Abstract

The photostability of dye molecules trapped in transparent solid matrices synthesized by the solgel technique was studied both experimentally and theoretically using a model with numerical and approximate analytical solutions. The model is based on a one-photon photodestruction process with the creation of an absorbing bleached molecule. We give the number of photons that different trapped dye molecules can absorb on average before they are bleached. Dyes such as Perylene Red, Perylene Orange, Pyrromethenes 567 and 597, Rhodamines 6G and B, DCM, a Xanthylium salt, and Neon Red were investigated; significant differences were observed. Some dye molecules in solvents were also studied; increased stability resulted when the molecules were trapped in solid matrices.

© 1996 Optical Society of America

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References

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  1. M. Canva, A. Dubois, P. Georges, A. Brun, F. Chaput, A. Ranger, J. P. Boilot, “Perylene, Pyrromethene and grafted Rhodamine doped xerogels for tunable solid state laser,” in Sol-Gel Optics III, J. D. Mackenzie, ed., Proc. SPIE2288, 298–309 (1994).
    [CrossRef]
  2. M. Canva, A. Dubois, P. Georges, A. Brun, F. Chaput, J. P. Boilot, “Dye doped xerogels for tunable lasers,” Adv. Solid-State Lasers 20, 291–295 (1994).
  3. B. Dunn, F. Nishida, R. Toda, J. J. Zink, T. H. Allik, S. Chandra, J. A. Hutchinson, “Advances in dye-doped sol-gel lasers,” Mater. Res. Soc. Symp. Proc. 329, 267–277 (1994).
  4. M. D. Rahn, T. A. King, “Solid state dye doped sol-gel glass composite lasers,” in Conference on Lasers and Electro-Optics, Vol. 8 of OSA 1994 Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 389–390.
  5. R. E. Hermes, T. H. Allik, S. Chandra, J. A. Hutchinson, “High-efficiency pyrromethene doped solid-state dye lasers,” Appl. Phys. Lett. 63, 877–879 (1993).
    [CrossRef]
  6. M. D. Rahn, T. A. King, “Lasers based on doped sol-gel composite glasses,” in Sol-Gel Optics III, J. D. Mackenzie, ed., Proc. SPIE2288, 382–391 (1994).
    [CrossRef]
  7. D. Avnir, D. Lévy, R. Reisfeld, “The nature of the silica cage as reflected by spectral changes and enhanced photostability of trapped Rhodamine 6G,” J. Phys. Chem. 88, 5956–5959 (1984).
    [CrossRef]
  8. E. T. Knobbe, B. Dunn, P. D. Fuqua, F. Nishida, “Laser behavior and photostability characteristics of organic dye doped silicate gel materials,” Appl. Opt. 29, 2729–2733 (1990).
    [CrossRef] [PubMed]
  9. I. P. Kaminow, L. W. Stulz, E. A. Chandross, C. A. Pryde, “Photobleaching of organic laser dyes in solid matrices,” Appl. Opt. 11, 1563–1567 (1972).
    [CrossRef] [PubMed]
  10. J. C. Newell, L. Solymar, A. A. Ward, “Holograms in dichromated gelatin: real-time effects,” Appl. Opt. 24, 4460–4466 (1985).
    [CrossRef] [PubMed]
  11. W. J. Tomlinson, E. A. Chandross, R. L. Fork, C. A. Pryde, A. A. Lamola, “Reversible photodimerization: a new type of photochromism,” Appl. Opt. 11, 533–548 (1972).
    [CrossRef] [PubMed]
  12. N. Capolla, R. A. Lessard, “Real time bleaching of methylene blue or thionine sensitized gelatin,” Appl. Opt. 30, 1196–1200 (1991).
    [CrossRef] [PubMed]
  13. A. N. Fletcher, “Laser dye stability. Part 4,” Appl. Phys. 16, 93–97 (1978).
    [CrossRef]
  14. B. Liphardt, B. Liphardt, W. Luttke, “Laser dyes III: concepts to increase the photostability of laser dyes,” Opt. Commun. 48, 129–133 (1983).
    [CrossRef]
  15. M. Canva, G. Le Saux, P. Georges, A. Brun, F. Chaput, J. P. Boilot, “All-optical gel memory,” Opt. Lett. 17, 218–220 (1992).
    [CrossRef] [PubMed]
  16. D. Beer, J. Weber, “Photobleaching of organic laser dyes,” Opt. Commun. 5, 307–309 (1972).
    [CrossRef]
  17. E. P. Ippen, C. V. Shank, A. Dienes, “Rapid photobleaching of organic laser dyes in continuously operated devices,” IEEE J. Quantum Electron. QE-7, 178–179 (1971).
    [CrossRef]

1994 (2)

M. Canva, A. Dubois, P. Georges, A. Brun, F. Chaput, J. P. Boilot, “Dye doped xerogels for tunable lasers,” Adv. Solid-State Lasers 20, 291–295 (1994).

B. Dunn, F. Nishida, R. Toda, J. J. Zink, T. H. Allik, S. Chandra, J. A. Hutchinson, “Advances in dye-doped sol-gel lasers,” Mater. Res. Soc. Symp. Proc. 329, 267–277 (1994).

1993 (1)

R. E. Hermes, T. H. Allik, S. Chandra, J. A. Hutchinson, “High-efficiency pyrromethene doped solid-state dye lasers,” Appl. Phys. Lett. 63, 877–879 (1993).
[CrossRef]

1992 (1)

1991 (1)

1990 (1)

1985 (1)

1984 (1)

D. Avnir, D. Lévy, R. Reisfeld, “The nature of the silica cage as reflected by spectral changes and enhanced photostability of trapped Rhodamine 6G,” J. Phys. Chem. 88, 5956–5959 (1984).
[CrossRef]

1983 (1)

B. Liphardt, B. Liphardt, W. Luttke, “Laser dyes III: concepts to increase the photostability of laser dyes,” Opt. Commun. 48, 129–133 (1983).
[CrossRef]

1978 (1)

A. N. Fletcher, “Laser dye stability. Part 4,” Appl. Phys. 16, 93–97 (1978).
[CrossRef]

1972 (3)

1971 (1)

E. P. Ippen, C. V. Shank, A. Dienes, “Rapid photobleaching of organic laser dyes in continuously operated devices,” IEEE J. Quantum Electron. QE-7, 178–179 (1971).
[CrossRef]

Allik, T. H.

B. Dunn, F. Nishida, R. Toda, J. J. Zink, T. H. Allik, S. Chandra, J. A. Hutchinson, “Advances in dye-doped sol-gel lasers,” Mater. Res. Soc. Symp. Proc. 329, 267–277 (1994).

R. E. Hermes, T. H. Allik, S. Chandra, J. A. Hutchinson, “High-efficiency pyrromethene doped solid-state dye lasers,” Appl. Phys. Lett. 63, 877–879 (1993).
[CrossRef]

Avnir, D.

D. Avnir, D. Lévy, R. Reisfeld, “The nature of the silica cage as reflected by spectral changes and enhanced photostability of trapped Rhodamine 6G,” J. Phys. Chem. 88, 5956–5959 (1984).
[CrossRef]

Beer, D.

D. Beer, J. Weber, “Photobleaching of organic laser dyes,” Opt. Commun. 5, 307–309 (1972).
[CrossRef]

Boilot, J. P.

M. Canva, A. Dubois, P. Georges, A. Brun, F. Chaput, J. P. Boilot, “Dye doped xerogels for tunable lasers,” Adv. Solid-State Lasers 20, 291–295 (1994).

M. Canva, G. Le Saux, P. Georges, A. Brun, F. Chaput, J. P. Boilot, “All-optical gel memory,” Opt. Lett. 17, 218–220 (1992).
[CrossRef] [PubMed]

M. Canva, A. Dubois, P. Georges, A. Brun, F. Chaput, A. Ranger, J. P. Boilot, “Perylene, Pyrromethene and grafted Rhodamine doped xerogels for tunable solid state laser,” in Sol-Gel Optics III, J. D. Mackenzie, ed., Proc. SPIE2288, 298–309 (1994).
[CrossRef]

Brun, A.

M. Canva, A. Dubois, P. Georges, A. Brun, F. Chaput, J. P. Boilot, “Dye doped xerogels for tunable lasers,” Adv. Solid-State Lasers 20, 291–295 (1994).

M. Canva, G. Le Saux, P. Georges, A. Brun, F. Chaput, J. P. Boilot, “All-optical gel memory,” Opt. Lett. 17, 218–220 (1992).
[CrossRef] [PubMed]

M. Canva, A. Dubois, P. Georges, A. Brun, F. Chaput, A. Ranger, J. P. Boilot, “Perylene, Pyrromethene and grafted Rhodamine doped xerogels for tunable solid state laser,” in Sol-Gel Optics III, J. D. Mackenzie, ed., Proc. SPIE2288, 298–309 (1994).
[CrossRef]

Canva, M.

M. Canva, A. Dubois, P. Georges, A. Brun, F. Chaput, J. P. Boilot, “Dye doped xerogels for tunable lasers,” Adv. Solid-State Lasers 20, 291–295 (1994).

M. Canva, G. Le Saux, P. Georges, A. Brun, F. Chaput, J. P. Boilot, “All-optical gel memory,” Opt. Lett. 17, 218–220 (1992).
[CrossRef] [PubMed]

M. Canva, A. Dubois, P. Georges, A. Brun, F. Chaput, A. Ranger, J. P. Boilot, “Perylene, Pyrromethene and grafted Rhodamine doped xerogels for tunable solid state laser,” in Sol-Gel Optics III, J. D. Mackenzie, ed., Proc. SPIE2288, 298–309 (1994).
[CrossRef]

Capolla, N.

Chandra, S.

B. Dunn, F. Nishida, R. Toda, J. J. Zink, T. H. Allik, S. Chandra, J. A. Hutchinson, “Advances in dye-doped sol-gel lasers,” Mater. Res. Soc. Symp. Proc. 329, 267–277 (1994).

R. E. Hermes, T. H. Allik, S. Chandra, J. A. Hutchinson, “High-efficiency pyrromethene doped solid-state dye lasers,” Appl. Phys. Lett. 63, 877–879 (1993).
[CrossRef]

Chandross, E. A.

Chaput, F.

M. Canva, A. Dubois, P. Georges, A. Brun, F. Chaput, J. P. Boilot, “Dye doped xerogels for tunable lasers,” Adv. Solid-State Lasers 20, 291–295 (1994).

M. Canva, G. Le Saux, P. Georges, A. Brun, F. Chaput, J. P. Boilot, “All-optical gel memory,” Opt. Lett. 17, 218–220 (1992).
[CrossRef] [PubMed]

M. Canva, A. Dubois, P. Georges, A. Brun, F. Chaput, A. Ranger, J. P. Boilot, “Perylene, Pyrromethene and grafted Rhodamine doped xerogels for tunable solid state laser,” in Sol-Gel Optics III, J. D. Mackenzie, ed., Proc. SPIE2288, 298–309 (1994).
[CrossRef]

Dienes, A.

E. P. Ippen, C. V. Shank, A. Dienes, “Rapid photobleaching of organic laser dyes in continuously operated devices,” IEEE J. Quantum Electron. QE-7, 178–179 (1971).
[CrossRef]

Dubois, A.

M. Canva, A. Dubois, P. Georges, A. Brun, F. Chaput, J. P. Boilot, “Dye doped xerogels for tunable lasers,” Adv. Solid-State Lasers 20, 291–295 (1994).

M. Canva, A. Dubois, P. Georges, A. Brun, F. Chaput, A. Ranger, J. P. Boilot, “Perylene, Pyrromethene and grafted Rhodamine doped xerogels for tunable solid state laser,” in Sol-Gel Optics III, J. D. Mackenzie, ed., Proc. SPIE2288, 298–309 (1994).
[CrossRef]

Dunn, B.

B. Dunn, F. Nishida, R. Toda, J. J. Zink, T. H. Allik, S. Chandra, J. A. Hutchinson, “Advances in dye-doped sol-gel lasers,” Mater. Res. Soc. Symp. Proc. 329, 267–277 (1994).

E. T. Knobbe, B. Dunn, P. D. Fuqua, F. Nishida, “Laser behavior and photostability characteristics of organic dye doped silicate gel materials,” Appl. Opt. 29, 2729–2733 (1990).
[CrossRef] [PubMed]

Fletcher, A. N.

A. N. Fletcher, “Laser dye stability. Part 4,” Appl. Phys. 16, 93–97 (1978).
[CrossRef]

Fork, R. L.

Fuqua, P. D.

Georges, P.

M. Canva, A. Dubois, P. Georges, A. Brun, F. Chaput, J. P. Boilot, “Dye doped xerogels for tunable lasers,” Adv. Solid-State Lasers 20, 291–295 (1994).

M. Canva, G. Le Saux, P. Georges, A. Brun, F. Chaput, J. P. Boilot, “All-optical gel memory,” Opt. Lett. 17, 218–220 (1992).
[CrossRef] [PubMed]

M. Canva, A. Dubois, P. Georges, A. Brun, F. Chaput, A. Ranger, J. P. Boilot, “Perylene, Pyrromethene and grafted Rhodamine doped xerogels for tunable solid state laser,” in Sol-Gel Optics III, J. D. Mackenzie, ed., Proc. SPIE2288, 298–309 (1994).
[CrossRef]

Hermes, R. E.

R. E. Hermes, T. H. Allik, S. Chandra, J. A. Hutchinson, “High-efficiency pyrromethene doped solid-state dye lasers,” Appl. Phys. Lett. 63, 877–879 (1993).
[CrossRef]

Hutchinson, J. A.

B. Dunn, F. Nishida, R. Toda, J. J. Zink, T. H. Allik, S. Chandra, J. A. Hutchinson, “Advances in dye-doped sol-gel lasers,” Mater. Res. Soc. Symp. Proc. 329, 267–277 (1994).

R. E. Hermes, T. H. Allik, S. Chandra, J. A. Hutchinson, “High-efficiency pyrromethene doped solid-state dye lasers,” Appl. Phys. Lett. 63, 877–879 (1993).
[CrossRef]

Ippen, E. P.

E. P. Ippen, C. V. Shank, A. Dienes, “Rapid photobleaching of organic laser dyes in continuously operated devices,” IEEE J. Quantum Electron. QE-7, 178–179 (1971).
[CrossRef]

Kaminow, I. P.

King, T. A.

M. D. Rahn, T. A. King, “Lasers based on doped sol-gel composite glasses,” in Sol-Gel Optics III, J. D. Mackenzie, ed., Proc. SPIE2288, 382–391 (1994).
[CrossRef]

M. D. Rahn, T. A. King, “Solid state dye doped sol-gel glass composite lasers,” in Conference on Lasers and Electro-Optics, Vol. 8 of OSA 1994 Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 389–390.

Knobbe, E. T.

Lamola, A. A.

Le Saux, G.

Lessard, R. A.

Lévy, D.

D. Avnir, D. Lévy, R. Reisfeld, “The nature of the silica cage as reflected by spectral changes and enhanced photostability of trapped Rhodamine 6G,” J. Phys. Chem. 88, 5956–5959 (1984).
[CrossRef]

Liphardt, B.

B. Liphardt, B. Liphardt, W. Luttke, “Laser dyes III: concepts to increase the photostability of laser dyes,” Opt. Commun. 48, 129–133 (1983).
[CrossRef]

B. Liphardt, B. Liphardt, W. Luttke, “Laser dyes III: concepts to increase the photostability of laser dyes,” Opt. Commun. 48, 129–133 (1983).
[CrossRef]

Luttke, W.

B. Liphardt, B. Liphardt, W. Luttke, “Laser dyes III: concepts to increase the photostability of laser dyes,” Opt. Commun. 48, 129–133 (1983).
[CrossRef]

Newell, J. C.

Nishida, F.

B. Dunn, F. Nishida, R. Toda, J. J. Zink, T. H. Allik, S. Chandra, J. A. Hutchinson, “Advances in dye-doped sol-gel lasers,” Mater. Res. Soc. Symp. Proc. 329, 267–277 (1994).

E. T. Knobbe, B. Dunn, P. D. Fuqua, F. Nishida, “Laser behavior and photostability characteristics of organic dye doped silicate gel materials,” Appl. Opt. 29, 2729–2733 (1990).
[CrossRef] [PubMed]

Pryde, C. A.

Rahn, M. D.

M. D. Rahn, T. A. King, “Solid state dye doped sol-gel glass composite lasers,” in Conference on Lasers and Electro-Optics, Vol. 8 of OSA 1994 Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 389–390.

M. D. Rahn, T. A. King, “Lasers based on doped sol-gel composite glasses,” in Sol-Gel Optics III, J. D. Mackenzie, ed., Proc. SPIE2288, 382–391 (1994).
[CrossRef]

Ranger, A.

M. Canva, A. Dubois, P. Georges, A. Brun, F. Chaput, A. Ranger, J. P. Boilot, “Perylene, Pyrromethene and grafted Rhodamine doped xerogels for tunable solid state laser,” in Sol-Gel Optics III, J. D. Mackenzie, ed., Proc. SPIE2288, 298–309 (1994).
[CrossRef]

Reisfeld, R.

D. Avnir, D. Lévy, R. Reisfeld, “The nature of the silica cage as reflected by spectral changes and enhanced photostability of trapped Rhodamine 6G,” J. Phys. Chem. 88, 5956–5959 (1984).
[CrossRef]

Shank, C. V.

E. P. Ippen, C. V. Shank, A. Dienes, “Rapid photobleaching of organic laser dyes in continuously operated devices,” IEEE J. Quantum Electron. QE-7, 178–179 (1971).
[CrossRef]

Solymar, L.

Stulz, L. W.

Toda, R.

B. Dunn, F. Nishida, R. Toda, J. J. Zink, T. H. Allik, S. Chandra, J. A. Hutchinson, “Advances in dye-doped sol-gel lasers,” Mater. Res. Soc. Symp. Proc. 329, 267–277 (1994).

Tomlinson, W. J.

Ward, A. A.

Weber, J.

D. Beer, J. Weber, “Photobleaching of organic laser dyes,” Opt. Commun. 5, 307–309 (1972).
[CrossRef]

Zink, J. J.

B. Dunn, F. Nishida, R. Toda, J. J. Zink, T. H. Allik, S. Chandra, J. A. Hutchinson, “Advances in dye-doped sol-gel lasers,” Mater. Res. Soc. Symp. Proc. 329, 267–277 (1994).

Adv. Solid-State Lasers (1)

M. Canva, A. Dubois, P. Georges, A. Brun, F. Chaput, J. P. Boilot, “Dye doped xerogels for tunable lasers,” Adv. Solid-State Lasers 20, 291–295 (1994).

Appl. Opt. (5)

Appl. Phys. (1)

A. N. Fletcher, “Laser dye stability. Part 4,” Appl. Phys. 16, 93–97 (1978).
[CrossRef]

Appl. Phys. Lett. (1)

R. E. Hermes, T. H. Allik, S. Chandra, J. A. Hutchinson, “High-efficiency pyrromethene doped solid-state dye lasers,” Appl. Phys. Lett. 63, 877–879 (1993).
[CrossRef]

IEEE J. Quantum Electron. (1)

E. P. Ippen, C. V. Shank, A. Dienes, “Rapid photobleaching of organic laser dyes in continuously operated devices,” IEEE J. Quantum Electron. QE-7, 178–179 (1971).
[CrossRef]

J. Phys. Chem. (1)

D. Avnir, D. Lévy, R. Reisfeld, “The nature of the silica cage as reflected by spectral changes and enhanced photostability of trapped Rhodamine 6G,” J. Phys. Chem. 88, 5956–5959 (1984).
[CrossRef]

Mater. Res. Soc. Symp. Proc. (1)

B. Dunn, F. Nishida, R. Toda, J. J. Zink, T. H. Allik, S. Chandra, J. A. Hutchinson, “Advances in dye-doped sol-gel lasers,” Mater. Res. Soc. Symp. Proc. 329, 267–277 (1994).

Opt. Commun. (2)

D. Beer, J. Weber, “Photobleaching of organic laser dyes,” Opt. Commun. 5, 307–309 (1972).
[CrossRef]

B. Liphardt, B. Liphardt, W. Luttke, “Laser dyes III: concepts to increase the photostability of laser dyes,” Opt. Commun. 48, 129–133 (1983).
[CrossRef]

Opt. Lett. (1)

Other (3)

M. Canva, A. Dubois, P. Georges, A. Brun, F. Chaput, A. Ranger, J. P. Boilot, “Perylene, Pyrromethene and grafted Rhodamine doped xerogels for tunable solid state laser,” in Sol-Gel Optics III, J. D. Mackenzie, ed., Proc. SPIE2288, 298–309 (1994).
[CrossRef]

M. D. Rahn, T. A. King, “Solid state dye doped sol-gel glass composite lasers,” in Conference on Lasers and Electro-Optics, Vol. 8 of OSA 1994 Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 389–390.

M. D. Rahn, T. A. King, “Lasers based on doped sol-gel composite glasses,” in Sol-Gel Optics III, J. D. Mackenzie, ed., Proc. SPIE2288, 382–391 (1994).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Comparison of the approximate analytical and exact numerical calculations of the transmission as a function of integrated input energy (σ2 N 0 L = 0.5).

Fig. 2
Fig. 2

Simulations of the transmission versus integrated input energy: a) influence of the dye concentration, absorption cross sections of b) the dye molecule and c) the bleached molecule, and d) the B factor (average number of photons absorbed by each molecule before bleaching).

Fig. 3
Fig. 3

Experimental setup.

Fig. 4
Fig. 4

Evolution of the transmission of a Pyrromethene 567-doped solgel sample versus the integrated input energy for three different power levels.

Fig. 5
Fig. 5

Fit of the transmission versus integrated input energy for a Pyrromethene 567-doped solgel sample: B = 2 × 105.

Fig. 6
Fig. 6

Fit of the transmission versus integrated input energy for the Pyrromethene 567 dye solution: B = 1.5 × 103 (after a volume correction).

Tables (1)

Tables Icon

Table 1 B Factors of Different Doped Solgel Samples and the Corresponding Absorbed Energy a

Equations (23)

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N 1 ( z , t ) + N 2 ( z , t ) = N 0 .
J 1 ( z , t ) = 0 z N 1 ( z , t ) d z , J 2 ( z , t ) = 0 z N 2 ( z , t ) d z ,
J 1 ( z , t ) + J 2 ( z , t ) = N 0 z .
J 1 ( L , t ) + J 2 ( L , t ) = N 0 L = J 0 .
n ( z , t ) = n 0 exp [ σ 1 J 1 ( z , t ) σ 2 J 2 ( z , t ) ] ,
n ( z , t ) = n 0 exp [ Δ σ J 1 ( z , t ) ] exp ( σ 2 N 0 z ) ,
N 1 ( z , t ) t = σ 1 N 1 ( z , t ) B 1 n ( z , t ) .
d J 1 ( L , t ) d t = 0 L σ 1 N 1 ( z , t ) B 1 n ( z , t ) d z .
d J 1 ( L , t ) d t = σ 1 B 1 n 0 0 L N 1 ( z , t ) × exp [ Δ σ J 1 ( z , t ) ] exp ( σ 2 N 0 z ) d z ,
d J 1 ( L , t ) d t = σ 1 B 1 n 0 Δ σ 0 L z [ Δ σ J 1 ( z , t ) ] × exp [ Δ σ J 1 ( z , t ) ] exp ( σ 2 N 0 z ) d z .
d J 1 ( L , t ) d t = σ 1 B 1 n 0 Δ σ { exp [ Δ σ J 1 ( L , t ) ] 1 } .
d x x ( 1 x ) = σ 1 B 1 n 0 d t
ln ( x 1 x ) = σ 1 B 1 n 0 t + const .
exp [ Δ σ J 1 ( L , t ) ] = 1 1 + exp ( Δ σ J 0 1 ) exp ( σ 1 B 1 n 0 t ) .
J 1 ( L , t ) J 1 ( L , 0 ) = In [ 1 + exp ( Δ σ J 0 1 ) exp ( σ 1 B 1 n 0 t ) ] In [ 1 + exp ( Δ σ J 0 1 ) ] .
J 1 ( L , E ) J 1 ( L , 0 ) = ln [ 1 + exp ( Δ σ J 0 1 ) exp ( β E ) ] ln [ 1 + exp ( Δ σ J 0 1 ) ] ,
T ( t ) = n ( L , t ) n 0 .
T ( t ) = exp [ Δ σ J 1 ( L , t ) ] exp ( σ 2 J 0 ) .
T ( t ) = exp ( σ 2 J 0 ) 1 + exp ( Δ σ J 0 1 ) exp ( σ 1 B 1 n 0 t ) .
T ( E ) = exp ( σ 2 J 0 ) 1 + exp ( Δ σ J 0 1 ) exp ( β E ) .
T ( E ) = T ( ) 1 + [ T ( ) / T ( 0 ) 1 ] exp ( β E ) ,
T ( 0 ) = exp ( σ 1 J 0 ) , T ( ) = exp ( σ 2 J 0 ) .
E 1 / 2 = B h υ σ 1 × ( Δ σ J 0 1 ln { [ 1 + exp ( Δ σ J 0 1 ) ] 1 / 2 1 } ) .

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