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. 184-186

Section 1.7.2.2. Symmetry properties

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. Symmetry properties

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1.7.2.2.1. Intrinsic permutation symmetry

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1.7.2.2.1.1. ABDP and Kleinman symmetries

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Intrinsic permutation symmetry, as already discussed, imposes the condition that the nth order susceptibility [\chi^{(n)}_{\mu\alpha_1\alpha_2\ldots\alpha_n}(-\omega_\sigma;] [\omega_1,\omega_2,\ldots,\omega_n)] be invariant under the [n!] permutations of the ([\alpha_i,\omega_i]) pairs as a result of time invariance and causality. Furthermore, the overall permutation symmetry, i.e. the invariance over the [(n+1)!] permutations of the ([\alpha_i,\omega_i]) and ([\mu,-\omega_\sigma]) pairs, may be valid when all the optical frequencies occuring in the susceptibility and combinations of these appearing in the denominators of quantum expressions are far removed from the transitions, making the medium transparent at these frequencies. This property is termed ABDP symmetry, from the initials of the authors of the pioneering article by Armstrong et al. (1962[link]).

Let us consider as an application the quantum expression of the quadratic susceptibility (with damping factors neglected), the derivation of which being beyond the scope of this chapter, but which can be found in nonlinear optics treatises dealing with microscopic interactions, such as in Boyd (1992[link]):[\eqalignno{&\chi^{(2)}_{\mu\alpha\beta}(-\omega_\sigma\semi\omega_1,\omega_2)&\cr&\quad ={Ne^3 \over \varepsilon_o^2\hbar^2}S_T\displaystyle\sum\limits_{abc}\rho_o(a){r_{ab}^{\mu}r_{bc}^\alpha r_{ca}^{\beta}\over (\Omega_{ba}-\omega_1-\omega_2)(\Omega_{ca}-\omega_1)},&\cr&&(1.7.2.36)}]where N is the number of microscopic units (e.g. molecules in the case of organic crystals) per unit volume, a, b and c are the eigen states of the system, Ωba and Ωca are transition energies, [r_{ab}^\mu] is the μ component of the transition dipole connecting states a and b, and [\rho_o(a)] is the population of level a as given by the corresponding diagonal term of the density operator. ST is the summation operator over the six permutations of the ([\mu, -\omega_\sigma]), ([\alpha, \omega_1]), ([\beta, \omega_2]). Provided all frequencies at the denominator are much smaller than the transition frequencies Ωba and Ωca, the optical frequencies [-\omega_\sigma], [\omega_1], [\omega_2] can be permuted without significant variation of the susceptibility. It follows correspondingly that the susceptibility is invariant with respect to the permutation of Cartesian indices appearing only in the numerator of (1.7.2.36)[link], regardless of frequency. This property, which can be generalized to higher-order susceptibilities, is known as Kleinman symmetry. Its validity can help reduce the number of non-vanishing terms in the susceptibility, as will be shown later.

1.7.2.2.1.2. Manley–Rowe relations

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An important consequence of overall permutation symmetry is the Manley–Rowe power relations, which account for energy exchange between electromagnetic waves in a purely reactive (e.g. non-dissipative) medium. Calling Wi the power input at frequency ωi into a unit volume of a dielectric polarizable medium,[W_i=\left\langle {\bf E}(t)\cdot{{\rm d}{\bf P} \over {\rm d}t}(t)\right\rangle,\eqno(1.7.2.37)]where the averaging is performed over a cycle and[\eqalignno{{\bf E}(t)&=Re[E_{\omega_i}\exp(-j\omega_i t)]&\cr {\bf P}(t)&=Re[P_{\omega_i}\exp(-j\omega_i t)].&(1.7.2.38)}]The following expressions can be derived straightforwardly:[W_i=\textstyle{1 \over 2}\omega_i \,Re(iE_{\omega_i}\cdot P_{\omega_i})=\textstyle{1 \over 2}\omega_i \,Im(E_{\omega_i}^* \cdot P_{\omega_i}).\eqno(1.7.2.39)]Introducing the quadratic induced polarization P(2), Manley–Rowe relations for sum-frequency generation state[{W_1 \over \omega_1}={W_2 \over \omega_2}=-{W_3 \over \omega_3}.\eqno(1.7.2.40)]Since [\omega_1+\omega_2=\omega_3], (1.7.2.40)[link] leads to an energy conservation condition, namely [W_3+W_1+W_2=0], which expresses that the power generated at ω3 is equal to the sum of the powers lost at ω1 and ω2.

A quantum mechanical interpretation of these expressions in terms of photon fusion or splitting can be given, remembering that [W_i/\hbar\omega_i] is precisely the number of photons generated or annihilated per unit volume in unit time in the course of the nonlinear interactions.

1.7.2.2.1.3. Contracted notation for susceptibility tensors

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The tensors [\chi^{(2)}_{\mu\alpha\beta}(-2\omega;\omega,\omega)] or [d^{(2)}_{\mu\alpha\beta}(-2\omega;\omega,\omega)] are invariant with respect to (α, β) permutation as a consequence of the intrinsic permutation symmetry. Independently, it is not possible to distinguish the coefficients [\chi^{(2)}_{ijk}(-2\omega;\omega,\omega)] and [\chi^{(2)}_{ikj}(-2\omega;\omega,\omega)] by SHG experiments, even if the two fundamental waves have different directions of polarization.

Therefore, these third-rank tensors can be represented in contracted form as [3\times 6] matrices [\chi_{\mu m}(-2\omega;\omega,\omega)] and [d_{\mu m}(-2\omega;\omega,\omega)], where the suffix m runs over the six possible (α, β) Cartesian index pairs according to the classical convention of contraction:[\eqalign{\hbox{for }\mu\hbox{: } &x\rightarrow 1\quad y\rightarrow 2\quad z\rightarrow 3\hfill\cr \hbox{for }m\hbox{: } &xx\rightarrow 1\quad yy\rightarrow 2\quad zz\rightarrow 3\quad yz=zy\rightarrow 4\hfill\cr& xz=zx\rightarrow 5\quad xy=yx\rightarrow 6.\hfill}]The 27 elements of [\chi^{(2)}_{\mu\alpha\beta}(-2\omega;\omega,\omega)] are then reduced to 18 in the [\chi_{\mu m}] contracted tensor notation (see Section 1.1.4.10[link] ).

For example, (1.7.2.35)[link] can be written[\eqalignno{P_y^{(2)}(2\omega)&=\varepsilon_o\chi_{25}(-2\omega\semi\omega,\omega)[e_x^+(\omega){\bf E}^+(\omega)e_z^-(\omega){\bf E}^-(\omega)&\cr&\quad +e_z^+(\omega){\bf E}^+(\omega)e_x^-(\omega){\bf E}^-(\omega)].&(1.7.2.41)}]The same considerations can be applied to THG. Then the 81 elements of [\chi^{(3)}_{\mu\alpha\beta\gamma}(-3\omega;\omega,\omega,\omega)] can be reduced to 30 in the [\chi_{\mu m}] contracted tensor notation with the following contraction convention:[\eqalign{\hbox{for }\mu\hbox{: } &x\rightarrow 1\quad y\rightarrow 2\quad z\rightarrow 3\hfill\cr \hbox{for }m\hbox{: } &xxx\rightarrow 1\quad yyy\rightarrow 2\quad zzz\rightarrow 3\quad yzz \rightarrow 4\quad yyz\rightarrow 5\hfill\cr& xzz\rightarrow 6\quad xxz\rightarrow 7\quad xyy\rightarrow 8\quad xxy\rightarrow 9\quad xyz\rightarrow 0.\hfill}]If Kleinman symmetry holds, the contracted tensor can be further extended beyond SHG and THG to any other processes where all the frequencies are different.

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]

References

Armstrong, J. A., Bloembergen, N., Ducuing, J. & Pershan, P. (1962). Interactions between light waves in a nonlinear dielectric. Phys. Rev. 127, 1918–1939.
Boyd, R. W. (1992). Nonlinear optics. San Diego: Academic Press.








































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