Abstract

A new technique has been developed that employs highly focused laser beams for both generating and detecting thermal waves in the megahertz frequency regime. This technique includes a comprehensive 3-D depth-profiling theoretical model; it has been used to measure the thickness of both transparent and opaque thin films with high spatial resolution. Thickness sensitivities of ±2% over the 500–25,000-Å range have been obtained for Al and SiO2 films on Si substrates.

© 1983 Optical Society of America

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References

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  1. A. Rosencwaig, Photoacoustics and Photoacoustic Spectroscopy (Wiley, New York, 1980).
  2. R. L. Swofford, M. E. Long, A. C. Albrecht, J. Chem. Phys. 65, 179 (1979).
    [CrossRef]
  3. P. Korpiun, R. Tilgner, J. Appl. Phys. 51, 6115 (1980).
    [CrossRef]
  4. A. Rosencwaig, Science 218, 223 (1982).
    [CrossRef] [PubMed]
  5. Y. H. Pao, Ed., Optoacoustic Spectroscopy and Detection (Academic, New York, 1977).
  6. M. Luukkala, in Scanned Image Microscopy, E. A. Ash, Ed. (Academic, London, 1980), p. 273.
  7. G. Busse, in Ref. 6, p. 341.
  8. P.-E. Nordal and S. O. Kanstad, in Ref. 6, p. 331.
  9. D. Fournier and A. C. Boccara, in Ref. 6, p. 347.
  10. W. B. Jackson, N. M. Amer, A. C. Boccara, D. Fournier, Appl-Opt. 20, 1333 (1981).
    [CrossRef] [PubMed]
  11. J. C. Murphy, L. C. Aamodt, Appl. Phys. Lett. 38, 196 (1981).
    [CrossRef]
  12. S. Ameri, E. A. Ash, V. Neuman, C. R. Petts, Electron. Lett. 17, 337 (1981).
    [CrossRef]
  13. M. A. Olmstead, S. E. Kohn, N. M. Amer, Bull. Am. Phys. Soc. 27, 227 (1982).
  14. M. A. Olmstead, N. M. Amer, S. Kohn, D. Fournier, A. C. Boccara, Appl. Phys. A, accepted for publication.
  15. M. A. Olmstead, N. M. Amer, J. Vac. Sci. Technol., accepted for publication.
  16. A. Rosencwaig, J. Opsal, D. L. Willenburg, 1983 Photoacoustics Conference.
  17. A. Hordvick, H. Schlossberg, Appl. Opt. 16, 101 (1977).
    [CrossRef]
  18. C. K. N. Patel, A. C. Tarn, Rev. Mod. Phys. 53, 517 (1981).
    [CrossRef]
  19. G. S. Cargill, Phys. Today 34, 27 (Oct.1981).
    [CrossRef]
  20. A. Rosencwaig, Solid State Technol. 25, 91 (1982).
    [CrossRef]
  21. G. Busse, A. Rosencwaig, Appl. Phys. Lett. 36, 815 (1980).
    [CrossRef]
  22. R. M. de la Rue, R. F. Humphreys, I. M. Mason, E. A. Ash, Proc. Inst. Electr. Eng. 119, 117 (1972).
    [CrossRef]
  23. R. L. Whitman, A. Korpel, Appl. Opt. 8, 1567 (1969).
    [CrossRef] [PubMed]
  24. J. Opsal, A. Rosencwaig, J. Appl. Phys. 53, 4240 (1982).
    [CrossRef]
  25. L. C. Aamodt, J. C. Murphy, J. Appl. Phys. 52, 4903 (1981).
    [CrossRef]
  26. F. A. McDonald, J. Appl. Phys. 52, 381 (1981).
    [CrossRef]
  27. P. Morse, H. J. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953), p. 942.
  28. Thermal lens effects in the air appear to play a much smaller role in the laser probe method described in Refs. 13–15 because the probe beam in this method is not normal to the sample surface, and thus there is considerable cancellation of the thermal lens deflection as the probe beam traverses different regions of the thermal lens.
  29. D. E. Gray, Ed., American Institute of Physics Handbook, (McGraw-Hill, New York, 1972), Table 6e-5.
  30. D. L. Kwong, D. M. Kim, J. Appl. Phys. 54, 366 (1983).
    [CrossRef]

1983 (1)

D. L. Kwong, D. M. Kim, J. Appl. Phys. 54, 366 (1983).
[CrossRef]

1982 (4)

A. Rosencwaig, Solid State Technol. 25, 91 (1982).
[CrossRef]

J. Opsal, A. Rosencwaig, J. Appl. Phys. 53, 4240 (1982).
[CrossRef]

A. Rosencwaig, Science 218, 223 (1982).
[CrossRef] [PubMed]

M. A. Olmstead, S. E. Kohn, N. M. Amer, Bull. Am. Phys. Soc. 27, 227 (1982).

1981 (7)

C. K. N. Patel, A. C. Tarn, Rev. Mod. Phys. 53, 517 (1981).
[CrossRef]

G. S. Cargill, Phys. Today 34, 27 (Oct.1981).
[CrossRef]

W. B. Jackson, N. M. Amer, A. C. Boccara, D. Fournier, Appl-Opt. 20, 1333 (1981).
[CrossRef] [PubMed]

J. C. Murphy, L. C. Aamodt, Appl. Phys. Lett. 38, 196 (1981).
[CrossRef]

S. Ameri, E. A. Ash, V. Neuman, C. R. Petts, Electron. Lett. 17, 337 (1981).
[CrossRef]

L. C. Aamodt, J. C. Murphy, J. Appl. Phys. 52, 4903 (1981).
[CrossRef]

F. A. McDonald, J. Appl. Phys. 52, 381 (1981).
[CrossRef]

1980 (2)

G. Busse, A. Rosencwaig, Appl. Phys. Lett. 36, 815 (1980).
[CrossRef]

P. Korpiun, R. Tilgner, J. Appl. Phys. 51, 6115 (1980).
[CrossRef]

1979 (1)

R. L. Swofford, M. E. Long, A. C. Albrecht, J. Chem. Phys. 65, 179 (1979).
[CrossRef]

1977 (1)

1972 (1)

R. M. de la Rue, R. F. Humphreys, I. M. Mason, E. A. Ash, Proc. Inst. Electr. Eng. 119, 117 (1972).
[CrossRef]

1969 (1)

Aamodt, L. C.

J. C. Murphy, L. C. Aamodt, Appl. Phys. Lett. 38, 196 (1981).
[CrossRef]

L. C. Aamodt, J. C. Murphy, J. Appl. Phys. 52, 4903 (1981).
[CrossRef]

Albrecht, A. C.

R. L. Swofford, M. E. Long, A. C. Albrecht, J. Chem. Phys. 65, 179 (1979).
[CrossRef]

Amer, N. M.

M. A. Olmstead, S. E. Kohn, N. M. Amer, Bull. Am. Phys. Soc. 27, 227 (1982).

W. B. Jackson, N. M. Amer, A. C. Boccara, D. Fournier, Appl-Opt. 20, 1333 (1981).
[CrossRef] [PubMed]

M. A. Olmstead, N. M. Amer, S. Kohn, D. Fournier, A. C. Boccara, Appl. Phys. A, accepted for publication.

M. A. Olmstead, N. M. Amer, J. Vac. Sci. Technol., accepted for publication.

Ameri, S.

S. Ameri, E. A. Ash, V. Neuman, C. R. Petts, Electron. Lett. 17, 337 (1981).
[CrossRef]

Ash, E. A.

S. Ameri, E. A. Ash, V. Neuman, C. R. Petts, Electron. Lett. 17, 337 (1981).
[CrossRef]

R. M. de la Rue, R. F. Humphreys, I. M. Mason, E. A. Ash, Proc. Inst. Electr. Eng. 119, 117 (1972).
[CrossRef]

Boccara, A. C.

W. B. Jackson, N. M. Amer, A. C. Boccara, D. Fournier, Appl-Opt. 20, 1333 (1981).
[CrossRef] [PubMed]

M. A. Olmstead, N. M. Amer, S. Kohn, D. Fournier, A. C. Boccara, Appl. Phys. A, accepted for publication.

Busse, G.

G. Busse, A. Rosencwaig, Appl. Phys. Lett. 36, 815 (1980).
[CrossRef]

Cargill, G. S.

G. S. Cargill, Phys. Today 34, 27 (Oct.1981).
[CrossRef]

de la Rue, R. M.

R. M. de la Rue, R. F. Humphreys, I. M. Mason, E. A. Ash, Proc. Inst. Electr. Eng. 119, 117 (1972).
[CrossRef]

Feshbach, H. J.

P. Morse, H. J. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953), p. 942.

Fournier, D.

W. B. Jackson, N. M. Amer, A. C. Boccara, D. Fournier, Appl-Opt. 20, 1333 (1981).
[CrossRef] [PubMed]

M. A. Olmstead, N. M. Amer, S. Kohn, D. Fournier, A. C. Boccara, Appl. Phys. A, accepted for publication.

Hordvick, A.

Humphreys, R. F.

R. M. de la Rue, R. F. Humphreys, I. M. Mason, E. A. Ash, Proc. Inst. Electr. Eng. 119, 117 (1972).
[CrossRef]

Jackson, W. B.

W. B. Jackson, N. M. Amer, A. C. Boccara, D. Fournier, Appl-Opt. 20, 1333 (1981).
[CrossRef] [PubMed]

Kim, D. M.

D. L. Kwong, D. M. Kim, J. Appl. Phys. 54, 366 (1983).
[CrossRef]

Kohn, S.

M. A. Olmstead, N. M. Amer, S. Kohn, D. Fournier, A. C. Boccara, Appl. Phys. A, accepted for publication.

Kohn, S. E.

M. A. Olmstead, S. E. Kohn, N. M. Amer, Bull. Am. Phys. Soc. 27, 227 (1982).

Korpel, A.

Korpiun, P.

P. Korpiun, R. Tilgner, J. Appl. Phys. 51, 6115 (1980).
[CrossRef]

Kwong, D. L.

D. L. Kwong, D. M. Kim, J. Appl. Phys. 54, 366 (1983).
[CrossRef]

Long, M. E.

R. L. Swofford, M. E. Long, A. C. Albrecht, J. Chem. Phys. 65, 179 (1979).
[CrossRef]

Luukkala, M.

M. Luukkala, in Scanned Image Microscopy, E. A. Ash, Ed. (Academic, London, 1980), p. 273.

Mason, I. M.

R. M. de la Rue, R. F. Humphreys, I. M. Mason, E. A. Ash, Proc. Inst. Electr. Eng. 119, 117 (1972).
[CrossRef]

McDonald, F. A.

F. A. McDonald, J. Appl. Phys. 52, 381 (1981).
[CrossRef]

Morse, P.

P. Morse, H. J. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953), p. 942.

Murphy, J. C.

L. C. Aamodt, J. C. Murphy, J. Appl. Phys. 52, 4903 (1981).
[CrossRef]

J. C. Murphy, L. C. Aamodt, Appl. Phys. Lett. 38, 196 (1981).
[CrossRef]

Neuman, V.

S. Ameri, E. A. Ash, V. Neuman, C. R. Petts, Electron. Lett. 17, 337 (1981).
[CrossRef]

Olmstead, M. A.

M. A. Olmstead, S. E. Kohn, N. M. Amer, Bull. Am. Phys. Soc. 27, 227 (1982).

M. A. Olmstead, N. M. Amer, S. Kohn, D. Fournier, A. C. Boccara, Appl. Phys. A, accepted for publication.

M. A. Olmstead, N. M. Amer, J. Vac. Sci. Technol., accepted for publication.

Opsal, J.

J. Opsal, A. Rosencwaig, J. Appl. Phys. 53, 4240 (1982).
[CrossRef]

A. Rosencwaig, J. Opsal, D. L. Willenburg, 1983 Photoacoustics Conference.

Patel, C. K. N.

C. K. N. Patel, A. C. Tarn, Rev. Mod. Phys. 53, 517 (1981).
[CrossRef]

Petts, C. R.

S. Ameri, E. A. Ash, V. Neuman, C. R. Petts, Electron. Lett. 17, 337 (1981).
[CrossRef]

Rosencwaig, A.

A. Rosencwaig, Solid State Technol. 25, 91 (1982).
[CrossRef]

J. Opsal, A. Rosencwaig, J. Appl. Phys. 53, 4240 (1982).
[CrossRef]

A. Rosencwaig, Science 218, 223 (1982).
[CrossRef] [PubMed]

G. Busse, A. Rosencwaig, Appl. Phys. Lett. 36, 815 (1980).
[CrossRef]

A. Rosencwaig, Photoacoustics and Photoacoustic Spectroscopy (Wiley, New York, 1980).

A. Rosencwaig, J. Opsal, D. L. Willenburg, 1983 Photoacoustics Conference.

Schlossberg, H.

Swofford, R. L.

R. L. Swofford, M. E. Long, A. C. Albrecht, J. Chem. Phys. 65, 179 (1979).
[CrossRef]

Tarn, A. C.

C. K. N. Patel, A. C. Tarn, Rev. Mod. Phys. 53, 517 (1981).
[CrossRef]

Tilgner, R.

P. Korpiun, R. Tilgner, J. Appl. Phys. 51, 6115 (1980).
[CrossRef]

Whitman, R. L.

Willenburg, D. L.

A. Rosencwaig, J. Opsal, D. L. Willenburg, 1983 Photoacoustics Conference.

Appl-Opt. (1)

W. B. Jackson, N. M. Amer, A. C. Boccara, D. Fournier, Appl-Opt. 20, 1333 (1981).
[CrossRef] [PubMed]

Appl. Opt. (2)

Appl. Phys. Lett. (2)

G. Busse, A. Rosencwaig, Appl. Phys. Lett. 36, 815 (1980).
[CrossRef]

J. C. Murphy, L. C. Aamodt, Appl. Phys. Lett. 38, 196 (1981).
[CrossRef]

Bull. Am. Phys. Soc. (1)

M. A. Olmstead, S. E. Kohn, N. M. Amer, Bull. Am. Phys. Soc. 27, 227 (1982).

Electron. Lett. (1)

S. Ameri, E. A. Ash, V. Neuman, C. R. Petts, Electron. Lett. 17, 337 (1981).
[CrossRef]

J. Appl. Phys. (5)

P. Korpiun, R. Tilgner, J. Appl. Phys. 51, 6115 (1980).
[CrossRef]

D. L. Kwong, D. M. Kim, J. Appl. Phys. 54, 366 (1983).
[CrossRef]

J. Opsal, A. Rosencwaig, J. Appl. Phys. 53, 4240 (1982).
[CrossRef]

L. C. Aamodt, J. C. Murphy, J. Appl. Phys. 52, 4903 (1981).
[CrossRef]

F. A. McDonald, J. Appl. Phys. 52, 381 (1981).
[CrossRef]

J. Chem. Phys. (1)

R. L. Swofford, M. E. Long, A. C. Albrecht, J. Chem. Phys. 65, 179 (1979).
[CrossRef]

Phys. Today (1)

G. S. Cargill, Phys. Today 34, 27 (Oct.1981).
[CrossRef]

Proc. Inst. Electr. Eng. (1)

R. M. de la Rue, R. F. Humphreys, I. M. Mason, E. A. Ash, Proc. Inst. Electr. Eng. 119, 117 (1972).
[CrossRef]

Rev. Mod. Phys. (1)

C. K. N. Patel, A. C. Tarn, Rev. Mod. Phys. 53, 517 (1981).
[CrossRef]

Science (1)

A. Rosencwaig, Science 218, 223 (1982).
[CrossRef] [PubMed]

Solid State Technol. (1)

A. Rosencwaig, Solid State Technol. 25, 91 (1982).
[CrossRef]

Other (12)

M. A. Olmstead, N. M. Amer, S. Kohn, D. Fournier, A. C. Boccara, Appl. Phys. A, accepted for publication.

M. A. Olmstead, N. M. Amer, J. Vac. Sci. Technol., accepted for publication.

A. Rosencwaig, J. Opsal, D. L. Willenburg, 1983 Photoacoustics Conference.

Y. H. Pao, Ed., Optoacoustic Spectroscopy and Detection (Academic, New York, 1977).

M. Luukkala, in Scanned Image Microscopy, E. A. Ash, Ed. (Academic, London, 1980), p. 273.

G. Busse, in Ref. 6, p. 341.

P.-E. Nordal and S. O. Kanstad, in Ref. 6, p. 331.

D. Fournier and A. C. Boccara, in Ref. 6, p. 347.

A. Rosencwaig, Photoacoustics and Photoacoustic Spectroscopy (Wiley, New York, 1980).

P. Morse, H. J. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953), p. 942.

Thermal lens effects in the air appear to play a much smaller role in the laser probe method described in Refs. 13–15 because the probe beam in this method is not normal to the sample surface, and thus there is considerable cancellation of the thermal lens deflection as the probe beam traverses different regions of the thermal lens.

D. E. Gray, Ed., American Institute of Physics Handbook, (McGraw-Hill, New York, 1972), Table 6e-5.

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Figures (7)

Fig. 1
Fig. 1

Schematic depiction of laser beam deflection technique used for the thin-film thickness measurement experiments.

Fig. 2
Fig. 2

Schematic depiction of physical processes affecting the laser probe beam for an opaque homogeneous sample including thermoelastic deformation of the air–sample interface and thermal lens effects in the air above the sample.

Fig. 3
Fig. 3

Surface temperatures of the center of the heating beam (a) per milliwatt of power absorbed for a variety of materials at zero modulation frequency and (b) ac temperature normalized to the dc temperature as a function of the ratio of beam radius to the thermal diffusion length.

Fig. 4
Fig. 4

Relative amplitude of laser beam deflection signal as a function of thermal-wave (modulation) frequency for Al and Si under air (with thermal lens) and vacuum (no thermal lens) conditions. Experimental data are plotted as open (vacuum) and closed (air) circles and theoretical results as dashed (vacuum) and solid (air) curves.

Fig. 5
Fig. 5

Schematic depiction of physical processes affecting the laser probe beam for SiO2 on Si, including thermoelastic deformations of Si–SiO2 and SiO2–air interfaces, thermal lenses of opposite sign in air and SiO2, and optical interference effects in the SiO2 film.

Fig. 6
Fig. 6

Relative amplitude at 1 MHz of laser beam deflection signal as a function of Al film thickness for a series of Al-on-Si and Al-on- SiO2-on-Si films. Circles are experimental data, and curves are from the extended Opsal-Rosencwaig model.

Fig. 7
Fig. 7

Relative amplitude at 1 MHz of laser beam deflection signal as a function of SiO2 film thickness for a series of SiO2-on-Si films. Circles are experimental data, and curves are from the extended Opsal-Rosencwaig model.

Equations (35)

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T ( r ) = T 0 + T 1 ( r ) + T 2 ( r ) cos ( ω t + ϕ ) ,
T 1 ( r ) = P 0 2 π κ dq exp ( q 2 a 2 / 4 ) J 0 ( qr ) ,
T 2 ( r ) = P 0 μ 2 π κ | qdq exp ( q 2 a 2 / 4 ) J 0 ( qr ) q 2 μ 2 2 i | ,
T 1 ( 0 ) P 0 = 1 2 π κ a ,
T 2 ( 0 ) T 1 ( 0 ) = a μ π | xdx exp ( a 2 x 2 / 4 μ 2 ) x 2 2 i | ,
T 2 ( r ) = a 2 2 | qdq exp ( q 2 a 2 / 4 ) J 0 ( qr ) T 2 ( q ) | ,
T 2 ( q ) = Q 0 Z 1 [ Z 1 + Z 2 tanh ( q 1 z d ) Z 2 + Z 1 tanh ( q 1 z d ) ] .
q jz = q 2 μ j 2 2 i / μ j ,
T 2 ( r ) T 1 ( r ) = a 2 2 qdq exp ( q 2 a 2 / 4 ) J 0 ( qr ) T 1 ( q ) ,
T 1 ( q ) = Q 0 q κ 1 [ κ 1 + κ 2 tanh ( qd ) κ 2 + κ 1 tanh ( qd ) ] .
T 2 ( r ) T 1 ( r ) = P 0 2 π κ 2 a [ exp ( r 2 / 2 a 2 ) I 0 ( r 2 / 2 a 2 ) + 2 π ( d a ) × exp ( r 2 / a 2 ) ( κ 2 κ 1 κ 1 κ 2 ) ] ,
T 2 ( r ) T 1 ( r ) = P 0 2 π κ 2 a [ 1 + 2 π ( d a ) ( κ 2 κ 1 κ 1 κ 2 ) 0 ( r 2 / a 2 ) ]
T 2 ( r ) T 1 ( r ) = P 0 2 π κ 2 r [ 1 + 2 ( d a ) ( r a ) exp ( r 2 / a 2 ) ( κ 2 κ 1 κ 1 κ 2 ) ] ,
δ ( r ) = δ 1 ( r ) + δ 2 ( r ) cos ( ω t + ψ ) .
δ ( r ) = 2 du dr ,
u ( r ) = a 2 2 q d q exp ( q 2 a 2 / 4 ) J 0 ( q r ) ũ ( q ) ,
δ 1 ( r ) = a 2 q 2 dq exp ( q 2 a 2 / 4 ) J 1 ( qr ) ũ 1 ( q ) ,
δ 2 ( r ) = a 2 q 2 dq exp ( q 2 a 2 / 4 ) J 1 ( qr ) ũ 2 ( q ) ,
ũ 2 ( q ) = γ 2 Q 0 ( q + q 2 z ) Z 2 { 1 + q 1 z d [ ( q + q 2 z q 1 z ) ( γ 1 γ 2 γ 2 ) ( q 2 z q 1 z ) ( q 1 z Z 1 q 2 z Z 2 q 2 z Z 2 ) ] } ,
ũ 1 ( q ) = γ 2 O 0 2 q 2 κ 2 { 1 + dq [ 2 ( γ 1 γ 2 γ 2 ) ( κ 1 κ 2 κ 2 ) ] } .
δ 2 ( r ) δ 1 ( r ) = γ 2 P 0 2 π κ 2 a { a r [ 1 exp ( r 2 / a 2 ) ] + π ( d a ) ( r a ) [ 2 ( γ 1 γ 2 γ 2 ) ( κ 1 κ 2 κ 2 ) ] × exp ( r 2 / 2 a 2 ) [ I 0 ( r 2 / 2 a 2 ) I 1 ( r 2 / 2 a 2 ) ] ,
δ 2 ( r ) δ 1 ( r ) = γ 2 P 0 2 π κ 2 a ( r a ) { 1 + π ( d a ) [ 2 ( γ 1 γ 2 γ 2 ) ( κ 1 κ 2 κ 2 ) ] }
δ 2 ( r ) δ 1 ( r ) = γ 2 P 0 2 π κ 2 r { 1 + ( d r ) [ 2 ( γ 1 γ 2 γ 2 ) ( κ 1 κ 2 κ 2 ) ] } .
ξ ( r ) = ξ 1 ( r ) + ξ 2 ( r ) cos ( ω t + χ ) .
ξ ( r ) = 2 ε dz dT dr ,
ξ 1 ( r ) = ε a 2 qdq exp ( q 2 a 2 / 4 ) J 1 ( qr ) T 1 ( q ) ,
ξ 2 ( r ) = ε a 2 q 2 dq exp ( q 2 a 2 / 4 ) J 1 ( qr ) T 2 ( q ) q 0 z ,
ξ 2 ( r ) ξ 1 ( r ) = ε P 0 π κ 2 a ( r a ) [ 1 + π ( d a ) ( κ 2 κ 1 κ 1 κ 2 ) ] ,
ξ 2 ( r ) ξ 1 ( r ) = ε P 0 π κ 2 r [ 1 + ( d r ) ( κ 2 κ 1 κ 1 κ 2 ) ] .
δ 1 ( r ) = γ T 1 ( 0 ) π ( a r ) [ 1 exp ( r 2 / a 2 ) ] ,
δ 2 ( r ) δ 1 ( r ) = 2 ( r / μ ) 1 exp ( r 2 / a 2 ) × | x 2 dx J 1 ( rx / μ ) exp ( a 2 x 2 / 4 μ 2 ) x 2 2 i ( x + x 2 2 i ) | .
ξ 1 ( r ) = 2 ε γ δ 1 ( r ) ,
ξ 2 ( r ) ξ 1 ( r ) = r / μ 1 exp ( r 2 / a 2 ) | x 2 dx J 1 ( rx / μ ) exp ( a 2 x 2 / 4 μ 2 ) x 2 2 i | .
n = 1 + n 0 1 1 + α υ T P ,
κ ( T ) = κ 0 1 + β κ T ,

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