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

International Tables for Crystallography (2006). Vol. D, ch. 1.5, p. 133

Table 1.5.7.1 

A. S. Borovik-Romanova and H. Grimmerb*

aP. 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 classMatrix representation [\Lambda_{i\alpha}] of the piezomagnetic tensor
SchoenfliesHermann–Mauguin
[{\bi C}_1] [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] [222] [\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}] [mmm]
[{\bi D}_2({\bi C}_2)] [2'2'2] [\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)] [m'm'2]
[{\bi C}_{2v}({\bi C}_s)] [m'2'm\,[2'm'm]]
[{\bi D}_{2h}({\bi C}_{2h})] [m'm'm]
[{\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)] [4'] [\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})] [4'/m]
[{\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)] [4'22'] [\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})] [4'mm']
[{\bi D}_{2d}({\bi D}_2),\,{\bi D}_{2d}({\bi C}_{2v})] [\bar{4}'2m',\,\bar{4}'m2']
[{\bi D}_{4h}({\bi D}_{2h})] [4'/mmm']
[{\bi C}_3] [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) ] [6'] [\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)] [6'/m']
[{\bi D}_6({\bi D}_3)] [6'22'] [\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})] [6'mm']
[{\bi D}_{3h}({\bi D}_3),\,{\bi D}_{3h}({\bi C}_{3v})] [\bar{6}'2m',\,\bar{6}'m2']
[{\bi D}_{6h}({\bi D}_{3d})] [6'/m'mm']
[{\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})] [4'32']
[{\bi T}_d({\bi T})] [\bar{4}'3m']
[{\bi O}_h({\bi T}_h)] [m\bar{3}m']