Abstract

We present a theoretical study of the influence of a single spherical metal nanoparticle (MNP) on the fluorescence intensity of nearby emitters with two-level and multi-level energy systems. The enhancement factors of the excitation and relaxation processes are deduced. To reveal the interrelationship between the excitation and relaxation processes we adopt the rate equations of two-level fluorescent systems and upconversion fluorescent systems, and deduce the expression for the fluorescence enhancement factor. Our calculated results for the two-level systems agree well with reported experimental data. As to the upconversion fluorescent systems, our numerical results provide the first theoretical prediction showing that the MNP may selectively enhance a certain fluorescence process among various ones.

© 2010 OSA

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2009 (3)

M. H. Chowdhury, K. Ray, S. K. Gray, J. Pond, and J. R. Lakowicz, “Aluminum nanoparticles as substrates for metal-enhanced fluorescence in the ultraviolet for the label-free detection of biomolecules,” Anal. Chem. 81(4), 1397–1403 (2009).
[CrossRef] [PubMed]

R. Esteban, M. Laroche, and J.-J. Greffet, “Influence of metallic nanoparticles on upconversion processes,” J. Appl. Phys. 105(3), 033107 (2009).
[CrossRef]

E. Verhagen, L. Kuipers, and A. Polman, “Field enhancement in metallic subwavelength aperture arrays probed by erbium upconversion luminescence,” Opt. Express 17(17), 14586–14598 (2009).
[CrossRef] [PubMed]

2008 (2)

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100(20), 203002 (2008).
[CrossRef] [PubMed]

C. Hagglund, M. Zach, and B. Kasemo, “Enhanced charge carrier generation in dye sensitized solar cells by nanoparticle plasmons,” Appl. Phys. Lett. 92(1), 013113 (2008).
[CrossRef]

2007 (6)

P. Bharadwaj and L. Novotny, “Spectral dependence of single molecule fluorescence enhancement,” Opt. Express 15(21), 14266–14274 (2007).
[CrossRef] [PubMed]

P. Bharadwaj, P. Anger, and L. Novotny, “Nanoplasmonic enhancement of single-molecule fluorescence,” Nanotechnology 18(4), 044017 (2007).
[CrossRef]

T. Härtling, P. Reichenbach, and L. M. Eng, “Near-field coupling of a single fluorescent molecule and a spherical gold nanoparticle,” Opt. Express 15(20), 12806–12817 (2007).
[CrossRef] [PubMed]

Y. Ito, K. Matsuda, and Y. Kanemitsu, “Mechanism of photoluminescence enhancement in single semiconductor nanocrystals on metal surfaces,” Phys. Rev. B 75(3), 033309 (2007).
[CrossRef]

K. L. Shuford, M. A. Ratner, S. K. Gray, and G. C. Schatz, “Electric field enhancement and light transmission in cylindrical nanoholes,” J. Comput. Theor. Nanosci. 4, 239–246 (2007).

E. Verhagen, L. Kuipers, and A. Polman, “Enhanced nonlinear optical effects with a tapered plasmonic waveguide,” Nano Lett. 7(2), 334–337 (2007).
[CrossRef] [PubMed]

2006 (3)

S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97(1), 017402 (2006).
[CrossRef] [PubMed]

Y. Wang and Z. P. Zhou, “Strong enhancement of erbium ion emission by a metallic double grating,” Appl. Phys. Lett. 89(25), 253122 (2006).
[CrossRef]

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett. 96(11), 113002 (2006).
[CrossRef] [PubMed]

2005 (1)

2004 (1)

M. Thomas, J.-J. Greffet, R. Carminati, and J. R. Arias-Gonzalez, “Single-molecule spontaneous emission close to absorbing nanostructures,” Appl. Phys. Lett. 85(17), 3863–3865 (2004).
[CrossRef]

2003 (1)

J. Kalkman, C. Strohhöfer, B. Gralak, and A. Polman, “Surface plasmon polariton modified emission of erbium in a metallodielectric grating,” Appl. Phys. Lett. 83(1), 30–32 (2003).
[CrossRef]

2002 (1)

K. T. Shimizu, W. K. Woo, B. R. Fisher, H. J. Eisler, and M. G. Bawendi, “Surface-enhanced emission from single semiconductor nanocrystals,” Phys. Rev. Lett. 89(11), 117401 (2002).
[CrossRef] [PubMed]

2001 (2)

1995 (1)

1987 (1)

1982 (1)

R. Ruppin, “Decay of an excited molecule near a small metal sphere,” J. Chem. Phys. 76(4), 1681–1684 (1982).
[CrossRef]

1980 (1)

1972 (1)

P. B. Johnson and R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

1946 (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Andrew, P.

P. Andrew and W. L. Barnes, “Molecular fluorescence above metallic gratings,” Phys. Rev. B 64(12), 125405 (2001).
[CrossRef]

Anger, P.

P. Bharadwaj, P. Anger, and L. Novotny, “Nanoplasmonic enhancement of single-molecule fluorescence,” Nanotechnology 18(4), 044017 (2007).
[CrossRef]

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett. 96(11), 113002 (2006).
[CrossRef] [PubMed]

Arias-Gonzalez, J. R.

M. Thomas, J.-J. Greffet, R. Carminati, and J. R. Arias-Gonzalez, “Single-molecule spontaneous emission close to absorbing nanostructures,” Appl. Phys. Lett. 85(17), 3863–3865 (2004).
[CrossRef]

Barnes, W. L.

P. Andrew and W. L. Barnes, “Molecular fluorescence above metallic gratings,” Phys. Rev. B 64(12), 125405 (2001).
[CrossRef]

Bawendi, M. G.

K. T. Shimizu, W. K. Woo, B. R. Fisher, H. J. Eisler, and M. G. Bawendi, “Surface-enhanced emission from single semiconductor nanocrystals,” Phys. Rev. Lett. 89(11), 117401 (2002).
[CrossRef] [PubMed]

Becker, P. C.

Bharadwaj, P.

P. Bharadwaj, P. Anger, and L. Novotny, “Nanoplasmonic enhancement of single-molecule fluorescence,” Nanotechnology 18(4), 044017 (2007).
[CrossRef]

P. Bharadwaj and L. Novotny, “Spectral dependence of single molecule fluorescence enhancement,” Opt. Express 15(21), 14266–14274 (2007).
[CrossRef] [PubMed]

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett. 96(11), 113002 (2006).
[CrossRef] [PubMed]

Carminati, R.

M. Thomas, J.-J. Greffet, R. Carminati, and J. R. Arias-Gonzalez, “Single-molecule spontaneous emission close to absorbing nanostructures,” Appl. Phys. Lett. 85(17), 3863–3865 (2004).
[CrossRef]

Chew, H.

Chowdhury, M. H.

M. H. Chowdhury, K. Ray, S. K. Gray, J. Pond, and J. R. Lakowicz, “Aluminum nanoparticles as substrates for metal-enhanced fluorescence in the ultraviolet for the label-free detection of biomolecules,” Anal. Chem. 81(4), 1397–1403 (2009).
[CrossRef] [PubMed]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

Desurvire, E.

Eisler, H. J.

K. T. Shimizu, W. K. Woo, B. R. Fisher, H. J. Eisler, and M. G. Bawendi, “Surface-enhanced emission from single semiconductor nanocrystals,” Phys. Rev. Lett. 89(11), 117401 (2002).
[CrossRef] [PubMed]

Eng, L. M.

Esteban, R.

R. Esteban, M. Laroche, and J.-J. Greffet, “Influence of metallic nanoparticles on upconversion processes,” J. Appl. Phys. 105(3), 033107 (2009).
[CrossRef]

Feldmann, J.

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100(20), 203002 (2008).
[CrossRef] [PubMed]

Fisher, B. R.

K. T. Shimizu, W. K. Woo, B. R. Fisher, H. J. Eisler, and M. G. Bawendi, “Surface-enhanced emission from single semiconductor nanocrystals,” Phys. Rev. Lett. 89(11), 117401 (2002).
[CrossRef] [PubMed]

Gralak, B.

J. Kalkman, C. Strohhöfer, B. Gralak, and A. Polman, “Surface plasmon polariton modified emission of erbium in a metallodielectric grating,” Appl. Phys. Lett. 83(1), 30–32 (2003).
[CrossRef]

Gray, S. K.

M. H. Chowdhury, K. Ray, S. K. Gray, J. Pond, and J. R. Lakowicz, “Aluminum nanoparticles as substrates for metal-enhanced fluorescence in the ultraviolet for the label-free detection of biomolecules,” Anal. Chem. 81(4), 1397–1403 (2009).
[CrossRef] [PubMed]

K. L. Shuford, M. A. Ratner, S. K. Gray, and G. C. Schatz, “Electric field enhancement and light transmission in cylindrical nanoholes,” J. Comput. Theor. Nanosci. 4, 239–246 (2007).

Greffet, J.-J.

R. Esteban, M. Laroche, and J.-J. Greffet, “Influence of metallic nanoparticles on upconversion processes,” J. Appl. Phys. 105(3), 033107 (2009).
[CrossRef]

M. Thomas, J.-J. Greffet, R. Carminati, and J. R. Arias-Gonzalez, “Single-molecule spontaneous emission close to absorbing nanostructures,” Appl. Phys. Lett. 85(17), 3863–3865 (2004).
[CrossRef]

Hagglund, C.

C. Hagglund, M. Zach, and B. Kasemo, “Enhanced charge carrier generation in dye sensitized solar cells by nanoparticle plasmons,” Appl. Phys. Lett. 92(1), 013113 (2008).
[CrossRef]

Håkanson, U.

S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97(1), 017402 (2006).
[CrossRef] [PubMed]

Härtling, T.

Ito, Y.

Y. Ito, K. Matsuda, and Y. Kanemitsu, “Mechanism of photoluminescence enhancement in single semiconductor nanocrystals on metal surfaces,” Phys. Rev. B 75(3), 033309 (2007).
[CrossRef]

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

Kalkman, J.

J. Kalkman, C. Strohhöfer, B. Gralak, and A. Polman, “Surface plasmon polariton modified emission of erbium in a metallodielectric grating,” Appl. Phys. Lett. 83(1), 30–32 (2003).
[CrossRef]

Kanemitsu, Y.

Y. Ito, K. Matsuda, and Y. Kanemitsu, “Mechanism of photoluminescence enhancement in single semiconductor nanocrystals on metal surfaces,” Phys. Rev. B 75(3), 033309 (2007).
[CrossRef]

Kasemo, B.

C. Hagglund, M. Zach, and B. Kasemo, “Enhanced charge carrier generation in dye sensitized solar cells by nanoparticle plasmons,” Appl. Phys. Lett. 92(1), 013113 (2008).
[CrossRef]

Kerker, M.

Klar, T. A.

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100(20), 203002 (2008).
[CrossRef] [PubMed]

Kühn, S.

S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97(1), 017402 (2006).
[CrossRef] [PubMed]

Kuipers, L.

Kürzinger, K.

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100(20), 203002 (2008).
[CrossRef] [PubMed]

Lakowicz, J. R.

M. H. Chowdhury, K. Ray, S. K. Gray, J. Pond, and J. R. Lakowicz, “Aluminum nanoparticles as substrates for metal-enhanced fluorescence in the ultraviolet for the label-free detection of biomolecules,” Anal. Chem. 81(4), 1397–1403 (2009).
[CrossRef] [PubMed]

Laroche, M.

R. Esteban, M. Laroche, and J.-J. Greffet, “Influence of metallic nanoparticles on upconversion processes,” J. Appl. Phys. 105(3), 033107 (2009).
[CrossRef]

Lin, H.

Liu, X. M.

Liu, X. R.

Lu, C.

Lu, F. Y.

Man, S. Q.

Matsuda, K.

Y. Ito, K. Matsuda, and Y. Kanemitsu, “Mechanism of photoluminescence enhancement in single semiconductor nanocrystals on metal surfaces,” Phys. Rev. B 75(3), 033309 (2007).
[CrossRef]

Ng, J. H.

Nichtl, A.

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100(20), 203002 (2008).
[CrossRef] [PubMed]

Novotny, L.

P. Bharadwaj and L. Novotny, “Spectral dependence of single molecule fluorescence enhancement,” Opt. Express 15(21), 14266–14274 (2007).
[CrossRef] [PubMed]

P. Bharadwaj, P. Anger, and L. Novotny, “Nanoplasmonic enhancement of single-molecule fluorescence,” Nanotechnology 18(4), 044017 (2007).
[CrossRef]

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett. 96(11), 113002 (2006).
[CrossRef] [PubMed]

Polman, A.

E. Verhagen, L. Kuipers, and A. Polman, “Field enhancement in metallic subwavelength aperture arrays probed by erbium upconversion luminescence,” Opt. Express 17(17), 14586–14598 (2009).
[CrossRef] [PubMed]

E. Verhagen, L. Kuipers, and A. Polman, “Enhanced nonlinear optical effects with a tapered plasmonic waveguide,” Nano Lett. 7(2), 334–337 (2007).
[CrossRef] [PubMed]

J. Kalkman, C. Strohhöfer, B. Gralak, and A. Polman, “Surface plasmon polariton modified emission of erbium in a metallodielectric grating,” Appl. Phys. Lett. 83(1), 30–32 (2003).
[CrossRef]

Pond, J.

M. H. Chowdhury, K. Ray, S. K. Gray, J. Pond, and J. R. Lakowicz, “Aluminum nanoparticles as substrates for metal-enhanced fluorescence in the ultraviolet for the label-free detection of biomolecules,” Anal. Chem. 81(4), 1397–1403 (2009).
[CrossRef] [PubMed]

Pun, E. Y. B.

Purcell, E. M.

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Ratner, M. A.

K. L. Shuford, M. A. Ratner, S. K. Gray, and G. C. Schatz, “Electric field enhancement and light transmission in cylindrical nanoholes,” J. Comput. Theor. Nanosci. 4, 239–246 (2007).

Ray, K.

M. H. Chowdhury, K. Ray, S. K. Gray, J. Pond, and J. R. Lakowicz, “Aluminum nanoparticles as substrates for metal-enhanced fluorescence in the ultraviolet for the label-free detection of biomolecules,” Anal. Chem. 81(4), 1397–1403 (2009).
[CrossRef] [PubMed]

Reichenbach, P.

Ringler, M.

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100(20), 203002 (2008).
[CrossRef] [PubMed]

Rogobete, L.

S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97(1), 017402 (2006).
[CrossRef] [PubMed]

Ruppin, R.

R. Ruppin, “Decay of an excited molecule near a small metal sphere,” J. Chem. Phys. 76(4), 1681–1684 (1982).
[CrossRef]

Sandoghdar, V.

S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97(1), 017402 (2006).
[CrossRef] [PubMed]

Schatz, G. C.

K. L. Shuford, M. A. Ratner, S. K. Gray, and G. C. Schatz, “Electric field enhancement and light transmission in cylindrical nanoholes,” J. Comput. Theor. Nanosci. 4, 239–246 (2007).

Schwemer, A.

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100(20), 203002 (2008).
[CrossRef] [PubMed]

Shimizu, K. T.

K. T. Shimizu, W. K. Woo, B. R. Fisher, H. J. Eisler, and M. G. Bawendi, “Surface-enhanced emission from single semiconductor nanocrystals,” Phys. Rev. Lett. 89(11), 117401 (2002).
[CrossRef] [PubMed]

Shuford, K. L.

K. L. Shuford, M. A. Ratner, S. K. Gray, and G. C. Schatz, “Electric field enhancement and light transmission in cylindrical nanoholes,” J. Comput. Theor. Nanosci. 4, 239–246 (2007).

Simpson, J. R.

Strohhöfer, C.

J. Kalkman, C. Strohhöfer, B. Gralak, and A. Polman, “Surface plasmon polariton modified emission of erbium in a metallodielectric grating,” Appl. Phys. Lett. 83(1), 30–32 (2003).
[CrossRef]

Thomas, M.

M. Thomas, J.-J. Greffet, R. Carminati, and J. R. Arias-Gonzalez, “Single-molecule spontaneous emission close to absorbing nanostructures,” Appl. Phys. Lett. 85(17), 3863–3865 (2004).
[CrossRef]

Verhagen, E.

Wang, D.-S.

Wang, Y.

Y. Wang and Z. P. Zhou, “Strong enhancement of erbium ion emission by a metallic double grating,” Appl. Phys. Lett. 89(25), 253122 (2006).
[CrossRef]

Woo, W. K.

K. T. Shimizu, W. K. Woo, B. R. Fisher, H. J. Eisler, and M. G. Bawendi, “Surface-enhanced emission from single semiconductor nanocrystals,” Phys. Rev. Lett. 89(11), 117401 (2002).
[CrossRef] [PubMed]

Wunderlich, M.

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100(20), 203002 (2008).
[CrossRef] [PubMed]

Xu, Y.-L.

Yang, X. F.

Zach, M.

C. Hagglund, M. Zach, and B. Kasemo, “Enhanced charge carrier generation in dye sensitized solar cells by nanoparticle plasmons,” Appl. Phys. Lett. 92(1), 013113 (2008).
[CrossRef]

Zhou, X. Q.

Zhou, Z. P.

Y. Wang and Z. P. Zhou, “Strong enhancement of erbium ion emission by a metallic double grating,” Appl. Phys. Lett. 89(25), 253122 (2006).
[CrossRef]

Anal. Chem. (1)

M. H. Chowdhury, K. Ray, S. K. Gray, J. Pond, and J. R. Lakowicz, “Aluminum nanoparticles as substrates for metal-enhanced fluorescence in the ultraviolet for the label-free detection of biomolecules,” Anal. Chem. 81(4), 1397–1403 (2009).
[CrossRef] [PubMed]

Appl. Opt. (2)

Appl. Phys. Lett. (4)

C. Hagglund, M. Zach, and B. Kasemo, “Enhanced charge carrier generation in dye sensitized solar cells by nanoparticle plasmons,” Appl. Phys. Lett. 92(1), 013113 (2008).
[CrossRef]

M. Thomas, J.-J. Greffet, R. Carminati, and J. R. Arias-Gonzalez, “Single-molecule spontaneous emission close to absorbing nanostructures,” Appl. Phys. Lett. 85(17), 3863–3865 (2004).
[CrossRef]

J. Kalkman, C. Strohhöfer, B. Gralak, and A. Polman, “Surface plasmon polariton modified emission of erbium in a metallodielectric grating,” Appl. Phys. Lett. 83(1), 30–32 (2003).
[CrossRef]

Y. Wang and Z. P. Zhou, “Strong enhancement of erbium ion emission by a metallic double grating,” Appl. Phys. Lett. 89(25), 253122 (2006).
[CrossRef]

J. Appl. Phys. (1)

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Figures (5)

Fig. 1
Fig. 1

Configuration of the theoretical model. The sphere represents the metal particle, and the dipole represents the fluorescent emitter.

Fig. 2
Fig. 2

Schematic diagram of a two-level system.

Fig. 3
Fig. 3

(a) Excitation enhancement factor of the dipole in the vicinity of the MNP with a radius of 40nm. (b) Enhancement factor of quantum efficiency. (c) Fluorescence enhancement factor.

Fig. 4
Fig. 4

Schematic diagram of energy levels for erbium ions.

Fig. 5
Fig. 5

The red dashed curve and the green solid curve in (a) (c) (e) denote the infrared and green fluorescence enhancement factors of erbium ions in the vicinity of MNP with radius 50nm, 100nm and 150nm, respectively. The blue solid curve in (b) (d) (f) is the relative enhancement of the 537nm upconversion fluorescence with respect to the 1533nm infrared fluorescence of erbium ions near MNP with radius 50nm, 100nm and 150nm, separately.

Equations (35)

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h e x ( ω ) = | p E ( r ) | 2 | p E i ( r ) | 2
E ( r ) = E i ( r ) + E s ( r ) .
h e x ( ω ) = | E ( r ) | 2 | E i ( r ) | 2
P t o t a l = P r + P n r 0 + P a b s .
γ 0 = P 0 ω .
γ t o t a l 0 = γ 0 η ,
γ n r 0 = P n r 0 ω = 1 η η γ 0 .
γ r = P r ω ,
γ t o t a l = P t o t a l ω .
h d e c a y = γ t o t a l γ t o t a l 0 = η P t o t a l P 0 .
γ E T = P a b s ω .
η = γ r γ t o t a l = P r P t o t a l .
h q = η η = 1 η P r P t o t a l .
d N 0 d t = N 0 R + N 1 W , d N 1 d t = N 0 R N 1 W , N 0 + N 1 = N .
N 0 = N W W + R , N 1 = N R W + R .
Γ f l u o = N 1 W η .
N 1 / N 0 = exp ( E 1 E 0 k B T ) = exp ( ω k B T ) ,
N 1 << N 0 .
R << W ,
N 1 = N R W ,
Γ f l u o = N R η .
h f l u o = h e x h q .
d N 0 d t = R 01 N 0 + W 1 N 1 + W 2 N 2 + W 3 N 3 , d N 1 d t = R 01 N 0 ( W 1 + R 12 ) N 1 , d N 2 d t = α 2 R 12 N 1 W 2 N 2 , d N 3 d t = α 3 R 12 N 1 W 3 N 3 , N 0 + N 1 + N 2 + N 3 = N ,
N 1 = N W 1 + R 12 R 01 + 1 + α 2 R 12 W 2 + α 3 R 12 W 3 , N 2 = α 2 R 12 W 2 N 1 , N 3 = α 3 R 12 W 3 N 1 .
Γ f l u o i = N i W i η i ,
Γ f l u o i Γ f l u o 1 = N i W i η i N 1 W 1 η 1 .
Γ f l u o i Γ f l u o 1 = α i R 12 η i W 1 η 1 .
h f l u o i h f l u o 1 = h e x h q i h d e c a y 1 h q 1 .
N i / N 1 = exp ( E i E 1 k B T ) , N i / N 1 W i / W 1 = W i / W 1 exp ( E i E 1 k B T ) .
N i / N 1 << 1 , N i / N 1 W i / W 1 << 1.
R 12 / W i << 1 , R 12 / W 1 << 1.
R 01 R 12 .
N 1 = N R 01 W 1 , N i = α i N R 01 R 12 W 1 W i . ( i = 2 , 3 )
Γ f l u o 1 = N R 01 η 1 , Γ f l u o i = α i N R 01 R 12 η i / W 1 .
h f l u o 1 = h e x h q 1 , h f l u o i = h e x 2 h q i h d e c a y 1 .

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