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

International Tables for Crystallography (2006). Vol. D, ch. 1.7, p. 183

Table 1.7.2.3 

B. Boulangera* and J. Zyssb

aLaboratoire 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 1.7.2.3| top | pdf |
Nonzero χ(2) coefficients and equalities between them under the Kleinman symmetry assumption

Symmetry classIndependent nonzero χ(2) elements under Kleinman symmetry
Triclinic  
C1 (1) [xxx], [xyy=yxy=yyx], [xzz=zxz=zzx], [xyz=xzy=yxz=yzx=zxy=zyx], [xxz=xzx=zxx], [xxy=xyx=yxx], [yyy], [yzz=zyz=zzy], [yyz=] [yzy=] [zyy], [zzz]
   
Monoclinic  
C2 (2) (twofold axis parallel to z) [xyz=xzy=yxz=yzx=zxy=zyx], [xxz=xzx=zxx], [yyz=yzy=zyy], [zzz]
Cs (m) (mirror perpendicular to z) [xxx], [xyy=yxy=yyx], [xzz=zxz=zzx], [xxy=xyx=yxx], [yyy], [yzz=zyz=zzy]
   
Orthorhombic  
C2v (mm2) (twofold axis parallel to z) [xzx=xxz=zxx], [yyz=yzy=zyy], [zzz]
D2 (222) [xyz=xzy=yzx=yxz=zxy=zyx]
   
Tetragonal  
C4 (4) [xzx=xxz=zxx=yzy=yyz=zyy], [zzz]
S4 ([\bar 4]) [xyz=xzy=yzx=yzx=zxy=zyx], [xzx=] [xxz=] [zxx=] [-yzy=] [-yyz=] [-zyy]
D4 (422) All elements are nil
C4v (4mm) [xzx=xxz=zxx=yyz=yzy=zyy], [zzz]
D2d ([\bar 4 2 m]) [xyz=xzy=yzx=yxz=zxy=zyx]
   
Hexagonal  
C6 (6) [xzx=xxz=zxx=yyz=yzy=zyy], [zzz]
C3h ([\bar 6]) [xxx=-xyy=-yxy=-yyx], [yyy=] [-yxx=] [-xyx=] [-xxy]
D6 (622) All elements are nil
C6v (6mm) [xzx=xxz=zxx=yyz=yzy=zyy], [zzz]
D3h ([\bar 6 2 m]) (mirror perpendicular to x) [yyy=-yxx=-xxy=-xyx]
   
Trigonal  
C3 (3) [xxx=-xyy=-yyx=-yxy], [xzx] [=] [xxz] [=] [zxx] [=] [yyz] [=] [yzy] [=] [zyy], [yyy] [=] [-yxx] [=] [-xxy] [=] [-xyx], [zzz]
D3 (32) [xxx=-xyy=-yyx=-yxy]
C3v (3m) (mirror perpendicular to x) [yyy=-yxx=-xxy=-xyx], [xzx] [=] [xxz] [=] [zxx] [=] [yyz] [=] [yzy] [=] [zyy], [zzz]
   
Cubic  
T (23), Td ([\bar 4 3 m]) [xyz=xzy=yzx=yxz=zxy=zyx]
O (432) All elements are nil