Abstract

Simplified equations are developed for computing the emissivity of the carbon dioxide 4.3-μ band. The harmonic oscillator model is assumed in computing band intensities, but anharmonicity is considered in computing the spectral distribution of emitted radiation. Calculated values of S¯/d and 2α012S¯12/d are presented for temperatures ranging from 300° to 3000°K, from which emissivities may be readily computed for a given pressure and optical path in both the weak-line and strong-line approximations. In the absence of significant Doppler broadening, both approximations provide upper limits to the emissivity. Emissivities are also computed for the case of a pure Doppler line shape for temperatures ranging from 300° to 1500°K. Doppler broadening is shown to be an important factor in setting a lower limit to the possible values of emissivity obtained at low pressures and long path lengths. Comparisons are made with published experimental data using the weak-line approximation; in several cases, the strong-line approximation as well as a mixed strong-line–weak-line approximation is also shown. The results of the present work are shown to be in good agreement with published experimental data and in definite disagreement with other theoretical calculations.

© 1963 Optical Society of America

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References

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  1. G. N. Plass, J. Opt. Soc. Am. 49, 821 (1959).
    [CrossRef]
  2. C. C. Ferriso, Astronautics Report AE 61-0910, “High Temperature Infrared Emission and Absorption Studies,” September1961; also published in J. Chem. Phys. 37, 1955 (1962).
    [CrossRef]
  3. R. H. Tourin, J. Opt. Soc. Am. 51, 175 (1961).
    [CrossRef]
  4. D. E. Burch and D. A. Gryvnak, “Infrared Radiation Emitted by Hot Gases and its Transmission Through Synthetic Atmospheres,” Aeronutronic Rep. U-1929, October1962.
  5. M. Steinberg and W. O. Davies, J. Chem. Phys. 34, 1373 (1961).
    [CrossRef]
  6. G. Herzberg, Infrared and Raman Spectra (D. Van Nostrand Inc., Princeton, New Jersey, 1945), p. 380.
  7. W. Malkmus and A. Thomson, Convair Report ZPh-095, “Infrared Emissivity of Diatomic Gases for the Anharmonic Vibrating Rotator Model,” May1961; also published in J. Quant. Spectry. Radiative Transfer 2, 17 (1962).
    [CrossRef]
  8. G. N. Plass, J. Opt. Soc. Am. 48, 690 (1958).
    [CrossRef]
  9. W. Malkmus, “Infrared Emissivity of Diatomic Gases with Doppler Line Shape,” Convair Report ZPh-119, September1961; also to be published in another journal.
  10. Reference 5, p. 211.
  11. Reference 5, p. 215.
  12. This approximation is based on Dennison’s values (x13= −21.9, x23= −11.0) as given by Herzberg,13 and used by Plass.8 Courtoy’s14 more recent measurements differ somewhat (x13= 19.37, x23= 12.53); however, the exactness of the approximation x13≈ 2x23is not critical since x13υ1+x23υ2≡x¯υ+12(12x13−x23)(2ν1−ν2). The second term is dropped in Eq. (6); its coefficient is small compared with x¯(using either set of data), and the factor |2υ1− υ2| ≤ υ. The approximation ω1≈ 2ω2 is very close; however, the same remarks apply to its noncriticality.
  13. Reference 6, p. 276.
  14. C. P. Courtoy, Can. J. Phys. 35, 608 (1957).
    [CrossRef]
  15. Reference 5, p. 503.
  16. S. S. Penner, Quantitative Molecular Spectroscopy and Gas Emissivities (Addison-Wesley Publishing Company, Inc., Reading, Massachusetts, 1959), p. 152.
  17. D. M. Dennison, Rev. Mod. Phys. 3, 280 (1931).
    [CrossRef]
  18. W. S. Benedict and E. K. Plyler, “High-Resolution Spectra of Hydrocarbon Flames,” in Energy Transfer in Hot Gases, pp. 57–73, Natl. Bur. Std. Circ. No. 523, 1954.
  19. L. D. Kaplan and D. F. Eggers, J. Chem. Phys. 25, 876 (1956).
    [CrossRef]
  20. R. Herman and R. F. Wallis, J. Chem. Phys. 23, 635 (1955).
    [CrossRef]
  21. R. P. Madden, J. Chem. Phys. 35, 2083 (1961).
    [CrossRef]
  22. D. W. G. Ballentyne and L. E. Q. Walker, A Dictionary of Named Effects and Laws in Chemistry, Physics and Mathematics (The Macmillan Company, New York, 1961), 2nd ed., p. 14; Van Nostrand’s Scientific Encyclopedia (D. Van Nostrand Inc., New York, 1958), 3rd ed., p. 181; Encyclopaedic Dictionary of Physics (Pergamon Press, Inc., New York, 1961), p. 388. Also known as Beer–Bouguer or Beer–Lambert Law.
  23. Assume a statistical distribution of lines of intensity S with an arbitrary probability distribution P(S). Then we have8ln(I0I)=d−1∫0∞∫0∞{1−exp[−cplSb(p,ω)]}dωP(S)dS, where pis the total pressure, c the mole fraction of absorbing gas, and b(p,ω)the normalized line shape factor. If we assume the line shape to be linearly pressure broadened, but otherwise arbitrary we have b(p,ω)=p−1f[(ω−ω0)/p],where f is an arbitrary function of its argument. If we change the lower limit of the integral over ω from 0 to −∞ for mathematical convenience and make the substitution u=(ω−ω0)/p, we findln(I0I)=pd−1∫0∞∫−∞∞{1−exp[−clSf(u)]}duP(S)dS. Thus ln (I0/I) is linear in p, although Beer’s law is not satisfied in general, as is seen from the complicated functional dependence on l.
  24. A. Thomson (private communication).
  25. H. J. Babrov, P. M. Henry, and R. H. Tourin, The Warner & Swasey Company Scientific Report No. 2 under Contract AF19 (604)-6106, “Methods of Predicting Infrared Radiance of Flames by Extrapolation from Laboratory Measurements,” October1961; also published in J. Chem. Phys. 37, 581 (1962).
    [CrossRef]
  26. U. P. Oppenheim and Y. Ben-Aryeh, J. Opt. Soc. Am. 53, 344 (1963).
    [CrossRef]

1963 (1)

1961 (3)

R. H. Tourin, J. Opt. Soc. Am. 51, 175 (1961).
[CrossRef]

M. Steinberg and W. O. Davies, J. Chem. Phys. 34, 1373 (1961).
[CrossRef]

R. P. Madden, J. Chem. Phys. 35, 2083 (1961).
[CrossRef]

1959 (1)

1958 (1)

1957 (1)

C. P. Courtoy, Can. J. Phys. 35, 608 (1957).
[CrossRef]

1956 (1)

L. D. Kaplan and D. F. Eggers, J. Chem. Phys. 25, 876 (1956).
[CrossRef]

1955 (1)

R. Herman and R. F. Wallis, J. Chem. Phys. 23, 635 (1955).
[CrossRef]

1931 (1)

D. M. Dennison, Rev. Mod. Phys. 3, 280 (1931).
[CrossRef]

Babrov, H. J.

H. J. Babrov, P. M. Henry, and R. H. Tourin, The Warner & Swasey Company Scientific Report No. 2 under Contract AF19 (604)-6106, “Methods of Predicting Infrared Radiance of Flames by Extrapolation from Laboratory Measurements,” October1961; also published in J. Chem. Phys. 37, 581 (1962).
[CrossRef]

Ballentyne, D. W. G.

D. W. G. Ballentyne and L. E. Q. Walker, A Dictionary of Named Effects and Laws in Chemistry, Physics and Mathematics (The Macmillan Company, New York, 1961), 2nd ed., p. 14; Van Nostrand’s Scientific Encyclopedia (D. Van Nostrand Inc., New York, 1958), 3rd ed., p. 181; Encyclopaedic Dictionary of Physics (Pergamon Press, Inc., New York, 1961), p. 388. Also known as Beer–Bouguer or Beer–Lambert Law.

Ben-Aryeh, Y.

Benedict, W. S.

W. S. Benedict and E. K. Plyler, “High-Resolution Spectra of Hydrocarbon Flames,” in Energy Transfer in Hot Gases, pp. 57–73, Natl. Bur. Std. Circ. No. 523, 1954.

Burch, D. E.

D. E. Burch and D. A. Gryvnak, “Infrared Radiation Emitted by Hot Gases and its Transmission Through Synthetic Atmospheres,” Aeronutronic Rep. U-1929, October1962.

Courtoy, C. P.

C. P. Courtoy, Can. J. Phys. 35, 608 (1957).
[CrossRef]

Davies, W. O.

M. Steinberg and W. O. Davies, J. Chem. Phys. 34, 1373 (1961).
[CrossRef]

Dennison, D. M.

D. M. Dennison, Rev. Mod. Phys. 3, 280 (1931).
[CrossRef]

Eggers, D. F.

L. D. Kaplan and D. F. Eggers, J. Chem. Phys. 25, 876 (1956).
[CrossRef]

Ferriso, C. C.

C. C. Ferriso, Astronautics Report AE 61-0910, “High Temperature Infrared Emission and Absorption Studies,” September1961; also published in J. Chem. Phys. 37, 1955 (1962).
[CrossRef]

Gryvnak, D. A.

D. E. Burch and D. A. Gryvnak, “Infrared Radiation Emitted by Hot Gases and its Transmission Through Synthetic Atmospheres,” Aeronutronic Rep. U-1929, October1962.

Henry, P. M.

H. J. Babrov, P. M. Henry, and R. H. Tourin, The Warner & Swasey Company Scientific Report No. 2 under Contract AF19 (604)-6106, “Methods of Predicting Infrared Radiance of Flames by Extrapolation from Laboratory Measurements,” October1961; also published in J. Chem. Phys. 37, 581 (1962).
[CrossRef]

Herman, R.

R. Herman and R. F. Wallis, J. Chem. Phys. 23, 635 (1955).
[CrossRef]

Herzberg, G.

G. Herzberg, Infrared and Raman Spectra (D. Van Nostrand Inc., Princeton, New Jersey, 1945), p. 380.

Kaplan, L. D.

L. D. Kaplan and D. F. Eggers, J. Chem. Phys. 25, 876 (1956).
[CrossRef]

Madden, R. P.

R. P. Madden, J. Chem. Phys. 35, 2083 (1961).
[CrossRef]

Malkmus, W.

W. Malkmus, “Infrared Emissivity of Diatomic Gases with Doppler Line Shape,” Convair Report ZPh-119, September1961; also to be published in another journal.

W. Malkmus and A. Thomson, Convair Report ZPh-095, “Infrared Emissivity of Diatomic Gases for the Anharmonic Vibrating Rotator Model,” May1961; also published in J. Quant. Spectry. Radiative Transfer 2, 17 (1962).
[CrossRef]

Oppenheim, U. P.

Penner, S. S.

S. S. Penner, Quantitative Molecular Spectroscopy and Gas Emissivities (Addison-Wesley Publishing Company, Inc., Reading, Massachusetts, 1959), p. 152.

Plass, G. N.

Plyler, E. K.

W. S. Benedict and E. K. Plyler, “High-Resolution Spectra of Hydrocarbon Flames,” in Energy Transfer in Hot Gases, pp. 57–73, Natl. Bur. Std. Circ. No. 523, 1954.

Steinberg, M.

M. Steinberg and W. O. Davies, J. Chem. Phys. 34, 1373 (1961).
[CrossRef]

Thomson, A.

A. Thomson (private communication).

W. Malkmus and A. Thomson, Convair Report ZPh-095, “Infrared Emissivity of Diatomic Gases for the Anharmonic Vibrating Rotator Model,” May1961; also published in J. Quant. Spectry. Radiative Transfer 2, 17 (1962).
[CrossRef]

Tourin, R. H.

R. H. Tourin, J. Opt. Soc. Am. 51, 175 (1961).
[CrossRef]

H. J. Babrov, P. M. Henry, and R. H. Tourin, The Warner & Swasey Company Scientific Report No. 2 under Contract AF19 (604)-6106, “Methods of Predicting Infrared Radiance of Flames by Extrapolation from Laboratory Measurements,” October1961; also published in J. Chem. Phys. 37, 581 (1962).
[CrossRef]

Walker, L. E. Q.

D. W. G. Ballentyne and L. E. Q. Walker, A Dictionary of Named Effects and Laws in Chemistry, Physics and Mathematics (The Macmillan Company, New York, 1961), 2nd ed., p. 14; Van Nostrand’s Scientific Encyclopedia (D. Van Nostrand Inc., New York, 1958), 3rd ed., p. 181; Encyclopaedic Dictionary of Physics (Pergamon Press, Inc., New York, 1961), p. 388. Also known as Beer–Bouguer or Beer–Lambert Law.

Wallis, R. F.

R. Herman and R. F. Wallis, J. Chem. Phys. 23, 635 (1955).
[CrossRef]

Can. J. Phys. (1)

C. P. Courtoy, Can. J. Phys. 35, 608 (1957).
[CrossRef]

J. Chem. Phys. (4)

M. Steinberg and W. O. Davies, J. Chem. Phys. 34, 1373 (1961).
[CrossRef]

L. D. Kaplan and D. F. Eggers, J. Chem. Phys. 25, 876 (1956).
[CrossRef]

R. Herman and R. F. Wallis, J. Chem. Phys. 23, 635 (1955).
[CrossRef]

R. P. Madden, J. Chem. Phys. 35, 2083 (1961).
[CrossRef]

J. Opt. Soc. Am. (4)

Rev. Mod. Phys. (1)

D. M. Dennison, Rev. Mod. Phys. 3, 280 (1931).
[CrossRef]

Other (16)

W. S. Benedict and E. K. Plyler, “High-Resolution Spectra of Hydrocarbon Flames,” in Energy Transfer in Hot Gases, pp. 57–73, Natl. Bur. Std. Circ. No. 523, 1954.

W. Malkmus, “Infrared Emissivity of Diatomic Gases with Doppler Line Shape,” Convair Report ZPh-119, September1961; also to be published in another journal.

Reference 5, p. 211.

Reference 5, p. 215.

This approximation is based on Dennison’s values (x13= −21.9, x23= −11.0) as given by Herzberg,13 and used by Plass.8 Courtoy’s14 more recent measurements differ somewhat (x13= 19.37, x23= 12.53); however, the exactness of the approximation x13≈ 2x23is not critical since x13υ1+x23υ2≡x¯υ+12(12x13−x23)(2ν1−ν2). The second term is dropped in Eq. (6); its coefficient is small compared with x¯(using either set of data), and the factor |2υ1− υ2| ≤ υ. The approximation ω1≈ 2ω2 is very close; however, the same remarks apply to its noncriticality.

Reference 6, p. 276.

D. E. Burch and D. A. Gryvnak, “Infrared Radiation Emitted by Hot Gases and its Transmission Through Synthetic Atmospheres,” Aeronutronic Rep. U-1929, October1962.

C. C. Ferriso, Astronautics Report AE 61-0910, “High Temperature Infrared Emission and Absorption Studies,” September1961; also published in J. Chem. Phys. 37, 1955 (1962).
[CrossRef]

D. W. G. Ballentyne and L. E. Q. Walker, A Dictionary of Named Effects and Laws in Chemistry, Physics and Mathematics (The Macmillan Company, New York, 1961), 2nd ed., p. 14; Van Nostrand’s Scientific Encyclopedia (D. Van Nostrand Inc., New York, 1958), 3rd ed., p. 181; Encyclopaedic Dictionary of Physics (Pergamon Press, Inc., New York, 1961), p. 388. Also known as Beer–Bouguer or Beer–Lambert Law.

Assume a statistical distribution of lines of intensity S with an arbitrary probability distribution P(S). Then we have8ln(I0I)=d−1∫0∞∫0∞{1−exp[−cplSb(p,ω)]}dωP(S)dS, where pis the total pressure, c the mole fraction of absorbing gas, and b(p,ω)the normalized line shape factor. If we assume the line shape to be linearly pressure broadened, but otherwise arbitrary we have b(p,ω)=p−1f[(ω−ω0)/p],where f is an arbitrary function of its argument. If we change the lower limit of the integral over ω from 0 to −∞ for mathematical convenience and make the substitution u=(ω−ω0)/p, we findln(I0I)=pd−1∫0∞∫−∞∞{1−exp[−clSf(u)]}duP(S)dS. Thus ln (I0/I) is linear in p, although Beer’s law is not satisfied in general, as is seen from the complicated functional dependence on l.

A. Thomson (private communication).

H. J. Babrov, P. M. Henry, and R. H. Tourin, The Warner & Swasey Company Scientific Report No. 2 under Contract AF19 (604)-6106, “Methods of Predicting Infrared Radiance of Flames by Extrapolation from Laboratory Measurements,” October1961; also published in J. Chem. Phys. 37, 581 (1962).
[CrossRef]

G. Herzberg, Infrared and Raman Spectra (D. Van Nostrand Inc., Princeton, New Jersey, 1945), p. 380.

W. Malkmus and A. Thomson, Convair Report ZPh-095, “Infrared Emissivity of Diatomic Gases for the Anharmonic Vibrating Rotator Model,” May1961; also published in J. Quant. Spectry. Radiative Transfer 2, 17 (1962).
[CrossRef]

Reference 5, p. 503.

S. S. Penner, Quantitative Molecular Spectroscopy and Gas Emissivities (Addison-Wesley Publishing Company, Inc., Reading, Massachusetts, 1959), p. 152.

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Figures (20)

Fig. 1
Fig. 1

S ¯ / d vs wavenumber for T = 300°, 600°, and 1200°K.

Fig. 2
Fig. 2

S ¯ / d vs wavenumber for T = 1800°, 2400°, and 3000°K.

Fig. 3
Fig. 3

2 α 0 1 2 S ¯ 1 2 / d vs wavenumber for T = 300°, 600°, and 1200°K.

Fig. 4
Fig. 4

2 α 0 1 2 S ¯ 1 2 / d vs wavenumber for T = 1800°, 2400°, and 3000°K.

Fig. 5
Fig. 5

S ¯ / d vs T for 1900 ≤ ω ≤ 2330 cm−1.

Fig. 6
Fig. 6

S ¯ / d vs T for 2340 ≤ ω ≤ 2390 cm−1.

Fig. 7
Fig. 7

Emissivity vs wavenumber for pure Doppler line shape and weak-line approximation for T = 300°K.

Fig. 8
Fig. 8

Emissivity vs wavenumber for pure Doppler line shape and weak-line approximation for T = 600°K.

Fig. 9
Fig. 9

Emissivity vs wavenumber for pure Doppler line shape and weak-line approximation for T = 1200°K.

Fig. 10
Fig. 10

Emissivity vs wavenumber for pure Doppler line shape and weak-line approximation for T = 1500°K.

Fig. 11
Fig. 11

Comparison of S ¯ / d with Plass at 1200°K.

Fig. 12
Fig. 12

Comparison of S/d with Plass at 2400°K.

Fig. 13
Fig. 13

Comparison of calculated emissivity with Ferriso’s measurements at 1200°K.

Fig. 14
Fig. 14

Comparison of calculated emissivity with Ferriso’s measurements at 1800°K.

Fig. 15
Fig. 15

Comparison of calculated emissivity with Ferriso’s measurements at 2400°K.

Fig. 16
Fig. 16

Comparison of calculated emissivity with Tourin’s measurements of pure CO2 at 1200°K.

Fig. 17
Fig. 17

Comparison of calculated emissivity with Tourin’s measurements of N2-broadened CO2 at 1200°K.

Fig. 18
Fig. 18

Comparison of calculated emissivity with Burch’s measurements at 1200°K.

Fig. 19
Fig. 19

Comparison of S ¯ / d with Oppenheim and Ben-Aryeh’s extrapolated experimental measurements.

Fig. 20
Fig. 20

Comparison of 2 α 0 1 2 S ¯ 1 2 / d with Oppenheim and Ben-Aryeh’s extrapolated experimental measurements.

Equations (56)

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G ( υ 1 υ 2 υ 3 l ) = ω 1 ( υ 1 + 1 2 ) + ω 2 ( υ 2 + 1 ) + ω 3 ( υ 3 + 1 2 ) + x 11 ( υ 1 + 1 2 ) 2 + x 22 ( υ 2 + 1 ) 2 + g 22 l 2 + x 33 ( υ 3 + 1 2 ) 2 + x 12 ( υ 1 + 1 2 ) ( υ 2 + 1 ) + x 13 ( υ 1 + 1 2 ) ( υ 3 + 1 2 ) + x 23 ( υ 2 + 1 ) ( υ 3 + 1 2 ) + ,
ω ( υ 1 υ 2 υ 3 l υ 1 υ 2 υ 3 + 1 l ) = G ( υ 1 υ 2 υ 3 + 1 l ) G ( υ 1 υ 2 υ 3 l ) = ω 3 + 1 2 x 13 + x 23 + 2 x 33 + x 13 υ 1 + x 23 υ 2 + 2 x 33 υ 3 + .
ω ¯ = 1 2 ( 1 2 ω 1 + ω 2 )
x ¯ = 1 2 ( 1 2 x 13 + x 23 ) .
υ = 2 υ 1 + υ 2 .
x ¯ υ = x 13 υ 1 + x 23 υ 2
ω ¯ υ = ω 1 υ 1 + ω 2 υ 2 .
ω υ υ 3 = ω 3 + 1 2 x 13 + x 23 + 2 x 33 + x ¯ υ + 2 x 33 υ 3
E ( υ 1 υ 2 υ 3 l ) = G ( υ 1 υ 2 υ 3 l ) G ( 0000 ) = ω 1 υ 1 + ω 2 υ 2 + ω 3 υ 3
E ( υ υ 3 ) = ω ¯ υ + ω 3 υ 3 .
Q = υ 1 = 0 υ 2 = 0 υ 3 = 0 l υ 2 g l exp [ E ( υ 1 υ 2 υ 3 l ) h c k T ] ,
g l = 2 l > 0 = 1 l = 0 ,
Q ( T ) = Q 0 ( T ) Q 3 ( T ) ,
Q 3 ( T ) = υ 3 = 0 exp ( υ 3 ω 3 h c k T ) = [ 1 exp ( ω 3 h c k T ) ] 1
Q 0 ( T ) = υ 1 = 0 υ 2 = 0 l υ 2 g l exp [ ( ω 1 υ 1 + ω 2 υ 2 ) h c k T ] ,
υ * f ( υ 1 , υ 2 , l ) ,
υ * f ( υ 1 , υ 2 , l ) = even υ 2 = 0 υ even l = 0 υ 2 f [ 1 2 ( υ υ 2 ) , υ 2 , l ] ( υ even ) ,
= odd υ 2 = 1 υ odd l = 1 υ 2 f [ 1 2 ( υ υ 2 ) , υ 2 , l ] ( υ odd ) .
Q 0 ( T ) = υ = 0 υ * g l exp ( υ ω ¯ h c k T ) ,
Q 0 ( T ) = υ = 0 exp ( υ ω ¯ h c k T ) υ * g l .
υ * g l = 1 4 ( υ + 2 ) 2 ( υ even )
= 1 4 ( υ + 1 ) ( υ + 3 ) ( υ odd ) .
Q 0 ( T ) = υ = 0 q υ ( T ) ,
q υ ( T ) = g υ exp ( υ h c ω ¯ / k T ) ,
g υ = 1 4 ( υ + 2 ) 2 ( υ even )
= 1 4 ( υ + 1 ) ( υ + 3 ) ( υ odd ) .
Q 0 ( T ) = [ 1 exp ( ω ¯ h c / k T ) ] 3 + [ 1 + exp ( ω ¯ h c / k T ) ] 1 ,
Q 0 ( T ) = [ 1 exp ( 2 ω ¯ h c / k T ) ] 1 + [ 1 exp ( ω ¯ h c / k T ) ] 2 ,
n υ υ 3 ( T ) = q υ ( T ) exp ( υ 3 ω 3 h c / k T ) Q ( T ) = g υ exp ( υ ω ¯ h c k T ) exp ( υ 3 ω 3 h c k T ) × [ 1 exp ( ω ¯ h c k T ) ] 3 [ 1 + exp ( ω ¯ h c k T ) ] × [ 1 exp ( ω 3 h c k T ) ] ,
α ( υ 1 υ 2 υ 3 l υ 1 υ 2 υ 3 l ) = 8 π 3 3 h c N T β 2 ω g l exp [ W υ ( υ 1 υ 2 υ 3 l ) k T ] × Q 1 [ 1 exp ( ω h c k T ) ] ,
α υ υ 3 ( T ) = υ * α ( υ 1 υ 2 υ 3 l υ 1 υ 2 υ 3 + 1 l ) .
α υ 0 ( T ) / α 00 ( T ) = ( υ * g l ) exp ( υ ω ¯ h c / k T ) = g υ exp ( υ ω ¯ h c / k T ) = q υ ( T ) ,
α υ 0 ( T ) / α υ 0 ( T ) = q υ ( T ) / q υ ( T ) = n υ 0 ( T ) / n υ 0 ( T ) ,
α υ υ 3 ( T ) / α υ 0 ( T ) = ( υ 3 + 1 ) exp ( υ 3 ω 3 h c / k T ) = ( υ 3 + 1 ) n υ υ 3 ( T ) / n υ 0 ( T ) .
α υ 0 ( T ) α υ 0 ( T 0 ) = T 0 T n υ 0 ( T ) n υ 0 ( T 0 ) 1 exp ( ω 3 h c / k T ) 1 exp ( ω 3 h c / k T 0 ) .
ω = 1 exp [ p l S ¯ ( ω ) / d ( ω ) ] ,
S ¯ ( ω ) d ( ω ) = υ = 0 υ 3 = 0 [ S υ υ 3 ( ) ( ω ) + S υ υ 3 ( + ) ( ω ) ] d υ υ 3 ( ω ) ,
S υ υ 3 ( ) ( ω ) = α υ υ 3 B e h c ω ¯ υ υ 3 k T ω | B e α 3 ( υ 3 + 1 ) { [ B e α 3 ( υ 3 + 1 ) ] 2 α 3 ( ω ω υ υ 3 ) } 1 2 α 3 | × exp { h c [ B e α 3 ( υ 3 + 1 ) ] k α 3 2 T [ 2 [ B e α 3 ( υ 3 + 1 ) ] ( B e α 3 ( υ 3 + 1 ) { [ B e α 3 ( υ 3 + 1 ) ] 2 α 3 ( ω ω υ υ 3 ) } 1 2 ) [ 1 + 1 2 α 3 B e α 3 ( υ 3 + 1 ) ] α 3 ( ω ω υ υ 3 ) ] } × [ 1 + C α 3 ( B e α 3 ( υ 3 + 1 ) { [ B e α 3 ( υ 3 + 1 ) ] 2 α 3 ( ω ω υ υ 3 ) } 1 2 ) ] [ 1 exp ( ω h c k T ) ] ,
d υ υ 3 ( ω ) = 2 { [ B e α 3 ( υ 3 + 1 ) ] 2 α 3 ( ω ω υ υ 3 ) } 1 2 ,
ω ¯ υ υ 3 = ω υ υ 3 [ 1 exp ( h c ω υ υ 3 / k T ) ] .
1 / d * = υ * 1 / d l ,
d l = 2 d υ υ 3 for l = 0
= d υ υ 3 for l > 0 ,
d l = 2 d υ υ 3 / g l .
1 / d * = 1 2 υ * g l / d υ υ 3 = 1 2 g υ / d υ υ 3 ,
d * = 2 d υ υ 3 / g υ .
S * = 2 S υ υ 3 ( ) ( ω ) / g υ .
( S * ) 1 2 / d * = [ 2 S υ υ 3 ( ) ( ω ) / g υ ] 1 2 / [ 2 d υ υ 3 ( ω ) / g υ ] = ( 1 2 g υ ) 1 2 [ S υ υ 3 ( ) ( ω ) ] 1 2 / d υ υ 3 ( ω ) .
ω = 1 exp [ 2 α 0 1 2 ( p 2 l ) 1 2 S ¯ 1 2 ( ω ) / d ( ω ) ] ,
S ¯ 1 2 ( ω ) d ( ω ) = υ = 0 υ 3 = 0 ( 1 2 g υ ) 1 2 { [ S υ υ 3 ( ) ( ω ) ] 1 2 + [ S υ υ 3 ( + ) ( ω ) ] 1 2 } d υ υ 3 ( ω ) .
W s l = S X n = 0 ( P X ) n ( n + 1 ) ! ( n + 1 ) 1 2 ,
P = ( S / ω 0 ) ( m c 2 / 2 π k T ) 1 2
X = p l .
ω = 1 exp [ υ = 0 υ 3 = 0 ( W * υ υ 3 sl / d * υ υ 3 ) ] .
ln(I0I)=d100{1exp[cplSb(p,ω)]}dωP(S)dS,
ln(I0I)=pd10{1exp[clSf(u)]}duP(S)dS.