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

International Tables for Crystallography (2013). Vol. D, ch. 1.7, pp. 185-186

Section 1.7.2.2.2. Implications of spatial symmetry on the susceptibility tensors

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

1.7.2.2.2. Implications of spatial symmetry on the susceptibility tensors

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Centrosymmetry is the most detrimental crystalline symmetry constraint that will fully cancel all odd-rank tensors such as the d(2) [or χ(2)] susceptibilities. Intermediate situations, corresponding to noncentrosymmetric crystalline point groups, will reduce the number of nonzero coefficients without fully depleting the tensors.

Tables 1.7.2.2[link] to 1.7.2.5[link][link][link] detail, for each crystal point group, the remaining nonzero χ(2) and χ(3) coefficients and the eventual connections between them. χ(2) and χ(3) are expressed in the principal axes x, y and z of the second-rank χ(1) tensor. ([x,y,z]) is usually called the optical frame; it is linked to the crystallographical frame by the standard conventions given in Chapter 1.6[link] .

Table 1.7.2.2| top | pdf |
Nonzero χ(2) coefficients and equalities between them in the general case

Symmetry classχ(2) nonzero elements
Triclinic  
C1 (1) All 27 elements are independent and nonzero
   
Monoclinic  
C2 (2) (twofold axis parallel to z) [xyz], [xzy], [xxz], [xzx], [yyz], [yzy], [yxz], [yzx], [zxx], [zyy], [zzz], [zxy], [zyx]
Cs (m) (mirror perpendicular to z) [xxx], [xyy], [xzz], [xxy], [xyx], [yxx], [yyy], [yzz], [yxy], [yyx], [zyz], [zzy], [zxz], [zzx]
   
Orthorhombic  
C2v (mm2) (twofold axis parallel to z) [xzx], [xxz], [yyz], [yzy], [zxx], [zyy], [zzz]
D2 (222) [xyz], [xzy], [yzx], [yxz], [zxy], [zyx]
   
Tetragonal  
C4 (4) [xyz=-yxz], [xzy=-yzx], [xzx=yzy], [xxz=yyz], [zxx=zyy], [zzz], [zxy=-zyx]
S4 ([\bar 4]) [xyz=yxz], [xzy=yzx], [xzx=-yzy], [xxz=-yyz], [zxx=-zyy], [zxy=zyx]
D4 (422) [xyz=-yxz], [xzy=-yzx], [zxy=-zyx]
C4v (4mm) [xzx=yzy], [xxz=yyz], [zxx=zyy], [zzz]
D2d ([\bar 4 2 m]) [xyz=yxz], [xzy=yzx], [zxy=zyx]
   
Hexagonal  
C6 (6) [xyz=-yxz], [xzy=-yzx], [xzx=yzy], [xxz=yyz], [zxx=zyy], [zzz], [zxy=-zyx]
C3h ([\bar 6]) [xxx=-xyy=-yxy=-yyx], [yyy=-yxx=] [-xyx=-xxy]
D6 (622) [xyz=-yxz], [xzy=-yzx], [zxy=-zyx]
C6v (6mm) [xzx=yzy], [xxz=yyz], [zxx=zyy], [zzz]
D3h ([\bar 6 2 m]) (mirror perpendicular to x) [yyy=-yxx=-xxy=-xyx]
   
Trigonal  
C3 (3) [xxx=-xyy=-yyx=-yxy], [xyz=-yxz], [xzy=-yzx], [xzx=yzy], [xxz=yyz], [yyy=-yxx=-xxy=-xyx], [zxx=zyy], [zzz], [zxy=-zyx]
D3 (32) [xxx=-xyy=-yyx=-yxy], [xyz=-yxz], [xzy=-yzx], [zxy=-zyx]
C3v (3m) (mirror perpendicular to x) [yyy=-yxx=-xxy=-xyx], [xzx=yzy], [xxz=yyz], [zxx=zyy, zzz]
   
Cubic  
T (23), Td ([\bar 4 3 m]) [xyz=xzy=yzx=yxz=zxy=zyx]
O (432) [xyz=-xzy=yzx=-yxz=zxy=-zyx]

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

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]

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]








































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