Abstract

Detailed rate equation modeling of the laser-excited sodium–molecular nitrogen system was conducted to investigate the applicability of the steady-state three-level model to prediction of sodium laser-induced fluorescence intensities. Redistribution of vibrational population in the nitrogen molecules can, under some conditions, produce a transition to a second-state condition not predicted by the simple three-level model. The feasibility of a major species number density measurement using tracer gas impurity laser-induced fluorescence is discussed in light of the sodium–nitrogen modeling results.

© 1981 Optical Society of America

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References

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  1. J. W. Daily, C. Chan, Combust. Flame 33, 47 (1978).
    [CrossRef]
  2. L. Pasternack, A. P. Baronavski, J. R. McDonald, J. Chem. Phys. 69, 4830 (1978).
    [CrossRef]
  3. B. Smith, J. D. Winefordner, N. Omenetto, J. Appl. Phys. 48, 2676 (1977).
    [CrossRef]
  4. B. L. Sharp, A. Goldwasser, Spectrochim. Acta Part B 31, 431 (1976).
    [CrossRef]
  5. R. A. Van Calcar, M. J. M. Van de Ven, B. K. Van Uitert, K. J. Biewenga, Tj. Hollander, C. Th. J. Alkemade, J. Quant. Spectrosc. Radiat. Transfer 21, 11 (1979).
    [CrossRef]
  6. J. M. Salter, D. D. Burgess, N. A. Ebrahim, J. Phy. B 12, L759 (1979).
    [CrossRef]
  7. J. M. Salter, J. Phys. B 12, L763 (1979).
    [CrossRef]
  8. M. Allegrini, G. Alzetta, A. Kopystynska, L. Moi, G. Orriols, Opt. Commun. 19, 96 (1976).
    [CrossRef]
  9. M. Allegrini, G. Alzetta, A. Kopystynska, L. Moi, G. Orriols, Opt. Commun. 22, 329 (1977).
    [CrossRef]
  10. A. Kopystynska, P. Kowalczyk, Opt. Commun. 25, 351 (1978).
    [CrossRef]
  11. J. Krasinski, T. Stacewicz, C. R. Stroud, Opt. Commun. 33, 158 (1980).
    [CrossRef]
  12. W. L. Starr, J. Chem. Phys. 43, 73 (1965).
    [CrossRef]
  13. P. L. Lijnse, R. J. Elsenaar, J. Quant. Spectrosc. Radiat. Transfer 12, 1115 (1972).
    [CrossRef]
  14. H. F. Krause, J. Fricke, W. L. Fite, J. Chem. Phys. 56, 4593 (1972).
    [CrossRef]
  15. E. Bauer, E. R. Fisher, F. R. Gilmore, J. Chem. Phys. 51, 4173 (1969).
    [CrossRef]
  16. K. F. Herzfeld, T. A. Litovitz, Absorption and Dispersion of Ultrasonic Waves (Academic, New York, 1959).
  17. D. H. Campbell, “Computer Modeling of Laser-Excited Molecular and Atomic-Molecular Systems,” AEDC-TR-80-47 (1981).
  18. C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations (Prentice-Hall, Englewood Cliffs, N.J., 1971).
  19. I. V. Hertel, H. Hofmann, K. J. Rost, Phys. Rev. Lett. 36, 861 (1976).
    [CrossRef]
  20. J. Apt, D. E. Pritchard, J. Phys. B 12, 83 (1979).
    [CrossRef]
  21. M. Stupauski, L. Krause, Can. J. Phys. 46, 2127 (1968).
    [CrossRef]
  22. B. P. Kibble, G. Copley, L. Krause, Phys. Rev. 153, 9 (1967).
    [CrossRef]
  23. J. Pitre, L. Krause, Can. J. Phys. 46, 125 (1968).
    [CrossRef]
  24. H. G. Kuhn, Atomic Spectra (Academic, New York, 1961).
  25. G. Zizak, J. D. Bradshaw, J. D. Winefordner, Appl. Spectrosc. 35, 59 (1981).
    [CrossRef]

1981

1980

J. Krasinski, T. Stacewicz, C. R. Stroud, Opt. Commun. 33, 158 (1980).
[CrossRef]

1979

J. Apt, D. E. Pritchard, J. Phys. B 12, 83 (1979).
[CrossRef]

R. A. Van Calcar, M. J. M. Van de Ven, B. K. Van Uitert, K. J. Biewenga, Tj. Hollander, C. Th. J. Alkemade, J. Quant. Spectrosc. Radiat. Transfer 21, 11 (1979).
[CrossRef]

J. M. Salter, D. D. Burgess, N. A. Ebrahim, J. Phy. B 12, L759 (1979).
[CrossRef]

J. M. Salter, J. Phys. B 12, L763 (1979).
[CrossRef]

1978

J. W. Daily, C. Chan, Combust. Flame 33, 47 (1978).
[CrossRef]

L. Pasternack, A. P. Baronavski, J. R. McDonald, J. Chem. Phys. 69, 4830 (1978).
[CrossRef]

A. Kopystynska, P. Kowalczyk, Opt. Commun. 25, 351 (1978).
[CrossRef]

1977

M. Allegrini, G. Alzetta, A. Kopystynska, L. Moi, G. Orriols, Opt. Commun. 22, 329 (1977).
[CrossRef]

B. Smith, J. D. Winefordner, N. Omenetto, J. Appl. Phys. 48, 2676 (1977).
[CrossRef]

1976

B. L. Sharp, A. Goldwasser, Spectrochim. Acta Part B 31, 431 (1976).
[CrossRef]

M. Allegrini, G. Alzetta, A. Kopystynska, L. Moi, G. Orriols, Opt. Commun. 19, 96 (1976).
[CrossRef]

I. V. Hertel, H. Hofmann, K. J. Rost, Phys. Rev. Lett. 36, 861 (1976).
[CrossRef]

1972

P. L. Lijnse, R. J. Elsenaar, J. Quant. Spectrosc. Radiat. Transfer 12, 1115 (1972).
[CrossRef]

H. F. Krause, J. Fricke, W. L. Fite, J. Chem. Phys. 56, 4593 (1972).
[CrossRef]

1969

E. Bauer, E. R. Fisher, F. R. Gilmore, J. Chem. Phys. 51, 4173 (1969).
[CrossRef]

1968

M. Stupauski, L. Krause, Can. J. Phys. 46, 2127 (1968).
[CrossRef]

J. Pitre, L. Krause, Can. J. Phys. 46, 125 (1968).
[CrossRef]

1967

B. P. Kibble, G. Copley, L. Krause, Phys. Rev. 153, 9 (1967).
[CrossRef]

1965

W. L. Starr, J. Chem. Phys. 43, 73 (1965).
[CrossRef]

Alkemade, C. Th. J.

R. A. Van Calcar, M. J. M. Van de Ven, B. K. Van Uitert, K. J. Biewenga, Tj. Hollander, C. Th. J. Alkemade, J. Quant. Spectrosc. Radiat. Transfer 21, 11 (1979).
[CrossRef]

Allegrini, M.

M. Allegrini, G. Alzetta, A. Kopystynska, L. Moi, G. Orriols, Opt. Commun. 22, 329 (1977).
[CrossRef]

M. Allegrini, G. Alzetta, A. Kopystynska, L. Moi, G. Orriols, Opt. Commun. 19, 96 (1976).
[CrossRef]

Alzetta, G.

M. Allegrini, G. Alzetta, A. Kopystynska, L. Moi, G. Orriols, Opt. Commun. 22, 329 (1977).
[CrossRef]

M. Allegrini, G. Alzetta, A. Kopystynska, L. Moi, G. Orriols, Opt. Commun. 19, 96 (1976).
[CrossRef]

Apt, J.

J. Apt, D. E. Pritchard, J. Phys. B 12, 83 (1979).
[CrossRef]

Baronavski, A. P.

L. Pasternack, A. P. Baronavski, J. R. McDonald, J. Chem. Phys. 69, 4830 (1978).
[CrossRef]

Bauer, E.

E. Bauer, E. R. Fisher, F. R. Gilmore, J. Chem. Phys. 51, 4173 (1969).
[CrossRef]

Biewenga, K. J.

R. A. Van Calcar, M. J. M. Van de Ven, B. K. Van Uitert, K. J. Biewenga, Tj. Hollander, C. Th. J. Alkemade, J. Quant. Spectrosc. Radiat. Transfer 21, 11 (1979).
[CrossRef]

Bradshaw, J. D.

Burgess, D. D.

J. M. Salter, D. D. Burgess, N. A. Ebrahim, J. Phy. B 12, L759 (1979).
[CrossRef]

Campbell, D. H.

D. H. Campbell, “Computer Modeling of Laser-Excited Molecular and Atomic-Molecular Systems,” AEDC-TR-80-47 (1981).

Chan, C.

J. W. Daily, C. Chan, Combust. Flame 33, 47 (1978).
[CrossRef]

Copley, G.

B. P. Kibble, G. Copley, L. Krause, Phys. Rev. 153, 9 (1967).
[CrossRef]

Daily, J. W.

J. W. Daily, C. Chan, Combust. Flame 33, 47 (1978).
[CrossRef]

Ebrahim, N. A.

J. M. Salter, D. D. Burgess, N. A. Ebrahim, J. Phy. B 12, L759 (1979).
[CrossRef]

Elsenaar, R. J.

P. L. Lijnse, R. J. Elsenaar, J. Quant. Spectrosc. Radiat. Transfer 12, 1115 (1972).
[CrossRef]

Fisher, E. R.

E. Bauer, E. R. Fisher, F. R. Gilmore, J. Chem. Phys. 51, 4173 (1969).
[CrossRef]

Fite, W. L.

H. F. Krause, J. Fricke, W. L. Fite, J. Chem. Phys. 56, 4593 (1972).
[CrossRef]

Fricke, J.

H. F. Krause, J. Fricke, W. L. Fite, J. Chem. Phys. 56, 4593 (1972).
[CrossRef]

Gear, C. W.

C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations (Prentice-Hall, Englewood Cliffs, N.J., 1971).

Gilmore, F. R.

E. Bauer, E. R. Fisher, F. R. Gilmore, J. Chem. Phys. 51, 4173 (1969).
[CrossRef]

Goldwasser, A.

B. L. Sharp, A. Goldwasser, Spectrochim. Acta Part B 31, 431 (1976).
[CrossRef]

Hertel, I. V.

I. V. Hertel, H. Hofmann, K. J. Rost, Phys. Rev. Lett. 36, 861 (1976).
[CrossRef]

Herzfeld, K. F.

K. F. Herzfeld, T. A. Litovitz, Absorption and Dispersion of Ultrasonic Waves (Academic, New York, 1959).

Hofmann, H.

I. V. Hertel, H. Hofmann, K. J. Rost, Phys. Rev. Lett. 36, 861 (1976).
[CrossRef]

Hollander, Tj.

R. A. Van Calcar, M. J. M. Van de Ven, B. K. Van Uitert, K. J. Biewenga, Tj. Hollander, C. Th. J. Alkemade, J. Quant. Spectrosc. Radiat. Transfer 21, 11 (1979).
[CrossRef]

Kibble, B. P.

B. P. Kibble, G. Copley, L. Krause, Phys. Rev. 153, 9 (1967).
[CrossRef]

Kopystynska, A.

A. Kopystynska, P. Kowalczyk, Opt. Commun. 25, 351 (1978).
[CrossRef]

M. Allegrini, G. Alzetta, A. Kopystynska, L. Moi, G. Orriols, Opt. Commun. 22, 329 (1977).
[CrossRef]

M. Allegrini, G. Alzetta, A. Kopystynska, L. Moi, G. Orriols, Opt. Commun. 19, 96 (1976).
[CrossRef]

Kowalczyk, P.

A. Kopystynska, P. Kowalczyk, Opt. Commun. 25, 351 (1978).
[CrossRef]

Krasinski, J.

J. Krasinski, T. Stacewicz, C. R. Stroud, Opt. Commun. 33, 158 (1980).
[CrossRef]

Krause, H. F.

H. F. Krause, J. Fricke, W. L. Fite, J. Chem. Phys. 56, 4593 (1972).
[CrossRef]

Krause, L.

M. Stupauski, L. Krause, Can. J. Phys. 46, 2127 (1968).
[CrossRef]

J. Pitre, L. Krause, Can. J. Phys. 46, 125 (1968).
[CrossRef]

B. P. Kibble, G. Copley, L. Krause, Phys. Rev. 153, 9 (1967).
[CrossRef]

Kuhn, H. G.

H. G. Kuhn, Atomic Spectra (Academic, New York, 1961).

Lijnse, P. L.

P. L. Lijnse, R. J. Elsenaar, J. Quant. Spectrosc. Radiat. Transfer 12, 1115 (1972).
[CrossRef]

Litovitz, T. A.

K. F. Herzfeld, T. A. Litovitz, Absorption and Dispersion of Ultrasonic Waves (Academic, New York, 1959).

McDonald, J. R.

L. Pasternack, A. P. Baronavski, J. R. McDonald, J. Chem. Phys. 69, 4830 (1978).
[CrossRef]

Moi, L.

M. Allegrini, G. Alzetta, A. Kopystynska, L. Moi, G. Orriols, Opt. Commun. 22, 329 (1977).
[CrossRef]

M. Allegrini, G. Alzetta, A. Kopystynska, L. Moi, G. Orriols, Opt. Commun. 19, 96 (1976).
[CrossRef]

Omenetto, N.

B. Smith, J. D. Winefordner, N. Omenetto, J. Appl. Phys. 48, 2676 (1977).
[CrossRef]

Orriols, G.

M. Allegrini, G. Alzetta, A. Kopystynska, L. Moi, G. Orriols, Opt. Commun. 22, 329 (1977).
[CrossRef]

M. Allegrini, G. Alzetta, A. Kopystynska, L. Moi, G. Orriols, Opt. Commun. 19, 96 (1976).
[CrossRef]

Pasternack, L.

L. Pasternack, A. P. Baronavski, J. R. McDonald, J. Chem. Phys. 69, 4830 (1978).
[CrossRef]

Pitre, J.

J. Pitre, L. Krause, Can. J. Phys. 46, 125 (1968).
[CrossRef]

Pritchard, D. E.

J. Apt, D. E. Pritchard, J. Phys. B 12, 83 (1979).
[CrossRef]

Rost, K. J.

I. V. Hertel, H. Hofmann, K. J. Rost, Phys. Rev. Lett. 36, 861 (1976).
[CrossRef]

Salter, J. M.

J. M. Salter, D. D. Burgess, N. A. Ebrahim, J. Phy. B 12, L759 (1979).
[CrossRef]

J. M. Salter, J. Phys. B 12, L763 (1979).
[CrossRef]

Sharp, B. L.

B. L. Sharp, A. Goldwasser, Spectrochim. Acta Part B 31, 431 (1976).
[CrossRef]

Smith, B.

B. Smith, J. D. Winefordner, N. Omenetto, J. Appl. Phys. 48, 2676 (1977).
[CrossRef]

Stacewicz, T.

J. Krasinski, T. Stacewicz, C. R. Stroud, Opt. Commun. 33, 158 (1980).
[CrossRef]

Starr, W. L.

W. L. Starr, J. Chem. Phys. 43, 73 (1965).
[CrossRef]

Stroud, C. R.

J. Krasinski, T. Stacewicz, C. R. Stroud, Opt. Commun. 33, 158 (1980).
[CrossRef]

Stupauski, M.

M. Stupauski, L. Krause, Can. J. Phys. 46, 2127 (1968).
[CrossRef]

Van Calcar, R. A.

R. A. Van Calcar, M. J. M. Van de Ven, B. K. Van Uitert, K. J. Biewenga, Tj. Hollander, C. Th. J. Alkemade, J. Quant. Spectrosc. Radiat. Transfer 21, 11 (1979).
[CrossRef]

Van de Ven, M. J. M.

R. A. Van Calcar, M. J. M. Van de Ven, B. K. Van Uitert, K. J. Biewenga, Tj. Hollander, C. Th. J. Alkemade, J. Quant. Spectrosc. Radiat. Transfer 21, 11 (1979).
[CrossRef]

Van Uitert, B. K.

R. A. Van Calcar, M. J. M. Van de Ven, B. K. Van Uitert, K. J. Biewenga, Tj. Hollander, C. Th. J. Alkemade, J. Quant. Spectrosc. Radiat. Transfer 21, 11 (1979).
[CrossRef]

Winefordner, J. D.

G. Zizak, J. D. Bradshaw, J. D. Winefordner, Appl. Spectrosc. 35, 59 (1981).
[CrossRef]

B. Smith, J. D. Winefordner, N. Omenetto, J. Appl. Phys. 48, 2676 (1977).
[CrossRef]

Zizak, G.

Appl. Spectrosc.

Can. J. Phys.

M. Stupauski, L. Krause, Can. J. Phys. 46, 2127 (1968).
[CrossRef]

J. Pitre, L. Krause, Can. J. Phys. 46, 125 (1968).
[CrossRef]

Combust. Flame

J. W. Daily, C. Chan, Combust. Flame 33, 47 (1978).
[CrossRef]

J. Appl. Phys.

B. Smith, J. D. Winefordner, N. Omenetto, J. Appl. Phys. 48, 2676 (1977).
[CrossRef]

J. Chem. Phys.

L. Pasternack, A. P. Baronavski, J. R. McDonald, J. Chem. Phys. 69, 4830 (1978).
[CrossRef]

H. F. Krause, J. Fricke, W. L. Fite, J. Chem. Phys. 56, 4593 (1972).
[CrossRef]

E. Bauer, E. R. Fisher, F. R. Gilmore, J. Chem. Phys. 51, 4173 (1969).
[CrossRef]

W. L. Starr, J. Chem. Phys. 43, 73 (1965).
[CrossRef]

J. Phy. B

J. M. Salter, D. D. Burgess, N. A. Ebrahim, J. Phy. B 12, L759 (1979).
[CrossRef]

J. Phys. B

J. M. Salter, J. Phys. B 12, L763 (1979).
[CrossRef]

J. Apt, D. E. Pritchard, J. Phys. B 12, 83 (1979).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer

P. L. Lijnse, R. J. Elsenaar, J. Quant. Spectrosc. Radiat. Transfer 12, 1115 (1972).
[CrossRef]

R. A. Van Calcar, M. J. M. Van de Ven, B. K. Van Uitert, K. J. Biewenga, Tj. Hollander, C. Th. J. Alkemade, J. Quant. Spectrosc. Radiat. Transfer 21, 11 (1979).
[CrossRef]

Opt. Commun.

M. Allegrini, G. Alzetta, A. Kopystynska, L. Moi, G. Orriols, Opt. Commun. 19, 96 (1976).
[CrossRef]

M. Allegrini, G. Alzetta, A. Kopystynska, L. Moi, G. Orriols, Opt. Commun. 22, 329 (1977).
[CrossRef]

A. Kopystynska, P. Kowalczyk, Opt. Commun. 25, 351 (1978).
[CrossRef]

J. Krasinski, T. Stacewicz, C. R. Stroud, Opt. Commun. 33, 158 (1980).
[CrossRef]

Phys. Rev.

B. P. Kibble, G. Copley, L. Krause, Phys. Rev. 153, 9 (1967).
[CrossRef]

Phys. Rev. Lett.

I. V. Hertel, H. Hofmann, K. J. Rost, Phys. Rev. Lett. 36, 861 (1976).
[CrossRef]

Spectrochim. Acta Part B

B. L. Sharp, A. Goldwasser, Spectrochim. Acta Part B 31, 431 (1976).
[CrossRef]

Other

K. F. Herzfeld, T. A. Litovitz, Absorption and Dispersion of Ultrasonic Waves (Academic, New York, 1959).

D. H. Campbell, “Computer Modeling of Laser-Excited Molecular and Atomic-Molecular Systems,” AEDC-TR-80-47 (1981).

C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations (Prentice-Hall, Englewood Cliffs, N.J., 1971).

H. G. Kuhn, Atomic Spectra (Academic, New York, 1961).

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Figures (14)

Fig. 1
Fig. 1

Three-level model (sodium) schematic showing the various radiative and collisional energy transfer processes for laser excitation of level 2(32P1/2 electronic level).

Fig. 2
Fig. 2

Relative quenching probabilities for sodium quenched by nitrogen as a function of the change in nitrogen vibrational quantum number (Δv).

Fig. 3
Fig. 3

Quenching and reverse quenching rate constants for nitrogen vibrational transitions originating or ending in v = 0.

Fig. 4
Fig. 4

3P-level mixing exchange rate constant and total cross sections.

Fig. 5
Fig. 5

Normalized number densities in sodium levels: NT = 1.0 × 1019 cm−3, N 1 0 = 1.0 × 10 13 cm - 3, T = 300 K, ρν 1.0 × 10−10 erg cm−3 Hz−1, 32P1/2 level excited. Laser is turned on at t = 0 and is on at all times.

Fig. 6
Fig. 6

Nitrogen vibrational population distributions: NT = 1.0 × 1019 cm−3, N 1 0 = 1.0 × 10 13 cm - 3, T = 300 K, ρν 1.0 × 10−10 erg cm−3 Hz−1, 32P1/2 level excited. Laser is turned on at t = 0 and is on at all times.

Fig. 7
Fig. 7

Sodium 32P3/2–32P1/2 number density ratio for conditions of Figs. 5 and 6.

Fig. 8
Fig. 8

Normalized number densities in sodium levels: NT = 1.0 × 1019 cm−3, N 1 0 = 1.0 × 10 13 cm - 3, T = 300 K, ρν 1.0 × 10−10 erg cm−3 Hz−1, 32P1/2 level excited.

Fig. 9
Fig. 9

Nitrogen vibrational population distributions: NT = 1.0 × 1019 cm−3, N 1 0 = 1.0 × 10 13 cm - 3, T = 300 K, ρν 1.0 × 10−10 erg cm−3 Hz−1, 32P1/2 level excited.

Fig. 10
Fig. 10

Normalized number densities in sodium levels: NT = 1.0 × 1019 cm−3, N 1 0 = 1.0 × 10 10 cm - 3, T = 300 K, ρν 1.0 × 10−10 erg cm−3 Hz−1, 32P1/2 level excited.

Fig. 11
Fig. 11

Normalized number densities in sodium levels: NT = 1.0 × 1019 cm−3, N 1 0 = 1.0 × 10 10 cm - 3, T = 300 K, ρν 1.0 × 10−10 erg cm−3 Hz−1, 32P1/2 level excited.

Fig. 12
Fig. 12

Nitrogen vibrational population distributions: NT = 1.0 × 1019 cm−3, N 1 0 = 1.0 × 10 10 cm 3, T = 300 K, ρν 1.0 × 10−10 erg cm−3 Hz−1, 32P1/2 level excited.

Fig. 13
Fig. 13

32P3/2–32P1/2 population ratio for 32P1/2 level excited.

Fig. 14
Fig. 14

32P3/2–32P1/2 population ratio for 32P1/2 level excited.

Tables (2)

Tables Icon

Table I Constants for Sodium–Nitrogen Model

Tables Icon

Table II Rate Constants for Three-Level Model Analysis

Equations (15)

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

d N 2 d t = - ( B 21 ρ ν + A 31 + Q 21 + Q 23 ) N 2 + ( B 12 ρ ν + Q 12 ) N 1 + Q 32 N 3
d N 3 d t = - ( Q 31 + A 21 + Q 32 ) N 3 + N 2 Q 23 + N 1 Q 13 ,
N = N 1 + N 2 + N 3 = N 1 0 ,
N 1 0 = [ ( B 21 + B 12 ) ρ ν + A 21 + Q 21 + Q 12 + Q 23 ] N 2 - ( Q 32 - Q 12 - B 12 ρ ν ) N 3 B 12 ρ ν + Q 12 .
N 0 1 = N 3 + ( B 12 + B 21 B 12 ) N 2 ,
N 3 N 2 = ( Q 31 + A 31 Q 23 + Q 32 Q 23 ) - 1 ,
G 32 = Q 32 Q 23 = g 2 g 3 exp ( Δ E 32 / k T )
I R = ( Q 31 + A 31 Q 23 + G 32 ) - 1 .
Q 31 = k 31 N Q             Q 23 = k 23 N Q ,
N T = A 31 k 23 ( 1 IR - G 32 ) - k 31 .
Q 21 = v N u w v k v w                 Q 12 = w N w u w k w v ,
k w v = k v w g 2 g 1 exp [ - ( Δ E 21 - Δ E v w ) / k T ] ,
d N 1 d t - - N 1 [ B ρ 12 ν w N w v w ( k w v + k w v ) ] + N 2 ( B 21 ρ ν + A 21 + v n v w v k v w ) + N 3 ( A 31 + v N v w v k v w ) , d N 2 d t = N 1 ( B 12 ρ ν + w N w v w k v w ) - N 2 ( B 21 ρ ν + A 21 + N 3 k 23 + N T k 23 + v N v w v k v w + N 3 ( N 2 k 32 + N T k 32 ) , d N 3 d t = N 1 v N v v w k w v + N 2 ( N 3 k 23 + N T k 23 ) - N 3 ( A 31 + N 2 k 32 + N T k 32 + u N v w v k v w ) ,
d N v d t = - N v [ w N w { R v , w ; v + 1 , w - 1 + R v , w ; v - 1 , w + 1 } depleting V V + N T w R v w ] depleting V t + N v [ w N w { R v + 1 , w ; v , w - 1 + R v - 1 , w ; v , w + 1 } populating V V + N T w R w v ] populating V T - N v [ N 1 w v ( k v w + k v w ) depleting reverse Q + ( N 2 + N 3 ) w v k v w ] depleting Q + N 1 w v N w ( k w v + k w v ) populating reverse Q + ( N 2 + N 3 ) w v N w k w v populating Q .
v N v w v k w v = Q T

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