Abstract

This paper shows that spectrally resolved thermal radiation from silica aggregate particles can be used to extract an emissivity and a temperature in the visible regime. Emissivity of silica aggregate particles at temperatures above 2000K is measured by the analysis of emission radiation spectra from the particles. Temperature is estimated from the relation between the emission intensity and the wavenumber. Relative emissivities at temperatures from 2150 to 2919K are presented. Proper knowledge of optical properties for silica aggregate particles will help further the understanding of thermophysics at high temperature.

© 2011 Optical Society of America

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References

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  1. J. Y. Yin and L. H. Liu, “Influence of complex component and particle polydispersity on radiative properties of soot aggregate in atmosphere,” J. Quant. Spectrosc. Radiat. Transfer 111, 2115–2126 (2010).
    [CrossRef]
  2. M. Ojanen, P. Kärhä, and E. Ikonen, “Spectral irradiance model for tungsten halogen lamps in 340–850 nm wavelength range,” Appl. Opt. 49, 880–886 (2010).
    [CrossRef] [PubMed]
  3. M. Freitag, H.-Y. Chiu, M. Steiner, V. Perebeinos, and P. Avouris, “Thermal infrared emission from biased graphene,” Nat. Nanotechnol. 5, 497–501 (2010).
    [CrossRef] [PubMed]
  4. N. M. Ravindra, B. Sopori, O. H. Gokce, S. X. Cheng, A. Shenoy, L. Jin, S. Abedrabbo, W. Chen, and Y. Zhang, “Emissivity measurements and modeling of silicon-related materials: an overview,” Int. J. Thermophys. 22, 1593–1611(2001).
    [CrossRef]
  5. B. Rousseau, M. D. Michiel, A. Canizares, D. D. S. Meneses, P. Echegut, and J. F. Thovert, “Temperature effect (300–1500 K) on the infrared photon transport inside an x-ray microtomographic reconstructed porous silica glass,” J. Quant. Spectrosc. Radiat. Transfer 104, 257–265 (2007).
    [CrossRef]
  6. E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, and J.-J. Greffet, “Radiative heat transfer at the nanoscale,” Nat. Photonics 3, 514–517 (2009).
    [CrossRef]
  7. E. Takasuka, E. Tokizaki, K. Terashima, and S. Kimura, “Emissivity of liquid silicon in visible and infrared regions,” J. Appl. Phys. 81, 6384–6389 (1997).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  12. H. D. Jang, “Generation of silica nanoparticles from tetraethylorthodilicate (TEOS) vapor in a diffusion flame,” Aerosol Sci. Technol. 30, 477–488 (1999).
    [CrossRef]
  13. H. W. Coleman and W. G. Steele Jr., Experimentation and Uncertainty Analysis for Engineers (Wiley, 1989).

2010 (3)

J. Y. Yin and L. H. Liu, “Influence of complex component and particle polydispersity on radiative properties of soot aggregate in atmosphere,” J. Quant. Spectrosc. Radiat. Transfer 111, 2115–2126 (2010).
[CrossRef]

M. Ojanen, P. Kärhä, and E. Ikonen, “Spectral irradiance model for tungsten halogen lamps in 340–850 nm wavelength range,” Appl. Opt. 49, 880–886 (2010).
[CrossRef] [PubMed]

M. Freitag, H.-Y. Chiu, M. Steiner, V. Perebeinos, and P. Avouris, “Thermal infrared emission from biased graphene,” Nat. Nanotechnol. 5, 497–501 (2010).
[CrossRef] [PubMed]

2009 (1)

E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, and J.-J. Greffet, “Radiative heat transfer at the nanoscale,” Nat. Photonics 3, 514–517 (2009).
[CrossRef]

2007 (1)

B. Rousseau, M. D. Michiel, A. Canizares, D. D. S. Meneses, P. Echegut, and J. F. Thovert, “Temperature effect (300–1500 K) on the infrared photon transport inside an x-ray microtomographic reconstructed porous silica glass,” J. Quant. Spectrosc. Radiat. Transfer 104, 257–265 (2007).
[CrossRef]

2004 (1)

R. Mueller, H. K. Kammler, S. E. Pratsinis, A. Vital, G. Beaucage, and P. Burtscher, “Non-agglomerated dry silica nanoparticles,” Powder Technol. 140, 40–48 (2004).
[CrossRef]

2001 (3)

P. V. Pikhitsa and I. S. Altman, “Anomalies in light absorption coefficient of silica nanoparticles generated within flame,” J. Nanopart. Res. 3, 303–308 (2001).
[CrossRef]

I. S. Altman, D. Lee, J. D. Chung, J. Song, and M. Choi, “Light absorption of silica nanoparticles,” Phys. Rev. B 63, 161402(2001).
[CrossRef]

N. M. Ravindra, B. Sopori, O. H. Gokce, S. X. Cheng, A. Shenoy, L. Jin, S. Abedrabbo, W. Chen, and Y. Zhang, “Emissivity measurements and modeling of silicon-related materials: an overview,” Int. J. Thermophys. 22, 1593–1611(2001).
[CrossRef]

1999 (1)

H. D. Jang, “Generation of silica nanoparticles from tetraethylorthodilicate (TEOS) vapor in a diffusion flame,” Aerosol Sci. Technol. 30, 477–488 (1999).
[CrossRef]

1997 (1)

E. Takasuka, E. Tokizaki, K. Terashima, and S. Kimura, “Emissivity of liquid silicon in visible and infrared regions,” J. Appl. Phys. 81, 6384–6389 (1997).
[CrossRef]

1989 (2)

Abedrabbo, S.

N. M. Ravindra, B. Sopori, O. H. Gokce, S. X. Cheng, A. Shenoy, L. Jin, S. Abedrabbo, W. Chen, and Y. Zhang, “Emissivity measurements and modeling of silicon-related materials: an overview,” Int. J. Thermophys. 22, 1593–1611(2001).
[CrossRef]

Altman, I. S.

P. V. Pikhitsa and I. S. Altman, “Anomalies in light absorption coefficient of silica nanoparticles generated within flame,” J. Nanopart. Res. 3, 303–308 (2001).
[CrossRef]

I. S. Altman, D. Lee, J. D. Chung, J. Song, and M. Choi, “Light absorption of silica nanoparticles,” Phys. Rev. B 63, 161402(2001).
[CrossRef]

Avouris, P.

M. Freitag, H.-Y. Chiu, M. Steiner, V. Perebeinos, and P. Avouris, “Thermal infrared emission from biased graphene,” Nat. Nanotechnol. 5, 497–501 (2010).
[CrossRef] [PubMed]

Beaucage, G.

R. Mueller, H. K. Kammler, S. E. Pratsinis, A. Vital, G. Beaucage, and P. Burtscher, “Non-agglomerated dry silica nanoparticles,” Powder Technol. 140, 40–48 (2004).
[CrossRef]

Burtscher, P.

R. Mueller, H. K. Kammler, S. E. Pratsinis, A. Vital, G. Beaucage, and P. Burtscher, “Non-agglomerated dry silica nanoparticles,” Powder Technol. 140, 40–48 (2004).
[CrossRef]

Canizares, A.

B. Rousseau, M. D. Michiel, A. Canizares, D. D. S. Meneses, P. Echegut, and J. F. Thovert, “Temperature effect (300–1500 K) on the infrared photon transport inside an x-ray microtomographic reconstructed porous silica glass,” J. Quant. Spectrosc. Radiat. Transfer 104, 257–265 (2007).
[CrossRef]

Chen, W.

N. M. Ravindra, B. Sopori, O. H. Gokce, S. X. Cheng, A. Shenoy, L. Jin, S. Abedrabbo, W. Chen, and Y. Zhang, “Emissivity measurements and modeling of silicon-related materials: an overview,” Int. J. Thermophys. 22, 1593–1611(2001).
[CrossRef]

Cheng, S. X.

N. M. Ravindra, B. Sopori, O. H. Gokce, S. X. Cheng, A. Shenoy, L. Jin, S. Abedrabbo, W. Chen, and Y. Zhang, “Emissivity measurements and modeling of silicon-related materials: an overview,” Int. J. Thermophys. 22, 1593–1611(2001).
[CrossRef]

Chevrier, J.

E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, and J.-J. Greffet, “Radiative heat transfer at the nanoscale,” Nat. Photonics 3, 514–517 (2009).
[CrossRef]

Chin, D.

Chiu, H.-Y.

M. Freitag, H.-Y. Chiu, M. Steiner, V. Perebeinos, and P. Avouris, “Thermal infrared emission from biased graphene,” Nat. Nanotechnol. 5, 497–501 (2010).
[CrossRef] [PubMed]

Choi, M.

I. S. Altman, D. Lee, J. D. Chung, J. Song, and M. Choi, “Light absorption of silica nanoparticles,” Phys. Rev. B 63, 161402(2001).
[CrossRef]

Chung, J. D.

I. S. Altman, D. Lee, J. D. Chung, J. Song, and M. Choi, “Light absorption of silica nanoparticles,” Phys. Rev. B 63, 161402(2001).
[CrossRef]

Coleman, H. W.

H. W. Coleman and W. G. Steele Jr., Experimentation and Uncertainty Analysis for Engineers (Wiley, 1989).

Comin, F.

E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, and J.-J. Greffet, “Radiative heat transfer at the nanoscale,” Nat. Photonics 3, 514–517 (2009).
[CrossRef]

Echegut, P.

B. Rousseau, M. D. Michiel, A. Canizares, D. D. S. Meneses, P. Echegut, and J. F. Thovert, “Temperature effect (300–1500 K) on the infrared photon transport inside an x-ray microtomographic reconstructed porous silica glass,” J. Quant. Spectrosc. Radiat. Transfer 104, 257–265 (2007).
[CrossRef]

Freitag, M.

M. Freitag, H.-Y. Chiu, M. Steiner, V. Perebeinos, and P. Avouris, “Thermal infrared emission from biased graphene,” Nat. Nanotechnol. 5, 497–501 (2010).
[CrossRef] [PubMed]

Gokce, O. H.

N. M. Ravindra, B. Sopori, O. H. Gokce, S. X. Cheng, A. Shenoy, L. Jin, S. Abedrabbo, W. Chen, and Y. Zhang, “Emissivity measurements and modeling of silicon-related materials: an overview,” Int. J. Thermophys. 22, 1593–1611(2001).
[CrossRef]

Greffet, J.-J.

E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, and J.-J. Greffet, “Radiative heat transfer at the nanoscale,” Nat. Photonics 3, 514–517 (2009).
[CrossRef]

Ikonen, E.

Jang, H. D.

H. D. Jang, “Generation of silica nanoparticles from tetraethylorthodilicate (TEOS) vapor in a diffusion flame,” Aerosol Sci. Technol. 30, 477–488 (1999).
[CrossRef]

Jin, L.

N. M. Ravindra, B. Sopori, O. H. Gokce, S. X. Cheng, A. Shenoy, L. Jin, S. Abedrabbo, W. Chen, and Y. Zhang, “Emissivity measurements and modeling of silicon-related materials: an overview,” Int. J. Thermophys. 22, 1593–1611(2001).
[CrossRef]

Jourdan, G.

E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, and J.-J. Greffet, “Radiative heat transfer at the nanoscale,” Nat. Photonics 3, 514–517 (2009).
[CrossRef]

Kammler, H. K.

R. Mueller, H. K. Kammler, S. E. Pratsinis, A. Vital, G. Beaucage, and P. Burtscher, “Non-agglomerated dry silica nanoparticles,” Powder Technol. 140, 40–48 (2004).
[CrossRef]

Kärhä, P.

Katz, J. L.

Kimura, S.

E. Takasuka, E. Tokizaki, K. Terashima, and S. Kimura, “Emissivity of liquid silicon in visible and infrared regions,” J. Appl. Phys. 81, 6384–6389 (1997).
[CrossRef]

Lee, D.

I. S. Altman, D. Lee, J. D. Chung, J. Song, and M. Choi, “Light absorption of silica nanoparticles,” Phys. Rev. B 63, 161402(2001).
[CrossRef]

Liu, L. H.

J. Y. Yin and L. H. Liu, “Influence of complex component and particle polydispersity on radiative properties of soot aggregate in atmosphere,” J. Quant. Spectrosc. Radiat. Transfer 111, 2115–2126 (2010).
[CrossRef]

Meneses, D. D. S.

B. Rousseau, M. D. Michiel, A. Canizares, D. D. S. Meneses, P. Echegut, and J. F. Thovert, “Temperature effect (300–1500 K) on the infrared photon transport inside an x-ray microtomographic reconstructed porous silica glass,” J. Quant. Spectrosc. Radiat. Transfer 104, 257–265 (2007).
[CrossRef]

Michiel, M. D.

B. Rousseau, M. D. Michiel, A. Canizares, D. D. S. Meneses, P. Echegut, and J. F. Thovert, “Temperature effect (300–1500 K) on the infrared photon transport inside an x-ray microtomographic reconstructed porous silica glass,” J. Quant. Spectrosc. Radiat. Transfer 104, 257–265 (2007).
[CrossRef]

Mueller, R.

R. Mueller, H. K. Kammler, S. E. Pratsinis, A. Vital, G. Beaucage, and P. Burtscher, “Non-agglomerated dry silica nanoparticles,” Powder Technol. 140, 40–48 (2004).
[CrossRef]

Ojanen, M.

Perebeinos, V.

M. Freitag, H.-Y. Chiu, M. Steiner, V. Perebeinos, and P. Avouris, “Thermal infrared emission from biased graphene,” Nat. Nanotechnol. 5, 497–501 (2010).
[CrossRef] [PubMed]

Pikhitsa, P. V.

P. V. Pikhitsa and I. S. Altman, “Anomalies in light absorption coefficient of silica nanoparticles generated within flame,” J. Nanopart. Res. 3, 303–308 (2001).
[CrossRef]

Pratsinis, S. E.

R. Mueller, H. K. Kammler, S. E. Pratsinis, A. Vital, G. Beaucage, and P. Burtscher, “Non-agglomerated dry silica nanoparticles,” Powder Technol. 140, 40–48 (2004).
[CrossRef]

Ravindra, N. M.

N. M. Ravindra, B. Sopori, O. H. Gokce, S. X. Cheng, A. Shenoy, L. Jin, S. Abedrabbo, W. Chen, and Y. Zhang, “Emissivity measurements and modeling of silicon-related materials: an overview,” Int. J. Thermophys. 22, 1593–1611(2001).
[CrossRef]

Rousseau, B.

B. Rousseau, M. D. Michiel, A. Canizares, D. D. S. Meneses, P. Echegut, and J. F. Thovert, “Temperature effect (300–1500 K) on the infrared photon transport inside an x-ray microtomographic reconstructed porous silica glass,” J. Quant. Spectrosc. Radiat. Transfer 104, 257–265 (2007).
[CrossRef]

Rousseau, E.

E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, and J.-J. Greffet, “Radiative heat transfer at the nanoscale,” Nat. Photonics 3, 514–517 (2009).
[CrossRef]

Semerjian, H. G.

Shenoy, A.

N. M. Ravindra, B. Sopori, O. H. Gokce, S. X. Cheng, A. Shenoy, L. Jin, S. Abedrabbo, W. Chen, and Y. Zhang, “Emissivity measurements and modeling of silicon-related materials: an overview,” Int. J. Thermophys. 22, 1593–1611(2001).
[CrossRef]

Siria, A.

E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, and J.-J. Greffet, “Radiative heat transfer at the nanoscale,” Nat. Photonics 3, 514–517 (2009).
[CrossRef]

Song, J.

I. S. Altman, D. Lee, J. D. Chung, J. Song, and M. Choi, “Light absorption of silica nanoparticles,” Phys. Rev. B 63, 161402(2001).
[CrossRef]

Sopori, B.

N. M. Ravindra, B. Sopori, O. H. Gokce, S. X. Cheng, A. Shenoy, L. Jin, S. Abedrabbo, W. Chen, and Y. Zhang, “Emissivity measurements and modeling of silicon-related materials: an overview,” Int. J. Thermophys. 22, 1593–1611(2001).
[CrossRef]

Steele, W. G.

H. W. Coleman and W. G. Steele Jr., Experimentation and Uncertainty Analysis for Engineers (Wiley, 1989).

Steiner, M.

M. Freitag, H.-Y. Chiu, M. Steiner, V. Perebeinos, and P. Avouris, “Thermal infrared emission from biased graphene,” Nat. Nanotechnol. 5, 497–501 (2010).
[CrossRef] [PubMed]

Takasuka, E.

E. Takasuka, E. Tokizaki, K. Terashima, and S. Kimura, “Emissivity of liquid silicon in visible and infrared regions,” J. Appl. Phys. 81, 6384–6389 (1997).
[CrossRef]

Terashima, K.

E. Takasuka, E. Tokizaki, K. Terashima, and S. Kimura, “Emissivity of liquid silicon in visible and infrared regions,” J. Appl. Phys. 81, 6384–6389 (1997).
[CrossRef]

Thovert, J. F.

B. Rousseau, M. D. Michiel, A. Canizares, D. D. S. Meneses, P. Echegut, and J. F. Thovert, “Temperature effect (300–1500 K) on the infrared photon transport inside an x-ray microtomographic reconstructed porous silica glass,” J. Quant. Spectrosc. Radiat. Transfer 104, 257–265 (2007).
[CrossRef]

Tokizaki, E.

E. Takasuka, E. Tokizaki, K. Terashima, and S. Kimura, “Emissivity of liquid silicon in visible and infrared regions,” J. Appl. Phys. 81, 6384–6389 (1997).
[CrossRef]

Vital, A.

R. Mueller, H. K. Kammler, S. E. Pratsinis, A. Vital, G. Beaucage, and P. Burtscher, “Non-agglomerated dry silica nanoparticles,” Powder Technol. 140, 40–48 (2004).
[CrossRef]

Volz, S.

E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, and J.-J. Greffet, “Radiative heat transfer at the nanoscale,” Nat. Photonics 3, 514–517 (2009).
[CrossRef]

Yin, J. Y.

J. Y. Yin and L. H. Liu, “Influence of complex component and particle polydispersity on radiative properties of soot aggregate in atmosphere,” J. Quant. Spectrosc. Radiat. Transfer 111, 2115–2126 (2010).
[CrossRef]

Zachariah, M. R.

Zhang, Y.

N. M. Ravindra, B. Sopori, O. H. Gokce, S. X. Cheng, A. Shenoy, L. Jin, S. Abedrabbo, W. Chen, and Y. Zhang, “Emissivity measurements and modeling of silicon-related materials: an overview,” Int. J. Thermophys. 22, 1593–1611(2001).
[CrossRef]

Aerosol Sci. Technol. (1)

H. D. Jang, “Generation of silica nanoparticles from tetraethylorthodilicate (TEOS) vapor in a diffusion flame,” Aerosol Sci. Technol. 30, 477–488 (1999).
[CrossRef]

Appl. Opt. (2)

Int. J. Thermophys. (1)

N. M. Ravindra, B. Sopori, O. H. Gokce, S. X. Cheng, A. Shenoy, L. Jin, S. Abedrabbo, W. Chen, and Y. Zhang, “Emissivity measurements and modeling of silicon-related materials: an overview,” Int. J. Thermophys. 22, 1593–1611(2001).
[CrossRef]

J. Appl. Phys. (1)

E. Takasuka, E. Tokizaki, K. Terashima, and S. Kimura, “Emissivity of liquid silicon in visible and infrared regions,” J. Appl. Phys. 81, 6384–6389 (1997).
[CrossRef]

J. Nanopart. Res. (1)

P. V. Pikhitsa and I. S. Altman, “Anomalies in light absorption coefficient of silica nanoparticles generated within flame,” J. Nanopart. Res. 3, 303–308 (2001).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (2)

B. Rousseau, M. D. Michiel, A. Canizares, D. D. S. Meneses, P. Echegut, and J. F. Thovert, “Temperature effect (300–1500 K) on the infrared photon transport inside an x-ray microtomographic reconstructed porous silica glass,” J. Quant. Spectrosc. Radiat. Transfer 104, 257–265 (2007).
[CrossRef]

J. Y. Yin and L. H. Liu, “Influence of complex component and particle polydispersity on radiative properties of soot aggregate in atmosphere,” J. Quant. Spectrosc. Radiat. Transfer 111, 2115–2126 (2010).
[CrossRef]

Nat. Nanotechnol. (1)

M. Freitag, H.-Y. Chiu, M. Steiner, V. Perebeinos, and P. Avouris, “Thermal infrared emission from biased graphene,” Nat. Nanotechnol. 5, 497–501 (2010).
[CrossRef] [PubMed]

Nat. Photonics (1)

E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, and J.-J. Greffet, “Radiative heat transfer at the nanoscale,” Nat. Photonics 3, 514–517 (2009).
[CrossRef]

Phys. Rev. B (1)

I. S. Altman, D. Lee, J. D. Chung, J. Song, and M. Choi, “Light absorption of silica nanoparticles,” Phys. Rev. B 63, 161402(2001).
[CrossRef]

Powder Technol. (1)

R. Mueller, H. K. Kammler, S. E. Pratsinis, A. Vital, G. Beaucage, and P. Burtscher, “Non-agglomerated dry silica nanoparticles,” Powder Technol. 140, 40–48 (2004).
[CrossRef]

Other (1)

H. W. Coleman and W. G. Steele Jr., Experimentation and Uncertainty Analysis for Engineers (Wiley, 1989).

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Figures (8)

Fig. 1
Fig. 1

Schematic of experimental apparatus. (a) Spectroscopy experiment: emission intensity from the particles is measured by a spectrometer. (b) Visualization experiment: a coherent He–Ne laser and a cylindrical lens construct a sheet beam, which visualizes the particle stream line. Positions of flame position and stagnation position are determined by the equivalent ratio and momentum ratio.

Fig. 2
Fig. 2

Hydrogen/oxygen counterflow flame is generated where the SiO 2 particles are formed via chemical reaction. Radiation emission is clearly observed above the stagnation plane. Particles flowing out of the burner are clearly visualized.

Fig. 3
Fig. 3

Emission radiation intensity at various positions as a function of wavelength. The intensity increases with the distance from the burner exit.

Fig. 4
Fig. 4

Emission intensity relative to the reference position ( z = 4.6 mm ) versus wavenumber for the various positions from the burner exit.

Fig. 5
Fig. 5

Slope of tangential line at the inflection point of the fitting curve for the case of z = 7.0 mm . The slope is used to estimate the temperature profiles. Please see the contents for more detail about how to estimate the temperature.

Fig. 6
Fig. 6

Emissivity of silica particles in flames at various positions above the hydrogen exit. Error bars were plotted using the results of uncertainty analysis.

Fig. 7
Fig. 7

Temperature distribution estimated from the flame radiation experiment. Temperature increases with the distance from the burner exit. Error bars were plotted using the value obtained through the uncertainty analysis.

Fig. 8
Fig. 8

Relative emissivity of silica as a function of temperature with respect to the emissivity at temperature 2100 K .

Equations (7)

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

E λ = A ε ( λ , T ) f v λ 5 ( exp ( h c λ k T ) 1 ) ,
E λ = A ε ( λ , T ) f v λ 5 exp ( h c λ k T ) .
ln ( E λ E λ , ref ) = h c k B 1 λ ( 1 T ref 1 T ) + ln ( ε ( λ , T ) ε ( λ , T ref ) ) + ln ( f v f v , ref ) .
d d ( 1 / λ ) ln ( E λ E λ , ref ) | inflect , pt = h c k B ( 1 T ref 1 T ) .
U ln ( ε / ε ref ) = ( U ln ( E / E ref ) ) 2 + ( U 1 / λ ) 2 ,
U ln ( ε / ε ref ) = ( ln ( ε / ε ref ) E U E ) 2 + ( ln ( ε / ε ref ) ( 1 / λ ) U 1 / λ ) 2 = ( U E E ) 2 + ( h c k B ( 1 T ref 1 T ) U 1 / λ ) 2 .
U T = ( 1 T ref k B h c B ) 2 k B h c B E U E .

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