Abstract

The time behavior of a two- and a three-level molecular system under narrowband pulsed-laser excitation is discussed, and relations are presented for the temporal variation of the concentrations of the levels involved. It is shown that if saturation is approached, the response time of the molecular system is strictly related to the pumping time, which is given by the reciprocal of the product of the laser spectral density and the Einstein coefficient of stimulated absorption. Typical calculations of the fractional number densities of the levels are presented as a function of the value of the laser spectral density required to achieve saturation for a few typical molecules such as benzophenone, fluorene, and rhodamine 6G.

© 1980 Optical Society of America

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  1. N. Omenetto, J. D. Winefordner, Prog. Anal. At. Spectrosc. 2(1,2), 0000 (1979).
  2. E. H. Piepmeier, Spectrochim. Acta 27B, 431 (1972).
  3. E. H. Piepmeier, Spectrochim. Acta 27B, 445 (1972).
  4. N. Omenetto, P. Benetti, L. P. Hart, J. D. Winefordner, C. Th. J. Alkemade, Spectrochim. Acta 28B, 289 (1973).
  5. J. Kuhl, S. Neumann, M. Kriese, Z. Naturforsch. 28a, 273 (1973).
  6. N. Omenetto, J. D. Winefordner in Analytical Laser Spectroscopy, N. Omenetto, Ed. (Wiley, New York, 1979), Chap. 4.
  7. G. D. Boutilier, M. B. Blackburn, J. M. Mermet, S. J. Weeks, H. Haraguchi, J. D. Winefordner, N. Omenetto, Appl. Opt. 17, 229 (1978).
  8. D. R. Olivares, G. M. Hieftje, Spectrochim. Acta 33B, 79 (1978)
  9. B. L. Sharp, A. Goldwasser, Spectrochim. Acta 31B, 431 (1976).
  10. J. W. Daily, Appl. Opt. 15, 955 (1976).
    [CrossRef] [PubMed]
  11. J. W. Daily, Appl. Opt. 16, 568 (1977).
    [CrossRef] [PubMed]
  12. A. P. Baronavski, J. R. McDonald, Appl. Opt. 16, 1897 (1977).
    [CrossRef] [PubMed]
  13. R. A. Van Calcar, M. J. M. Van DeVen, B. K. Van Uitert, K. J. Biewenge, T. Hollander, C. T. J. Alkemade, J. Quant. Spectrosc. Radiat. Transfer 21, 11 (1979).
    [CrossRef]
  14. M. B. Blackburn, J. M. Mermet, G. D. Boutilier, J. D. Winefordner, Appl. Opt. 18, 1804 (1979).
    [CrossRef] [PubMed]
  15. M. Mailander, J. Appl. Phys 49, 1256 (1978).
    [CrossRef]
  16. G. D. Boutilier, J. D. Winefordner, N. Omenetto, Appl. Opt. 17, 3482 (1978).
    [CrossRef] [PubMed]
  17. M. Yamashita, H. Ikeda, H. Kashiwagi, J. Chem. Phys. 63, 1127 (1975).
    [CrossRef]
  18. D. K. Killinger, C. C. Wang, M. Hanabusa, Phys. Rev. A13, 2145 (1976).
  19. B. W. Smith, F. W. Plankey, N. Omenetto, L. P. Hart, J. D. Winefordner, Spectrochim. Acta 30A, 1459 (1974).
  20. J. W. Daily, Appl. Opt. 16, 2322 (1977).
    [CrossRef] [PubMed]
  21. R. M. Measures, J. Appl. Phys. 39, 5232 (1968).
    [CrossRef]
  22. M. A. Bolshov, A. V. Zybin, V. G. Koloshnikov, K. N. Koshelev, Spectrochim. Acta 32B, 279 (1977).
  23. K. G. Müller, M. Stania, J. Appl. Phys. 49, 5801 (1978).
    [CrossRef]
  24. F. R. Lipsett, in Progress in Dielectrics (Iliffe, London, 1968), Vol. 7.
  25. S. J. Strickler, R. A. Berg, J. Chem. Phys. 37, 814 (1962).
    [CrossRef]
  26. N. Turro, Molecular Photochemistry (W. A. Benjamin, New York, 1967).
  27. R. S. Becker, Theory and Interpretation of Fluorescence and Phosphorescence (Wiley, New York, 1969).
  28. M. Jamashita, H. Kashiwagi, J. Phys. Chem. 78, 2006 (1974).
    [CrossRef]

1979 (3)

N. Omenetto, J. D. Winefordner, Prog. Anal. At. Spectrosc. 2(1,2), 0000 (1979).

R. A. Van Calcar, M. J. M. Van DeVen, B. K. Van Uitert, K. J. Biewenge, T. Hollander, C. T. J. Alkemade, J. Quant. Spectrosc. Radiat. Transfer 21, 11 (1979).
[CrossRef]

M. B. Blackburn, J. M. Mermet, G. D. Boutilier, J. D. Winefordner, Appl. Opt. 18, 1804 (1979).
[CrossRef] [PubMed]

1978 (5)

M. Mailander, J. Appl. Phys 49, 1256 (1978).
[CrossRef]

G. D. Boutilier, J. D. Winefordner, N. Omenetto, Appl. Opt. 17, 3482 (1978).
[CrossRef] [PubMed]

G. D. Boutilier, M. B. Blackburn, J. M. Mermet, S. J. Weeks, H. Haraguchi, J. D. Winefordner, N. Omenetto, Appl. Opt. 17, 229 (1978).

D. R. Olivares, G. M. Hieftje, Spectrochim. Acta 33B, 79 (1978)

K. G. Müller, M. Stania, J. Appl. Phys. 49, 5801 (1978).
[CrossRef]

1977 (4)

1976 (3)

D. K. Killinger, C. C. Wang, M. Hanabusa, Phys. Rev. A13, 2145 (1976).

B. L. Sharp, A. Goldwasser, Spectrochim. Acta 31B, 431 (1976).

J. W. Daily, Appl. Opt. 15, 955 (1976).
[CrossRef] [PubMed]

1975 (1)

M. Yamashita, H. Ikeda, H. Kashiwagi, J. Chem. Phys. 63, 1127 (1975).
[CrossRef]

1974 (2)

B. W. Smith, F. W. Plankey, N. Omenetto, L. P. Hart, J. D. Winefordner, Spectrochim. Acta 30A, 1459 (1974).

M. Jamashita, H. Kashiwagi, J. Phys. Chem. 78, 2006 (1974).
[CrossRef]

1973 (2)

N. Omenetto, P. Benetti, L. P. Hart, J. D. Winefordner, C. Th. J. Alkemade, Spectrochim. Acta 28B, 289 (1973).

J. Kuhl, S. Neumann, M. Kriese, Z. Naturforsch. 28a, 273 (1973).

1972 (2)

E. H. Piepmeier, Spectrochim. Acta 27B, 431 (1972).

E. H. Piepmeier, Spectrochim. Acta 27B, 445 (1972).

1968 (1)

R. M. Measures, J. Appl. Phys. 39, 5232 (1968).
[CrossRef]

1962 (1)

S. J. Strickler, R. A. Berg, J. Chem. Phys. 37, 814 (1962).
[CrossRef]

Alkemade, C. T. J.

R. A. Van Calcar, M. J. M. Van DeVen, B. K. Van Uitert, K. J. Biewenge, T. Hollander, C. T. J. Alkemade, J. Quant. Spectrosc. Radiat. Transfer 21, 11 (1979).
[CrossRef]

Alkemade, C. Th. J.

N. Omenetto, P. Benetti, L. P. Hart, J. D. Winefordner, C. Th. J. Alkemade, Spectrochim. Acta 28B, 289 (1973).

Baronavski, A. P.

Becker, R. S.

R. S. Becker, Theory and Interpretation of Fluorescence and Phosphorescence (Wiley, New York, 1969).

Benetti, P.

N. Omenetto, P. Benetti, L. P. Hart, J. D. Winefordner, C. Th. J. Alkemade, Spectrochim. Acta 28B, 289 (1973).

Berg, R. A.

S. J. Strickler, R. A. Berg, J. Chem. Phys. 37, 814 (1962).
[CrossRef]

Biewenge, K. J.

R. A. Van Calcar, M. J. M. Van DeVen, B. K. Van Uitert, K. J. Biewenge, T. Hollander, C. T. J. Alkemade, J. Quant. Spectrosc. Radiat. Transfer 21, 11 (1979).
[CrossRef]

Blackburn, M. B.

M. B. Blackburn, J. M. Mermet, G. D. Boutilier, J. D. Winefordner, Appl. Opt. 18, 1804 (1979).
[CrossRef] [PubMed]

G. D. Boutilier, M. B. Blackburn, J. M. Mermet, S. J. Weeks, H. Haraguchi, J. D. Winefordner, N. Omenetto, Appl. Opt. 17, 229 (1978).

Bolshov, M. A.

M. A. Bolshov, A. V. Zybin, V. G. Koloshnikov, K. N. Koshelev, Spectrochim. Acta 32B, 279 (1977).

Boutilier, G. D.

Daily, J. W.

Goldwasser, A.

B. L. Sharp, A. Goldwasser, Spectrochim. Acta 31B, 431 (1976).

Hanabusa, M.

D. K. Killinger, C. C. Wang, M. Hanabusa, Phys. Rev. A13, 2145 (1976).

Haraguchi, H.

G. D. Boutilier, M. B. Blackburn, J. M. Mermet, S. J. Weeks, H. Haraguchi, J. D. Winefordner, N. Omenetto, Appl. Opt. 17, 229 (1978).

Hart, L. P.

B. W. Smith, F. W. Plankey, N. Omenetto, L. P. Hart, J. D. Winefordner, Spectrochim. Acta 30A, 1459 (1974).

N. Omenetto, P. Benetti, L. P. Hart, J. D. Winefordner, C. Th. J. Alkemade, Spectrochim. Acta 28B, 289 (1973).

Hieftje, G. M.

D. R. Olivares, G. M. Hieftje, Spectrochim. Acta 33B, 79 (1978)

Hollander, T.

R. A. Van Calcar, M. J. M. Van DeVen, B. K. Van Uitert, K. J. Biewenge, T. Hollander, C. T. J. Alkemade, J. Quant. Spectrosc. Radiat. Transfer 21, 11 (1979).
[CrossRef]

Ikeda, H.

M. Yamashita, H. Ikeda, H. Kashiwagi, J. Chem. Phys. 63, 1127 (1975).
[CrossRef]

Jamashita, M.

M. Jamashita, H. Kashiwagi, J. Phys. Chem. 78, 2006 (1974).
[CrossRef]

Kashiwagi, H.

M. Yamashita, H. Ikeda, H. Kashiwagi, J. Chem. Phys. 63, 1127 (1975).
[CrossRef]

M. Jamashita, H. Kashiwagi, J. Phys. Chem. 78, 2006 (1974).
[CrossRef]

Killinger, D. K.

D. K. Killinger, C. C. Wang, M. Hanabusa, Phys. Rev. A13, 2145 (1976).

Koloshnikov, V. G.

M. A. Bolshov, A. V. Zybin, V. G. Koloshnikov, K. N. Koshelev, Spectrochim. Acta 32B, 279 (1977).

Koshelev, K. N.

M. A. Bolshov, A. V. Zybin, V. G. Koloshnikov, K. N. Koshelev, Spectrochim. Acta 32B, 279 (1977).

Kriese, M.

J. Kuhl, S. Neumann, M. Kriese, Z. Naturforsch. 28a, 273 (1973).

Kuhl, J.

J. Kuhl, S. Neumann, M. Kriese, Z. Naturforsch. 28a, 273 (1973).

Lipsett, F. R.

F. R. Lipsett, in Progress in Dielectrics (Iliffe, London, 1968), Vol. 7.

Mailander, M.

M. Mailander, J. Appl. Phys 49, 1256 (1978).
[CrossRef]

McDonald, J. R.

Measures, R. M.

R. M. Measures, J. Appl. Phys. 39, 5232 (1968).
[CrossRef]

Mermet, J. M.

M. B. Blackburn, J. M. Mermet, G. D. Boutilier, J. D. Winefordner, Appl. Opt. 18, 1804 (1979).
[CrossRef] [PubMed]

G. D. Boutilier, M. B. Blackburn, J. M. Mermet, S. J. Weeks, H. Haraguchi, J. D. Winefordner, N. Omenetto, Appl. Opt. 17, 229 (1978).

Müller, K. G.

K. G. Müller, M. Stania, J. Appl. Phys. 49, 5801 (1978).
[CrossRef]

Neumann, S.

J. Kuhl, S. Neumann, M. Kriese, Z. Naturforsch. 28a, 273 (1973).

Olivares, D. R.

D. R. Olivares, G. M. Hieftje, Spectrochim. Acta 33B, 79 (1978)

Omenetto, N.

N. Omenetto, J. D. Winefordner, Prog. Anal. At. Spectrosc. 2(1,2), 0000 (1979).

G. D. Boutilier, M. B. Blackburn, J. M. Mermet, S. J. Weeks, H. Haraguchi, J. D. Winefordner, N. Omenetto, Appl. Opt. 17, 229 (1978).

G. D. Boutilier, J. D. Winefordner, N. Omenetto, Appl. Opt. 17, 3482 (1978).
[CrossRef] [PubMed]

B. W. Smith, F. W. Plankey, N. Omenetto, L. P. Hart, J. D. Winefordner, Spectrochim. Acta 30A, 1459 (1974).

N. Omenetto, P. Benetti, L. P. Hart, J. D. Winefordner, C. Th. J. Alkemade, Spectrochim. Acta 28B, 289 (1973).

N. Omenetto, J. D. Winefordner in Analytical Laser Spectroscopy, N. Omenetto, Ed. (Wiley, New York, 1979), Chap. 4.

Piepmeier, E. H.

E. H. Piepmeier, Spectrochim. Acta 27B, 431 (1972).

E. H. Piepmeier, Spectrochim. Acta 27B, 445 (1972).

Plankey, F. W.

B. W. Smith, F. W. Plankey, N. Omenetto, L. P. Hart, J. D. Winefordner, Spectrochim. Acta 30A, 1459 (1974).

Sharp, B. L.

B. L. Sharp, A. Goldwasser, Spectrochim. Acta 31B, 431 (1976).

Smith, B. W.

B. W. Smith, F. W. Plankey, N. Omenetto, L. P. Hart, J. D. Winefordner, Spectrochim. Acta 30A, 1459 (1974).

Stania, M.

K. G. Müller, M. Stania, J. Appl. Phys. 49, 5801 (1978).
[CrossRef]

Strickler, S. J.

S. J. Strickler, R. A. Berg, J. Chem. Phys. 37, 814 (1962).
[CrossRef]

Turro, N.

N. Turro, Molecular Photochemistry (W. A. Benjamin, New York, 1967).

Van Calcar, R. A.

R. A. Van Calcar, M. J. M. Van DeVen, B. K. Van Uitert, K. J. Biewenge, T. Hollander, C. T. J. Alkemade, J. Quant. Spectrosc. Radiat. Transfer 21, 11 (1979).
[CrossRef]

Van DeVen, M. J. M.

R. A. Van Calcar, M. J. M. Van DeVen, B. K. Van Uitert, K. J. Biewenge, T. Hollander, C. T. J. Alkemade, J. Quant. Spectrosc. Radiat. Transfer 21, 11 (1979).
[CrossRef]

Van Uitert, B. K.

R. A. Van Calcar, M. J. M. Van DeVen, B. K. Van Uitert, K. J. Biewenge, T. Hollander, C. T. J. Alkemade, J. Quant. Spectrosc. Radiat. Transfer 21, 11 (1979).
[CrossRef]

Wang, C. C.

D. K. Killinger, C. C. Wang, M. Hanabusa, Phys. Rev. A13, 2145 (1976).

Weeks, S. J.

G. D. Boutilier, M. B. Blackburn, J. M. Mermet, S. J. Weeks, H. Haraguchi, J. D. Winefordner, N. Omenetto, Appl. Opt. 17, 229 (1978).

Winefordner, J. D.

N. Omenetto, J. D. Winefordner, Prog. Anal. At. Spectrosc. 2(1,2), 0000 (1979).

M. B. Blackburn, J. M. Mermet, G. D. Boutilier, J. D. Winefordner, Appl. Opt. 18, 1804 (1979).
[CrossRef] [PubMed]

G. D. Boutilier, M. B. Blackburn, J. M. Mermet, S. J. Weeks, H. Haraguchi, J. D. Winefordner, N. Omenetto, Appl. Opt. 17, 229 (1978).

G. D. Boutilier, J. D. Winefordner, N. Omenetto, Appl. Opt. 17, 3482 (1978).
[CrossRef] [PubMed]

B. W. Smith, F. W. Plankey, N. Omenetto, L. P. Hart, J. D. Winefordner, Spectrochim. Acta 30A, 1459 (1974).

N. Omenetto, P. Benetti, L. P. Hart, J. D. Winefordner, C. Th. J. Alkemade, Spectrochim. Acta 28B, 289 (1973).

N. Omenetto, J. D. Winefordner in Analytical Laser Spectroscopy, N. Omenetto, Ed. (Wiley, New York, 1979), Chap. 4.

Yamashita, M.

M. Yamashita, H. Ikeda, H. Kashiwagi, J. Chem. Phys. 63, 1127 (1975).
[CrossRef]

Zybin, A. V.

M. A. Bolshov, A. V. Zybin, V. G. Koloshnikov, K. N. Koshelev, Spectrochim. Acta 32B, 279 (1977).

Appl. Opt. (7)

J. Appl. Phys (1)

M. Mailander, J. Appl. Phys 49, 1256 (1978).
[CrossRef]

J. Appl. Phys. (2)

R. M. Measures, J. Appl. Phys. 39, 5232 (1968).
[CrossRef]

K. G. Müller, M. Stania, J. Appl. Phys. 49, 5801 (1978).
[CrossRef]

J. Chem. Phys. (2)

S. J. Strickler, R. A. Berg, J. Chem. Phys. 37, 814 (1962).
[CrossRef]

M. Yamashita, H. Ikeda, H. Kashiwagi, J. Chem. Phys. 63, 1127 (1975).
[CrossRef]

J. Phys. Chem. (1)

M. Jamashita, H. Kashiwagi, J. Phys. Chem. 78, 2006 (1974).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (1)

R. A. Van Calcar, M. J. M. Van DeVen, B. K. Van Uitert, K. J. Biewenge, T. Hollander, C. T. J. Alkemade, J. Quant. Spectrosc. Radiat. Transfer 21, 11 (1979).
[CrossRef]

Phys. Rev. (1)

D. K. Killinger, C. C. Wang, M. Hanabusa, Phys. Rev. A13, 2145 (1976).

Prog. Anal. At. Spectrosc. (1)

N. Omenetto, J. D. Winefordner, Prog. Anal. At. Spectrosc. 2(1,2), 0000 (1979).

Spectrochim. Acta (7)

E. H. Piepmeier, Spectrochim. Acta 27B, 431 (1972).

E. H. Piepmeier, Spectrochim. Acta 27B, 445 (1972).

N. Omenetto, P. Benetti, L. P. Hart, J. D. Winefordner, C. Th. J. Alkemade, Spectrochim. Acta 28B, 289 (1973).

B. W. Smith, F. W. Plankey, N. Omenetto, L. P. Hart, J. D. Winefordner, Spectrochim. Acta 30A, 1459 (1974).

D. R. Olivares, G. M. Hieftje, Spectrochim. Acta 33B, 79 (1978)

B. L. Sharp, A. Goldwasser, Spectrochim. Acta 31B, 431 (1976).

M. A. Bolshov, A. V. Zybin, V. G. Koloshnikov, K. N. Koshelev, Spectrochim. Acta 32B, 279 (1977).

Z. Naturforsch. (1)

J. Kuhl, S. Neumann, M. Kriese, Z. Naturforsch. 28a, 273 (1973).

Other (4)

N. Omenetto, J. D. Winefordner in Analytical Laser Spectroscopy, N. Omenetto, Ed. (Wiley, New York, 1979), Chap. 4.

F. R. Lipsett, in Progress in Dielectrics (Iliffe, London, 1968), Vol. 7.

N. Turro, Molecular Photochemistry (W. A. Benjamin, New York, 1967).

R. S. Becker, Theory and Interpretation of Fluorescence and Phosphorescence (Wiley, New York, 1969).

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Figures (3)

Fig. 1
Fig. 1

Temporal behavior of levels 2 and 3 for benzophenone. Values of E increase in same manner for n3/nT as given for n2/nT. Values used in calculations are A31= 1.0 × 106 sec−1;k31= 1.0 × 107 sec−1;k32= 1.0 × 1010 sec−1;A21= 1.8 × 102 sec−1;k21= 1.8 × 101 sec−1;ξ3j,10= 1.0 × 10−3;λ= 337.1 nm;δνabs= 2.6 × 1013 Hz;Es= 6.8 × 108 W m−2.Values for A31, k31, k32, A21, and k21 taken from Turro.26

Fig. 2
Fig. 2

Temporal behavior of levels 2 and 3 for fluorene. Values of E increase in same manner for n3/nT as given for n2/nT. Values used in calculations are A31= 5.4 × 107 sec−1;k31= 2.2 × 107 sec−1;k32= 4.5 × 107 sec−1;A21= 2.0 × 10−1 sec−1;k21= 4.0 × 10−3 sec−1;ξ3j,10= 1.0 × 10−1;λ= 250 nm;δνabs= 2.6 × 1013 Hz;Es= 8.6 × 102 W m−2.Values for A31, k31, k32, A21, and k21 taken from Becker.27

Fig. 3
Fig. 3

Temporal behavior of levels 2 and 3 for rhodamine 6G. Values of E increase in same manner for n3/nT as given for n2/nT. Values used in calculations are A31= 1.2 × 108 sec−1;k31= 1.0 × 105 sec−1;k32= 1.1 × 105 sec−1;A21= 5.0 × 10−1 sec−1;k21= 8.0 × 10−2 sec−1;ξ3j,10= 5.0 × 10−2;λ= 514 nm;δνabs= 2.6 × 1013;Es= 1.0 × 105 W m−2.Values for A31, k31, k32, A21, and k21 taken from Ref. 28.

Equations (40)

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n 2 s s = E ν ( ν 10 , 2 j ) [ n T ( 1 + g 1 g 2 ) E ν ( ν 10 , 2 j ) + g 1 g 2 E ν * ( ν 10 , 2 j ) ξ 2 j , 10 ] ,
E ν * ( ν 10 , 2 j )
n 2 s s = n T E ν ( ν 10 , 2 j ) c B ( i A 20 , 1 i + k 21 ) + 2 B E ν ( ν 10 , 2 j ) c ,
n 2 ( t ) = n 2 s s { 1 exp [ ( i A 20 , 1 i + k 21 + 2 B E ν 0 c ) t ] }
n 2 ( t ) = n 2 s s [ 1 exp ( t / t r ) ] .
t r ( i A 20 , 1 i + k 21 + 2 B E ν 0 c ) 1
B L ( t ) = n 2 ( t ) h ν 20 , 1 i ( l 4 π ) ( i A 20 , 1 i ) ,
d n 3 ( t ) d t = ( B 31 E ν 0 c + i A 30 , 1 i + k 31 + k 32 ) n 3 + ( B 13 E ν 0 c + k 13 ) n 1 + k 23 n 2 ,
d n 2 ( t ) d t = ( i A 20 , 1 i + k 21 + k 23 ) n 2 + k 32 n 3 + k 12 n 1 .
a 3 = i A 30 , 1 i + k 31 + k 32 ,
a 2 = i A 20 , 1 i + k 21 ,
b 3 = B 31 E ν 0 c + B 13 E ν 0 c = 2 B E ν 0 c ( if g 1 = g 2 ) ,
β = B 13 E ν 0 c .
n T = n 1 + n 2 + n 3 ,
( D + b 3 + a 3 ) n 3 + β n 2 = β n T ,
[ ( D + a 2 ) / k 32 ] n 2 = n 3 .
( D + b 3 + a 3 ) ( D + a 2 ) n 2 + β k 32 n 2 β k 32 n T = 0 .
n 2 ( t ) = C 1 exp ( α 2 t ) + C 2 exp ( α 3 t ) + C 0 ,
α 2 = X ( X 2 4 Y ) 1 / 2 2 ,
α 3 = X + ( X 2 4 Y ) 1 / 2 2 ,
X = b 3 + a 3 + a 2 ,
Y = ( b 3 + a 3 ) a 2 + β k 32 .
C 0 = ( k 32 a 2 ) ( k 32 a 2 + a 3 + b 3 β ¯ n T ) = n 2 s s .
n 2 ( t ) = n 2 s s [ α 3 α 3 α 2 exp ( α 2 t ) + α 2 α 3 α 2 exp ( α 3 t ) + 1 ] ,
n 3 ( t ) = n 3 s s [ α 3 ( α 2 α 2 ) α 2 ( α 3 α 2 ) exp ( α 2 t ) + α 2 ( α 2 α 3 ) α 2 ( α 3 α 2 ) exp ( α 3 t ) + 1 ] .
n 2 s s = B ( E ν 0 c ) n T k 32 2 B ( E ν 0 c ) ( i A 20 , 1 i + k 21 ) + B ( E ν 0 c ) k 32 + ( i A 30 , 1 i + k 31 + k 32 ) ( i A 20 , 1 i + k 21 ) ,
n 2 s s = B ( E ν 0 c ) n T B ( E ν 0 c ) ( 2 + k 32 i A 20 , 1 i + k 21 ) + ( i A 30 , 1 i + k 32 + k 32 ) .
( n 2 s s ) E ν = n T k 32 2 ( i A 20 , 1 i + k 21 ) + k 32 ,
( n 2 s s ) E ν = n T 2 + k 32 i A 20 , 1 i + k 21 .
α 2 Y / X ,
α 3 X ( Y / X ) X .
C 1 Y a 2 X 1 Y a 2 X ,
C 2 Y a 2 X ,
Y B ( E ν 0 c ) [ 2 ( i A 20 , 1 i + k 21 ) + k 32 ]
Y = B ( E ν 0 c ) k 32 if k 32 ( i A 20 , 1 i + k 21 ) ,
X 2 B ( E ν 0 c ) .
n 3 ( t ) = ( n 3 s s ) E ν { C 1 exp [ ( Y X ) t ] + C 2 exp ( X t ) + 1 }
n 3 ( t ) = ( n 3 s s ) E ν [ C 1 exp ( t τ 3 ) + C 2 exp ( t t r ) + 1 ] .
t r τ 3 .
[ n 3 ( t ) ] peak = ( n 3 s s ) E C 1 = ( n 3 s s ) ( Y a 2 X ) = n T 2 ,

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