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. 184

Table 1.7.2.4 

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 1.7.2.4| top | pdf |
Nonzero χ(3) coefficients and equalities between them in the general case

Symmetry classχ(3) nonzero elements
Triclinic  
C1 (1), Ci ([\bar 1]) All 81 elements are independent and nonzero
   
Monoclinic  
Cs (m), C2 (2), C2h [\left(2 \over m\right)] (twofold axis parallel to z) [xxxx], [xyyy], [xyzz], [xzyz], [xzzy], [xxzz], [xzxz], [xzzx], [xxyy], [xyxy], [xyyx], [xxxy], [xxyx], [xyxx], [yxxx], [yyyy], [yyzz], [yzyz], [yzzy], [yxzz], [yzxz], [yzzx], [yxyy], [yyxy], [yyyx], [yxxy], [yxyx], [yyxx], [zzzz], [zyyz], [zyzy], [zzyy], [zxxz], [zxzx], [zzxx], [zxyz], [zxzy], [zyxz], [zzxy], [zyzx], [zzyx]
   
Orthorhombic  
C2v (mm2), D2 (222), D2h (mmm) (twofold axis parallel to z) [xxxx], [xxzz], [xzxz], [xzzx], [xxyy], [xyxy], [xyyx], [yyyy], [yyzz], [yzyz], [yzzy], [yxxy], [yxyx], [yyxx], [zzzz], [zyyz], [zyzy], [zzyy], [zxxz], [zxzx], [zzxx]
   
Tetragonal  
S4 ([\bar 4]), C4 (4), C4h [\left(4\over m\right)] [xxxx=yyyy], [xyyy=-yxxx], [xyzz=-yxzz], [xzyz=-yzxz], [xzzy=-yzzx], [xxzz=yyzz], [xzxz=yzyz], [xzzx=yzzy], [xxyy=yyxx], [xyxy=yxyx], [xyyx=yxxy], [xxxy=-yyyx], [xxyx=-yyxy], [xyxx=-yxyy], [zzzz], [zyyz=zxxz], [zyzy=zxzx], [zzyy=zzxx], [zxyz=-zyxz], [zxzy=-zyzx], [zzxy=-zzyx]
C4v (4mm), D2d ([\bar 4 2 m]), D4 (422), D4h [\left({4 \over m}mm\right)] [xxxx=yyyy], [xxzz=yyzz], [xzxz=yzyz], [xzzx=yzzy], [xxyy=yyxx], [xyxy=yxyx], [xyyx=yxxy], [zzzz], [zyyz=zxxz], [zyzy=zxzx], [zzyy=zzxx]
   
Hexagonal  
C3h ([\bar 6]), C6 (6), C6h [\left(6\over m\right)] [xxxx=yyyy=xxyy+xyxy+xyyx], [xyyy=xxxy+xxyx+xyxx=-yxxx], [xyzz=-yxzz], [xzyz=-yzxz], [xzzy=-yzzx], [xxzz=yyzz], [xzxz=yzyz], [xzzx=yzzy], [xxyy=yyxx], [xyxy=yxyx], [xyyx=yxxy], [xxxy=-yyyx], [xxyx=-yyxy], [xyxx=-yxyy], [zzzz], [zyyz=zxxz], [zyzy=zxzx], [zzyy=zzxx], [zxyz=-zyxz], [zxzy=-zyzx], [zzxy=-zzyx]
C6v (6mm), D3h ([\bar 6 2 m]), D6 (622), D6h [\left({6 \over m}mm\right)] [xxxx=yyyy=xxyy+xyxy+xyyx], [xxzz=yyzz], [xzxz=yzyz], [xzzx=yzzy], [xxyy=yyxx], [xyxy=yxyx], [xyyx=yxxy], [zzzz], [zyyz=zxxz], [zyzy=zxzx], [zzyy=zzxx]
   
Trigonal  
C3 (3), C3i ([\bar 3]) [xxxx=yyyy=xxyy+xyxy+xyyx], [xyyy=xxxy+xxyx+xyxx=-yxxx], [xyzz=-yxzz], [xzyz=-yzxz], [xzzy=-yzzx], [xyyz=yxyz=yyxz=-xxxz], [xyzy=yyzx=yxzy=-xxzx], [xzyy=yzxy=yzyx=-xzxx], [xxzz=yyzz], [xzxz=yzyz], [xzzx=yzzy], [xxyy=yyxx], [xyxy=yxyx], [xyyx=yxxy], [xxxy=-yyyx], [xxyx=-yyxy], [xyxx=-yxyy], [xxyz] [=] [xyxz] [=] [yxxz] [=] [-yyyz], [xxzy] [=] [xyzx] [=] [yxzx] [=] [-yyzy], [xzxy] [=] [xzyx] [=] [yzxx] [=] [-yzyy], [-zxxx] [=] [zxyy] [=] [zyxy] [=] [zyyx], [-zyyy] [=] [zxxy] [=] [zxyx] [=] [zyxx], [zzzz], [zyyz=zxxz], [zyzy=zxzx], [zzyy=zzxx], [zxyz=-zyxz], [zxzy=-zyzx], [zzxy=-zzyx]
C3v (3m), D3 (32), D3d ([\bar 3 m]) (mirror perpendicular to x) (twofold axis parallel to x) [xxxx=yyyy=xxyy+xyxy+xyyx], [xxzz=yyzz], [xzxz=yzyz], [xzzx=yzzy], [xxyy=yyxx], [xyxy=yxyx], [xyyx=yxxy], [xxyz=xyxz=yxxz=-yyyz], [xxzy=xyzx=yxzx=-yyzy], [xzxy=xzyx=yzxx=-yzyy], [-zyyy] [=] [zxxy] [=] [zxyx] [=] [zyxx], [zzzz], [zyyz=zxxz], [zyzy=zxzx], [zzyy=zzxx]
   
Cubic  
T (23), Th (m3) [xxxx=yyyy=zzzz], [xxzz=yyxx=zzyy], [xzxz=yxyx=zyzy], [xzzx=yxxy=zyyz], [xxyy=yyzz=zzxx], [xyxy] [=] [yzyz] [=] [zxzx], [xyyx] [=] [yzzy] [=] [zxxz]
Td ([\bar 4 3 m]), O (432), Oh (m3m) [xxxx=yyyy=zzzz], [xxzz=xxyy=yyzz=yyxx=zzyy=zzxx], [xzxz=xyxy=yzyz=yxyx=zyzy=zxzx], [xzzx=xyyx=yzzy=yxxy=zyyz=zxxz]