Abstract

A number of explanations have been put forward to account for systematic discrepancies between the results of surface Brillouin scattering spectroscopy and those of ultrasonic measurements. These include the effect of surface imperfections, geometrical aperture effects, and variation in the scattering cross section. We have developed an analytical procedure for correcting the errors associated with the aperture and the cross section, taking into account issues such as sample anisotropy and out-of-plane scattering that were not previously considered. These new computational tools can be applied to ripple scattering for arbitrary polarization from any smooth opaque surface, including films and multilayers. The power of the approach is demonstrated for the case of single-crystal silicon, in which some features of the observed spectra are more accurately predicted. Experimental methods that minimize the errors are also discussed.

© 1998 Optical Society of America

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References

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  1. F. Nizzoli and J. R. Sandercock, “Surface Brillouin scattering from phonons,” in Dynamical Properties of Solids, G. K. Horton and A. A. Maradudin, eds. (North-Holland, Amsterdam, 1990), pp. 281–335.
  2. M. Mendik, S. Sathish, A. Kulik, G. Gremaud, and P. Wachter, “Surface acoustic wave studies on single-crystal nickel using Brillouin scattering and scanning acoustic microscopy,” J. Appl. Phys. 71, 2830–2834 (1992).
    [CrossRef]
  3. M. W. Elmiger, “Raman scattering under high pressure in samarium selenide and Brillouin spectroscopy from surface acoustic waves,” Doctoral dissertation (Eidgenössische Technische Hochschule, Zurich, Switzerland, 1988).
  4. P. R. Stoddart, J. D. Comins, and A. G. Every, “Brillouin-scattering measurements of surface-acoustic-wave velocities in silicon at high temperatures,” Phys. Rev. B 51, 17, 574–17, 578 (1995).
    [CrossRef]
  5. J. R. Sandercock, “Light scattering from surface acoustic phonons in metals and semiconductors,” Solid State Commun. 26, 547–551 (1978).
    [CrossRef]
  6. V. R. Velasco and F. Garcia-Moliner, “Brillouin scattering from surface waves,” Solid State Commun. 33, 1–5 (1980).
    [CrossRef]
  7. P. Mutti, C. E. Bottani, G. Ghislotti, M. Beghi, G. A. D. Briggs, and J. R. Sandercock, “Surface Brillouin scattering—extending surface wave measurements to 20 GHz,” in Advances in Acoustic Microscopy, A. Briggs, ed. (Plenum, New York, 1995), Vol. 1, pp. 249–300.
  8. M. Ezz-el-Arab, B. Galperin, J. Brielles, and B. Vodar, “Variation des vitesses de propagation des ultrasons dans le silicium monocrystallin entre 25° et 830 °C,” Solid State Commun. 6, 387–390 (1968).
    [CrossRef]
  9. M. G. Beghi, C. E. Bottani, P. M. Ossi, T. A. Lafford, and B. K. Tanner, “Combined surface Brillouin scattering and x-ray reflectivity characterization of thin metallic films,” J. Appl. Phys. 81, 672–678 (1997); R. Jorna, D. Visser, V. Bortolani, and F. Nizzoli, “Elastic and vibrational properties of nickel films measured by surface Brillouin scattering,” J. Appl. Phys. 65, 718–825 (1989).
    [CrossRef]
  10. R. Loudon and J. R. Sandercock, “Analysis of the light scattering cross section for surface ripples on solids,” J. Phys. C 13, 2609–2622 (1980).
    [CrossRef]
  11. G. S. Agarwal, “Interaction of electromagnetic waves at rough dielectric surfaces,” Phys. Rev. B 15, 2371–2383 (1977).
    [CrossRef]
  12. R. Loudon, “Theory of surface-ripple Brillouin scattering by solids,” Phys. Rev. Lett. 40, 581–583 (1978).
    [CrossRef]
  13. K. R. Subbaswamy and A. A. Maradudin, “Photoelastic and surface-corrugation contributions to Brillouin scattering from an opaque crystal,” Phys. Rev. B 18, 4181–4199 (1978).
    [CrossRef]
  14. V. Bortolani, F. Nizzoli, and G. Santoro, “Surface density of acoustic phonons in GaAs,” Phys. Rev. Lett. 41, 39–42 (1978).
    [CrossRef]
  15. J. A. Bell, R. J. Zanoni, C. T. Seaton, G. I. Stegeman, W. R. Bennet, and C. Falco, “Brillouin scattering from Love waves in Cu/Nb metallic superlattices,” Appl. Phys. Lett. 51, 652–654 (1987).
    [CrossRef]
  16. G. W. Farnell, “Properties of elastic surface waves,” in Physical Acoustics–Principles and Methods, W. P. Mason and R. N. Thurston, eds. (Academic, New York, 1970), Vol. VI, pp. 109–166.
  17. J. O. Kim and R. D. Weglein, “Comment on ‘Surface acoustic wave studies on single crystal nickel using Brillouin scattering and scanning acoustic microscope, ’ ” J. Appl. Phys. 75, 5459–5460 (1994).
    [CrossRef]
  18. M. M. Puentes-Heras and O. Kolosov, Department of Materials, University of Oxford, Parks Road, Oxford OX13PH, England (personal communication, 1997).
  19. A. E. Kadyshevich, V. M. Beilin, Yu. Kh. Vekilov, O. M. Krasil’nikov, and V. N. Podd’yakov, “Investigation of the influence of free current carriers on the elastic constants of germanium and silicon,” Fiz. Tverd. Tela Leningrad 9, 1861–1867 (1967) [Sov. Phys. Solid State 9, 1467–1472 (1968)];R. W. Keyes, “Device implications of the electronic effect in the elastic constants of silicon,” IEEE Trans. Sonics Ultrason. SU-29, 99–103 (1982).
    [CrossRef]
  20. A. G. Eguiluz and A. A. Maradudin, “Frequency shift and attenuation length of a Rayleigh wave due to surface roughness,” Phys. Rev. B 28, 728–747 (1983).
    [CrossRef]

1995

P. R. Stoddart, J. D. Comins, and A. G. Every, “Brillouin-scattering measurements of surface-acoustic-wave velocities in silicon at high temperatures,” Phys. Rev. B 51, 17, 574–17, 578 (1995).
[CrossRef]

1994

J. O. Kim and R. D. Weglein, “Comment on ‘Surface acoustic wave studies on single crystal nickel using Brillouin scattering and scanning acoustic microscope, ’ ” J. Appl. Phys. 75, 5459–5460 (1994).
[CrossRef]

1992

M. Mendik, S. Sathish, A. Kulik, G. Gremaud, and P. Wachter, “Surface acoustic wave studies on single-crystal nickel using Brillouin scattering and scanning acoustic microscopy,” J. Appl. Phys. 71, 2830–2834 (1992).
[CrossRef]

1987

J. A. Bell, R. J. Zanoni, C. T. Seaton, G. I. Stegeman, W. R. Bennet, and C. Falco, “Brillouin scattering from Love waves in Cu/Nb metallic superlattices,” Appl. Phys. Lett. 51, 652–654 (1987).
[CrossRef]

1983

A. G. Eguiluz and A. A. Maradudin, “Frequency shift and attenuation length of a Rayleigh wave due to surface roughness,” Phys. Rev. B 28, 728–747 (1983).
[CrossRef]

1980

V. R. Velasco and F. Garcia-Moliner, “Brillouin scattering from surface waves,” Solid State Commun. 33, 1–5 (1980).
[CrossRef]

R. Loudon and J. R. Sandercock, “Analysis of the light scattering cross section for surface ripples on solids,” J. Phys. C 13, 2609–2622 (1980).
[CrossRef]

1978

R. Loudon, “Theory of surface-ripple Brillouin scattering by solids,” Phys. Rev. Lett. 40, 581–583 (1978).
[CrossRef]

K. R. Subbaswamy and A. A. Maradudin, “Photoelastic and surface-corrugation contributions to Brillouin scattering from an opaque crystal,” Phys. Rev. B 18, 4181–4199 (1978).
[CrossRef]

V. Bortolani, F. Nizzoli, and G. Santoro, “Surface density of acoustic phonons in GaAs,” Phys. Rev. Lett. 41, 39–42 (1978).
[CrossRef]

J. R. Sandercock, “Light scattering from surface acoustic phonons in metals and semiconductors,” Solid State Commun. 26, 547–551 (1978).
[CrossRef]

1977

G. S. Agarwal, “Interaction of electromagnetic waves at rough dielectric surfaces,” Phys. Rev. B 15, 2371–2383 (1977).
[CrossRef]

1968

M. Ezz-el-Arab, B. Galperin, J. Brielles, and B. Vodar, “Variation des vitesses de propagation des ultrasons dans le silicium monocrystallin entre 25° et 830 °C,” Solid State Commun. 6, 387–390 (1968).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal, “Interaction of electromagnetic waves at rough dielectric surfaces,” Phys. Rev. B 15, 2371–2383 (1977).
[CrossRef]

Bell, J. A.

J. A. Bell, R. J. Zanoni, C. T. Seaton, G. I. Stegeman, W. R. Bennet, and C. Falco, “Brillouin scattering from Love waves in Cu/Nb metallic superlattices,” Appl. Phys. Lett. 51, 652–654 (1987).
[CrossRef]

Bennet, W. R.

J. A. Bell, R. J. Zanoni, C. T. Seaton, G. I. Stegeman, W. R. Bennet, and C. Falco, “Brillouin scattering from Love waves in Cu/Nb metallic superlattices,” Appl. Phys. Lett. 51, 652–654 (1987).
[CrossRef]

Bortolani, V.

V. Bortolani, F. Nizzoli, and G. Santoro, “Surface density of acoustic phonons in GaAs,” Phys. Rev. Lett. 41, 39–42 (1978).
[CrossRef]

Brielles, J.

M. Ezz-el-Arab, B. Galperin, J. Brielles, and B. Vodar, “Variation des vitesses de propagation des ultrasons dans le silicium monocrystallin entre 25° et 830 °C,” Solid State Commun. 6, 387–390 (1968).
[CrossRef]

Comins, J. D.

P. R. Stoddart, J. D. Comins, and A. G. Every, “Brillouin-scattering measurements of surface-acoustic-wave velocities in silicon at high temperatures,” Phys. Rev. B 51, 17, 574–17, 578 (1995).
[CrossRef]

Eguiluz, A. G.

A. G. Eguiluz and A. A. Maradudin, “Frequency shift and attenuation length of a Rayleigh wave due to surface roughness,” Phys. Rev. B 28, 728–747 (1983).
[CrossRef]

Every, A. G.

P. R. Stoddart, J. D. Comins, and A. G. Every, “Brillouin-scattering measurements of surface-acoustic-wave velocities in silicon at high temperatures,” Phys. Rev. B 51, 17, 574–17, 578 (1995).
[CrossRef]

Ezz-el-Arab, M.

M. Ezz-el-Arab, B. Galperin, J. Brielles, and B. Vodar, “Variation des vitesses de propagation des ultrasons dans le silicium monocrystallin entre 25° et 830 °C,” Solid State Commun. 6, 387–390 (1968).
[CrossRef]

Falco, C.

J. A. Bell, R. J. Zanoni, C. T. Seaton, G. I. Stegeman, W. R. Bennet, and C. Falco, “Brillouin scattering from Love waves in Cu/Nb metallic superlattices,” Appl. Phys. Lett. 51, 652–654 (1987).
[CrossRef]

Galperin, B.

M. Ezz-el-Arab, B. Galperin, J. Brielles, and B. Vodar, “Variation des vitesses de propagation des ultrasons dans le silicium monocrystallin entre 25° et 830 °C,” Solid State Commun. 6, 387–390 (1968).
[CrossRef]

Garcia-Moliner, F.

V. R. Velasco and F. Garcia-Moliner, “Brillouin scattering from surface waves,” Solid State Commun. 33, 1–5 (1980).
[CrossRef]

Gremaud, G.

M. Mendik, S. Sathish, A. Kulik, G. Gremaud, and P. Wachter, “Surface acoustic wave studies on single-crystal nickel using Brillouin scattering and scanning acoustic microscopy,” J. Appl. Phys. 71, 2830–2834 (1992).
[CrossRef]

Kim, J. O.

J. O. Kim and R. D. Weglein, “Comment on ‘Surface acoustic wave studies on single crystal nickel using Brillouin scattering and scanning acoustic microscope, ’ ” J. Appl. Phys. 75, 5459–5460 (1994).
[CrossRef]

Kulik, A.

M. Mendik, S. Sathish, A. Kulik, G. Gremaud, and P. Wachter, “Surface acoustic wave studies on single-crystal nickel using Brillouin scattering and scanning acoustic microscopy,” J. Appl. Phys. 71, 2830–2834 (1992).
[CrossRef]

Loudon, R.

R. Loudon and J. R. Sandercock, “Analysis of the light scattering cross section for surface ripples on solids,” J. Phys. C 13, 2609–2622 (1980).
[CrossRef]

R. Loudon, “Theory of surface-ripple Brillouin scattering by solids,” Phys. Rev. Lett. 40, 581–583 (1978).
[CrossRef]

Maradudin, A. A.

A. G. Eguiluz and A. A. Maradudin, “Frequency shift and attenuation length of a Rayleigh wave due to surface roughness,” Phys. Rev. B 28, 728–747 (1983).
[CrossRef]

K. R. Subbaswamy and A. A. Maradudin, “Photoelastic and surface-corrugation contributions to Brillouin scattering from an opaque crystal,” Phys. Rev. B 18, 4181–4199 (1978).
[CrossRef]

Mendik, M.

M. Mendik, S. Sathish, A. Kulik, G. Gremaud, and P. Wachter, “Surface acoustic wave studies on single-crystal nickel using Brillouin scattering and scanning acoustic microscopy,” J. Appl. Phys. 71, 2830–2834 (1992).
[CrossRef]

Nizzoli, F.

V. Bortolani, F. Nizzoli, and G. Santoro, “Surface density of acoustic phonons in GaAs,” Phys. Rev. Lett. 41, 39–42 (1978).
[CrossRef]

Sandercock, J. R.

R. Loudon and J. R. Sandercock, “Analysis of the light scattering cross section for surface ripples on solids,” J. Phys. C 13, 2609–2622 (1980).
[CrossRef]

J. R. Sandercock, “Light scattering from surface acoustic phonons in metals and semiconductors,” Solid State Commun. 26, 547–551 (1978).
[CrossRef]

Santoro, G.

V. Bortolani, F. Nizzoli, and G. Santoro, “Surface density of acoustic phonons in GaAs,” Phys. Rev. Lett. 41, 39–42 (1978).
[CrossRef]

Sathish, S.

M. Mendik, S. Sathish, A. Kulik, G. Gremaud, and P. Wachter, “Surface acoustic wave studies on single-crystal nickel using Brillouin scattering and scanning acoustic microscopy,” J. Appl. Phys. 71, 2830–2834 (1992).
[CrossRef]

Seaton, C. T.

J. A. Bell, R. J. Zanoni, C. T. Seaton, G. I. Stegeman, W. R. Bennet, and C. Falco, “Brillouin scattering from Love waves in Cu/Nb metallic superlattices,” Appl. Phys. Lett. 51, 652–654 (1987).
[CrossRef]

Stegeman, G. I.

J. A. Bell, R. J. Zanoni, C. T. Seaton, G. I. Stegeman, W. R. Bennet, and C. Falco, “Brillouin scattering from Love waves in Cu/Nb metallic superlattices,” Appl. Phys. Lett. 51, 652–654 (1987).
[CrossRef]

Stoddart, P. R.

P. R. Stoddart, J. D. Comins, and A. G. Every, “Brillouin-scattering measurements of surface-acoustic-wave velocities in silicon at high temperatures,” Phys. Rev. B 51, 17, 574–17, 578 (1995).
[CrossRef]

Subbaswamy, K. R.

K. R. Subbaswamy and A. A. Maradudin, “Photoelastic and surface-corrugation contributions to Brillouin scattering from an opaque crystal,” Phys. Rev. B 18, 4181–4199 (1978).
[CrossRef]

Velasco, V. R.

V. R. Velasco and F. Garcia-Moliner, “Brillouin scattering from surface waves,” Solid State Commun. 33, 1–5 (1980).
[CrossRef]

Vodar, B.

M. Ezz-el-Arab, B. Galperin, J. Brielles, and B. Vodar, “Variation des vitesses de propagation des ultrasons dans le silicium monocrystallin entre 25° et 830 °C,” Solid State Commun. 6, 387–390 (1968).
[CrossRef]

Wachter, P.

M. Mendik, S. Sathish, A. Kulik, G. Gremaud, and P. Wachter, “Surface acoustic wave studies on single-crystal nickel using Brillouin scattering and scanning acoustic microscopy,” J. Appl. Phys. 71, 2830–2834 (1992).
[CrossRef]

Weglein, R. D.

J. O. Kim and R. D. Weglein, “Comment on ‘Surface acoustic wave studies on single crystal nickel using Brillouin scattering and scanning acoustic microscope, ’ ” J. Appl. Phys. 75, 5459–5460 (1994).
[CrossRef]

Zanoni, R. J.

J. A. Bell, R. J. Zanoni, C. T. Seaton, G. I. Stegeman, W. R. Bennet, and C. Falco, “Brillouin scattering from Love waves in Cu/Nb metallic superlattices,” Appl. Phys. Lett. 51, 652–654 (1987).
[CrossRef]

Appl. Phys. Lett.

J. A. Bell, R. J. Zanoni, C. T. Seaton, G. I. Stegeman, W. R. Bennet, and C. Falco, “Brillouin scattering from Love waves in Cu/Nb metallic superlattices,” Appl. Phys. Lett. 51, 652–654 (1987).
[CrossRef]

J. Appl. Phys.

J. O. Kim and R. D. Weglein, “Comment on ‘Surface acoustic wave studies on single crystal nickel using Brillouin scattering and scanning acoustic microscope, ’ ” J. Appl. Phys. 75, 5459–5460 (1994).
[CrossRef]

M. Mendik, S. Sathish, A. Kulik, G. Gremaud, and P. Wachter, “Surface acoustic wave studies on single-crystal nickel using Brillouin scattering and scanning acoustic microscopy,” J. Appl. Phys. 71, 2830–2834 (1992).
[CrossRef]

J. Phys. C

R. Loudon and J. R. Sandercock, “Analysis of the light scattering cross section for surface ripples on solids,” J. Phys. C 13, 2609–2622 (1980).
[CrossRef]

Phys. Rev. B

G. S. Agarwal, “Interaction of electromagnetic waves at rough dielectric surfaces,” Phys. Rev. B 15, 2371–2383 (1977).
[CrossRef]

K. R. Subbaswamy and A. A. Maradudin, “Photoelastic and surface-corrugation contributions to Brillouin scattering from an opaque crystal,” Phys. Rev. B 18, 4181–4199 (1978).
[CrossRef]

P. R. Stoddart, J. D. Comins, and A. G. Every, “Brillouin-scattering measurements of surface-acoustic-wave velocities in silicon at high temperatures,” Phys. Rev. B 51, 17, 574–17, 578 (1995).
[CrossRef]

A. G. Eguiluz and A. A. Maradudin, “Frequency shift and attenuation length of a Rayleigh wave due to surface roughness,” Phys. Rev. B 28, 728–747 (1983).
[CrossRef]

Phys. Rev. Lett.

V. Bortolani, F. Nizzoli, and G. Santoro, “Surface density of acoustic phonons in GaAs,” Phys. Rev. Lett. 41, 39–42 (1978).
[CrossRef]

R. Loudon, “Theory of surface-ripple Brillouin scattering by solids,” Phys. Rev. Lett. 40, 581–583 (1978).
[CrossRef]

Solid State Commun.

J. R. Sandercock, “Light scattering from surface acoustic phonons in metals and semiconductors,” Solid State Commun. 26, 547–551 (1978).
[CrossRef]

V. R. Velasco and F. Garcia-Moliner, “Brillouin scattering from surface waves,” Solid State Commun. 33, 1–5 (1980).
[CrossRef]

M. Ezz-el-Arab, B. Galperin, J. Brielles, and B. Vodar, “Variation des vitesses de propagation des ultrasons dans le silicium monocrystallin entre 25° et 830 °C,” Solid State Commun. 6, 387–390 (1968).
[CrossRef]

Other

M. G. Beghi, C. E. Bottani, P. M. Ossi, T. A. Lafford, and B. K. Tanner, “Combined surface Brillouin scattering and x-ray reflectivity characterization of thin metallic films,” J. Appl. Phys. 81, 672–678 (1997); R. Jorna, D. Visser, V. Bortolani, and F. Nizzoli, “Elastic and vibrational properties of nickel films measured by surface Brillouin scattering,” J. Appl. Phys. 65, 718–825 (1989).
[CrossRef]

M. W. Elmiger, “Raman scattering under high pressure in samarium selenide and Brillouin spectroscopy from surface acoustic waves,” Doctoral dissertation (Eidgenössische Technische Hochschule, Zurich, Switzerland, 1988).

P. Mutti, C. E. Bottani, G. Ghislotti, M. Beghi, G. A. D. Briggs, and J. R. Sandercock, “Surface Brillouin scattering—extending surface wave measurements to 20 GHz,” in Advances in Acoustic Microscopy, A. Briggs, ed. (Plenum, New York, 1995), Vol. 1, pp. 249–300.

M. M. Puentes-Heras and O. Kolosov, Department of Materials, University of Oxford, Parks Road, Oxford OX13PH, England (personal communication, 1997).

A. E. Kadyshevich, V. M. Beilin, Yu. Kh. Vekilov, O. M. Krasil’nikov, and V. N. Podd’yakov, “Investigation of the influence of free current carriers on the elastic constants of germanium and silicon,” Fiz. Tverd. Tela Leningrad 9, 1861–1867 (1967) [Sov. Phys. Solid State 9, 1467–1472 (1968)];R. W. Keyes, “Device implications of the electronic effect in the elastic constants of silicon,” IEEE Trans. Sonics Ultrason. SU-29, 99–103 (1982).
[CrossRef]

G. W. Farnell, “Properties of elastic surface waves,” in Physical Acoustics–Principles and Methods, W. P. Mason and R. N. Thurston, eds. (Academic, New York, 1970), Vol. VI, pp. 109–166.

F. Nizzoli and J. R. Sandercock, “Surface Brillouin scattering from phonons,” in Dynamical Properties of Solids, G. K. Horton and A. A. Maradudin, eds. (North-Holland, Amsterdam, 1990), pp. 281–335.

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Figures (8)

Fig. 1
Fig. 1

The scattering geometry.

Fig. 2
Fig. 2

Calculations of the purely geometrical shift in average scattering wave vector qG for collection systems of apertures f/5.5 and f/2.3. When the weighting that is due to the cross-section variation is included (see Section 3), the f/2.3 result changes to that labeled q.

Fig. 3
Fig. 3

Cross section dσx/dΩ for backscattering from silicon in the isotropic approximation. Polarization in the sagittal plane is labeled p, and s polarization lies perpendicular to this. The continuous curves were calculated for θi=θs and ϕ=0; the fragmented curves are shown over a range of θs for the listed value of θi.

Fig. 4
Fig. 4

Calculated spectra for scattering along [100] on the silicon (001) surface for f/2.3 and f/5.5 lenses. The dashed curves indicate the best Gaussian fits to the spectra.

Fig. 5
Fig. 5

Results for surface-acoustic-wave velocity along [100] on the silicon (001) surface as a function of incident angle. The calculated results were obtained with ultrasonically measured elastic constants from Ref. 14; the short-dashed curve includes the effect of a 3-nm oxide layer upon the surface. The f/2.3 data were corrected by the fitting procedure described in the text.

Fig. 6
Fig. 6

The spectrum at the left was calculated with an f/5.5 aperture for scattering in the direction α=30° from [100] upon the silicon (001) surface. The effect of instrumental broadening can be seen at the right, after convolution with a line of width 350 MHz.

Fig. 7
Fig. 7

Calculated angular dependence of the Brillouin frequency shift (solid curve) compared with data from Ref. 4. The dashed curves are calculated frequencies of PSAW’s and SAW’s that contribute to the scattering.

Fig. 8
Fig. 8

Angular dependence of the silicon cross section for PSAW’s and SAW’s upon the (001) surface. The Green’s function approach was used to calculate the surface displacement power spectrum.

Equations (25)

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

q1=ki sin θi+ks sin θs cos ϕ,
q2=ks sin θs sin ϕ.
q=ki[sin2 θi+2 sin θi sin θs cos ϕ+sin2 θs]1/2.
q0=2ki sin θi.
Δqq0=sin Δθtan θi=12.3%.
Δqq0=1-[(1/2)(1+cos ϕ+)]1/2=0.8%.
tan-1q2q1=7.2°.
qG=Ωq(θs, ϕ)dΩΩdΩ,
ϕ±=tan-1±R2-X2X cos θi+f sin θi,
X=ftan θi 1-cos θscos θi R2+f2f21/2
d2σxdΩdωcos θi cos2 θsFx(θi, θs, ϕ, )|u3(0)|2q,ω,
Fss(θi, θs, ϕ, )=cos2 ϕ(-1)[cos θs+(-sin2 θs)1/2][cos θi+(-sin2 θi)1/2]2,
Fsp(θi, θs, ϕ, )=sin2 ϕ(-1)(-sin2 θs)1/2[cos θi+(-sin2 θi)1/2][ cos θs+(-sin2 θs)1/2]2,
Fps(θi, θs, ϕ, )=sin2 ϕ(-1)(-sin2 θi)1/2[ cos θi+(-sin2 θi)1/2][cos θs+(-sin2 θs)1/2]2,
Fpp(θi, θs, ϕ, )=(-1)[cos ϕ(-sin2 θi)1/2(-sin2 θs)1/2+ sin θs sin θi][ cos θi+(-sin2 θi)1/2][ cos θs+(-sin2 θs)1/2]2.
|u3(0)|2q,ωdω1q(θi, θs, ϕ).
dσxdΩcos θi cos2 θsq Fx(θi, θs, ϕ, ).
q=Ω dσdΩ q(θs, ϕ)dΩΩ dσdΩ dΩ.
|u3(0)|2q,ωkBTπω Im[G33(q;0;ω)].
G33(q; 0; ω)=n=13Γ(n)U3(n),
τl3(q; 0; ω)=-δl3(2π)3.
G33(q; 0; ω)=i(2π)3 n=13 U3(n) adj(B)3(n)|B|,
|u3(0)|2q,ωkBT8π4ω Ren=13 U3(n) adj(B)3(n)|B|.
fB=vq2π,
Ix(ω)=Ω d2σxdΩdω dΩ.

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