Abstract

Gas temperatures and centrifugal distortion constants can be determined by means of a computer program which simulates the Fabry-Perot interferograms of rotational Raman spectra. This computer program takes into account the instrumental broadening factors of the Fabry-Perot interferometer along with Doppler and pressure broadening of the spectral lines. Variations in the profiles of the computed interferograms for nitrogen were studied as a function of gas temperature and centrifugal distortion constant D. Comparison with experimental interferometric data for nitrogen yielded the gas temperature and a value of the centrifugal distortion constant equal to (5.80 ± 0.23) × 10−6 cm−1.

© 1976 Optical Society of America

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  1. J. J. Barrett and S. A. Myers, J. Opt. Soc. Am. 61, 1246 (1971).
    [Crossref]
  2. J. J. Barrett, in Laser Raman Gas Diagnostics, edited by M. Lapp and C. M. Penney (Plenum, New York, 1974), pp. 63–85.
    [Crossref]
  3. R. L. Armstrong, J. Opt. Soc. Am. 64, 871 (1974).
    [Crossref]
  4. P. J. Hargis and R. A. Hill, J. Opt. Soc. Am. 65, 219 (1975).
    [Crossref]
  5. T. Sundius, J. Raman Spectrosc. 1, 471 (1973).
    [Crossref]
  6. E. A. Ballik, Appl. Opt. 5, 170 (1966).
    [Crossref] [PubMed]
  7. G. Hernandez, Appl. Opt. 5, 1745 (1966).
    [Crossref] [PubMed]
  8. G. T. Best, Appl. Opt. 6, 287 (1967).
    [Crossref] [PubMed]
  9. H. P. Larson and K. L. Andrew, Appl. Opt. 6, 1701 (1967).
    [Crossref] [PubMed]
  10. G. Hernandez, Appl. Opt. 9, 1591 (1970).
    [Crossref] [PubMed]
  11. V. G. Cooper, Appl. Opt. 10, 525 (1971).
    [Crossref] [PubMed]
  12. K. Kaminishi and S. Nawata, Jpn. J. Appl. Phys. 13, 1640 (1974).
    [Crossref]
  13. J. Cooney, J. Appl. Meteorol. 11, 108 (1972).
    [Crossref]
  14. R. S. Hickman and L. H. Liang, Rev. Sci. Instrum. 43, 796 (1972).
    [Crossref]
  15. T. A. Coney and J. A. Salzman, , 1973.
  16. J. A. Salzman and T. A. Coney, , 1973.
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    [Crossref]
  19. M. Lapp, in Ref. 2, pp. 107–145.
  20. A. S. Gilbert and H. J. Bernstein, in Ref. 2, pp. 161–169.
  21. J. R. Smith, in Ref. 2, pp. 171–178.
  22. J. A. Salzman, in Ref. 2, pp. 179–188.
  23. G. Herzberg, Molecular Spectra and Molecular Structure I. Spectra of Diatomic Molecules (Van Nostrand, Princeton, 1950).
  24. B. P. Stoicheff, in Advances in Spectroscopy, edited by H. W. Thompson (Interscience, New York, 1959), p. 113.
  25. G. O. Neely, L. Y. Nelson, and A. B. Harvey, Appl. Spectrosc. 26, 553 (1972).
    [Crossref]
  26. M. Lapp, C. M. Penney, and L. M. Goldman, Opt. Commun. 9, 195 (1973).
    [Crossref]
  27. A. B. Harvey, in Ref. 2, pp. 147–152.
  28. J. J. Barrett and A. B. Harvey, J. Opt. Soc. Am. 65, 392 (1975).
    [Crossref]
  29. R. L. Armstrong, Appl. Opt. 14, 383 (1975).
    [Crossref] [PubMed]
  30. R. Bracewell, The Fourier Transform and Its Applications, (McGraw-Hill, New York, 1965), p. 110.
  31. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), pp. 323–333.
  32. W. H. Steel, Interferometry (Cambridge University, Cambridge England, 1967), p. 112.
  33. A. Weber, “High Resolution Raman Studies of Gases,” in The Raman Effect, edited by A. Anderson (Marcel Dekker, New York, 1973), Vol. 2, pp. 543–757.
  34. C. M. Penney, R. L. St. Peters, and M. Lapp, J. Opt. Soc. Am. 64, 712 (1974).
    [Crossref]
  35. J. J. Barrett and A. Weber, J. Mol. Spectrosc. 50, 205 (1974).
    [Crossref]
  36. J. Fintak and J. VanKranendonk, Can. J. Phys. 40, 1085 (1962); Can. J. Phys. 41, 21 (1963).
    [Crossref]
  37. J. VanKranendonk, Can. J. Phys. 41, 433 (1963).
    [Crossref]
  38. C. G. Gray and J. VanKranendonk, Can. J. Phys. 44, 2411 (1966).
    [Crossref]
  39. K. S. Jammu, G. E. St. John, and H. L. Welsh, Can. J. Phys. 44, 797 (1966).
    [Crossref]
  40. A. D. May, E. G. Rawson, and H. L. Welsh, in Physics of Quantum Electronics, edited by P. L. Kelley, B. Lax, and P. E. Tannenwald (McGraw-Hill, New York, 1966), p. 260.
  41. J. Bendtsen, J. Raman Spectrosc. 2, 133 (1974).
    [Crossref]
  42. R. J. Butcher, P. V. Willetts, and W. J. Jones, Proc. E. Soc. London Ser. A 324, 231 (1971).
    [Crossref]
  43. A. Weber (private communication).

1975 (3)

1974 (6)

C. M. Penney, R. L. St. Peters, and M. Lapp, J. Opt. Soc. Am. 64, 712 (1974).
[Crossref]

J. J. Barrett and A. Weber, J. Mol. Spectrosc. 50, 205 (1974).
[Crossref]

K. Kaminishi and S. Nawata, Jpn. J. Appl. Phys. 13, 1640 (1974).
[Crossref]

R. L. Armstrong, J. Opt. Soc. Am. 64, 871 (1974).
[Crossref]

R. S. Hickman and L. Liang, Rev. Sci. Instrum. 45, 1580 (1974).
[Crossref]

J. Bendtsen, J. Raman Spectrosc. 2, 133 (1974).
[Crossref]

1973 (2)

M. Lapp, C. M. Penney, and L. M. Goldman, Opt. Commun. 9, 195 (1973).
[Crossref]

T. Sundius, J. Raman Spectrosc. 1, 471 (1973).
[Crossref]

1972 (3)

J. Cooney, J. Appl. Meteorol. 11, 108 (1972).
[Crossref]

R. S. Hickman and L. H. Liang, Rev. Sci. Instrum. 43, 796 (1972).
[Crossref]

G. O. Neely, L. Y. Nelson, and A. B. Harvey, Appl. Spectrosc. 26, 553 (1972).
[Crossref]

1971 (3)

1970 (1)

1967 (2)

1966 (4)

E. A. Ballik, Appl. Opt. 5, 170 (1966).
[Crossref] [PubMed]

G. Hernandez, Appl. Opt. 5, 1745 (1966).
[Crossref] [PubMed]

C. G. Gray and J. VanKranendonk, Can. J. Phys. 44, 2411 (1966).
[Crossref]

K. S. Jammu, G. E. St. John, and H. L. Welsh, Can. J. Phys. 44, 797 (1966).
[Crossref]

1963 (1)

J. VanKranendonk, Can. J. Phys. 41, 433 (1963).
[Crossref]

1962 (1)

J. Fintak and J. VanKranendonk, Can. J. Phys. 40, 1085 (1962); Can. J. Phys. 41, 21 (1963).
[Crossref]

Andrew, K. L.

Armstrong, R. L.

Ballik, E. A.

Barrett, J. J.

J. J. Barrett and A. B. Harvey, J. Opt. Soc. Am. 65, 392 (1975).
[Crossref]

J. J. Barrett and A. Weber, J. Mol. Spectrosc. 50, 205 (1974).
[Crossref]

J. J. Barrett and S. A. Myers, J. Opt. Soc. Am. 61, 1246 (1971).
[Crossref]

J. J. Barrett, in Laser Raman Gas Diagnostics, edited by M. Lapp and C. M. Penney (Plenum, New York, 1974), pp. 63–85.
[Crossref]

Bendtsen, J.

J. Bendtsen, J. Raman Spectrosc. 2, 133 (1974).
[Crossref]

Bernstein, H. J.

A. S. Gilbert and H. J. Bernstein, in Ref. 2, pp. 161–169.

Best, G. T.

Born, M.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), pp. 323–333.

Bracewell, R.

R. Bracewell, The Fourier Transform and Its Applications, (McGraw-Hill, New York, 1965), p. 110.

Butcher, R. J.

R. J. Butcher, P. V. Willetts, and W. J. Jones, Proc. E. Soc. London Ser. A 324, 231 (1971).
[Crossref]

Coney, T. A.

J. A. Salzman and T. A. Coney, , 1974.

T. A. Coney and J. A. Salzman, , 1973.

J. A. Salzman and T. A. Coney, , 1973.

Cooney, J.

J. Cooney, J. Appl. Meteorol. 11, 108 (1972).
[Crossref]

Cooper, V. G.

Fintak, J.

J. Fintak and J. VanKranendonk, Can. J. Phys. 40, 1085 (1962); Can. J. Phys. 41, 21 (1963).
[Crossref]

Gilbert, A. S.

A. S. Gilbert and H. J. Bernstein, in Ref. 2, pp. 161–169.

Goldman, L. M.

M. Lapp, C. M. Penney, and L. M. Goldman, Opt. Commun. 9, 195 (1973).
[Crossref]

Gray, C. G.

C. G. Gray and J. VanKranendonk, Can. J. Phys. 44, 2411 (1966).
[Crossref]

Hargis, P. J.

Harvey, A. B.

Hernandez, G.

Herzberg, G.

G. Herzberg, Molecular Spectra and Molecular Structure I. Spectra of Diatomic Molecules (Van Nostrand, Princeton, 1950).

Hickman, R. S.

R. S. Hickman and L. Liang, Rev. Sci. Instrum. 45, 1580 (1974).
[Crossref]

R. S. Hickman and L. H. Liang, Rev. Sci. Instrum. 43, 796 (1972).
[Crossref]

Hill, R. A.

Jammu, K. S.

K. S. Jammu, G. E. St. John, and H. L. Welsh, Can. J. Phys. 44, 797 (1966).
[Crossref]

John, G. E. St.

K. S. Jammu, G. E. St. John, and H. L. Welsh, Can. J. Phys. 44, 797 (1966).
[Crossref]

Jones, W. J.

R. J. Butcher, P. V. Willetts, and W. J. Jones, Proc. E. Soc. London Ser. A 324, 231 (1971).
[Crossref]

Kaminishi, K.

K. Kaminishi and S. Nawata, Jpn. J. Appl. Phys. 13, 1640 (1974).
[Crossref]

Lapp, M.

C. M. Penney, R. L. St. Peters, and M. Lapp, J. Opt. Soc. Am. 64, 712 (1974).
[Crossref]

M. Lapp, C. M. Penney, and L. M. Goldman, Opt. Commun. 9, 195 (1973).
[Crossref]

M. Lapp, in Ref. 2, pp. 107–145.

Larson, H. P.

Liang, L.

R. S. Hickman and L. Liang, Rev. Sci. Instrum. 45, 1580 (1974).
[Crossref]

Liang, L. H.

R. S. Hickman and L. H. Liang, Rev. Sci. Instrum. 43, 796 (1972).
[Crossref]

May, A. D.

A. D. May, E. G. Rawson, and H. L. Welsh, in Physics of Quantum Electronics, edited by P. L. Kelley, B. Lax, and P. E. Tannenwald (McGraw-Hill, New York, 1966), p. 260.

Myers, S. A.

Nawata, S.

K. Kaminishi and S. Nawata, Jpn. J. Appl. Phys. 13, 1640 (1974).
[Crossref]

Neely, G. O.

Nelson, L. Y.

Penney, C. M.

C. M. Penney, R. L. St. Peters, and M. Lapp, J. Opt. Soc. Am. 64, 712 (1974).
[Crossref]

M. Lapp, C. M. Penney, and L. M. Goldman, Opt. Commun. 9, 195 (1973).
[Crossref]

Peters, R. L. St.

Rawson, E. G.

A. D. May, E. G. Rawson, and H. L. Welsh, in Physics of Quantum Electronics, edited by P. L. Kelley, B. Lax, and P. E. Tannenwald (McGraw-Hill, New York, 1966), p. 260.

Salzman, J. A.

J. A. Salzman, in Ref. 2, pp. 179–188.

J. A. Salzman and T. A. Coney, , 1974.

T. A. Coney and J. A. Salzman, , 1973.

J. A. Salzman and T. A. Coney, , 1973.

Smith, J. R.

J. R. Smith, in Ref. 2, pp. 171–178.

Steel, W. H.

W. H. Steel, Interferometry (Cambridge University, Cambridge England, 1967), p. 112.

Stoicheff, B. P.

B. P. Stoicheff, in Advances in Spectroscopy, edited by H. W. Thompson (Interscience, New York, 1959), p. 113.

Sundius, T.

T. Sundius, J. Raman Spectrosc. 1, 471 (1973).
[Crossref]

VanKranendonk, J.

C. G. Gray and J. VanKranendonk, Can. J. Phys. 44, 2411 (1966).
[Crossref]

J. VanKranendonk, Can. J. Phys. 41, 433 (1963).
[Crossref]

J. Fintak and J. VanKranendonk, Can. J. Phys. 40, 1085 (1962); Can. J. Phys. 41, 21 (1963).
[Crossref]

Weber, A.

J. J. Barrett and A. Weber, J. Mol. Spectrosc. 50, 205 (1974).
[Crossref]

A. Weber, “High Resolution Raman Studies of Gases,” in The Raman Effect, edited by A. Anderson (Marcel Dekker, New York, 1973), Vol. 2, pp. 543–757.

A. Weber (private communication).

Welsh, H. L.

K. S. Jammu, G. E. St. John, and H. L. Welsh, Can. J. Phys. 44, 797 (1966).
[Crossref]

A. D. May, E. G. Rawson, and H. L. Welsh, in Physics of Quantum Electronics, edited by P. L. Kelley, B. Lax, and P. E. Tannenwald (McGraw-Hill, New York, 1966), p. 260.

Willetts, P. V.

R. J. Butcher, P. V. Willetts, and W. J. Jones, Proc. E. Soc. London Ser. A 324, 231 (1971).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), pp. 323–333.

Appl. Opt. (7)

Appl. Spectrosc. (1)

Can. J. Phys. (4)

J. Fintak and J. VanKranendonk, Can. J. Phys. 40, 1085 (1962); Can. J. Phys. 41, 21 (1963).
[Crossref]

J. VanKranendonk, Can. J. Phys. 41, 433 (1963).
[Crossref]

C. G. Gray and J. VanKranendonk, Can. J. Phys. 44, 2411 (1966).
[Crossref]

K. S. Jammu, G. E. St. John, and H. L. Welsh, Can. J. Phys. 44, 797 (1966).
[Crossref]

J. Appl. Meteorol. (1)

J. Cooney, J. Appl. Meteorol. 11, 108 (1972).
[Crossref]

J. Mol. Spectrosc. (1)

J. J. Barrett and A. Weber, J. Mol. Spectrosc. 50, 205 (1974).
[Crossref]

J. Opt. Soc. Am. (5)

J. Raman Spectrosc. (2)

T. Sundius, J. Raman Spectrosc. 1, 471 (1973).
[Crossref]

J. Bendtsen, J. Raman Spectrosc. 2, 133 (1974).
[Crossref]

Jpn. J. Appl. Phys. (1)

K. Kaminishi and S. Nawata, Jpn. J. Appl. Phys. 13, 1640 (1974).
[Crossref]

Opt. Commun. (1)

M. Lapp, C. M. Penney, and L. M. Goldman, Opt. Commun. 9, 195 (1973).
[Crossref]

Proc. E. Soc. London Ser. A (1)

R. J. Butcher, P. V. Willetts, and W. J. Jones, Proc. E. Soc. London Ser. A 324, 231 (1971).
[Crossref]

Rev. Sci. Instrum. (2)

R. S. Hickman and L. H. Liang, Rev. Sci. Instrum. 43, 796 (1972).
[Crossref]

R. S. Hickman and L. Liang, Rev. Sci. Instrum. 45, 1580 (1974).
[Crossref]

Other (17)

M. Lapp, in Ref. 2, pp. 107–145.

A. S. Gilbert and H. J. Bernstein, in Ref. 2, pp. 161–169.

J. R. Smith, in Ref. 2, pp. 171–178.

J. A. Salzman, in Ref. 2, pp. 179–188.

G. Herzberg, Molecular Spectra and Molecular Structure I. Spectra of Diatomic Molecules (Van Nostrand, Princeton, 1950).

B. P. Stoicheff, in Advances in Spectroscopy, edited by H. W. Thompson (Interscience, New York, 1959), p. 113.

T. A. Coney and J. A. Salzman, , 1973.

J. A. Salzman and T. A. Coney, , 1973.

J. A. Salzman and T. A. Coney, , 1974.

J. J. Barrett, in Laser Raman Gas Diagnostics, edited by M. Lapp and C. M. Penney (Plenum, New York, 1974), pp. 63–85.
[Crossref]

A. Weber (private communication).

A. D. May, E. G. Rawson, and H. L. Welsh, in Physics of Quantum Electronics, edited by P. L. Kelley, B. Lax, and P. E. Tannenwald (McGraw-Hill, New York, 1966), p. 260.

A. B. Harvey, in Ref. 2, pp. 147–152.

R. Bracewell, The Fourier Transform and Its Applications, (McGraw-Hill, New York, 1965), p. 110.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), pp. 323–333.

W. H. Steel, Interferometry (Cambridge University, Cambridge England, 1967), p. 112.

A. Weber, “High Resolution Raman Studies of Gases,” in The Raman Effect, edited by A. Anderson (Marcel Dekker, New York, 1973), Vol. 2, pp. 543–757.

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Figures (11)

FIG. 1
FIG. 1

A portion of a rotational Raman interferogram for N2 gas showing the split Raman fringe. The N2 gas was excited by the 514.53 nm line of an argon ion laser. The more intense fringes are due to Rayleigh scattering and the abscissa denotes the interference order number for the 514.53 nm line. Between adjacent Rayleigh fringes are two Raman fringes. The more intense Raman fringe is produced by Stokes rotational lines, while the less intense Raman fringe is produced by anti-Stokes rotationl lines. The interferogram shown in this figure was hand-traced from the experimental data.

FIG. 2
FIG. 2

Schematic representation of the experimental arrangement used for generating Fabry-Perot interferograms of gases. The light which is scattered by the gas is collected and collimated by the lens L1 and analyzed by the Fabry-Perot interferometer FPI. The lens L2 images the circular interference fringes in the plane S, which has a circular aperture coincident with the center of the fringe pattern. A detector located behind the circular aperture is used to measure the fringe irradiance at the center of the interference pattern.

FIG. 3
FIG. 3

Graphical representation of the Fourier transforms of the FPI instrumental functions gA(x), gS(x), and gD(x), and a Gaussian line profile h(x). The Fourier transform of the Airy function gA(x) is nonzero only for arguments equal to integral multiples of the optical path γ. Because of this, the product An of all these Fourier transforms is also nonzero for the same discrete values of the argument. This fact greatly simplifies the numerical calculation since it is only necessary to evaluate An for a finite number of discrete points.

FIG. 4
FIG. 4

Computed rotational Raman interferograms of nitrogen gas for the 514.53 nm Rayleigh order numbers: (a) 12140 to 12142, (b) 12160 to 12162, and (c) 12180 to 12182. Figures 46 inclusive were drawn directly by an incremental digital plotter from the computed results. The vertical lines in these figures represent the positions and peak irradiances of the individual rotational Raman lines.

FIG. 5
FIG. 5

Computed rotational Raman interferograms of nitrogen gas for the 514.53 nm Rayleigh order numbers: (a) 12200 to 12202, (b) 12215 to 12217, and (c) 12240 to 12242.

FIG. 6
FIG. 6

Computed rotational Raman interferograms of nitrogen gas for the 514.53 nm Rayleigh order numbers: (a) 12260 to 12262, (b) 12280 to 12282, and (c) 12300 to 12302.

FIG. 7
FIG. 7

Experimental Fabry-Perot profile for a split Raman fringe of nitrogen at room temperature. The fringe components labelled S and A are composed of Stokes and anti-Stokes rotational Raman lines, respectively. The curves shown in this figure was hand-traced from the experimental data.

FIG. 8
FIG. 8

Computed temperature dependence of the Raman fringe splitting Δ and fringe irradiance ratio R for the 514.53 nm Rayleigh fringe number 12260 of nitrogen. Also shown in this figure are the temperature values which were obtained from the experimentally measured values of Δ and R.

FIG. 9
FIG. 9

Computed rotational Raman fringe profiles for the 514.53 nm Rayleigh fringe number 12260 of nitrogen for four different gas temperatures.

FIG. 10
FIG. 10

Computed rotational Raman fringe profiles for the central Raman fringe of nitrogen for different values of the centrifugal distortion constant D.

FIG. 11
FIG. 11

Experimental Fabry-Perot profile of the central rotational Raman fringe of nitrogen. The curve shown in this figure was hand-traced from the experimental data.

Tables (1)

Tables Icon

TABLE I Rotational constants of nitrogen gas.

Equations (38)

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h ( x ) = f ( x ) * g ( x ) - f ( u ) g ( x - u ) d u .
I t = I i [ T / ( 1 - R ) ] 2 [ 1 + F sin 2 ( ϕ / 2 ) ] - 1 ,
m = ω γ ,
Δ ω = ( 2 μ d ) - 1 ,
J ( ω ) = G A ( ω ) * G D ( ω ) * G S ( ω ) ,
j ( x ) F . T . [ J ( ω ) ] = g A ( x ) g D ( x ) g S ( x ) .
G A ( ω ) = [ T 2 / ( 1 - R 2 ) ] ( 1 + 2 n = 1 R n cos 2 π n ω γ ) ,
F . T . [ cos 2 π ω γ ] = - cos 2 π ω γ e - 2 π i ω x d ω = 1 2 [ δ ( x - γ ) + δ ( x + γ ) ] ,
g A ( x ) = [ T 2 / ( 1 - R 2 ) ] n = 0 R n [ δ ( x - n γ ) + δ ( x + n γ ) ] .
G D ( ω ) = 1 for - [ 2 γ N D ] - 1 ω [ 2 γ N D ] - 1 = 0 for all other ω .
g D ( x ) = sin [ π ( γ N D ) - 1 x ] / [ π ( γ N D ) - 1 x ] .
G s ( ω ) = 1 for - [ 2 γ N S ] - 1 ω [ 2 γ N S ] - 1 = 0 for all other ω .
g S ( x ) = sin [ π ( γ N S ) - 1 x ] / [ π ( γ N S ) - 1 x ] .
J ( ω ) = F . T . [ j ( x ) ] = - j ( x ) e 2 π i ω x d x = [ T 2 / ( 1 - R 2 ) ] ( 1 + 2 n = 1 j n cos 2 π n ω γ ) ,
j n = N D N S R n ( n π ) 2 [ sin ( n π / N D ) sin ( n π / N S ) ] .
H N ( ω ) = 2 Δ ω 0 ( ln 2 π ) 1 / 2 exp ( - 4 ln 2 ( Δ ω 0 ) 2 ω 2 ) .
h N ( x ) = exp ( - π 2 ( Δ ω 0 ) 2 4 ln 2 x 2 ) .
I ( J ) = P 0 ρ 0 l f J σ J ,
f J = Q R - 1 g J ( 2 J + 1 ) exp [ - F ( J ) h c / k T ] ,
F ( J ) = B J ( J + 1 ) - D J 2 ( J + 1 ) 2 .
ω ( J ) = F ( J + 2 ) - F ( J ) = ( 4 B - 6 D ) ( J + 3 2 ) - 8 D ( J + 3 2 ) 3 .
I S ( J ) = K 0 ( J + 1 ) ( J + 2 ) ( 2 J + 3 ) B T [ ω 0 - ω ( J ) ] 4 g J exp [ - F ( J ) h c / k T ]
I A ( J ) = K 0 ( J + 1 ) ( J + 2 ) ( 2 J + 3 ) B T [ ω 0 + ω ( J ) ] 4 g J exp [ - F ( J + 2 ) h c / k T ]
H D ( ω ) = 2 Δ ω D ( ln 2 π ) 1 / 2 exp ( - 4 ln 2 ( Δ ω D ) 2 ω 2 ) ,
Δ ω D = ( 2 / c ) ( 2 R T ln 2 / M ) 1 / 2 { 4 [ ω 0 2 + ω 0 ω ( J ) ] × sin 2 ( θ / 2 ) + ω 2 ( J ) } 1 / 2 ,
Δ ω D RAY = ( 4 ω 0 / c ) ( 2 R T ln 2 / M ) 1 / 2 sin ( θ / 2 ) .
h D ( x ) = exp ( - π 2 ( Δ ω D ) 2 4 ln 2 x 2 ) .
L ( ω ) = Δ ω P / ( 2 π ) ω 2 + ( Δ ω P / 2 ) 2 ,
Δ ω P ( 0.110 - 0.003 J ) P ,
l ( x ) = exp ( - π Δ ω P x ) .
I ( γ ) = T 2 1 - R 2 { P 0 ( 1 + 2 n = 1 n A n cos [ 2 π n ω 0 γ ] ) + J = 0 J m [ I S ( J ) ( 1 + 2 n = 1 n A n cos { 2 π n [ ω 0 - ω ( J ) ] γ } ) + I A ( J ) ( 1 + 2 n = 1 n A n cos { 2 π n [ ω 0 + ω ( J ) ] γ } ) ] } ,
A n = j n h N ( n γ ) h D ( n γ )
A n = j n h N ( n γ ) h D ( n γ ) l ( n γ )
I S ( J ) I A ( J ) = ( ω 0 - ω ( J ) ω 0 + ω ( J ) ) 4 exp ( 4 B ( J + 3 2 ) h c k T R ) ,
T R = 4 B ( J + 3 2 ) h c / k ln [ I S ( J ) I A ( J ) ( ω 0 + ω ( J ) ω 0 - ω ( J ) ) 4 ] .
T = 33 453.9 - 277 890 Δ + 865 104 Δ 2 - 1 192 540 Δ 3 + 616 188 Δ 4 .
T = 25 800.6 - 53 100.7 R + 37 793.8 R 2 - 9210.94 R 3 .
D = 399.091 - 4399.75 x + 15 542.5 x 2 - 17 120.3 x 3 .