Abstract

We have analyzed the depolarized hyper-Rayleigh scattering from a molecular system with partial macroscopic polar ordering. It is shown that the hyperpolarizability tensor ratio and the molecular dipole moment can be determined simultaneously by the measurement of depolarization as a function of external field strength. As an experimental example, we performed a quantitative analysis of the electric-field-dependent depolarized hyper-Rayleigh signal from a poly-γ-benzyl-l-glutamate solution, obtaining the dipole moment and the ratio of hyperpolarizability components as 4.0 D and β311/β333=-0.81, respectively.

© 2002 Optical Society of America

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  1. K. Clays and A. Persoons, “Hyper-Rayleigh scattering in solution,” Phys. Rev. Lett. 66, 2980–2983 (1991).
    [CrossRef] [PubMed]
  2. G. J. T. Heesink, A. G. Ruiter, N. F. van Hulst, and B. Boelger, “Determination of hyperpolarizability tensor components by depolarized hyper Rayleigh scattering,” Phys. Rev. Lett. 71, 999–1002 (1993).
    [CrossRef] [PubMed]
  3. T. Verbiest, M. Kauranen, and A. Persoon, “Parametric light scattering,” J. Chem. Phys. 101, 1745–1747 (1994).
    [CrossRef]
  4. R. Bersohn, Y.-H. Pao, and H. L. Frisch, “Double-quantum light scattering by molecules,” J. Chem. Phys. 45, 3184–3198 (1966).
    [CrossRef]
  5. S. J. Cyvin, J. E. Rauch, and J. C. Decius, “Theory of hyper-Raman effects (nonlinear inelastic light scattering): selection rules and depolarization ratios for the second-order polarizability,” J. Chem. Phys. 43, 4083–4095 (1965).
    [CrossRef]
  6. B. F. Levine and C. G. Bethea, “Second and third order hy-perpolarizabilities of organic molecules,” J. Chem. Phys. 63, 2666–2682 (1975).
    [CrossRef]
  7. M. Joffre, D. Yaron, and R. J. Silbey, “Second order optical nonlinearity in octupolar aromatic systems,” J. Chem. Phys. 97, 5607–5615 (1992).
    [CrossRef]
  8. B. F. Levine and C. G. Bethea, “Second order hyperpolarizability of a polypeptide α-helix: poly-γ-benzyl-L-glutamate,” J. Chem. Phys. 65, 1989–1993 (1976).
    [CrossRef]
  9. J. Zyss, “Octupolar organic systems in quadratic nonlinear optics: molecules and materials,” Nonlinear Opt. 1, 3–18 (1991).
  10. T. Verbiest, K. Clays, A. Persoons, F. Meyers, and J. L. Breda, “Determination of the hyperpolarizability of an octopolar molecular ion by hyper-Rayleigh scattering,” Opt. Lett. 18, 525–527 (1993).
    [CrossRef] [PubMed]
  11. W. M. Laidlaw, R. G. Denning, T. Verbiest, E. Chauchard, and A. Persoons, “Large second-order optical polarizabilities in mixed-valency metal complexes,” Nature 363, 58–60 (1993).
    [CrossRef]
  12. K. D. Singer, J. E. Sohn, L. A. King, H. M. Gordon, H. E. Katz, and C. W. Dirk, “Second-order nonlinear-optical properties of donor- and acceptor-substituted aromatic compounds,” J. Opt. Soc. Am. B 6, 1339–1350 (1989).
    [CrossRef]
  13. R. W. Terhune, P. D. Maker, and C. M. Savage, “Measurements of nonlinear light scattering,” Phys. Rev. Lett. 14, 681–684 (1965).
    [CrossRef]
  14. J. W. Wu, “Birefringent and electro-optic effects in poled polymer films: steady-state and transient properties,” J. Opt. Soc. Am. B 8, 142–152 (1991).
    [CrossRef]
  15. J. Watanabe, Y. Hirose, M. Tokita, T. Watanabe, and S. Miyata, “Polar structure in polypeptide cholesteric liquid crystals evidenced from observation of second-harmonic generation due to the helicoidal cavity effect,” Macromolecules 31, 5937–5939 (1998).
    [CrossRef]
  16. D. S. Chemla and J. Zyss, Nonlinear Optical Properties of Organic Molecules and Crystals (Academic, Orlando, Fla., 1987).
  17. D. R. Lide, Handbook of Organic Solvents (CRC Press, New York, 1995).
  18. B. Park, Y. Kinoshita, H. Takezoe, and J. Watanabe, “Ferroelectricity in the lyotropic cholesteric phase of poly L-glutamate,” Jpn. J. Appl. Phys. 37, L136–L138 (1998).
    [CrossRef]
  19. B. Park, J. W. Wu, and H. Takezoe, “Generalized mean-field potential description for ferroelectric ordering in nematic liquid crystals,” Phys. Rev. E 63, 21707–1–21707–7 (2001).
    [CrossRef]

2001 (1)

B. Park, J. W. Wu, and H. Takezoe, “Generalized mean-field potential description for ferroelectric ordering in nematic liquid crystals,” Phys. Rev. E 63, 21707–1–21707–7 (2001).
[CrossRef]

1998 (2)

J. Watanabe, Y. Hirose, M. Tokita, T. Watanabe, and S. Miyata, “Polar structure in polypeptide cholesteric liquid crystals evidenced from observation of second-harmonic generation due to the helicoidal cavity effect,” Macromolecules 31, 5937–5939 (1998).
[CrossRef]

B. Park, Y. Kinoshita, H. Takezoe, and J. Watanabe, “Ferroelectricity in the lyotropic cholesteric phase of poly L-glutamate,” Jpn. J. Appl. Phys. 37, L136–L138 (1998).
[CrossRef]

1994 (1)

T. Verbiest, M. Kauranen, and A. Persoon, “Parametric light scattering,” J. Chem. Phys. 101, 1745–1747 (1994).
[CrossRef]

1993 (3)

G. J. T. Heesink, A. G. Ruiter, N. F. van Hulst, and B. Boelger, “Determination of hyperpolarizability tensor components by depolarized hyper Rayleigh scattering,” Phys. Rev. Lett. 71, 999–1002 (1993).
[CrossRef] [PubMed]

W. M. Laidlaw, R. G. Denning, T. Verbiest, E. Chauchard, and A. Persoons, “Large second-order optical polarizabilities in mixed-valency metal complexes,” Nature 363, 58–60 (1993).
[CrossRef]

T. Verbiest, K. Clays, A. Persoons, F. Meyers, and J. L. Breda, “Determination of the hyperpolarizability of an octopolar molecular ion by hyper-Rayleigh scattering,” Opt. Lett. 18, 525–527 (1993).
[CrossRef] [PubMed]

1992 (1)

M. Joffre, D. Yaron, and R. J. Silbey, “Second order optical nonlinearity in octupolar aromatic systems,” J. Chem. Phys. 97, 5607–5615 (1992).
[CrossRef]

1991 (3)

J. Zyss, “Octupolar organic systems in quadratic nonlinear optics: molecules and materials,” Nonlinear Opt. 1, 3–18 (1991).

K. Clays and A. Persoons, “Hyper-Rayleigh scattering in solution,” Phys. Rev. Lett. 66, 2980–2983 (1991).
[CrossRef] [PubMed]

J. W. Wu, “Birefringent and electro-optic effects in poled polymer films: steady-state and transient properties,” J. Opt. Soc. Am. B 8, 142–152 (1991).
[CrossRef]

1989 (1)

1976 (1)

B. F. Levine and C. G. Bethea, “Second order hyperpolarizability of a polypeptide α-helix: poly-γ-benzyl-L-glutamate,” J. Chem. Phys. 65, 1989–1993 (1976).
[CrossRef]

1975 (1)

B. F. Levine and C. G. Bethea, “Second and third order hy-perpolarizabilities of organic molecules,” J. Chem. Phys. 63, 2666–2682 (1975).
[CrossRef]

1966 (1)

R. Bersohn, Y.-H. Pao, and H. L. Frisch, “Double-quantum light scattering by molecules,” J. Chem. Phys. 45, 3184–3198 (1966).
[CrossRef]

1965 (2)

S. J. Cyvin, J. E. Rauch, and J. C. Decius, “Theory of hyper-Raman effects (nonlinear inelastic light scattering): selection rules and depolarization ratios for the second-order polarizability,” J. Chem. Phys. 43, 4083–4095 (1965).
[CrossRef]

R. W. Terhune, P. D. Maker, and C. M. Savage, “Measurements of nonlinear light scattering,” Phys. Rev. Lett. 14, 681–684 (1965).
[CrossRef]

Bersohn, R.

R. Bersohn, Y.-H. Pao, and H. L. Frisch, “Double-quantum light scattering by molecules,” J. Chem. Phys. 45, 3184–3198 (1966).
[CrossRef]

Bethea, C. G.

B. F. Levine and C. G. Bethea, “Second order hyperpolarizability of a polypeptide α-helix: poly-γ-benzyl-L-glutamate,” J. Chem. Phys. 65, 1989–1993 (1976).
[CrossRef]

B. F. Levine and C. G. Bethea, “Second and third order hy-perpolarizabilities of organic molecules,” J. Chem. Phys. 63, 2666–2682 (1975).
[CrossRef]

Boelger, B.

G. J. T. Heesink, A. G. Ruiter, N. F. van Hulst, and B. Boelger, “Determination of hyperpolarizability tensor components by depolarized hyper Rayleigh scattering,” Phys. Rev. Lett. 71, 999–1002 (1993).
[CrossRef] [PubMed]

Breda, J. L.

Chauchard, E.

W. M. Laidlaw, R. G. Denning, T. Verbiest, E. Chauchard, and A. Persoons, “Large second-order optical polarizabilities in mixed-valency metal complexes,” Nature 363, 58–60 (1993).
[CrossRef]

Clays, K.

Cyvin, S. J.

S. J. Cyvin, J. E. Rauch, and J. C. Decius, “Theory of hyper-Raman effects (nonlinear inelastic light scattering): selection rules and depolarization ratios for the second-order polarizability,” J. Chem. Phys. 43, 4083–4095 (1965).
[CrossRef]

Decius, J. C.

S. J. Cyvin, J. E. Rauch, and J. C. Decius, “Theory of hyper-Raman effects (nonlinear inelastic light scattering): selection rules and depolarization ratios for the second-order polarizability,” J. Chem. Phys. 43, 4083–4095 (1965).
[CrossRef]

Denning, R. G.

W. M. Laidlaw, R. G. Denning, T. Verbiest, E. Chauchard, and A. Persoons, “Large second-order optical polarizabilities in mixed-valency metal complexes,” Nature 363, 58–60 (1993).
[CrossRef]

Dirk, C. W.

Frisch, H. L.

R. Bersohn, Y.-H. Pao, and H. L. Frisch, “Double-quantum light scattering by molecules,” J. Chem. Phys. 45, 3184–3198 (1966).
[CrossRef]

Gordon, H. M.

Heesink, G. J. T.

G. J. T. Heesink, A. G. Ruiter, N. F. van Hulst, and B. Boelger, “Determination of hyperpolarizability tensor components by depolarized hyper Rayleigh scattering,” Phys. Rev. Lett. 71, 999–1002 (1993).
[CrossRef] [PubMed]

Hirose, Y.

J. Watanabe, Y. Hirose, M. Tokita, T. Watanabe, and S. Miyata, “Polar structure in polypeptide cholesteric liquid crystals evidenced from observation of second-harmonic generation due to the helicoidal cavity effect,” Macromolecules 31, 5937–5939 (1998).
[CrossRef]

Joffre, M.

M. Joffre, D. Yaron, and R. J. Silbey, “Second order optical nonlinearity in octupolar aromatic systems,” J. Chem. Phys. 97, 5607–5615 (1992).
[CrossRef]

Katz, H. E.

Kauranen, M.

T. Verbiest, M. Kauranen, and A. Persoon, “Parametric light scattering,” J. Chem. Phys. 101, 1745–1747 (1994).
[CrossRef]

King, L. A.

Kinoshita, Y.

B. Park, Y. Kinoshita, H. Takezoe, and J. Watanabe, “Ferroelectricity in the lyotropic cholesteric phase of poly L-glutamate,” Jpn. J. Appl. Phys. 37, L136–L138 (1998).
[CrossRef]

Laidlaw, W. M.

W. M. Laidlaw, R. G. Denning, T. Verbiest, E. Chauchard, and A. Persoons, “Large second-order optical polarizabilities in mixed-valency metal complexes,” Nature 363, 58–60 (1993).
[CrossRef]

Levine, B. F.

B. F. Levine and C. G. Bethea, “Second order hyperpolarizability of a polypeptide α-helix: poly-γ-benzyl-L-glutamate,” J. Chem. Phys. 65, 1989–1993 (1976).
[CrossRef]

B. F. Levine and C. G. Bethea, “Second and third order hy-perpolarizabilities of organic molecules,” J. Chem. Phys. 63, 2666–2682 (1975).
[CrossRef]

Maker, P. D.

R. W. Terhune, P. D. Maker, and C. M. Savage, “Measurements of nonlinear light scattering,” Phys. Rev. Lett. 14, 681–684 (1965).
[CrossRef]

Meyers, F.

Miyata, S.

J. Watanabe, Y. Hirose, M. Tokita, T. Watanabe, and S. Miyata, “Polar structure in polypeptide cholesteric liquid crystals evidenced from observation of second-harmonic generation due to the helicoidal cavity effect,” Macromolecules 31, 5937–5939 (1998).
[CrossRef]

Pao, Y.-H.

R. Bersohn, Y.-H. Pao, and H. L. Frisch, “Double-quantum light scattering by molecules,” J. Chem. Phys. 45, 3184–3198 (1966).
[CrossRef]

Park, B.

B. Park, J. W. Wu, and H. Takezoe, “Generalized mean-field potential description for ferroelectric ordering in nematic liquid crystals,” Phys. Rev. E 63, 21707–1–21707–7 (2001).
[CrossRef]

B. Park, Y. Kinoshita, H. Takezoe, and J. Watanabe, “Ferroelectricity in the lyotropic cholesteric phase of poly L-glutamate,” Jpn. J. Appl. Phys. 37, L136–L138 (1998).
[CrossRef]

Persoon, A.

T. Verbiest, M. Kauranen, and A. Persoon, “Parametric light scattering,” J. Chem. Phys. 101, 1745–1747 (1994).
[CrossRef]

Persoons, A.

T. Verbiest, K. Clays, A. Persoons, F. Meyers, and J. L. Breda, “Determination of the hyperpolarizability of an octopolar molecular ion by hyper-Rayleigh scattering,” Opt. Lett. 18, 525–527 (1993).
[CrossRef] [PubMed]

W. M. Laidlaw, R. G. Denning, T. Verbiest, E. Chauchard, and A. Persoons, “Large second-order optical polarizabilities in mixed-valency metal complexes,” Nature 363, 58–60 (1993).
[CrossRef]

K. Clays and A. Persoons, “Hyper-Rayleigh scattering in solution,” Phys. Rev. Lett. 66, 2980–2983 (1991).
[CrossRef] [PubMed]

Rauch, J. E.

S. J. Cyvin, J. E. Rauch, and J. C. Decius, “Theory of hyper-Raman effects (nonlinear inelastic light scattering): selection rules and depolarization ratios for the second-order polarizability,” J. Chem. Phys. 43, 4083–4095 (1965).
[CrossRef]

Ruiter, A. G.

G. J. T. Heesink, A. G. Ruiter, N. F. van Hulst, and B. Boelger, “Determination of hyperpolarizability tensor components by depolarized hyper Rayleigh scattering,” Phys. Rev. Lett. 71, 999–1002 (1993).
[CrossRef] [PubMed]

Savage, C. M.

R. W. Terhune, P. D. Maker, and C. M. Savage, “Measurements of nonlinear light scattering,” Phys. Rev. Lett. 14, 681–684 (1965).
[CrossRef]

Silbey, R. J.

M. Joffre, D. Yaron, and R. J. Silbey, “Second order optical nonlinearity in octupolar aromatic systems,” J. Chem. Phys. 97, 5607–5615 (1992).
[CrossRef]

Singer, K. D.

Sohn, J. E.

Takezoe, H.

B. Park, J. W. Wu, and H. Takezoe, “Generalized mean-field potential description for ferroelectric ordering in nematic liquid crystals,” Phys. Rev. E 63, 21707–1–21707–7 (2001).
[CrossRef]

B. Park, Y. Kinoshita, H. Takezoe, and J. Watanabe, “Ferroelectricity in the lyotropic cholesteric phase of poly L-glutamate,” Jpn. J. Appl. Phys. 37, L136–L138 (1998).
[CrossRef]

Terhune, R. W.

R. W. Terhune, P. D. Maker, and C. M. Savage, “Measurements of nonlinear light scattering,” Phys. Rev. Lett. 14, 681–684 (1965).
[CrossRef]

Tokita, M.

J. Watanabe, Y. Hirose, M. Tokita, T. Watanabe, and S. Miyata, “Polar structure in polypeptide cholesteric liquid crystals evidenced from observation of second-harmonic generation due to the helicoidal cavity effect,” Macromolecules 31, 5937–5939 (1998).
[CrossRef]

van Hulst, N. F.

G. J. T. Heesink, A. G. Ruiter, N. F. van Hulst, and B. Boelger, “Determination of hyperpolarizability tensor components by depolarized hyper Rayleigh scattering,” Phys. Rev. Lett. 71, 999–1002 (1993).
[CrossRef] [PubMed]

Verbiest, T.

T. Verbiest, M. Kauranen, and A. Persoon, “Parametric light scattering,” J. Chem. Phys. 101, 1745–1747 (1994).
[CrossRef]

T. Verbiest, K. Clays, A. Persoons, F. Meyers, and J. L. Breda, “Determination of the hyperpolarizability of an octopolar molecular ion by hyper-Rayleigh scattering,” Opt. Lett. 18, 525–527 (1993).
[CrossRef] [PubMed]

W. M. Laidlaw, R. G. Denning, T. Verbiest, E. Chauchard, and A. Persoons, “Large second-order optical polarizabilities in mixed-valency metal complexes,” Nature 363, 58–60 (1993).
[CrossRef]

Watanabe, J.

B. Park, Y. Kinoshita, H. Takezoe, and J. Watanabe, “Ferroelectricity in the lyotropic cholesteric phase of poly L-glutamate,” Jpn. J. Appl. Phys. 37, L136–L138 (1998).
[CrossRef]

J. Watanabe, Y. Hirose, M. Tokita, T. Watanabe, and S. Miyata, “Polar structure in polypeptide cholesteric liquid crystals evidenced from observation of second-harmonic generation due to the helicoidal cavity effect,” Macromolecules 31, 5937–5939 (1998).
[CrossRef]

Watanabe, T.

J. Watanabe, Y. Hirose, M. Tokita, T. Watanabe, and S. Miyata, “Polar structure in polypeptide cholesteric liquid crystals evidenced from observation of second-harmonic generation due to the helicoidal cavity effect,” Macromolecules 31, 5937–5939 (1998).
[CrossRef]

Wu, J. W.

B. Park, J. W. Wu, and H. Takezoe, “Generalized mean-field potential description for ferroelectric ordering in nematic liquid crystals,” Phys. Rev. E 63, 21707–1–21707–7 (2001).
[CrossRef]

J. W. Wu, “Birefringent and electro-optic effects in poled polymer films: steady-state and transient properties,” J. Opt. Soc. Am. B 8, 142–152 (1991).
[CrossRef]

Yaron, D.

M. Joffre, D. Yaron, and R. J. Silbey, “Second order optical nonlinearity in octupolar aromatic systems,” J. Chem. Phys. 97, 5607–5615 (1992).
[CrossRef]

Zyss, J.

J. Zyss, “Octupolar organic systems in quadratic nonlinear optics: molecules and materials,” Nonlinear Opt. 1, 3–18 (1991).

J. Chem. Phys. (6)

T. Verbiest, M. Kauranen, and A. Persoon, “Parametric light scattering,” J. Chem. Phys. 101, 1745–1747 (1994).
[CrossRef]

R. Bersohn, Y.-H. Pao, and H. L. Frisch, “Double-quantum light scattering by molecules,” J. Chem. Phys. 45, 3184–3198 (1966).
[CrossRef]

S. J. Cyvin, J. E. Rauch, and J. C. Decius, “Theory of hyper-Raman effects (nonlinear inelastic light scattering): selection rules and depolarization ratios for the second-order polarizability,” J. Chem. Phys. 43, 4083–4095 (1965).
[CrossRef]

B. F. Levine and C. G. Bethea, “Second and third order hy-perpolarizabilities of organic molecules,” J. Chem. Phys. 63, 2666–2682 (1975).
[CrossRef]

M. Joffre, D. Yaron, and R. J. Silbey, “Second order optical nonlinearity in octupolar aromatic systems,” J. Chem. Phys. 97, 5607–5615 (1992).
[CrossRef]

B. F. Levine and C. G. Bethea, “Second order hyperpolarizability of a polypeptide α-helix: poly-γ-benzyl-L-glutamate,” J. Chem. Phys. 65, 1989–1993 (1976).
[CrossRef]

J. Opt. Soc. Am. B (2)

Jpn. J. Appl. Phys. (1)

B. Park, Y. Kinoshita, H. Takezoe, and J. Watanabe, “Ferroelectricity in the lyotropic cholesteric phase of poly L-glutamate,” Jpn. J. Appl. Phys. 37, L136–L138 (1998).
[CrossRef]

Macromolecules (1)

J. Watanabe, Y. Hirose, M. Tokita, T. Watanabe, and S. Miyata, “Polar structure in polypeptide cholesteric liquid crystals evidenced from observation of second-harmonic generation due to the helicoidal cavity effect,” Macromolecules 31, 5937–5939 (1998).
[CrossRef]

Nature (1)

W. M. Laidlaw, R. G. Denning, T. Verbiest, E. Chauchard, and A. Persoons, “Large second-order optical polarizabilities in mixed-valency metal complexes,” Nature 363, 58–60 (1993).
[CrossRef]

Nonlinear Opt. (1)

J. Zyss, “Octupolar organic systems in quadratic nonlinear optics: molecules and materials,” Nonlinear Opt. 1, 3–18 (1991).

Opt. Lett. (1)

Phys. Rev. E (1)

B. Park, J. W. Wu, and H. Takezoe, “Generalized mean-field potential description for ferroelectric ordering in nematic liquid crystals,” Phys. Rev. E 63, 21707–1–21707–7 (2001).
[CrossRef]

Phys. Rev. Lett. (3)

R. W. Terhune, P. D. Maker, and C. M. Savage, “Measurements of nonlinear light scattering,” Phys. Rev. Lett. 14, 681–684 (1965).
[CrossRef]

K. Clays and A. Persoons, “Hyper-Rayleigh scattering in solution,” Phys. Rev. Lett. 66, 2980–2983 (1991).
[CrossRef] [PubMed]

G. J. T. Heesink, A. G. Ruiter, N. F. van Hulst, and B. Boelger, “Determination of hyperpolarizability tensor components by depolarized hyper Rayleigh scattering,” Phys. Rev. Lett. 71, 999–1002 (1993).
[CrossRef] [PubMed]

Other (2)

D. S. Chemla and J. Zyss, Nonlinear Optical Properties of Organic Molecules and Crystals (Academic, Orlando, Fla., 1987).

D. R. Lide, Handbook of Organic Solvents (CRC Press, New York, 1995).

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Figures (2)

Fig. 1
Fig. 1

Laboratory coordinates for incident fundamental and scattered harmonics. An external electric field was applied to the sample cell along the Z direction.

Fig. 2
Fig. 2

Measured depolarization ratio DZZZX as a function of external electric field strength. Solid and dashed curves, β311/β333 -0.81 and β311/β333 +1.22, respectively. Inset, example of the lift of degeneracy in DZZZX.

Tables (2)

Tables Icon

Table 1 Six Quadratics of βIJK (I, J, and K; Laboratory Axes) Expressed in Terms of Coefficients of Quadratic Functions of β in Molecular Axes, p, q, r, All Differenta

Tables Icon

Table 2 Averaged Products of Six Direction Cosinesa in the Presence of Polar Ordering, x(μE/kT)

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

βIJKβLMN=ϕIpϕJqϕKrϕLsϕMtϕNuβpqrβstu,
βZZZ2f(θ)=p ApϕZp6f(θ)+p,q BpqϕZp4ϕZq2f(θ)+p,q,r CpqrϕZp2ϕZq2ϕZr2f(θ),
DZZZX(e-HRS)=IX2ωIZ2ω=βXZZ2f(θ)βZZZ2f(θ).
βXZZ2f(θ)=β3332135i0(x)-355i4(x)+277i6(x)+42105i0(x)+142i2(x)-23770i4(x)-177i6(x)β311β333+4-170i0(x)+3770i4(x)-177i6(x)β131+β113β333+4170i0(x)+156i2(x)-693080i4(x)×β131β113β3332+4-2105i0(x)-1168i2(x)+188i4(x)β311β131+β311β113β3332+2435i0(x)+27i2(x)+67770i4(x)+177i6(x)β311β3332+2170i0(x)+156i2(x)-693080i4(x)×β131β3332+β113β3332,
βZZZ2f(θ)=β333217i0(x)+1021i2(x)+2477i4(x)+16231i6(x)+4135i0(x)+121i2(x)-16385i4(x)-8231i6(x)×β311+β131+β113β333+24105i0(x)+121i2(x)-23385i4(x)-277i6(x)×β311+β131+β113β3332.
χZZZ(2)=15N3-2β311β333P1(cos θ)+21-β311β333×P3(cos θ)β333,
χZXX(2)=15N1+4β311β333P1(cos θ)-1-β311β333×P3(cos θ)β333.

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