Abstract

An interesting fluorescence intensity reverse photonic phenomenon between red and green fluorescence is investigated. The dynamic range ∑ of intensity reverse between red and green fluorescence of Er(0.5)Yb(3):FOV oxyfluoride nanophase vitroceramics, when excited by 378.5nm and 522.5nm light respectively, is about 4.32×102. It is calculated that the phonon-assistant energy transfer rate of the electric multi-dipole interaction of {4G11/2(Er3+)→4F9/2(Er3+), 2F7/2(Yb3+)→2F5/2(Yb3+)} energy transfer of Er(0.5)Yb(3):FOV is around 1.380×108s-1, which is much larger than the relative multiphonon nonradiative relaxation rates 3.20×105s-1. That energy transfer rate for general material with same rare earth ion’s concentration is about 1.194×105s-1. These are the reason to emerge the unusual intensity reverse phenomenon in Er(0.5)Yb(3):FOV.

© 2007 Optical Society of America

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2007 (2)

2006 (1)

2005 (4)

F. Bao, Y. S. Wang, and Z. J. Hu, "Influence of Er3+ doping on microstructure of oxyfluoride glass-ceramics," Mater. Res. Bull. 40, 1645-1653 (2005).
[CrossRef]

U. H. Kynast, M. M. Lezhnina, and H. Katker, "Potential of nano-sized rare earth fluorides in optical applications," Solid State Phenomena 106, 93-102 (2005).
[CrossRef]

E. De la Rosa, P. Salas, L. A. Diaz-Torres, A. Martinez, and C. Angeles, "Strong visible cooperative up-conversion emission in ZrO2 : Yb3+ nanocrystals," J. Nanosci. Nanotechnol. 5, 1480-1486 (2005).
[CrossRef] [PubMed]

S. L. Zhao and Z. Xu, "Donor concentration dependence of upconversion luminescence in YLiF4:Er3+Yb3+, " Spectrosc. Spect. Anal. 25, 1933-1937 (2005).

2003 (2)

M. Beggiora, I. M. Reaney, and M. S. Islam, "Structure of the nanocrystals in oxyfluoride glass ceramics," Appl. Phys. Lett. 83, 467-469 (2003).
[CrossRef]

S. Hinjosa, M. A. Meneses-Nava, O. Barbosa-Garcia, L. A. Diaz-Torres, M. A. Santoyo, and J. F. Mosino, "Energy back transfer, migration and energy transfer (Yb-to-Er and Er-to-Yb) processes in Yb,Er : YAG, " J. Lumin. 102, 694-698(2003).
[CrossRef]

2000 (3)

X. B. Chen, Y. L. Liu, and N. Sawanobori, "Raman and x-radiate diffraction study about material property of oxyfluoride vitroceramics and glass," Proc. SPIE 4221, 83-87 (2000).
[CrossRef]

Y. Z. Shen, C. S. Friend, Y. Jiang, D. Jakubczyk, J. Swiatkiewicz, and P. N. Prasad, "Nanophotonics: Interactions, materials, and applications," J. Phys. Chem. B 104, 7577-7587 (2000).
[CrossRef]

M. Tsuda, K. Soga, H. Inoue, and A. Makishima, "Comparison between the calculated and measured energy transfer rates of the 4S3/2 state in Er3+-doped fluorozirconate glasses," J. Appl. Phys. 88, 1900-1906 (2000).
[CrossRef]

1993 (1)

Y. H. Wang, and J. Ohwaki, "New transparent vitroceramics codoped with Er3 + and Yb3 + for efficient frequency upconversion," Appl. Phys. Lett. 63, 3268-3270 (1993).
[CrossRef]

1970 (2)

T. Miyakawa and D. L. Dexter, "Phonon Sidebands, Multiphonon Relaxation of Excited States, and Phonon-Assisted Energy Transfer between Ions in Solids," Phys. Rev. B 1, 2961-2969 (1970).
[CrossRef]

T. Miyakawa and D. L. Dexter, "Cooperative and Stepwise Excitation of Luminescence: Trivalent Rare-Earth Ions in Yb3+-Sensitized Crystals," Phys. Rev. B 1, 70-80 (1970).
[CrossRef]

1968 (1)

E. Cohen, L. A. Riseberg, and H. W. Moos, "Effective Density of Phonon States for NdCl3 from Vibronic Spectra and Applications to Ion-Lattice Interactions," Phys. Rev. 175,521-525 (1968).
[CrossRef]

1967 (1)

L. A. Riseberg, W. B. Gandrud, and H. W. Moos, "Multiphonon Relaxation of Near-Infrared Excited States of LaCl3:Dy3+," Phys. Rev. 159, 262-266 (1967).
[CrossRef]

1965 (1)

M. Inokuti and F. Hirayama, "Influence of Energy Transfer by the Exchange Mechanism on Donor Luminescence," J. Chem. Phys. 43,1978-1989 (1965).
[CrossRef]

1962 (2)

B. R. Judd, "Optical absorption intensities of rare-earth ions," Phys. Rev. 127,750-761 (1962).
[CrossRef]

G. S. Ofelt, "Intensities of crystal spectra of rare-earth ions," J. Chem. Phys. 37,511-520 (1962).
[CrossRef]

1960 (1)

R. P. Feynman, "There's plenty of room at the bottom," Engineering and Science 23, 22-23 (1960).

1953 (1)

D. L. Dexter, "A Theory of Sensitized Luminescence in Solids," J. Chem. Phys. 21, 836-850 (1953).
[CrossRef]

1931 (1)

J. Frenkel, "On the Transformation of light into Heat in Solids," Phys. Rev. 37, 17-44(1931).
[CrossRef]

Ahn, I. H.

Angeles, C.

E. De la Rosa, P. Salas, L. A. Diaz-Torres, A. Martinez, and C. Angeles, "Strong visible cooperative up-conversion emission in ZrO2 : Yb3+ nanocrystals," J. Nanosci. Nanotechnol. 5, 1480-1486 (2005).
[CrossRef] [PubMed]

Ballato, J.

Bao, F.

F. Bao, Y. S. Wang, and Z. J. Hu, "Influence of Er3+ doping on microstructure of oxyfluoride glass-ceramics," Mater. Res. Bull. 40, 1645-1653 (2005).
[CrossRef]

Barbosa-Garcia, O.

S. Hinjosa, M. A. Meneses-Nava, O. Barbosa-Garcia, L. A. Diaz-Torres, M. A. Santoyo, and J. F. Mosino, "Energy back transfer, migration and energy transfer (Yb-to-Er and Er-to-Yb) processes in Yb,Er : YAG, " J. Lumin. 102, 694-698(2003).
[CrossRef]

Beggiora, M.

M. Beggiora, I. M. Reaney, and M. S. Islam, "Structure of the nanocrystals in oxyfluoride glass ceramics," Appl. Phys. Lett. 83, 467-469 (2003).
[CrossRef]

Chen, H.

Chen, X. B.

X. B. Chen, Y. L. Liu, and N. Sawanobori, "Raman and x-radiate diffraction study about material property of oxyfluoride vitroceramics and glass," Proc. SPIE 4221, 83-87 (2000).
[CrossRef]

Cohen, E.

E. Cohen, L. A. Riseberg, and H. W. Moos, "Effective Density of Phonon States for NdCl3 from Vibronic Spectra and Applications to Ion-Lattice Interactions," Phys. Rev. 175,521-525 (1968).
[CrossRef]

De la Rosa, E.

E. De la Rosa, P. Salas, L. A. Diaz-Torres, A. Martinez, and C. Angeles, "Strong visible cooperative up-conversion emission in ZrO2 : Yb3+ nanocrystals," J. Nanosci. Nanotechnol. 5, 1480-1486 (2005).
[CrossRef] [PubMed]

Dexter, D. L.

T. Miyakawa and D. L. Dexter, "Phonon Sidebands, Multiphonon Relaxation of Excited States, and Phonon-Assisted Energy Transfer between Ions in Solids," Phys. Rev. B 1, 2961-2969 (1970).
[CrossRef]

T. Miyakawa and D. L. Dexter, "Cooperative and Stepwise Excitation of Luminescence: Trivalent Rare-Earth Ions in Yb3+-Sensitized Crystals," Phys. Rev. B 1, 70-80 (1970).
[CrossRef]

D. L. Dexter, "A Theory of Sensitized Luminescence in Solids," J. Chem. Phys. 21, 836-850 (1953).
[CrossRef]

Diaz-Torres, L. A.

E. De la Rosa, P. Salas, L. A. Diaz-Torres, A. Martinez, and C. Angeles, "Strong visible cooperative up-conversion emission in ZrO2 : Yb3+ nanocrystals," J. Nanosci. Nanotechnol. 5, 1480-1486 (2005).
[CrossRef] [PubMed]

S. Hinjosa, M. A. Meneses-Nava, O. Barbosa-Garcia, L. A. Diaz-Torres, M. A. Santoyo, and J. F. Mosino, "Energy back transfer, migration and energy transfer (Yb-to-Er and Er-to-Yb) processes in Yb,Er : YAG, " J. Lumin. 102, 694-698(2003).
[CrossRef]

DiMaio, J. R.

Feynman, R. P.

R. P. Feynman, "There's plenty of room at the bottom," Engineering and Science 23, 22-23 (1960).

Frenkel, J.

J. Frenkel, "On the Transformation of light into Heat in Solids," Phys. Rev. 37, 17-44(1931).
[CrossRef]

Friend, C. S.

Y. Z. Shen, C. S. Friend, Y. Jiang, D. Jakubczyk, J. Swiatkiewicz, and P. N. Prasad, "Nanophotonics: Interactions, materials, and applications," J. Phys. Chem. B 104, 7577-7587 (2000).
[CrossRef]

Gandrud, W. B.

L. A. Riseberg, W. B. Gandrud, and H. W. Moos, "Multiphonon Relaxation of Near-Infrared Excited States of LaCl3:Dy3+," Phys. Rev. 159, 262-266 (1967).
[CrossRef]

Gao, J. S.

Han, W. T.

Hinjosa, S.

S. Hinjosa, M. A. Meneses-Nava, O. Barbosa-Garcia, L. A. Diaz-Torres, M. A. Santoyo, and J. F. Mosino, "Energy back transfer, migration and energy transfer (Yb-to-Er and Er-to-Yb) processes in Yb,Er : YAG, " J. Lumin. 102, 694-698(2003).
[CrossRef]

Hirayama, F.

M. Inokuti and F. Hirayama, "Influence of Energy Transfer by the Exchange Mechanism on Donor Luminescence," J. Chem. Phys. 43,1978-1989 (1965).
[CrossRef]

Hu, Z. J.

F. Bao, Y. S. Wang, and Z. J. Hu, "Influence of Er3+ doping on microstructure of oxyfluoride glass-ceramics," Mater. Res. Bull. 40, 1645-1653 (2005).
[CrossRef]

Inokuti, M.

M. Inokuti and F. Hirayama, "Influence of Energy Transfer by the Exchange Mechanism on Donor Luminescence," J. Chem. Phys. 43,1978-1989 (1965).
[CrossRef]

Inoue, H.

M. Tsuda, K. Soga, H. Inoue, and A. Makishima, "Comparison between the calculated and measured energy transfer rates of the 4S3/2 state in Er3+-doped fluorozirconate glasses," J. Appl. Phys. 88, 1900-1906 (2000).
[CrossRef]

Islam, M. S.

M. Beggiora, I. M. Reaney, and M. S. Islam, "Structure of the nanocrystals in oxyfluoride glass ceramics," Appl. Phys. Lett. 83, 467-469 (2003).
[CrossRef]

Jakubczyk, D.

Y. Z. Shen, C. S. Friend, Y. Jiang, D. Jakubczyk, J. Swiatkiewicz, and P. N. Prasad, "Nanophotonics: Interactions, materials, and applications," J. Phys. Chem. B 104, 7577-7587 (2000).
[CrossRef]

Jiang, Y.

Y. Z. Shen, C. S. Friend, Y. Jiang, D. Jakubczyk, J. Swiatkiewicz, and P. N. Prasad, "Nanophotonics: Interactions, materials, and applications," J. Phys. Chem. B 104, 7577-7587 (2000).
[CrossRef]

Judd, B. R.

B. R. Judd, "Optical absorption intensities of rare-earth ions," Phys. Rev. 127,750-761 (1962).
[CrossRef]

Katker, H.

U. H. Kynast, M. M. Lezhnina, and H. Katker, "Potential of nano-sized rare earth fluorides in optical applications," Solid State Phenomena 106, 93-102 (2005).
[CrossRef]

Kokuoz, B.

Kynast, U. H.

U. H. Kynast, M. M. Lezhnina, and H. Katker, "Potential of nano-sized rare earth fluorides in optical applications," Solid State Phenomena 106, 93-102 (2005).
[CrossRef]

Lezhnina, M. M.

U. H. Kynast, M. M. Lezhnina, and H. Katker, "Potential of nano-sized rare earth fluorides in optical applications," Solid State Phenomena 106, 93-102 (2005).
[CrossRef]

Li, C. R.

Lin, A.

Liu, Y. L.

X. B. Chen, Y. L. Liu, and N. Sawanobori, "Raman and x-radiate diffraction study about material property of oxyfluoride vitroceramics and glass," Proc. SPIE 4221, 83-87 (2000).
[CrossRef]

Makishima, A.

M. Tsuda, K. Soga, H. Inoue, and A. Makishima, "Comparison between the calculated and measured energy transfer rates of the 4S3/2 state in Er3+-doped fluorozirconate glasses," J. Appl. Phys. 88, 1900-1906 (2000).
[CrossRef]

Martinez, A.

E. De la Rosa, P. Salas, L. A. Diaz-Torres, A. Martinez, and C. Angeles, "Strong visible cooperative up-conversion emission in ZrO2 : Yb3+ nanocrystals," J. Nanosci. Nanotechnol. 5, 1480-1486 (2005).
[CrossRef] [PubMed]

Meneses-Nava, M. A.

S. Hinjosa, M. A. Meneses-Nava, O. Barbosa-Garcia, L. A. Diaz-Torres, M. A. Santoyo, and J. F. Mosino, "Energy back transfer, migration and energy transfer (Yb-to-Er and Er-to-Yb) processes in Yb,Er : YAG, " J. Lumin. 102, 694-698(2003).
[CrossRef]

Miyakawa, T.

T. Miyakawa and D. L. Dexter, "Phonon Sidebands, Multiphonon Relaxation of Excited States, and Phonon-Assisted Energy Transfer between Ions in Solids," Phys. Rev. B 1, 2961-2969 (1970).
[CrossRef]

T. Miyakawa and D. L. Dexter, "Cooperative and Stepwise Excitation of Luminescence: Trivalent Rare-Earth Ions in Yb3+-Sensitized Crystals," Phys. Rev. B 1, 70-80 (1970).
[CrossRef]

Moos, H. W.

E. Cohen, L. A. Riseberg, and H. W. Moos, "Effective Density of Phonon States for NdCl3 from Vibronic Spectra and Applications to Ion-Lattice Interactions," Phys. Rev. 175,521-525 (1968).
[CrossRef]

L. A. Riseberg, W. B. Gandrud, and H. W. Moos, "Multiphonon Relaxation of Near-Infrared Excited States of LaCl3:Dy3+," Phys. Rev. 159, 262-266 (1967).
[CrossRef]

Mosino, J. F.

S. Hinjosa, M. A. Meneses-Nava, O. Barbosa-Garcia, L. A. Diaz-Torres, M. A. Santoyo, and J. F. Mosino, "Energy back transfer, migration and energy transfer (Yb-to-Er and Er-to-Yb) processes in Yb,Er : YAG, " J. Lumin. 102, 694-698(2003).
[CrossRef]

Ofelt, G. S.

G. S. Ofelt, "Intensities of crystal spectra of rare-earth ions," J. Chem. Phys. 37,511-520 (1962).
[CrossRef]

Ohwaki, J.

Y. H. Wang, and J. Ohwaki, "New transparent vitroceramics codoped with Er3 + and Yb3 + for efficient frequency upconversion," Appl. Phys. Lett. 63, 3268-3270 (1993).
[CrossRef]

Prasad, P. N.

Y. Z. Shen, C. S. Friend, Y. Jiang, D. Jakubczyk, J. Swiatkiewicz, and P. N. Prasad, "Nanophotonics: Interactions, materials, and applications," J. Phys. Chem. B 104, 7577-7587 (2000).
[CrossRef]

Reaney, I. M.

M. Beggiora, I. M. Reaney, and M. S. Islam, "Structure of the nanocrystals in oxyfluoride glass ceramics," Appl. Phys. Lett. 83, 467-469 (2003).
[CrossRef]

Riseberg, L. A.

E. Cohen, L. A. Riseberg, and H. W. Moos, "Effective Density of Phonon States for NdCl3 from Vibronic Spectra and Applications to Ion-Lattice Interactions," Phys. Rev. 175,521-525 (1968).
[CrossRef]

L. A. Riseberg, W. B. Gandrud, and H. W. Moos, "Multiphonon Relaxation of Near-Infrared Excited States of LaCl3:Dy3+," Phys. Rev. 159, 262-266 (1967).
[CrossRef]

Salas, P.

E. De la Rosa, P. Salas, L. A. Diaz-Torres, A. Martinez, and C. Angeles, "Strong visible cooperative up-conversion emission in ZrO2 : Yb3+ nanocrystals," J. Nanosci. Nanotechnol. 5, 1480-1486 (2005).
[CrossRef] [PubMed]

Santoyo, M. A.

S. Hinjosa, M. A. Meneses-Nava, O. Barbosa-Garcia, L. A. Diaz-Torres, M. A. Santoyo, and J. F. Mosino, "Energy back transfer, migration and energy transfer (Yb-to-Er and Er-to-Yb) processes in Yb,Er : YAG, " J. Lumin. 102, 694-698(2003).
[CrossRef]

Sawanobori, N.

X. B. Chen, Y. L. Liu, and N. Sawanobori, "Raman and x-radiate diffraction study about material property of oxyfluoride vitroceramics and glass," Proc. SPIE 4221, 83-87 (2000).
[CrossRef]

Shen, Y. Z.

Y. Z. Shen, C. S. Friend, Y. Jiang, D. Jakubczyk, J. Swiatkiewicz, and P. N. Prasad, "Nanophotonics: Interactions, materials, and applications," J. Phys. Chem. B 104, 7577-7587 (2000).
[CrossRef]

Soga, K.

M. Tsuda, K. Soga, H. Inoue, and A. Makishima, "Comparison between the calculated and measured energy transfer rates of the 4S3/2 state in Er3+-doped fluorozirconate glasses," J. Appl. Phys. 88, 1900-1906 (2000).
[CrossRef]

Son, D. H.

Song, C. L.

Song, G. H.

Song, Q.

Swiatkiewicz, J.

Y. Z. Shen, C. S. Friend, Y. Jiang, D. Jakubczyk, J. Swiatkiewicz, and P. N. Prasad, "Nanophotonics: Interactions, materials, and applications," J. Phys. Chem. B 104, 7577-7587 (2000).
[CrossRef]

Tsuda, M.

M. Tsuda, K. Soga, H. Inoue, and A. Makishima, "Comparison between the calculated and measured energy transfer rates of the 4S3/2 state in Er3+-doped fluorozirconate glasses," J. Appl. Phys. 88, 1900-1906 (2000).
[CrossRef]

Wang, T. T.

Wang, X. Y.

Wang, Y. H.

Y. H. Wang, and J. Ohwaki, "New transparent vitroceramics codoped with Er3 + and Yb3 + for efficient frequency upconversion," Appl. Phys. Lett. 63, 3268-3270 (1993).
[CrossRef]

Wang, Y. S.

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Figures (6)

Fig. 1.
Fig. 1.

The excitation spectra of 667.1nm and 543.7nm fluorescence of (a) Er(0.5)Yb(3):FOV and (b) Er(0.5):FOV oxyfluoride nanophase vitroceramics.

Fig. 2.
Fig. 2.

The emission spectra of 378.5nm and 522.5nm absorption energy levels of (a) Er(0.5)Yb(3):FOV and (b) Er(0.5):FOV oxyfluoride nanophase vitroceramics.

Fig. 3.
Fig. 3.

The schematic diagram of cross-energy-transfer processes in Yb3+ and Er3+ co-doped systems. The solid line is absorption or energy transfer. The dashed straight line is fluorescence.

Fig. 4.
Fig. 4.

The variation of the phonon-assistant energy transfer rate WPET of the dipole-dipole, dipole-quadruple and quadruple-quadruple interactions dependent on the distance R for (a) {4G11/2(Er3+)→4F9/2(Er3+), 2F7/2(Yb3+)→2F5/2(Yb3+)} and (b) {(2G4F2H)9/2(Er3+)→4F9/2(Er3+), 2F7/2(Yb3+)→2F5/2(Yb3+)} energy transfer.

Fig. 5.
Fig. 5.

The variation of the distance R between Er3+ and Yb3+ ions dependent on the mole concentration of Yb3+ ions when the mole concentration of Er3+ ion is fixed at 0.5%. The solid line, dash line and dotted line represent that their crystal volume fraction of nanocrystal phase to whole material is 10%, 1% and 30%.

Fig. 6.
Fig. 6.

The typical variation of the phonon-assistant energy transfer rate WPET of the electric multi-dipole interaction of {4G11/2(Er3+)→4F9/2(Er3+), 2F7/2(Yb3+)→2F5/2(Yb3+)}(Solid line) and {(2GF2H)9/2(Er3+)→4F9/2(Er3+), 2F7/2(Yb3+)→2F5/2(Yb3+)}(Dash line) energy transfers dependent on mole concentration of Yb3+ ions when the crystal volume fraction equals to (a) 10% and (b) 100%, meanwhile the mole concentration of Er3+ ion is fixed at 0.5%.

Tables (1)

Tables Icon

Table 1. Relative fluorescence intensity F of Stokes emission spectra, the common intensity ratio α=[F(4S3/2)/F(4F9/2)](2H11/2) between green and red fluorescence when the 2H11/2 level is excited, the unusual intensity reverse ratio γ=[F(4F9/2)/F(4S3/2)](4G11/2) between red and green fluorescence when the 4G11/2 level is excited, and the dynamic range ∑=γ×α of fluorescence intensity reverse between red and green fluorescence. The samples number N represent the used samples (1) Er(0.5)Yb(3):FOV, (2) Er(0.5)Yb(1):FOV, (3) Er(0.5):FOV, (4) Er(0.5)Yb(3):FOG, (5) Er(0.5)Yb(1):FOG, (6) Er(0.5):FOG, (7) Er(0.3)Yb(3):ZBLAN and (8) Er(0.3):ZBLAN respectively.

Equations (22)

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W E T ( d d ) = 2 3 1 g i g k π 2 e 4 h 2 c χ R 6 t = 2,4,6 Ω t Ψ i U ( t ) Ψ J 2 × t = 2,4,6 Ω t Ψ k U ( t ) Ψ e 2 S R 6 ρ 2
W E T ( d q ) = 11.4 1 g i g k π 2 e 4 h 2 c χ R 8 t = 2,4,6 Ω t Ψ i U ( t ) Ψ J 2 × 4 9 [ 4 f C ( 2 ) 4 f r Ψ k U ( 2 ) Ψ e ] 2 S R 8 ρ 8 3
W E T ( q q ) = 48.7 1 g i g k π 2 e 4 h 2 c χ R 10 4 9 [ 4 f C ( 2 ) 4 f r Ψ i U ( 2 ) Ψ J ] 2
× 4 9 [ 4 f C ( 2 ) 4 f r Ψ k U ( 2 ) Ψ e ] 2 S R 10 ρ 10 3
χ = ( n 2 + 2 3 n ) 4
f c 2 f = ( 28 15 ) 1 / 2
S = Γ Γ 2 + Δ υ 2
Ψ l r Ψ k 2 = Ω λ Ψ l U ( λ ) Ψ k 2
Ψ l r t r t Ψ k 2 = 4 9 [ f c 2 f r 2 Ψ l U ( 2 ) Ψ k ] 2
S ed ( aJ , bJ′ ) = e 2 t = 2,4,6 Ω t Ψ U ( t ) Ψ′ 2
A E D = 64 π 4 v 3 e 2 3 h ( 2 J + 1 ) c 3 × n ( n 2 + 2 ) 2 9 t = 2,4,6 Ω t Ψ U t Ψ′ 2
S m d ( Ψ J , Ψ′ J ) = e 2 4 m 2 c 2 f N Ψ J L + 2 S f N Ψ′ J 2
A M D ( Ψ J , Ψ′ J ) = 64 π 4 v ̄ 3 3 h c 3 ( 2 J + 1 ) n 3 S m d ( aj , bj′ )
f ( aj , bj′ ) = m c 2 N π e 2 O D ( λ ) × 2.303 d × λ 2 d λ
f ( aj , bj′ ) = 8 π 2 m v 3 h n 2 e 2 ( 2 J + 1 ) [ χ e d S e d ( aj , bj′ ) + χ m d S m d ( aj , bj′ ) ]
τ = 1 A + W
W p = W 0 e α Δ E
W PET = W E T e β Δ E
β = α γ
γ = 1 ħ ω ln ( 1 + g A g D )
R = 0.62 ( N E r + N Y b ) ( 1 / 3 )
N r e = ρ M 1 N a y

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