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

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. D. ch. 1.7, p. 182

Table 1.7.2.2 

B. Boulangera* and J. Zyssb

a Laboratoire de Spectrométrie Physique, Université Joseph Fourier, 140 avenue de la Physique, BP 87, 38 402 Saint-Martin-d'Hères, France, and bLaboratoire de Photonique Quantique et Moléculaire, Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, 94235 Cachan, France
Correspondence e-mail:  benoitb@satie-bourgogne.fr

Table 1.7.2.2 | top | pdf |
Nonzero χ(2) coefficients and equalities between them in the general case

Symmetry class χ (2) nonzero elements
Triclinic  
C 1 (1) All 27 elements are independent and nonzero
   
Monoclinic  
C 2 (2) (twofold axis parallel to z) [xyz], [xzy], [xxz], [xzx], [yyz], [yzy], [yxz], [yzx], [zxx], [zyy], [zzz], [zxy], [zyx]
C s (m) (mirror perpendicular to z) [xxx], [xyy], [xzz], [xxy], [xyx], [yxx], [yyy], [yzz], [yxy], [yyx], [zyz], [zzy], [zxz], [zzx]
   
Orthorhombic  
C 2v (mm2) (twofold axis parallel to z) [xzx], [xxz], [yyz], [yzy], [zxx], [zyy], [zzz]
D 2 (222) [xyz], [xzy], [yzx], [yxz], [zxy], [zyx]
   
Tetragonal  
C 4 (4) [xyz=-yxz], [xzy=-yzx], [xzx=yzy], [xxz=yyz], [zxx=zyy], [zzz], [zxy=-zyx]
S 4 ([\bar 4]) [xyz=yxz], [xzy=yzx], [xzx=-yzy], [xxz=-yyz], [zxx=-zyy], [zxy=zyx]
D 4 (422) [xyz=-yxz], [xzy=-yzx], [zxy=-zyx]
C 4v (4mm) [xzx=yzy], [xxz=yyz], [zxx=zyy], [zzz]
D 2d ([\bar 4 2 m]) [xyz=yxz], [xzy=yzx], [zxy=zyx]
   
Hexagonal  
C 6 (6) [xyz=-yxz], [xzy=-yzx], [xzx=yzy], [xxz=yyz], [zxx=zyy], [zzz], [zxy=-zyx]
C 3h ([\bar 6]) [xxx=-xyy=-yxy=-yyx], [yyy=-yxx=] [-xyx=-xxy]
D 6 (622) [xyz=-yxz], [xzy=-yzx], [zxy=-zyx]
C 6v (6mm) [xzx=yzy], [xxz=yyz], [zxx=zyy], [zzz]
D 3h ([\bar 6 2 m]) (mirror perpendicular to x) [yyy=-yxx=-xxy=-xyx]
   
Trigonal  
C 3 (3) [xxx=-xyy=-yyx=-yxy], [xyz=-yxz], [xzy=-yzx], [xzx=yzy], [xxz=yyz], [yyy=-yxx=-xxy=-xyx], [zxx=zyy], [zzz], [zxy=-zyx]
D 3 (32) [xxx=-xyy=-yyx=-yxy], [xyz=-yxz], [xzy=-yzx], [zxy=-zyx]
C 3v (3m) (mirror perpendicular to x) [yyy=-yxx=-xxy=-xyx], [xzx=yzy], [xxz=yyz], [zxx=zyy, zzz]
   
Cubic  
T (23), Td ([\bar 4 3 m]) [xyz=xzy=yzx=yxz=zxy=zyx]
O (432) [xyz=-xzy=yzx=-yxz=zxy=-zyx]