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

International Tables for Crystallography (2013). Vol. D, ch. 1.7, p. 186

Table 1.7.2.3 

B. Boulangera* and J. Zyssb

aInstitut Néel CNRS Université Joseph Fourier, 25 rue des Martyrs, BP 166, 38042 Grenoble Cedex 9, France, and bLaboratoire de Photonique Quantique et Moléculaire, Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, 94235 Cachan, France
Correspondence e-mail:  benoit.boulanger@grenoble.cnrs.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