Abstract

The determination of optical constants from spectra of low-symmetry single crystals or from spectra stemming from unfavorable crystal faces by dispersion analysis is a difficult task. Besides oscillator frequencies, band positions can additionally depend on oscillator strength as well as on the relative orientation of the transition moment. Therefore the results of the analysis are highly prone to errors. We present an easy method to validate the oscillator data resulting from dispersion analysis by the comparison of measured and simulated spectra of randomly oriented polycrystalline materials. Depending on the crystallite size, either average reflectance and transmittance theory (ARTT) or average refractive index theory (ARIT) can be applied to model the spectra of polycrystalline compounds. As an alternative to ARIT, effective medium theory (EMA) can also be employed. However, since principal dielectric functions do not exist in the general triclinic case, which are needed for the application of EMA, we suggest using the eigenvalues of the dielectric tensor function instead. Our method for the verification of optical constants of single crystals is validated using monoclinic CuO as an example.

© 2007 Optical Society of America

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References

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  1. T. G. Mayerhöfer, "Modelling IR spectra of single-phase polycrystalline materials with random orientation-a unified approach," Vib. Spectrosc. 35, 67-76 (2004), and references therein.
    [CrossRef]
  2. F. Wooten, Optical Properties of Solids (Academic, 1972).
  3. M. V. Belousov and V. F. Pavinich, "Infrared reflection spectra of monoclinic crystals," Opt. Spectrosc. 45, 771-774 (1978).
  4. J. R. Aronson, A. G. Emslie, and P. F. Strong, "Optical constants of triclinic anisotropic crystals: blue vitriol," Appl. Opt. 24, 1200-1203 (1985).
    [CrossRef] [PubMed]
  5. A. B. Kuz'menko, D. van der Marel, P. J. M. van Bentum, E. A. Tishchenko, C. Pressura, and A. A. Bush, "Infrared spectroscopic study of CuO: signatures of strong spin-phonon interaction and structural distortion," Phys. Rev. B 63, 094303 (2001).
    [CrossRef]
  6. J. R. Aronson, A. G. Emslie, E. V. Miseo, E. M. Smith, and P. F. Strong, "Optical constants of monoclinic anisotropic crystals: gypsum," Appl. Opt. 22, 4093-4098 (1983).
    [CrossRef] [PubMed]
  7. A. B. Kuz'menko, E. A. Tishchenko, and V. G. Orlov, "Transverse optic modes in monoclinic α-Bi2O3," J. Phys. Condens. Matter 8, 6199-6212 (1996).
    [CrossRef]
  8. S. Guha, D. Peebles, and T. J. Wieting, "Zone-center (q = 0) optical phonons in CuO studied by Raman and infrared spectroscopy," Phys. Rev. B 43, 13092-13101 (1991).
    [CrossRef]
  9. C. C. Homes, M. Ziaei, B. P. Clayman, and J. C. Irwin, "Softening of a reststrahlen band in CuO near the Néel transition," Phys. Rev. B 51, 3140-3151 (1995).
    [CrossRef]
  10. G. Emslie and J. R. Aronson, "Determination of the complex dielectric tensor of triclinic crystals: theory," J. Opt. Soc. Am. 73, 916-919 (1983).
    [CrossRef]
  11. T. G. Mayerhöfer and J. Popp, "Modeling IR-spectra of polycrystalline materials in the large crystallites limit-quantitative determination of orientation," J. Opt. A: Pure Appl. Opt. 8, 657-671 (2006).
    [CrossRef]
  12. P. Yeh, Optical Waves in Layered Media (Wiley, 1988).
  13. T. G. Mayerhöfer, "Symmetric Euler orientation representations for orientational averaging," Spectrochimica Acta A 61, 2611-2621 (2005).
    [CrossRef]
  14. T. G. Mayerhöfer and J. Popp, "Effective optical constants-a fundamental discrepancy," Vib. Spectrosc. 42(10), 118-123 (2006).
    [CrossRef]
  15. G. Kliche and Z. V. Popovic, "Far-infrared spectroscopic investigations on CuO," Phys. Rev. B 42, 10060-10066 (1990).
    [CrossRef]
  16. L. Degiorgi, E. Kaldis, and P. Wachter, "Electronic and phononic structure of La1.85Sr0.15CuO4," Physica C 153ߝ155, 657-658 (1988).
    [CrossRef]
  17. D. Stroud and F. P. Pan, "Self-consistent approach to electromagnetic wave propagation in composite media: application to model granular metals," Phys. Rev. B 17, 1602-1610 (1978).
    [CrossRef]

2006 (2)

T. G. Mayerhöfer and J. Popp, "Effective optical constants-a fundamental discrepancy," Vib. Spectrosc. 42(10), 118-123 (2006).
[CrossRef]

T. G. Mayerhöfer and J. Popp, "Modeling IR-spectra of polycrystalline materials in the large crystallites limit-quantitative determination of orientation," J. Opt. A: Pure Appl. Opt. 8, 657-671 (2006).
[CrossRef]

2005 (1)

T. G. Mayerhöfer, "Symmetric Euler orientation representations for orientational averaging," Spectrochimica Acta A 61, 2611-2621 (2005).
[CrossRef]

2004 (1)

T. G. Mayerhöfer, "Modelling IR spectra of single-phase polycrystalline materials with random orientation-a unified approach," Vib. Spectrosc. 35, 67-76 (2004), and references therein.
[CrossRef]

2001 (1)

A. B. Kuz'menko, D. van der Marel, P. J. M. van Bentum, E. A. Tishchenko, C. Pressura, and A. A. Bush, "Infrared spectroscopic study of CuO: signatures of strong spin-phonon interaction and structural distortion," Phys. Rev. B 63, 094303 (2001).
[CrossRef]

1996 (1)

A. B. Kuz'menko, E. A. Tishchenko, and V. G. Orlov, "Transverse optic modes in monoclinic α-Bi2O3," J. Phys. Condens. Matter 8, 6199-6212 (1996).
[CrossRef]

1995 (1)

C. C. Homes, M. Ziaei, B. P. Clayman, and J. C. Irwin, "Softening of a reststrahlen band in CuO near the Néel transition," Phys. Rev. B 51, 3140-3151 (1995).
[CrossRef]

1991 (1)

S. Guha, D. Peebles, and T. J. Wieting, "Zone-center (q = 0) optical phonons in CuO studied by Raman and infrared spectroscopy," Phys. Rev. B 43, 13092-13101 (1991).
[CrossRef]

1990 (1)

G. Kliche and Z. V. Popovic, "Far-infrared spectroscopic investigations on CuO," Phys. Rev. B 42, 10060-10066 (1990).
[CrossRef]

1988 (1)

L. Degiorgi, E. Kaldis, and P. Wachter, "Electronic and phononic structure of La1.85Sr0.15CuO4," Physica C 153ߝ155, 657-658 (1988).
[CrossRef]

1985 (1)

1983 (2)

G. Emslie and J. R. Aronson, "Determination of the complex dielectric tensor of triclinic crystals: theory," J. Opt. Soc. Am. 73, 916-919 (1983).
[CrossRef]

J. R. Aronson, A. G. Emslie, E. V. Miseo, E. M. Smith, and P. F. Strong, "Optical constants of monoclinic anisotropic crystals: gypsum," Appl. Opt. 22, 4093-4098 (1983).
[CrossRef] [PubMed]

1978 (2)

M. V. Belousov and V. F. Pavinich, "Infrared reflection spectra of monoclinic crystals," Opt. Spectrosc. 45, 771-774 (1978).

D. Stroud and F. P. Pan, "Self-consistent approach to electromagnetic wave propagation in composite media: application to model granular metals," Phys. Rev. B 17, 1602-1610 (1978).
[CrossRef]

Aronson, J. R.

J. R. Aronson, A. G. Emslie, and P. F. Strong, "Optical constants of triclinic anisotropic crystals: blue vitriol," Appl. Opt. 24, 1200-1203 (1985).
[CrossRef] [PubMed]

J. R. Aronson, A. G. Emslie, E. V. Miseo, E. M. Smith, and P. F. Strong, "Optical constants of monoclinic anisotropic crystals: gypsum," Appl. Opt. 22, 4093-4098 (1983).
[CrossRef] [PubMed]

G. Emslie and J. R. Aronson, "Determination of the complex dielectric tensor of triclinic crystals: theory," J. Opt. Soc. Am. 73, 916-919 (1983).
[CrossRef]

Belousov, M. V.

M. V. Belousov and V. F. Pavinich, "Infrared reflection spectra of monoclinic crystals," Opt. Spectrosc. 45, 771-774 (1978).

Bush, A. A.

A. B. Kuz'menko, D. van der Marel, P. J. M. van Bentum, E. A. Tishchenko, C. Pressura, and A. A. Bush, "Infrared spectroscopic study of CuO: signatures of strong spin-phonon interaction and structural distortion," Phys. Rev. B 63, 094303 (2001).
[CrossRef]

Clayman, B. P.

C. C. Homes, M. Ziaei, B. P. Clayman, and J. C. Irwin, "Softening of a reststrahlen band in CuO near the Néel transition," Phys. Rev. B 51, 3140-3151 (1995).
[CrossRef]

Degiorgi, L.

L. Degiorgi, E. Kaldis, and P. Wachter, "Electronic and phononic structure of La1.85Sr0.15CuO4," Physica C 153ߝ155, 657-658 (1988).
[CrossRef]

Emslie, A. G.

J. R. Aronson, A. G. Emslie, and P. F. Strong, "Optical constants of triclinic anisotropic crystals: blue vitriol," Appl. Opt. 24, 1200-1203 (1985).
[CrossRef] [PubMed]

J. R. Aronson, A. G. Emslie, E. V. Miseo, E. M. Smith, and P. F. Strong, "Optical constants of monoclinic anisotropic crystals: gypsum," Appl. Opt. 22, 4093-4098 (1983).
[CrossRef] [PubMed]

Emslie, G.

G. Emslie and J. R. Aronson, "Determination of the complex dielectric tensor of triclinic crystals: theory," J. Opt. Soc. Am. 73, 916-919 (1983).
[CrossRef]

Guha, S.

S. Guha, D. Peebles, and T. J. Wieting, "Zone-center (q = 0) optical phonons in CuO studied by Raman and infrared spectroscopy," Phys. Rev. B 43, 13092-13101 (1991).
[CrossRef]

Homes, C. C.

C. C. Homes, M. Ziaei, B. P. Clayman, and J. C. Irwin, "Softening of a reststrahlen band in CuO near the Néel transition," Phys. Rev. B 51, 3140-3151 (1995).
[CrossRef]

Irwin, J. C.

C. C. Homes, M. Ziaei, B. P. Clayman, and J. C. Irwin, "Softening of a reststrahlen band in CuO near the Néel transition," Phys. Rev. B 51, 3140-3151 (1995).
[CrossRef]

Kaldis, E.

L. Degiorgi, E. Kaldis, and P. Wachter, "Electronic and phononic structure of La1.85Sr0.15CuO4," Physica C 153ߝ155, 657-658 (1988).
[CrossRef]

Kliche, G.

G. Kliche and Z. V. Popovic, "Far-infrared spectroscopic investigations on CuO," Phys. Rev. B 42, 10060-10066 (1990).
[CrossRef]

Kuz'menko, A. B.

A. B. Kuz'menko, D. van der Marel, P. J. M. van Bentum, E. A. Tishchenko, C. Pressura, and A. A. Bush, "Infrared spectroscopic study of CuO: signatures of strong spin-phonon interaction and structural distortion," Phys. Rev. B 63, 094303 (2001).
[CrossRef]

A. B. Kuz'menko, E. A. Tishchenko, and V. G. Orlov, "Transverse optic modes in monoclinic α-Bi2O3," J. Phys. Condens. Matter 8, 6199-6212 (1996).
[CrossRef]

Mayerhöfer, T. G.

T. G. Mayerhöfer and J. Popp, "Effective optical constants-a fundamental discrepancy," Vib. Spectrosc. 42(10), 118-123 (2006).
[CrossRef]

T. G. Mayerhöfer and J. Popp, "Modeling IR-spectra of polycrystalline materials in the large crystallites limit-quantitative determination of orientation," J. Opt. A: Pure Appl. Opt. 8, 657-671 (2006).
[CrossRef]

T. G. Mayerhöfer, "Symmetric Euler orientation representations for orientational averaging," Spectrochimica Acta A 61, 2611-2621 (2005).
[CrossRef]

T. G. Mayerhöfer, "Modelling IR spectra of single-phase polycrystalline materials with random orientation-a unified approach," Vib. Spectrosc. 35, 67-76 (2004), and references therein.
[CrossRef]

Miseo, E. V.

J. R. Aronson, A. G. Emslie, E. V. Miseo, E. M. Smith, and P. F. Strong, "Optical constants of monoclinic anisotropic crystals: gypsum," Appl. Opt. 22, 4093-4098 (1983).
[CrossRef] [PubMed]

Orlov, V. G.

A. B. Kuz'menko, E. A. Tishchenko, and V. G. Orlov, "Transverse optic modes in monoclinic α-Bi2O3," J. Phys. Condens. Matter 8, 6199-6212 (1996).
[CrossRef]

Pan, F. P.

D. Stroud and F. P. Pan, "Self-consistent approach to electromagnetic wave propagation in composite media: application to model granular metals," Phys. Rev. B 17, 1602-1610 (1978).
[CrossRef]

Pavinich, V. F.

M. V. Belousov and V. F. Pavinich, "Infrared reflection spectra of monoclinic crystals," Opt. Spectrosc. 45, 771-774 (1978).

Peebles, D.

S. Guha, D. Peebles, and T. J. Wieting, "Zone-center (q = 0) optical phonons in CuO studied by Raman and infrared spectroscopy," Phys. Rev. B 43, 13092-13101 (1991).
[CrossRef]

Popovic, Z. V.

G. Kliche and Z. V. Popovic, "Far-infrared spectroscopic investigations on CuO," Phys. Rev. B 42, 10060-10066 (1990).
[CrossRef]

Popp, J.

T. G. Mayerhöfer and J. Popp, "Effective optical constants-a fundamental discrepancy," Vib. Spectrosc. 42(10), 118-123 (2006).
[CrossRef]

T. G. Mayerhöfer and J. Popp, "Modeling IR-spectra of polycrystalline materials in the large crystallites limit-quantitative determination of orientation," J. Opt. A: Pure Appl. Opt. 8, 657-671 (2006).
[CrossRef]

Pressura, C.

A. B. Kuz'menko, D. van der Marel, P. J. M. van Bentum, E. A. Tishchenko, C. Pressura, and A. A. Bush, "Infrared spectroscopic study of CuO: signatures of strong spin-phonon interaction and structural distortion," Phys. Rev. B 63, 094303 (2001).
[CrossRef]

Smith, E. M.

J. R. Aronson, A. G. Emslie, E. V. Miseo, E. M. Smith, and P. F. Strong, "Optical constants of monoclinic anisotropic crystals: gypsum," Appl. Opt. 22, 4093-4098 (1983).
[CrossRef] [PubMed]

Strong, P. F.

J. R. Aronson, A. G. Emslie, and P. F. Strong, "Optical constants of triclinic anisotropic crystals: blue vitriol," Appl. Opt. 24, 1200-1203 (1985).
[CrossRef] [PubMed]

J. R. Aronson, A. G. Emslie, E. V. Miseo, E. M. Smith, and P. F. Strong, "Optical constants of monoclinic anisotropic crystals: gypsum," Appl. Opt. 22, 4093-4098 (1983).
[CrossRef] [PubMed]

Stroud, D.

D. Stroud and F. P. Pan, "Self-consistent approach to electromagnetic wave propagation in composite media: application to model granular metals," Phys. Rev. B 17, 1602-1610 (1978).
[CrossRef]

Tishchenko, E. A.

A. B. Kuz'menko, D. van der Marel, P. J. M. van Bentum, E. A. Tishchenko, C. Pressura, and A. A. Bush, "Infrared spectroscopic study of CuO: signatures of strong spin-phonon interaction and structural distortion," Phys. Rev. B 63, 094303 (2001).
[CrossRef]

A. B. Kuz'menko, E. A. Tishchenko, and V. G. Orlov, "Transverse optic modes in monoclinic α-Bi2O3," J. Phys. Condens. Matter 8, 6199-6212 (1996).
[CrossRef]

van Bentum, P. J. M.

A. B. Kuz'menko, D. van der Marel, P. J. M. van Bentum, E. A. Tishchenko, C. Pressura, and A. A. Bush, "Infrared spectroscopic study of CuO: signatures of strong spin-phonon interaction and structural distortion," Phys. Rev. B 63, 094303 (2001).
[CrossRef]

van der Marel, D.

A. B. Kuz'menko, D. van der Marel, P. J. M. van Bentum, E. A. Tishchenko, C. Pressura, and A. A. Bush, "Infrared spectroscopic study of CuO: signatures of strong spin-phonon interaction and structural distortion," Phys. Rev. B 63, 094303 (2001).
[CrossRef]

Wachter, P.

L. Degiorgi, E. Kaldis, and P. Wachter, "Electronic and phononic structure of La1.85Sr0.15CuO4," Physica C 153ߝ155, 657-658 (1988).
[CrossRef]

Wieting, T. J.

S. Guha, D. Peebles, and T. J. Wieting, "Zone-center (q = 0) optical phonons in CuO studied by Raman and infrared spectroscopy," Phys. Rev. B 43, 13092-13101 (1991).
[CrossRef]

Wooten, F.

F. Wooten, Optical Properties of Solids (Academic, 1972).

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, 1988).

Ziaei, M.

C. C. Homes, M. Ziaei, B. P. Clayman, and J. C. Irwin, "Softening of a reststrahlen band in CuO near the Néel transition," Phys. Rev. B 51, 3140-3151 (1995).
[CrossRef]

Appl. Opt. (1)

J. R. Aronson, A. G. Emslie, E. V. Miseo, E. M. Smith, and P. F. Strong, "Optical constants of monoclinic anisotropic crystals: gypsum," Appl. Opt. 22, 4093-4098 (1983).
[CrossRef] [PubMed]

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

G. Emslie and J. R. Aronson, "Determination of the complex dielectric tensor of triclinic crystals: theory," J. Opt. Soc. Am. 73, 916-919 (1983).
[CrossRef]

J. Phys. Condens. Matter (1)

A. B. Kuz'menko, E. A. Tishchenko, and V. G. Orlov, "Transverse optic modes in monoclinic α-Bi2O3," J. Phys. Condens. Matter 8, 6199-6212 (1996).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

T. G. Mayerhöfer and J. Popp, "Modeling IR-spectra of polycrystalline materials in the large crystallites limit-quantitative determination of orientation," J. Opt. A: Pure Appl. Opt. 8, 657-671 (2006).
[CrossRef]

Opt. Spectrosc. (1)

M. V. Belousov and V. F. Pavinich, "Infrared reflection spectra of monoclinic crystals," Opt. Spectrosc. 45, 771-774 (1978).

Phys. Rev. B (2)

S. Guha, D. Peebles, and T. J. Wieting, "Zone-center (q = 0) optical phonons in CuO studied by Raman and infrared spectroscopy," Phys. Rev. B 43, 13092-13101 (1991).
[CrossRef]

A. B. Kuz'menko, D. van der Marel, P. J. M. van Bentum, E. A. Tishchenko, C. Pressura, and A. A. Bush, "Infrared spectroscopic study of CuO: signatures of strong spin-phonon interaction and structural distortion," Phys. Rev. B 63, 094303 (2001).
[CrossRef]

Phys. Rev. B (3)

C. C. Homes, M. Ziaei, B. P. Clayman, and J. C. Irwin, "Softening of a reststrahlen band in CuO near the Néel transition," Phys. Rev. B 51, 3140-3151 (1995).
[CrossRef]

D. Stroud and F. P. Pan, "Self-consistent approach to electromagnetic wave propagation in composite media: application to model granular metals," Phys. Rev. B 17, 1602-1610 (1978).
[CrossRef]

G. Kliche and Z. V. Popovic, "Far-infrared spectroscopic investigations on CuO," Phys. Rev. B 42, 10060-10066 (1990).
[CrossRef]

Physica C (1)

L. Degiorgi, E. Kaldis, and P. Wachter, "Electronic and phononic structure of La1.85Sr0.15CuO4," Physica C 153ߝ155, 657-658 (1988).
[CrossRef]

Spectrochimica Acta A (1)

T. G. Mayerhöfer, "Symmetric Euler orientation representations for orientational averaging," Spectrochimica Acta A 61, 2611-2621 (2005).
[CrossRef]

Vib. Spectrosc. (1)

T. G. Mayerhöfer, "Modelling IR spectra of single-phase polycrystalline materials with random orientation-a unified approach," Vib. Spectrosc. 35, 67-76 (2004), and references therein.
[CrossRef]

Vib. Spectrosc. (1)

T. G. Mayerhöfer and J. Popp, "Effective optical constants-a fundamental discrepancy," Vib. Spectrosc. 42(10), 118-123 (2006).
[CrossRef]

Other (2)

P. Yeh, Optical Waves in Layered Media (Wiley, 1988).

F. Wooten, Optical Properties of Solids (Academic, 1972).

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Figures (4)

Fig. 1
Fig. 1

Calculated reflectance spectra of single-crystalline CuO based on the dielectric tensor function ε from Refs. [5, 9] (angle of incidence α = 0°, linear polarized light, intrinsic coordinate system of the crystal: x a , y b , z a , b ).

Fig. 2
Fig. 2

Comparison between the experimental reflectance spectra of polycrystalline CuO according to Refs. [9] and [15].

Fig. 3
Fig. 3

Simulated reflectance spectra of randomly oriented polycrystalline CuO with crystallite diameters d λ / 10 based on EMA (dotted line) and ARIT (continuous line); angle of incidence α = 7 ° , unpolarized light. Upper panel: single-crystal data from Ref. [5]. Lower panel: single-crystal data from Ref. [9].

Fig. 4
Fig. 4

Comparison between experimental reflectance spectra and simulated reflectance spectra of randomly oriented polycrystalline CuO based on the dielectric tensor function ε from Refs. [5, 9]. Dotted curves: ε from Ref. [5]. Dashed curves: ε from Ref. [9]. Angle of incidence α = 7 ° , unpolarized light. Upper panel: ARTT ( d > λ / 10 ) ; solid curve, experimental spectra according to Ref. [9]. Lower panel: ARIT ( d λ / 10 ) ; solid curve, experimental spectra according to Ref. [15].

Tables (1)

Tables Icon

Table 1 Parameters of CuO from Dispersion Analysis According to Refs. [5, 9]

Equations (35)

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ε x , y , z ( ν ˜ ) = [ ε , x x ε , x y ε , x z ε , y x ε , y y ε , y z ε , z x ε , z y ε , z z ] + i = 1 N S i 2 ( ν ˜ i 2 ν ˜ 2 ) i ν ˜ γ i [ sin 2 Θ i cos 2 Φ i sin 2 Θ i   sin   Φ i   cos   Φ i sin   Θ i   cos   Θ i   cos   Φ i sin 2 Θ i   sin   Φ i   cos   Φ i sin 2 Θ i sin 2 Φ i sin   Θ i   cos   Θ i   sin   Φ i sin   Θ i   cos   Θ i   cos   Φ i sin   Θ i   cos   Θ i   sin   Φ i cos 2 Θ i ] .
D i n c = [ 1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 1 ] .
D c r y = [ γ + I 3 γ + I Δ 43 + Δ 23 0 γ + II 3 γ + I I Δ 43 + Δ 23 0 Δ 21 γ + I 2 + γ + I Δ 23 + Δ 23 2 Δ 21 Δ 43 0 Δ 21 γ + II 2 + γ + I I Δ 23 + Δ 23 2 Δ 21 Δ 43 0 γ + I 2 + γ + I Δ 23 Δ 21 0 γ + II 2 + γ + I I Δ 23 Δ 21 0 γ + I 3 + γ + I 2 Δ 23 γ + I Δ 21 0 γ + II 3 + γ + II 2 Δ 23 γ + I I Δ 21 0 ] ,
γ + I = Δ 21 4 Δ 23 2 + ( Δ 21 Δ 43 ) 2 + Δ 43 2 ,
γ + I I = Δ 21 + 4 Δ 23 2 + ( Δ 21 Δ 43 ) 2 + Δ 43 2 ,
Δ 21 = ε X X ε XZ 2 ε Z Z ,
Δ 23 = ε X Y ε X Z ε Y Z ε Z Z ,
Δ 43 = ε Y Y ε YZ 2 ε Z Z .
D c r y = [ 1 Δ 21 0 0 0 1 0 0 0 0 0 1 Δ 43 0 0 0 1 0 ] .
M ˜ = [ 1 2 ( γ + I 3 + γ + I 2 Δ 21 + Δ 23 + Δ 23 2    + γ + I ( Δ 23 Δ 43 ) Δ 21 Δ 43 ) 0 1 2 ( γ + II 3 + γ + II 2 Δ 21 + Δ 23 + Δ 23 2    + γ + I I ( Δ 23 Δ 43 ) Δ 21 Δ 43 ) 0 1 2 ( γ + I 3 γ + I 2 Δ 21 + Δ 23 Δ 23 2    - γ + I ( Δ 23 + Δ 43 ) + Δ 21 Δ 43 ) 0 1 2 ( γ + II 3 γ + II 2 Δ 21 + Δ 23 Δ 23 2    γ + I I ( Δ 23 + Δ 43 ) + Δ 21 Δ 43 ) 0 1 2 ( γ + I + 1 ) ( γ + I ( γ + I + Δ 23 ) Δ 21 ) 0 1 2 ( γ + I I + 1 ) ( γ + I I ( γ + I I + Δ 23 ) Δ 21 ) 0 1 2 ( γ + I 1 ) ( γ + I ( γ + I + Δ 23 ) Δ 21 ) 0 1 2 ( γ + I I 1 ) ( γ + I I ( γ + I I + Δ 23 ) Δ 21 ) 0 ] .
r X X = ( B X A X ) A Y = 0 = M ˜ 21 M ˜ 33 M ˜ 23 M ˜ 31 M ˜ 11 M ˜ 33 M ˜ 13 M ˜ 31 , r X Y = ( B Y A X ) A Y = 0 = M ˜ 41 M ˜ 33 M ˜ 43 M ˜ 31 M ˜ 11 M ˜ 33 M ˜ 13 M ˜ 31 , r Y X = ( B X A Y ) A X = 0 = M ˜ 11 M ˜ 23 M ˜ 21 M ˜ 13 M ˜ 11 M ˜ 33 M ˜ 13 M ˜ 31 , r Y Y = ( B Y A Y ) A X = 0 = M ˜ 11 M ˜ 43 M ˜ 41 M ˜ 13 M ˜ 11 M ˜ 33 M ˜ 13 M ˜ 31 ,
R = 1 2 R X + 1 2 R Y = N ( 3 ) Ω ( 3 ) [ R X ( Ω ) 2 + R Y ( Ω ) 2 ] d Ω .
ε X , Y , Z = A ( γ , φ , θ , ψ ) ε x , y , z A 1 ( γ , φ , θ , ψ ) ,
A ( γ , φ , θ , ψ ) = A z ( ψ ) A y ( θ ) A z ( φ ) A ( γ ) A z ( ψ ) = [ cos   ψ sin   ψ 0 sin   ψ cos   ψ 0 0 0 1 ] , A y ( θ ) = [ cos   θ 0 - sin   θ 0 1 0 sin   θ 0 cos   θ ] , A z ( φ ) = [ cos   φ sin   φ 0 sin   φ cos   φ 0 0 0 1 ] , A ( γ ) = [ 1 4 3 sin 2 γ 2 3 cos γ sin γ + 2 3 sin 2 γ 2 3 cos γ sin γ + 2 3 sin 2 γ 2 3 cos γ sin γ + 2 3 sin 2 γ 1 4 3 sin 2 γ 2 3 cos γ sin γ + 2 3 sin 2 γ 2 3 cos γ sin γ + 2 3 sin 2 γ 2 3 cos γ sin γ + 2 3 sin 2 γ 1 4 3 sin 2 γ ]  .
R i j = 1 2 π 3 0 π 0 π 0 π [ R i j ( φ = π 8 ) + R i j ( φ = 5 π 8 ) ] × d γ d θ d ψ .
n = N ( 3 ) Ω ( 3 ) [ n 1 ( Ω ) 2 + n 2 ( Ω ) 2 ] d Ω .
n 1 = K 1 1 2 ε Z Z K 2 4 K 3 ,
n 2 = K 1 + 1 2 ε Z Z K 2 4 K 3 ,
K 1 = 1 2 ( ε X X + ε Y Y ε XZ 2 + ε YZ 2 ε Z Z ) ,
K 2 = ( ε XZ 2 + ε YZ 2 ε X X ε Z Z ε Y Y ε Z Z ) 2 ,
K 3 = ε Z Z ( ε X X ε Y Y ε Z Z + 2 ε X Y ε Y Z ε X Z ε X X ε YZ 2 ε Z Z ε XZ 2 ε Y Y ε XZ 2 ) .
1 3 ε a ε ε a + 2 ε + 1 3 ε b ε ε b + 2 ε + 1 3 ε c ε ε c + 2 ε = 0 ,
ε 3 1 4 ( ε a ε b + ε b ε c + ε c ε a ) ε + 1 4 ε a ε b ε c = 0 ,
ε 1 = 3 1 / 3 K 1 + K 3 2 / 3 2 × 3 2 / 3 K 3 1 / 3 ,
ε 2 = 2 ( 3 ) 1 / 3 K 1 + ( 1 i 3 ) K 3 2 / 3 4 × 3 2 / 3 K 3 1 / 3 ,
ε 3 = 3 1 / 3 ( 1 i 3 ) K 1 + ( 1 + i 3 ) K 3 2 / 3 4 × 3 2 / 3 K 3 1 / 3 ,
K 1 = ε a ε b + ε b ε c + ε c ε a ,
K 2 = ε a ε b ε c ,
K 3 = 9 K 2 + 3 27 K 2 2 K 1 3 ,
Det ( ε ε i I ) = 0 i = 1 , 2 , 3.
ε 1 = 1 3 ( ε X X + ε Y Y + ε Z Z ) + 2 1 / 3 K 1 3 ( K 2 + K 2 2 + 4 K 1 3 ) 1 / 3 ( K 2 + K 2 2 + 4 K 1 3 ) 1 / 3 3 × 2 1 / 3 ,
ε 2 = 1 3 ( ε X X + ε Y Y + ε Z Z ) ( 1 + i 3 ) K 1 3 × 2 2 / 3 ( K 2 + K 2 2 + 4 K 1 3 ) 1 / 3 + ( 1 i 3 ) ( K 2 + K 2 2 + 4 K 1 3 ) 1 / 3 6 × 2 1 / 3 ,
ε 3 = 1 3 ( ε X X + ε Y Y + ε Z Z ) ( 1 i 3 ) K 1 3 × 2 2 / 3 ( K 2 + K 2 2 + 4 K 1 3 ) 1 / 3 + ( 1 + i 3 ) ( K 2 + K 2 2 + 4 K 1 3 ) 1 / 3 6 × 2 1 / 3 ,
K 1 = ( ε X X + ε Y Y + ε Z Z ) 2 3 ( ε XY 2 + ε XZ 2 + ε YZ 2 ε X X ε Y Y ε X X ε Z Z ε Y Y ε Z Z ) ,
K 2 = 2 ( ε XX 3 + ε YY 3 + ε ZZ 3 ) + 3 ( ε XX 2 ε Y Y + ε XX 2 ε Z Z + ε YY 2 ε X X + ε YY 2 ε Z Z + ε ZZ 2 ε X X + ε ZZ 2 ε Y Y ) 9 ( ε X X ε XY 2 + ε X X ε XZ 2 + ε Y Y ε XY 2 + ε Y Y ε YZ 2 + ε Z Z ε XZ 2 + ε Z Z ε YZ 2 ) + 18 ( ε X X ε YZ 2 + ε Y Y ε XZ 2 + ε Z Z ε XY 2 ) 12 ε X X ε Y Y ε Z Z 54 ε X Y ε X Z ε Y Z .

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