\,\,\,{\pmatrix{ {- 1}& 0 & 0 \cr 0 & {- 1}& 0 \cr 0 & 0 & 1 \cr }}]$	$[\,\,\,{\pmatrix{ 0 & 1 & 0 \cr {- 1}& 0 & 0 \cr 0 & 0 & 1 \cr }} ]$	$[\,\,\,{\pmatrix{ {- 1}& 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \cr }} ]$	$[\,\,\,{\pmatrix{ 1 & 0 & 0 \cr 0 & {- 1}& 0 \cr 0 & 0 & 1 \cr }}]$	$[\,\,\,{\pmatrix{ 0 & 1 & 0 \cr 1 & 0 & 0 \cr 0 & 0 & 1 \cr }} ]$	$[\,\,\,{\pmatrix{ 0 & -1 & 0 \cr -1 & 0 & 0 \cr 0 & 0 & 1 \cr }}\,\,\,\,\,\,]$
$[\chi _{\boldtau _v } ]$	3	1		1	1	1	1	1
$[\chi _{\boldtau _T } ]$	6	0	2	0	2	2	2	2

Using the multiplicities as calculated from (2.1.3.70), we obtain the decomposition of the tensor representation: $[\boldtau _T = 2 \tau _{}^{(1^ +)} + \tau _{}^{(3^ +)} + \tau _{}^{(3^ -)} + \boldtau ^{(2)}. ]$ Hence phonons corresponding to the representations $[\tau _{}^{(1^ +)} ]$ , $[\tau _{}^{(3^ +)}]$ , $[\tau _{}^{(3^ -)} ]$ and $[\boldtau ^{(2)}]$ are Raman active.

All lattice vibrations that belong to the representation $[\tau _{}^{(1^ -)} ]$ are neither infrared nor Raman active. They cannot be detected in (first-order) optical experiments and are therefore called silent modes.

References

International Tables for Crystallography (2006). Vol. D. ch. 2.1, p. 291