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. 188

Table 1.7.2.5 

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 1.7.2.5| top | pdf |
Nonzero χ(3) coefficients and equalities between them under the Kleinman symmetry assumption

Symmetry classIndependent nonzero elements of χ(3) under Kleinman symmetry
Triclinic  
C1 (1), Ci ([\bar 1]) [xxxx], [xyyy=yxyy =yyxy=yyyx], [xzzz=zxzz=zzxz=zzzx], [xyzz] [=] [xzyz] [=] [xzzy] [=] [yxzz] [=] [yzxz] [=] [yzzx] [=] [zxyz] [=] [zxzy] [=] [zyxz] [=] [zyzx] [=] [zzxy] [=] [zzyx], [xyyz] [=] [xyzy] [=] [xzyy] [=] [yxyz] [=] [yxzy] [=] [yyxz] [=] [yyzx] [=] [yzxy] [=] [yzyx] [=] [zxyy] [=] [zyxy] [=] [zyyx], [xxzz] [=] [xzxz] [=] [xzzx] [=] [zxxz] [=] [zxzx] [=] [zzxx], [xxxz] [=] [xxzx] [=] [xzxx] [=] [zxxx], [xxyy] [=] [xyxy] [=] [xyyx] [=] [yxxy] [=] [yxyx] [=] [yyxx], [xxxy=xxyx=xyxx=yxxx], [xxyz] [=] [xxzy] [=] [xyxz] [=] [xyzx] [=] [xzxy] [=] [xzyx] [=] [yxxz] [=] [yxzx] [=] [yzxx] [=] [zxxy] [=] [zxyx] [=] [zyxx], [yyyy], [yzzz=zyzz=zzyz=zzzy], [yyzz] [=] [yzyz] [=] [yzzy] [=] [zyyz] [=] [zyzy] [=] [zzyy], [yyyz] [=] [yyzy] [=] [yzyy] [=] [zyyy], [zzzz]
   
Monoclinic  
Cs (m), C2 (2), C2h [\left(2\over m\right)] (twofold axis parallel to z) [xxxx], [xyyy=yxyy=yyxy=yyyx], [xyzz] [=] [xzyz] [=] [xzzy] [=] [yxzz] [=] [yzxz] [=] [yzzx] [=] [zxyz] [=] [zxzy] [=] [zyxz] [=] [zyzx] [=] [zzxy] [=] [zzyx], [xxzz] [=] [xzxz] [=] [xzzx] [=] [zxxz] [=] [zxzx] [=] [zzxx], [xxyy] [=] [xyxy] [=] [xyyx] [=] [yxxy] [=] [yxyx] [=] [yyxx], [xxxy=xxyx=xyxx=yxxx], [yyyy], [yyzz] [=] [yzyz] [=] [yzzy] [=] [zyyz] [=] [zyzy] [=] [zzyy], [zzzz]
   
Orthorhombic  
C2v (mm2), D2 (222), D2h (mmm) (twofold axis parallel to z) [xxxx], [xxzz] [=] [xzxz] [=] [xzzx] [=] [zxxz] [=] [zxzx] [=] [zzxx], [xxyy] [=] [xyxy] [=] [xyyx] [=] [yxxy] [=] [yxyx] [=] [yyxx], [yyyy], [yyzz] [=] [yzyz] [=] [yzzy] [=] [zyyz] [=] [zyzy] [=] [zzyy], [zzzz]
   
Tetragonal  
S4 ([\bar 4]), C4 (4), C4h [\left(4\over m\right)] [xxxx=yyyy], [xyyy] [=] [yxyy] [=] [yyxy] [=] [yyyx] [=] [-xxxy] [=] [-xxyx] [=] [-xyxx] [=] [-yxxx], [xxzz] [=] [xzxz] [=] [xzzx] [=] [yyzz] [=] [yzyz] [=] [yzzy] [=] [zyyz] [=] [zyzy] [=] [zzyy] [=] [zxxz] [=] [zxzx] [=] [zzxx], [xxyy] [=] [xyxy] [=] [xyyx] [=] [yxxy] [=] [yxyx] [=] [yyxx], [zzzz]
C4v (4mm), D2d ([\bar 4 2 m]), D4 (422), D4h [\left({4 \over m}mm\right)] [xxxx=yyyy], [xxzz] [=] [xzxz] [=] [xzzx] [=] [yyzz] [=] [yzyz] [=] [yzzy] [=] [zyyz] [=] [zyzy] [=] [zzyy] [=] [zxxz] [=] [zxzx] [=] [zzxx], [xxyy] [=] [xyxy] [=] [xyyx] [=] [yxxy] [=] [yxyx] [=] [yyxx], [zzzz]
   
Hexagonal  
C3h ([\bar 6]), C6 (6), C6h [\,\left(6\over m\right)], C6v (6mm), D3h ([\bar 6 2 m]), D6 (622), D6h [\left({6\over m}mm\right)] [xxxx=yyyy=xxyy+xyxy+xyyx], [xxzz] [=] [xzxz] [=] [xzzx] [=] [yyzz] [=] [yzyz] [=] [yzzy] [=] [zyyz] [=] [zyzy] [=] [zzyy] [=] [zxxz] [=] [zxzx] [=] [zzxx], [xxyy] [=] [xyxy] [=] [xyyx] [=] [yxxy] [=] [yxyx] [=] [yyxx], [zzzz]
   
Trigonal  
C3 (3), C3i ([\bar 3]) [xxxx=yyyy=xxyy+xyxy+xyyx], [xyyz] [=] [xyzy] [=] [xzyy] [=] [-xxxz] [=] [-xxzx] [=] [-xzxx] [=] [yxyz] [=] [yxzy] [=] [yyxz] [=] [yyzx] [=] [yzxy] [=] [yzyx] [=] [-zxxx] [=] [zxyy] [=] [ zyxy] [=] [zyyx], [xxzz] [=] [xzxz] [=] [xzzx] [=] [yyzz] [=] [yzyz] [=] [yzzy] [=] [zyyz] [=] [zyzy] [=] [zzyy] [=] [zxxz] [=] [zxzx] [=] [zzxx], [xxyy] [=] [xyxy] [=] [xyyx] [=] [yxxy] [=] [yxyx] [=] [yyxx], [xxyz] [=] [xxzy] [=] [xyxz] [=] [xyzx] [=] [xzxy] [=] [xzyx] [=] [-yyyz] [=] [-yyzy] [=] [-yzyy] [=] [yxxz] [=] [yxzx] [=] [yzxx] [=] [-zyyy] [=] [zxxy] [=] [zxyx] [=] [zyxx], [zzzz]
C3v (3m), D3 (32), D3d ([\bar 3 m]) (mirror perpendicular to x) (twofold axis parallel to x) [xxxx=yyyy=xxyy+xyxy+xyyx], [xxzz] [=] [xzxz] [=] [xzzx] [=] [yyzz] [=] [yzyz] [=] [yzzy] [=] [zyyz] [=] [zyzy] [=] [zzyy] [=] [zxxz] [=] [zxzx] [=] [zzxx], [xxyy] [=] [xyxy] [=] [xyyx] [=] [yxxy] [=] [yxyx] [=] [yyxx], [xxyz] [=] [xxzy] [=] [xyxz] [=] [xyzx] [=] [xzxy] [=] [xzyx] [=] [-yyyz] [=] [-yyzy] [=] [-yzyy] [=] [yxxz] [=] [yxzx] [=] [yzxx] [=] [-zyyy] [=] [zxxy] [=] [zxyx] [=] [zyxx], [zzzz]
   
Cubic  
T (23), Th (m3), Td ([\bar 4 3 m]), O (432), Oh (m3m) [xxxx=yyyy=zzzz], [xxzz] [=] [xzxz] [=] [xzzx] [=] [xxyy] [=] [xyxy] [=] [xyyx] [=] [yyzz] [=] [yzyz] [=] [yzzy] [=] [yyxx] [=] [yxyx] [=] [yxxy] [=] [zzyy] [=] [zyzy] [=] [zyyz] [=] [zzxx] [=] [zxzx] [=] [zxxz]