Table 1.5.7.1

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. 133

Table 1.5.7.1

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 1.5.7.1 | top | pdf |
The forms of the matrix characterizing the piezomagnetic effect

Magnetic crystal class		Matrix representation $[\Lambda_{i\alpha}]$ of the piezomagnetic tensor
Schoenflies	Hermann–Mauguin
$[{\bi C}_1]$		$[\left[\matrix{\Lambda_{11} &\Lambda_{12} &\Lambda_{13} &\Lambda_{14} &\Lambda_{15} &\Lambda_{16}\cr \Lambda_{21} &\Lambda_{22} &\Lambda_{23} &\Lambda_{24} &\Lambda_{25} &\Lambda_{26}\cr \Lambda_{31} &\Lambda_{32} &\Lambda_{33} &\Lambda_{34} &\Lambda_{35} &\Lambda_{36} } \right]]$
$[{\bi C}_i]$	$[\bar{1}]$

$[{\bi C}_2]$	$[2\,(=121)]$	$[\left[\matrix{ 0 & 0 & 0 &\Lambda_{14} & 0 &\Lambda_{16}\cr \Lambda_{21} &\Lambda_{22} &\Lambda_{23} & 0 &\Lambda_{25} & 0 \cr 0 & 0 & 0 &\Lambda_{34} & 0 &\Lambda_{36} } \right]]$
$[{\bi C}_s]$	$[m\,(=1m1)]$
$[{\bi C}_{2h}]$	$[2/m\,(=1\,2/m\,1)]$
	(unique axis y)
$[{\bi C}_2({\bi C}_1)]$	$[2'\,(=12'1)]$	$[\left[\matrix{\Lambda_{11} &\Lambda_{12} &\Lambda_{13} & 0 &\Lambda_{15} & 0 \cr 0 & 0 & 0 &\Lambda_{24} & 0 &\Lambda_{26}\cr \Lambda_{31} &\Lambda_{32} &\Lambda_{33} & 0 &\Lambda_{35} & 0 } \right]]$
$[{\bi C}_s({\bi C}_1)]$	$[m'\,(=1m'1)]$
$[{\bi C}_{2h}({\bi C}_i)]$	$[2'/m'\,(=1\,2'/m'\,1)]$
	(unique axis y)
$[{\bi D}_2]$		$[\left[\matrix{ 0 & 0 & 0 &\Lambda_{14} & 0 & 0 \cr 0 & 0 & 0 & 0 &\Lambda_{25} & 0 \cr 0 & 0 & 0 & 0 & 0 &\Lambda_{36} } \right]]$
$[{\bi C}_{2v}]$	$[mm2\,[2mm,m2m]]$
$[{\bi D}_{2h}]$
$[{\bi D}_2({\bi C}_2)]$		$[\left[\matrix{ 0 & 0 & 0 & 0 &\Lambda_{15} & 0 \cr 0 & 0 & 0 &\Lambda_{24} & 0 & 0 \cr \Lambda_{31} &\Lambda_{32} &\Lambda_{33} & 0 & 0 & 0 }\right]]$
$[{\bi C}_{2v}({\bi C}_2)]$
$[{\bi C}_{2v}({\bi C}_s)]$	$[m'2'm\,[2'm'm]]$
$[{\bi D}_{2h}({\bi C}_{2h})]$
$[{\bi C}_4, \,{\bi C}_6]$	$[4,\, 6]$	$[\left[\matrix{ 0 & 0 & 0 &\Lambda_{14} & \Lambda_{15}& 0 \cr 0 & 0 & 0 &\Lambda_{15} &-\Lambda_{14}& 0 \cr \Lambda_{31} &\Lambda_{31} &\Lambda_{33} & 0 & 0 & 0 } \right]]$
$[{\bi S}_4,\,{\bi C}_{3h}]$	$[\bar{4},\,\bar{6}]$
$[{\bi C}_{4h},\,{\bi C}_{6h}]$	$[4/m,\,6/m]$
$[{\bi C}_4({\bi C}_2)]$		$[\left[\matrix{ 0 & 0 & 0 & \Lambda_{14}&\Lambda_{15} & 0 \cr 0 & 0 & 0 &-\Lambda_{15}&\Lambda_{14} & 0 \cr \Lambda_{31}&-\Lambda_{31} & 0 & 0 & 0 &\Lambda_{36} } \right]]$
$[{\bi S}_4({\bi C}_2)]$	$[\bar{4}']$
$[{\bi C}_{4h}({\bi C}_{2h})]$
$[{\bi D}_4,\,{\bi D}_6 ]$	$[422,\,622]$	$[\left[\matrix{ 0 & 0 & 0 &\Lambda_{14} & 0 & 0 \cr 0 & 0 & 0 & 0 &-\Lambda_{14} & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 } \right]]$
$[{\bi C}_{4v},\,{\bi C}_{6v}]$	$[4mm,\,6mm]$
$[{\bi D}_{2d},\,{\bi D}_{3h}]$	$[\bar{4}2m\,[\bar{4}m2],\,\bar{6}m2\,[\bar{6}2m]]$
$[{\bi D}_{4h},\,{\bi D}_{6h}]$	$[4/mmm,\,6/mmm]$
$[{\bi D}_4({\bi C}_4),\,{\bi D}_6({\bi C}_6)]$	$[42'2',\,62'2']$	$[\left[\matrix{ 0 & 0 & 0 & 0 &\Lambda_{15} & 0 \cr 0 & 0 & 0 &\Lambda_{15} & 0 & 0 \cr \Lambda_{31} &\Lambda_{31} &\Lambda_{33} & 0 & 0 & 0 } \right]]$
$[{\bi C}_{4v}({\bi C}_4),\,{\bi C}_{6v}({\bi C}_6)]$	$[4m'm',\,6m'm']$
$[{\bi D}_{2d}({\bi S}_4),\,{\bi D}_{3h}({\bi C}_{3h})]$	$[\bar{4}2'm'\,[\bar{4}m'2'],\,\bar{6}m'2'\,[\bar{6}2'm']]$
$[{\bi D}_{4h}({\bi C}_{4h}),\,{\bi D}_{6h}({\bi C}_{6h})]$	$[4/mm'm',\,6/mm'm']$
$[{\bi D}_4({\bi D}_2)]$		$[\left[\matrix{ 0 & 0 & 0 &\Lambda_{14} & 0 & 0 \cr 0 & 0 & 0 & 0 &\Lambda_{14} & 0 \cr 0 & 0 & 0 & 0 & 0 &\Lambda_{36} }\right]]$
$[{\bi C}_{4v}({\bi C}_{2v})]$
$[{\bi D}_{2d}({\bi D}_2),\,{\bi D}_{2d}({\bi C}_{2v})]$	$[\bar{4}'2m',\,\bar{4}'m2']$
$[{\bi D}_{4h}({\bi D}_{2h})]$
$[{\bi C}_3]$		$[\left[\matrix{ \Lambda_{11}&-\Lambda_{11} &0 &\Lambda_{14} & \Lambda_{15} &-2\Lambda_{22} \cr -\Lambda_{22} & \Lambda_{22} &0 &\Lambda_{15} &-\Lambda_{14} &-2\Lambda_{11} \cr \Lambda_{31} & \Lambda_{31} &\Lambda_{33}& 0 & 0 & 0 } \right]]$
$[{\bi S}_6]$	$[\bar{3}]$

$[{\bi D}_3]$	$[32\,(=321)]$	$[\left[\matrix{\Lambda_{11} &-\Lambda_{11}& 0 &\Lambda_{14} & 0 & 0 \cr 0 & 0 & 0 & 0 &-\Lambda_{14} &-2\Lambda_{11}\cr 0 & 0 & 0 & 0 & 0 & 0 } \right]]$
$[{\bi C}_{3v}]$	$[3m\,(=3m1)]$
$[{\bi D}_{3d}]$	$[\bar{3}m\,(=\bar{3}m1)]$
$[{\bi D}_3({\bi C}_3)]$	$[32'\,(=32'1)]$	$[\left[\matrix{ 0 & 0 & 0 & 0 & \Lambda_{15}&-2\Lambda_{22}\cr -\Lambda_{22} & \Lambda_{22} & 0 &\Lambda_{15} & 0 & 0 \cr \Lambda_{31} & \Lambda_{31} &\Lambda_{33} & 0 & 0 & 0 } \right]]$
$[{\bi C}_{3v}({\bi C}_3)]$	$[3m'\,(=3m'1)]$
$[{\bi D}_{3d}({\bi S}_6)]$	$[\bar{3}m'\,(=\bar{3}m'1)]$
$[{\bi C}_6({\bi C}_3) ]$		$[\left[\matrix{\Lambda_{11} &-\Lambda_{11} & 0 & 0 & 0 &-2\Lambda_{22} \cr -\Lambda_{22} & \Lambda_{22} & 0 & 0 & 0 &-2\Lambda_{11} \cr 0 & 0 & 0 & 0 & 0 & 0 } \right]]$
$[{\bi C}_{3h}({\bi C}_3)]$	$[\bar{6}']$
$[{\bi C}_{6h}({\bi S}_6)]$
$[{\bi D}_6({\bi D}_3)]$		$[\left[\matrix{\Lambda_{11} &-\Lambda_{11} & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 &-2\Lambda_{11}\cr 0 & 0 & 0 & 0 & 0 & 0 } \right]]$
$[{\bi C}_{6v}({\bi C}_{3v})]$
$[{\bi D}_{3h}({\bi D}_3),\,{\bi D}_{3h}({\bi C}_{3v})]$	$[\bar{6}'2m',\,\bar{6}'m2']$
$[{\bi D}_{6h}({\bi D}_{3d})]$
$[{\bi T},\,{\bi T}_h ]$	$[23,\,m\bar{3} ]$	$[\left[\matrix{ 0 & 0 & 0 &\Lambda_{14} & 0 & 0 \cr 0 & 0 & 0 & 0 &\Lambda_{14} & 0 \cr 0 & 0 & 0 & 0 & 0 &\Lambda_{14} }\right]]$
$[{\bi O}({\bi T})]$
$[{\bi T}_d({\bi T})]$	$[\bar{4}'3m']$
$[{\bi O}_h({\bi T}_h)]$	$[m\bar{3}m']$