Abstract

We report on spatially resolved simultaneous measurements of temperature and majority species concentrations along a line segment in a premixed laminar H2–air flame. The results are obtained from Raman and Rayleigh scattering by using a narrow-band KrF excimer laser and a spectrally and spatially resolving detector system that consists of a high-throughput spectrometer and a gated, intensified, two-dimensional CCD camera. The data presented here are integrated over 100 laser shots. Absolute density profiles of N2, O2, H2O, and H2, as well as temperature profiles at various heights through the flame, are presented. A discussion of the required calibration procedures and a summary of the necessary spectroscopic background are also included in this paper.

© 1993 Optical Society of America

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References

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  1. R. Cattolica, S. Vosen, “Combustion—torch—ignition: fluorescence imaging of OH concentration,” Combust. Flame 68, 267–281 (1987).
    [CrossRef]
  2. M. P. Lee, P. H. Paul, R. K. Hanson, “Quantitative imaging of temperature fields in air using planar laser induced fluorescence,” Opt. Lett. 12, 75–77 (1987).
    [CrossRef] [PubMed]
  3. L. Hüwel, A. Wodtke, P. Andresen, H. Voges, “Position sensitive detection with laser induced fluorescence,” AIP Conf. Proc. 205, 337–341 (1989).
    [CrossRef]
  4. A. Koch, A. Chryssostomou, P. Andresen, W. Bornscheuer, “Multispecies detection in spray flames with tunable eximer lasers,” Appl. Phys. B (to be published).
  5. P. Andresen, H. Schlüter, D. Wolff, H. Voges, A. Koch, W. Hentschel, W. Oppermann, E. Rothe, “Identification and imaging of OH (v″ = 0) and O2 (v″ = 6, 7) in an automobile spark ignition engine using an KrF excimer laser,” Appl. Opt. 31, 7684–7689 (1992).
    [CrossRef] [PubMed]
  6. U. Westblom, M. Aldén, “Simultaneous multiple species detection in a flame using laser-induced fluorescence,” Appl. Opt. 28, 2592–2599 (1989).
    [CrossRef] [PubMed]
  7. R. W. Dibble, W. Kollman, R. W. Schefer, “Conserved scalar fluxes measured in a turbulent nonpremixed flame by combined laser Doppler velocimetry and laser Raman scattering,” Combust. Flame 55, 307–321 (1984).
    [CrossRef]
  8. T.-S. Cheng, J.A. Wehrmeyer, R. W. Pitz, “Simultaneous temperature and multi-species measurement in a lifted hydrogen diffusion flame by a Krf excimer laser,” AIAA paper 91-0181, in Proceedings of the Twenty-Ninth Aerospace Sciences Meeting (American Institute of Aeronautics and Astronautics, New York, 1991); R. W. Pitz, J. A. Wehrmeyer, J. M. Bowling, T.-S. Cheng, “Single pulse vibrational Raman scattering by a KrF excimer laser in a hydrogen–air flame,” Appl. Opt. 29, 2325–2332 (1990).
    [CrossRef] [PubMed]
  9. F. Cignoli, S. Benecchi, G. Zizak, “Simultaneous one-dimensional visualization of OH, polycyclic aromatic hydrocarbons, and soot in a laminar diffusion flame,” Opt. Lett. 17, 229–231 (1992).
    [CrossRef] [PubMed]
  10. S. Vajda, H. Rabitz, R. A. Yetter, “Effects of thermal coupling and diffusion on the mechanism of H2 oxidation in steady premixed laminar flames,” Combust. Flame 82, 270–297 (1990).
    [CrossRef]
  11. A. C. Eckbreth, Laser Diagnostics for Combustion Temperature and Species, A. K. Gupta, D. G. Lilley, eds., Vol. 7 of Energy and Engineering Science Series (Abacus, Cambridge, Mass., 1988).
  12. A. Koch, “Verbrennungsanalyse in turbulenten und laminaren Flammen mit schmalbandigen Excimerlasern,” Ph.D. dissertation (University of Göttingen, Göttingen, Germany, 1991).
  13. P. Andresen, G. Meyer, H. Schlüter, H. Voges, A. Koch, W. Hentschel, W. Oppermann, E. Rothe, “Fluorescence imaging inside an internal combustion engine using tunable excimer lasers,” Appl. Opt. 29, 2392–2404 (1990).
    [CrossRef] [PubMed]
  14. P. Andresen, A. Bath, W. Gröger, H. W. Lülf, G. Meijer, J. J. ter Meulen, “Laser-induced fluorescence with tunable excimer lasers as a possible method for instantaneous temperature field measurements at high pressures: checks with an atmospheric flame,” Appl. Opt. 27, 365–378 (1988).
    [CrossRef] [PubMed]
  15. V. Engle, G. Meijer, A. Bath, P. Andresen, R. Schinke, “The C–A emission in water: theory and experiment,” J. Chern. Phys. 87, 4310–4314 (1987).
    [CrossRef]
  16. I. Namer, R. W. Schefer, “Error estimates for Rayleigh scattering density and temperature measurements in premixed flames,” Exp. Fluids 3, 1–9 (1985).
    [CrossRef]
  17. A. Koch, H. Voges, P. Andresen, H. Schlüter, D. Wolff, W. Hentschel, W. Oppermann, E. Rothe, “Planar imaging of a flame and of internal combustion in an automobile engine using UV Rayleigh and fluorescence light,” Appl. Phys. B (to be published).
  18. W. K. Bischel, G. Black, “Wavelength dependence of Raman scattering cross sections from 200 to 600 nm,” AIP Conf. Proc. 100, 181–187 (1983).
    [CrossRef]

1992 (2)

1990 (2)

P. Andresen, G. Meyer, H. Schlüter, H. Voges, A. Koch, W. Hentschel, W. Oppermann, E. Rothe, “Fluorescence imaging inside an internal combustion engine using tunable excimer lasers,” Appl. Opt. 29, 2392–2404 (1990).
[CrossRef] [PubMed]

S. Vajda, H. Rabitz, R. A. Yetter, “Effects of thermal coupling and diffusion on the mechanism of H2 oxidation in steady premixed laminar flames,” Combust. Flame 82, 270–297 (1990).
[CrossRef]

1989 (2)

L. Hüwel, A. Wodtke, P. Andresen, H. Voges, “Position sensitive detection with laser induced fluorescence,” AIP Conf. Proc. 205, 337–341 (1989).
[CrossRef]

U. Westblom, M. Aldén, “Simultaneous multiple species detection in a flame using laser-induced fluorescence,” Appl. Opt. 28, 2592–2599 (1989).
[CrossRef] [PubMed]

1988 (1)

1987 (3)

R. Cattolica, S. Vosen, “Combustion—torch—ignition: fluorescence imaging of OH concentration,” Combust. Flame 68, 267–281 (1987).
[CrossRef]

M. P. Lee, P. H. Paul, R. K. Hanson, “Quantitative imaging of temperature fields in air using planar laser induced fluorescence,” Opt. Lett. 12, 75–77 (1987).
[CrossRef] [PubMed]

V. Engle, G. Meijer, A. Bath, P. Andresen, R. Schinke, “The C–A emission in water: theory and experiment,” J. Chern. Phys. 87, 4310–4314 (1987).
[CrossRef]

1985 (1)

I. Namer, R. W. Schefer, “Error estimates for Rayleigh scattering density and temperature measurements in premixed flames,” Exp. Fluids 3, 1–9 (1985).
[CrossRef]

1984 (1)

R. W. Dibble, W. Kollman, R. W. Schefer, “Conserved scalar fluxes measured in a turbulent nonpremixed flame by combined laser Doppler velocimetry and laser Raman scattering,” Combust. Flame 55, 307–321 (1984).
[CrossRef]

1983 (1)

W. K. Bischel, G. Black, “Wavelength dependence of Raman scattering cross sections from 200 to 600 nm,” AIP Conf. Proc. 100, 181–187 (1983).
[CrossRef]

Aldén, M.

Andresen, P.

P. Andresen, H. Schlüter, D. Wolff, H. Voges, A. Koch, W. Hentschel, W. Oppermann, E. Rothe, “Identification and imaging of OH (v″ = 0) and O2 (v″ = 6, 7) in an automobile spark ignition engine using an KrF excimer laser,” Appl. Opt. 31, 7684–7689 (1992).
[CrossRef] [PubMed]

P. Andresen, G. Meyer, H. Schlüter, H. Voges, A. Koch, W. Hentschel, W. Oppermann, E. Rothe, “Fluorescence imaging inside an internal combustion engine using tunable excimer lasers,” Appl. Opt. 29, 2392–2404 (1990).
[CrossRef] [PubMed]

L. Hüwel, A. Wodtke, P. Andresen, H. Voges, “Position sensitive detection with laser induced fluorescence,” AIP Conf. Proc. 205, 337–341 (1989).
[CrossRef]

P. Andresen, A. Bath, W. Gröger, H. W. Lülf, G. Meijer, J. J. ter Meulen, “Laser-induced fluorescence with tunable excimer lasers as a possible method for instantaneous temperature field measurements at high pressures: checks with an atmospheric flame,” Appl. Opt. 27, 365–378 (1988).
[CrossRef] [PubMed]

V. Engle, G. Meijer, A. Bath, P. Andresen, R. Schinke, “The C–A emission in water: theory and experiment,” J. Chern. Phys. 87, 4310–4314 (1987).
[CrossRef]

A. Koch, H. Voges, P. Andresen, H. Schlüter, D. Wolff, W. Hentschel, W. Oppermann, E. Rothe, “Planar imaging of a flame and of internal combustion in an automobile engine using UV Rayleigh and fluorescence light,” Appl. Phys. B (to be published).

A. Koch, A. Chryssostomou, P. Andresen, W. Bornscheuer, “Multispecies detection in spray flames with tunable eximer lasers,” Appl. Phys. B (to be published).

Bath, A.

Benecchi, S.

Bischel, W. K.

W. K. Bischel, G. Black, “Wavelength dependence of Raman scattering cross sections from 200 to 600 nm,” AIP Conf. Proc. 100, 181–187 (1983).
[CrossRef]

Black, G.

W. K. Bischel, G. Black, “Wavelength dependence of Raman scattering cross sections from 200 to 600 nm,” AIP Conf. Proc. 100, 181–187 (1983).
[CrossRef]

Bornscheuer, W.

A. Koch, A. Chryssostomou, P. Andresen, W. Bornscheuer, “Multispecies detection in spray flames with tunable eximer lasers,” Appl. Phys. B (to be published).

Cattolica, R.

R. Cattolica, S. Vosen, “Combustion—torch—ignition: fluorescence imaging of OH concentration,” Combust. Flame 68, 267–281 (1987).
[CrossRef]

Cheng, T.-S.

T.-S. Cheng, J.A. Wehrmeyer, R. W. Pitz, “Simultaneous temperature and multi-species measurement in a lifted hydrogen diffusion flame by a Krf excimer laser,” AIAA paper 91-0181, in Proceedings of the Twenty-Ninth Aerospace Sciences Meeting (American Institute of Aeronautics and Astronautics, New York, 1991); R. W. Pitz, J. A. Wehrmeyer, J. M. Bowling, T.-S. Cheng, “Single pulse vibrational Raman scattering by a KrF excimer laser in a hydrogen–air flame,” Appl. Opt. 29, 2325–2332 (1990).
[CrossRef] [PubMed]

Chryssostomou, A.

A. Koch, A. Chryssostomou, P. Andresen, W. Bornscheuer, “Multispecies detection in spray flames with tunable eximer lasers,” Appl. Phys. B (to be published).

Cignoli, F.

Dibble, R. W.

R. W. Dibble, W. Kollman, R. W. Schefer, “Conserved scalar fluxes measured in a turbulent nonpremixed flame by combined laser Doppler velocimetry and laser Raman scattering,” Combust. Flame 55, 307–321 (1984).
[CrossRef]

Eckbreth, A. C.

A. C. Eckbreth, Laser Diagnostics for Combustion Temperature and Species, A. K. Gupta, D. G. Lilley, eds., Vol. 7 of Energy and Engineering Science Series (Abacus, Cambridge, Mass., 1988).

Engle, V.

V. Engle, G. Meijer, A. Bath, P. Andresen, R. Schinke, “The C–A emission in water: theory and experiment,” J. Chern. Phys. 87, 4310–4314 (1987).
[CrossRef]

Gröger, W.

Hanson, R. K.

Hentschel, W.

Hüwel, L.

L. Hüwel, A. Wodtke, P. Andresen, H. Voges, “Position sensitive detection with laser induced fluorescence,” AIP Conf. Proc. 205, 337–341 (1989).
[CrossRef]

Koch, A.

P. Andresen, H. Schlüter, D. Wolff, H. Voges, A. Koch, W. Hentschel, W. Oppermann, E. Rothe, “Identification and imaging of OH (v″ = 0) and O2 (v″ = 6, 7) in an automobile spark ignition engine using an KrF excimer laser,” Appl. Opt. 31, 7684–7689 (1992).
[CrossRef] [PubMed]

P. Andresen, G. Meyer, H. Schlüter, H. Voges, A. Koch, W. Hentschel, W. Oppermann, E. Rothe, “Fluorescence imaging inside an internal combustion engine using tunable excimer lasers,” Appl. Opt. 29, 2392–2404 (1990).
[CrossRef] [PubMed]

A. Koch, A. Chryssostomou, P. Andresen, W. Bornscheuer, “Multispecies detection in spray flames with tunable eximer lasers,” Appl. Phys. B (to be published).

A. Koch, H. Voges, P. Andresen, H. Schlüter, D. Wolff, W. Hentschel, W. Oppermann, E. Rothe, “Planar imaging of a flame and of internal combustion in an automobile engine using UV Rayleigh and fluorescence light,” Appl. Phys. B (to be published).

A. Koch, “Verbrennungsanalyse in turbulenten und laminaren Flammen mit schmalbandigen Excimerlasern,” Ph.D. dissertation (University of Göttingen, Göttingen, Germany, 1991).

Kollman, W.

R. W. Dibble, W. Kollman, R. W. Schefer, “Conserved scalar fluxes measured in a turbulent nonpremixed flame by combined laser Doppler velocimetry and laser Raman scattering,” Combust. Flame 55, 307–321 (1984).
[CrossRef]

Lee, M. P.

Lülf, H. W.

Meijer, G.

Meyer, G.

Namer, I.

I. Namer, R. W. Schefer, “Error estimates for Rayleigh scattering density and temperature measurements in premixed flames,” Exp. Fluids 3, 1–9 (1985).
[CrossRef]

Oppermann, W.

Paul, P. H.

Pitz, R. W.

T.-S. Cheng, J.A. Wehrmeyer, R. W. Pitz, “Simultaneous temperature and multi-species measurement in a lifted hydrogen diffusion flame by a Krf excimer laser,” AIAA paper 91-0181, in Proceedings of the Twenty-Ninth Aerospace Sciences Meeting (American Institute of Aeronautics and Astronautics, New York, 1991); R. W. Pitz, J. A. Wehrmeyer, J. M. Bowling, T.-S. Cheng, “Single pulse vibrational Raman scattering by a KrF excimer laser in a hydrogen–air flame,” Appl. Opt. 29, 2325–2332 (1990).
[CrossRef] [PubMed]

Rabitz, H.

S. Vajda, H. Rabitz, R. A. Yetter, “Effects of thermal coupling and diffusion on the mechanism of H2 oxidation in steady premixed laminar flames,” Combust. Flame 82, 270–297 (1990).
[CrossRef]

Rothe, E.

Schefer, R. W.

I. Namer, R. W. Schefer, “Error estimates for Rayleigh scattering density and temperature measurements in premixed flames,” Exp. Fluids 3, 1–9 (1985).
[CrossRef]

R. W. Dibble, W. Kollman, R. W. Schefer, “Conserved scalar fluxes measured in a turbulent nonpremixed flame by combined laser Doppler velocimetry and laser Raman scattering,” Combust. Flame 55, 307–321 (1984).
[CrossRef]

Schinke, R.

V. Engle, G. Meijer, A. Bath, P. Andresen, R. Schinke, “The C–A emission in water: theory and experiment,” J. Chern. Phys. 87, 4310–4314 (1987).
[CrossRef]

Schlüter, H.

ter Meulen, J. J.

Vajda, S.

S. Vajda, H. Rabitz, R. A. Yetter, “Effects of thermal coupling and diffusion on the mechanism of H2 oxidation in steady premixed laminar flames,” Combust. Flame 82, 270–297 (1990).
[CrossRef]

Voges, H.

P. Andresen, H. Schlüter, D. Wolff, H. Voges, A. Koch, W. Hentschel, W. Oppermann, E. Rothe, “Identification and imaging of OH (v″ = 0) and O2 (v″ = 6, 7) in an automobile spark ignition engine using an KrF excimer laser,” Appl. Opt. 31, 7684–7689 (1992).
[CrossRef] [PubMed]

P. Andresen, G. Meyer, H. Schlüter, H. Voges, A. Koch, W. Hentschel, W. Oppermann, E. Rothe, “Fluorescence imaging inside an internal combustion engine using tunable excimer lasers,” Appl. Opt. 29, 2392–2404 (1990).
[CrossRef] [PubMed]

L. Hüwel, A. Wodtke, P. Andresen, H. Voges, “Position sensitive detection with laser induced fluorescence,” AIP Conf. Proc. 205, 337–341 (1989).
[CrossRef]

A. Koch, H. Voges, P. Andresen, H. Schlüter, D. Wolff, W. Hentschel, W. Oppermann, E. Rothe, “Planar imaging of a flame and of internal combustion in an automobile engine using UV Rayleigh and fluorescence light,” Appl. Phys. B (to be published).

Vosen, S.

R. Cattolica, S. Vosen, “Combustion—torch—ignition: fluorescence imaging of OH concentration,” Combust. Flame 68, 267–281 (1987).
[CrossRef]

Wehrmeyer, J.A.

T.-S. Cheng, J.A. Wehrmeyer, R. W. Pitz, “Simultaneous temperature and multi-species measurement in a lifted hydrogen diffusion flame by a Krf excimer laser,” AIAA paper 91-0181, in Proceedings of the Twenty-Ninth Aerospace Sciences Meeting (American Institute of Aeronautics and Astronautics, New York, 1991); R. W. Pitz, J. A. Wehrmeyer, J. M. Bowling, T.-S. Cheng, “Single pulse vibrational Raman scattering by a KrF excimer laser in a hydrogen–air flame,” Appl. Opt. 29, 2325–2332 (1990).
[CrossRef] [PubMed]

Westblom, U.

Wodtke, A.

L. Hüwel, A. Wodtke, P. Andresen, H. Voges, “Position sensitive detection with laser induced fluorescence,” AIP Conf. Proc. 205, 337–341 (1989).
[CrossRef]

Wolff, D.

P. Andresen, H. Schlüter, D. Wolff, H. Voges, A. Koch, W. Hentschel, W. Oppermann, E. Rothe, “Identification and imaging of OH (v″ = 0) and O2 (v″ = 6, 7) in an automobile spark ignition engine using an KrF excimer laser,” Appl. Opt. 31, 7684–7689 (1992).
[CrossRef] [PubMed]

A. Koch, H. Voges, P. Andresen, H. Schlüter, D. Wolff, W. Hentschel, W. Oppermann, E. Rothe, “Planar imaging of a flame and of internal combustion in an automobile engine using UV Rayleigh and fluorescence light,” Appl. Phys. B (to be published).

Yetter, R. A.

S. Vajda, H. Rabitz, R. A. Yetter, “Effects of thermal coupling and diffusion on the mechanism of H2 oxidation in steady premixed laminar flames,” Combust. Flame 82, 270–297 (1990).
[CrossRef]

Zizak, G.

AIP Conf. Proc. (2)

L. Hüwel, A. Wodtke, P. Andresen, H. Voges, “Position sensitive detection with laser induced fluorescence,” AIP Conf. Proc. 205, 337–341 (1989).
[CrossRef]

W. K. Bischel, G. Black, “Wavelength dependence of Raman scattering cross sections from 200 to 600 nm,” AIP Conf. Proc. 100, 181–187 (1983).
[CrossRef]

Appl. Opt. (4)

Combust. Flame (3)

R. Cattolica, S. Vosen, “Combustion—torch—ignition: fluorescence imaging of OH concentration,” Combust. Flame 68, 267–281 (1987).
[CrossRef]

R. W. Dibble, W. Kollman, R. W. Schefer, “Conserved scalar fluxes measured in a turbulent nonpremixed flame by combined laser Doppler velocimetry and laser Raman scattering,” Combust. Flame 55, 307–321 (1984).
[CrossRef]

S. Vajda, H. Rabitz, R. A. Yetter, “Effects of thermal coupling and diffusion on the mechanism of H2 oxidation in steady premixed laminar flames,” Combust. Flame 82, 270–297 (1990).
[CrossRef]

Exp. Fluids (1)

I. Namer, R. W. Schefer, “Error estimates for Rayleigh scattering density and temperature measurements in premixed flames,” Exp. Fluids 3, 1–9 (1985).
[CrossRef]

J. Chern. Phys. (1)

V. Engle, G. Meijer, A. Bath, P. Andresen, R. Schinke, “The C–A emission in water: theory and experiment,” J. Chern. Phys. 87, 4310–4314 (1987).
[CrossRef]

Opt. Lett. (2)

Other (5)

A. Koch, H. Voges, P. Andresen, H. Schlüter, D. Wolff, W. Hentschel, W. Oppermann, E. Rothe, “Planar imaging of a flame and of internal combustion in an automobile engine using UV Rayleigh and fluorescence light,” Appl. Phys. B (to be published).

A. C. Eckbreth, Laser Diagnostics for Combustion Temperature and Species, A. K. Gupta, D. G. Lilley, eds., Vol. 7 of Energy and Engineering Science Series (Abacus, Cambridge, Mass., 1988).

A. Koch, “Verbrennungsanalyse in turbulenten und laminaren Flammen mit schmalbandigen Excimerlasern,” Ph.D. dissertation (University of Göttingen, Göttingen, Germany, 1991).

T.-S. Cheng, J.A. Wehrmeyer, R. W. Pitz, “Simultaneous temperature and multi-species measurement in a lifted hydrogen diffusion flame by a Krf excimer laser,” AIAA paper 91-0181, in Proceedings of the Twenty-Ninth Aerospace Sciences Meeting (American Institute of Aeronautics and Astronautics, New York, 1991); R. W. Pitz, J. A. Wehrmeyer, J. M. Bowling, T.-S. Cheng, “Single pulse vibrational Raman scattering by a KrF excimer laser in a hydrogen–air flame,” Appl. Opt. 29, 2325–2332 (1990).
[CrossRef] [PubMed]

A. Koch, A. Chryssostomou, P. Andresen, W. Bornscheuer, “Multispecies detection in spray flames with tunable eximer lasers,” Appl. Phys. B (to be published).

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Figures (9)

Fig. 1
Fig. 1

Experimental setup for recording one-dimensional Rayleigh and Raman profiles in a H2–air flame. OMA, optical multichannel analyzer.

Fig. 2
Fig. 2

Excitation-emission spectrum of oxygen-rich H2–air flame that was obtained by using a tunable KrF laser excimer laser.

Fig. 3
Fig. 3

(a) Measured density profiles of all majority species at a height of z = 2 mm above the burner nozzle. (b) Measured density profiles of all majority species at a height of z = 8 mm above the burner nozzle.

Fig. 4
Fig. 4

One-dimensional Rayleigh- and Raman-scattering images in air at atmospheric pressure used for density calibration of O2 and N2.

Fig. 5
Fig. 5

(a) Measured temperature profiles from the two different methods described in the text at a height of z = 2 mm above the burner nozzle. (b) Measured temperature profiles from the two different methods described in the text at a height of z = 8 mm above the burner nozzle.

Fig. 6
Fig. 6

Geometry of laser excitation in the H2–air flame.

Fig. 7
Fig. 7

One-dimensional Rayleigh/Raman scattering image in a premixed H2–air flame at a height of z = 2 mm above burner nozzle.

Fig. 8
Fig. 8

(a) One-dimensional Rayleigh- and Raman-scattering images in a premixed H2–air flame at a height of z = 8 mm above the burner nozzle. (b) Three-dimensional plot of Fig. 8(a) with a resolution corresponding to a 10-bit CCD camera. (c) Three-dimensional plot of Fig. 8(a) with a resolution corresponding to a 12-bit CCD camera.

Fig. 9
Fig. 9

One dimensional Rayleigh/Raman scattering image in a premixed H2–air flame at a height of z = 15 mm above burner nozzle.

Tables (1)

Tables Icon

Table 1 Rayleigh and Raman Cross Sections for λ = 248-nm Excitation

Equations (8)

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

N i ( j ) = τ t ω i σ ( j ) V i n ( j ) ( x i ) N L ,
τ = z o g q .
N k i ( j ) = α n k ( j ) ( x i ) ,             k = 1 , 2 ,
n 2 ( j ) ( x i ) = N 2 i ( j ) N 1 i ( j ) n 1 ( j ) ( x i ) .
T ( x ) = p 0 k B n ( x ) ,
n ( x ) = j n ( j ) ( x ) .
R ( x ) = α ( x ) N ( x ) j c ( j ) ( x ) σ Ray ( j ) ,
T ( x ) = T 0 R 0 ( x ) R ( x ) c ( j ) ( x ) σ Ray ( j ) c 0 ( j ) σ Ray ( j ) ,

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