Table 1.5.8.1

Borovik-Romanov, A. S.; Grimmer, H.

doi:10.1107/97809553602060000632

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. 1.5, p. 138

Table 1.5.8.1

A. S. Borovik-Romanov^a ^‡ and H. Grimmer^b ^*

^a P. L. Kapitza Institute for Physical Problems, Russian Academy of Sciences, Kosygin Street 2, 119334 Moscow, Russia, and ^bLabor für Neutronenstreuung, ETH Zurich, and Paul Scherrer Institute, CH-5234 Villigen PSI, Switzerland
Correspondence e-mail: hans.grimmer@psi.ch

Table 1.5.8.1 | top | pdf |
The forms of the tensor characterizing the linear magnetoelectric effect

Magnetic crystal class		Matrix representation of the property tensor $[\alpha_{ij}]$
Schoenflies	Hermann–Mauguin
$[{\bi C}_{1}]$		$[\left[\matrix { \alpha_{11} & \alpha_{12} & \alpha_{13}\cr \alpha_{21} & \alpha_{22} & \alpha_{23}\cr \alpha_{31} & \alpha_{32} & \alpha_{33} }\right]]$
$[{\bi C}_{i}({\bi C}_{1})]$	$[{\bar 1}']$

$[{\bi C}_{2}]$	$[2\,(=121)]$	$[\left[\matrix { \alpha_{11} & 0 & \alpha_{13}\cr 0 & \alpha_{22} & 0 \cr \alpha_{31} & 0 & \alpha_{33} }\right]]$
$[{\bi C}_{s}({\bi C}_{1})]$	$[m'\,(=1m'1)]$
$[{\bi C}_{2h}({\bi C}_{2})]$	$[2/m'\,(=1\,2/m'\,1)]$
	(unique axis y)
$[{\bi C}_{s} ]$	$[m\, (=1m1)]$	$[\left[\matrix { 0 & \alpha_{12} & 0 \cr \alpha_{21} & 0 & \alpha_{23}\cr 0 & \alpha_{32} & 0 }\right]]$
$[{\bi C}_{2}({\bi C}_{1})]$	$[2'\,(=12'1)]$
$[{\bi C}_{2h}({\bi C}_{s})]$	$[2'/m\,(=1\,2'/m\,1)]$
	(unique axis y)
$[{\bi D}_{2} ]$		$[\left[\matrix { \alpha_{11} & 0 & 0 \cr 0 & \alpha_{22} & 0 \cr 0 & 0 & \alpha_{33} }\right]]$
$[{\bi C}_{2v}({\bi C}_{2})]$	$[m'm'2\,[2m'm',\,m'2m']]$
$[{\bi D}_{2h}({\bi D}_{2})]$
$[{\bi C}_{2v}]$		$[\left[\matrix { 0 & \alpha_{12} & 0 \cr \alpha_{21} & 0 & 0 \cr 0 & 0 & 0 }\right]]$
$[{\bi D}_{2}({\bi C}_{2})]$
$[{\bi C}_{2v}({\bi C}_{s})]$	$[2'mm'\,[m2'm']]$
$[{\bi D}_{2h}({\bi C}_{2v})]$
$[{\bi C}_{4},\, {\bi S}_{4}({\bi C}_{2}),\, {\bi C}_{4h}({\bi C}_{4})]$	$[4,\, {\bar 4}',\,4/m' ]$	$[\left[\matrix { \alpha_{11}& \alpha_{12} & 0 \cr - \alpha_{12} & \alpha_{11} & 0 \cr 0 & 0 & \alpha_{33} }\right]]$
$[ {\bi C}_{3},\, {\bi S}_{6}({\bi C}_{3})]$	$[3,\, {\bar 3}' ]$
$[{\bi C}_{6},\, {\bi C}_{3h}({\bi C}_{3}),\, {\bi C}_{6h}({\bi C}_{6})]$	$[6,\, {\bar 6}',\,6/m']$
$[{\bi S}_{4} ]$	$[{\bar 4}]$	$[\left[\matrix { \alpha_{11}& \alpha_{12} & 0 \cr \alpha_{12}&- \alpha_{11} & 0 \cr 0 & 0 & 0 }\right]]$
$[{\bi C}_{4}({\bi C}_{2})]$
$[{\bi C}_{4h}({\bi S}_{4})]$
$[{\bi D}_{4},\, {\bi C}_{4v}({\bi C}_{4})]$	$[422,\,4m'm' ]$	$[\left[\matrix { \alpha_{11} & 0 & 0 \cr 0 & \alpha_{11} & 0 \cr 0 & 0 & \alpha_{33} }\right]]$
$[{\bi D}_{2d}({\bi D}_{2}),\, {\bi D}_{4h}({\bi D}_{4})]$	$[{\bar 4}'2m'\, [{\bar 4}'m'2],\,4/m'm'm' ]$
$[{\bi D}_{3},\, {\bi C}_{3v}({\bi C}_{3}),\, {\bi D}_{3d}({\bi D}_{3}) ]$	$[32,\,3m',\, {\bar 3}'m' ]$
$[{\bi D}_{6},\, {\bi C}_{6v}({\bi C}_{6})]$	$[622,\,6m'm']$
$[{\bi D}_{3h}({\bi D}_{3}),\, {\bi D}_{6h}({\bi D}_{6})]$	$[{\bar 6}'m'2\, [{\bar 6}'2m'],\, 6/m'm'm']$
$[{\bi C}_{4v},\, {\bi D}_{4}({\bi C}_{4})]$	$[4mm,\,42'2' ]$	$[\left[\matrix { 0 & \alpha_{12} & 0 \cr - \alpha_{12} & 0 & 0 \cr 0 & 0 & 0 }\right]]$
$[{\bi D}_{2d}({\bi C}_{2v}),\, {\bi D}_{4h}({\bi C}_{4v})]$	$[{\bar 4}'2'm\, [{\bar 4}'m2'],\,4/m'mm]$
$[{\bi C}_{3v},\, {\bi D}_{3}({\bi C}_{3}),\, {\bi D}_{3d}({\bi C}_{3v}) ]$	$[3m,\,32',\, {\bar 3}'m]$
$[{\bi C}_{6v},\, {\bi D}_{6}({\bi C}_{6}) ]$	$[6mm,\,62'2' ]$
$[{\bi D}_{3h}({\bi C}_{3v}),\, {\bi D}_{6h}({\bi C}_{6v})]$	$[{\bar 6}'m2'\, [{\bar 6}'2'm],\,6/m'mm]$
$[{\bi D}_{2d},\, {\bi D}_{2d}({\bi S}_{4})]$	$[{\bar 4}2m,\, {\bar 4}m'2']$	$[\left[\matrix { \alpha_{11} & 0 & 0 \cr 0 &- \alpha_{11} & 0 \cr 0 & 0 & 0 }\right]]$
$[{\bi D}_{4}({\bi D}_{2}),\, {\bi C}_{4v}({\bi C}_{2v})]$	$[4'22',\,4'm'm]$
$[{\bi D}_{4h}({\bi D}_{2d})]$
$[{\bi T},\, {\bi T}_{h}({\bi T})]$	$[23,\,m' {\bar 3}' ]$	$[\left[\matrix { \alpha_{11} & 0 & 0 \cr 0 & \alpha_{11} & 0 \cr 0 & 0 & \alpha_{11} }\right]]$
$[{\bi O},\, {\bi T}_{d}({\bi T}),\, {\bi O}_{h}({\bi O})]$	$[432,\, {\bar 4}'3m',\,m' {\bar 3}'m']$