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. 185

Table 1.7.2.5 

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 1.7.2.5| 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]