Abstract

The mechanism of the Yb3+Er3+ (PbOBi2O3Ga2O3) glasses by use of several combinations of Yb3+ and Er3+ concentrations. The measured luminescence decay curves of the donors are used to determine the experimental transfer rates; the best fittings for them are obtained with single exponentials. The microparameters are calculated and used to determine the theoretical rates. The real migration mechanism, involved in the energy transfer of each sample, diffusion, or hopping, is determined by means of the experimental transfer rates and the theoretical ones; fast excitation diffusion among donors, followed by a direct donor to acceptor energy transfer, predominates. A comparison is made with the energy-transfer mechanism of ZBLAN glass codoped with Yb3+ and Er3+, as fast-excitation diffusion among donors, followed by direct energy transfer, was also observed.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. W. H. Dumbaugh, "Heavy metal oxide glasses containing Bi2O3," Phys. Chem. 27, 119-122 (1986).
  2. D. Lezal, J. Pedlikova, P. Kostka, J. Bludska, M. Poulain, and J. Zavadil, "Heavy metal oxide glasses: preparation and physical properties," J. Non-Cryst. Solids 284, 288-295 (2001).
    [CrossRef]
  3. W. H. Dumbaugh, "Lead bismuthate glasses," Phys. Chem. Glasses 19, 121-125 (1978).
  4. Y. G. Choi and J. Heo, "1.3 µm emission and multiphonon relaxation phenomena in PbO-Bi2O3-Ga2O3 glasses doped with rare-earths," J. Non-Cryst. Solids 217, 199-207 (1997).
    [CrossRef]
  5. D. J. Coleman, S. D. Jackson, P. Golding, and T. King, "Measurements of the spectroscopic and energy transfer parameters for Er3+ doped and Er3+, Pr3+ codoped PbO-Bi2O3-Ga2O3 glasses," J. Opt. Soc. Am. B 19, 2927-2937 (2002).
    [CrossRef]
  6. J. Heo, Y. B. Shin, and J. N. Jang, "Spectroscopic analysis of Tm3+ in PbO-Bi2O3-Ga2O3 glass," Appl. Opt. 34, 4284-4289 (1995).
    [CrossRef] [PubMed]
  7. L. R. P. Kassab, L. C. Courrol, A. S. Morais, C. M. S. P. Mendes, S. H. Tatumi, N. U. Wetter, and L. Gomes, "Spectroscopic properties of lead fluoroborate and heavy metal oxide glasses doped with Yb3+," J. Non-Cryst. Solids 304, 233-237 (2002).
    [CrossRef]
  8. L. R. P. Kassab, S. H. Tatumi, A. S. Morais, L. C. Courrol, N. U. Wetter, and V. L. R. Salvador, "Optical properties of Nd doped Bi2O3-PbO-Ga2O3 glasses," Opt. Express 6, 104-108 (2000).
    [CrossRef] [PubMed]
  9. Y. B. Shin, H. T. Lim, Y. G. Choi, Y. S. Kim, and J. Heo, "2.0 µm emission properties and energy transfer between Ho3+ and Tm3+ in PbO-Bi2O3-Ga2O3 glasses," J. Am. Ceram. Soc. 83, 787-791 (2000).
    [CrossRef]
  10. L. D. da Vila, L. Gomes, L. V. G. Tarelho, S. J. L. Ribeiro, and Y. Messadeq, "Mechanism of the Yb-Er energy transfer in fluorozirconate glass," J. Appl. Phys. 93, 1-8 (2003).
    [CrossRef]
  11. D. E. McCumber, "Einstein relations connection broadband emission and absorption spectra," Phys. Rev. 136, 954-957 (1964).
    [CrossRef]
  12. R. C. Powell, Physics of Solid State Laser Material (Springer, New York, 1998), Chap. 5.
    [CrossRef]
  13. F. H. Jagosich, L. Gomes, L. V. G. Tarelho, L. C. Courrol, I. M. Ranieri, "Deactivation effects of the lowest excited states of Er3+ and Ho3+ introduced by Nd3+ ions in LiYF crystals," J. Appl. Phys. 91, 624-632 (2002).
    [CrossRef]
  14. R. K. Watts, Optical Properties of Ions in Solids (Plenum, New York, 1975), pp. 307.
    [CrossRef]

2003 (1)

L. D. da Vila, L. Gomes, L. V. G. Tarelho, S. J. L. Ribeiro, and Y. Messadeq, "Mechanism of the Yb-Er energy transfer in fluorozirconate glass," J. Appl. Phys. 93, 1-8 (2003).
[CrossRef]

2002 (3)

F. H. Jagosich, L. Gomes, L. V. G. Tarelho, L. C. Courrol, I. M. Ranieri, "Deactivation effects of the lowest excited states of Er3+ and Ho3+ introduced by Nd3+ ions in LiYF crystals," J. Appl. Phys. 91, 624-632 (2002).
[CrossRef]

L. R. P. Kassab, L. C. Courrol, A. S. Morais, C. M. S. P. Mendes, S. H. Tatumi, N. U. Wetter, and L. Gomes, "Spectroscopic properties of lead fluoroborate and heavy metal oxide glasses doped with Yb3+," J. Non-Cryst. Solids 304, 233-237 (2002).
[CrossRef]

D. J. Coleman, S. D. Jackson, P. Golding, and T. King, "Measurements of the spectroscopic and energy transfer parameters for Er3+ doped and Er3+, Pr3+ codoped PbO-Bi2O3-Ga2O3 glasses," J. Opt. Soc. Am. B 19, 2927-2937 (2002).
[CrossRef]

2001 (1)

D. Lezal, J. Pedlikova, P. Kostka, J. Bludska, M. Poulain, and J. Zavadil, "Heavy metal oxide glasses: preparation and physical properties," J. Non-Cryst. Solids 284, 288-295 (2001).
[CrossRef]

2000 (2)

Y. B. Shin, H. T. Lim, Y. G. Choi, Y. S. Kim, and J. Heo, "2.0 µm emission properties and energy transfer between Ho3+ and Tm3+ in PbO-Bi2O3-Ga2O3 glasses," J. Am. Ceram. Soc. 83, 787-791 (2000).
[CrossRef]

L. R. P. Kassab, S. H. Tatumi, A. S. Morais, L. C. Courrol, N. U. Wetter, and V. L. R. Salvador, "Optical properties of Nd doped Bi2O3-PbO-Ga2O3 glasses," Opt. Express 6, 104-108 (2000).
[CrossRef] [PubMed]

1997 (1)

Y. G. Choi and J. Heo, "1.3 µm emission and multiphonon relaxation phenomena in PbO-Bi2O3-Ga2O3 glasses doped with rare-earths," J. Non-Cryst. Solids 217, 199-207 (1997).
[CrossRef]

1995 (1)

1978 (1)

W. H. Dumbaugh, "Lead bismuthate glasses," Phys. Chem. Glasses 19, 121-125 (1978).

1964 (1)

D. E. McCumber, "Einstein relations connection broadband emission and absorption spectra," Phys. Rev. 136, 954-957 (1964).
[CrossRef]

Bludska, J.

D. Lezal, J. Pedlikova, P. Kostka, J. Bludska, M. Poulain, and J. Zavadil, "Heavy metal oxide glasses: preparation and physical properties," J. Non-Cryst. Solids 284, 288-295 (2001).
[CrossRef]

Choi, Y. G.

Y. B. Shin, H. T. Lim, Y. G. Choi, Y. S. Kim, and J. Heo, "2.0 µm emission properties and energy transfer between Ho3+ and Tm3+ in PbO-Bi2O3-Ga2O3 glasses," J. Am. Ceram. Soc. 83, 787-791 (2000).
[CrossRef]

Y. G. Choi and J. Heo, "1.3 µm emission and multiphonon relaxation phenomena in PbO-Bi2O3-Ga2O3 glasses doped with rare-earths," J. Non-Cryst. Solids 217, 199-207 (1997).
[CrossRef]

Coleman, D. J.

Courrol, L. C.

L. R. P. Kassab, L. C. Courrol, A. S. Morais, C. M. S. P. Mendes, S. H. Tatumi, N. U. Wetter, and L. Gomes, "Spectroscopic properties of lead fluoroborate and heavy metal oxide glasses doped with Yb3+," J. Non-Cryst. Solids 304, 233-237 (2002).
[CrossRef]

F. H. Jagosich, L. Gomes, L. V. G. Tarelho, L. C. Courrol, I. M. Ranieri, "Deactivation effects of the lowest excited states of Er3+ and Ho3+ introduced by Nd3+ ions in LiYF crystals," J. Appl. Phys. 91, 624-632 (2002).
[CrossRef]

L. R. P. Kassab, S. H. Tatumi, A. S. Morais, L. C. Courrol, N. U. Wetter, and V. L. R. Salvador, "Optical properties of Nd doped Bi2O3-PbO-Ga2O3 glasses," Opt. Express 6, 104-108 (2000).
[CrossRef] [PubMed]

da Vila, L. D.

L. D. da Vila, L. Gomes, L. V. G. Tarelho, S. J. L. Ribeiro, and Y. Messadeq, "Mechanism of the Yb-Er energy transfer in fluorozirconate glass," J. Appl. Phys. 93, 1-8 (2003).
[CrossRef]

Dumbaugh, W. H.

W. H. Dumbaugh, "Lead bismuthate glasses," Phys. Chem. Glasses 19, 121-125 (1978).

W. H. Dumbaugh, "Heavy metal oxide glasses containing Bi2O3," Phys. Chem. 27, 119-122 (1986).

Golding, P.

Gomes, L.

L. D. da Vila, L. Gomes, L. V. G. Tarelho, S. J. L. Ribeiro, and Y. Messadeq, "Mechanism of the Yb-Er energy transfer in fluorozirconate glass," J. Appl. Phys. 93, 1-8 (2003).
[CrossRef]

F. H. Jagosich, L. Gomes, L. V. G. Tarelho, L. C. Courrol, I. M. Ranieri, "Deactivation effects of the lowest excited states of Er3+ and Ho3+ introduced by Nd3+ ions in LiYF crystals," J. Appl. Phys. 91, 624-632 (2002).
[CrossRef]

L. R. P. Kassab, L. C. Courrol, A. S. Morais, C. M. S. P. Mendes, S. H. Tatumi, N. U. Wetter, and L. Gomes, "Spectroscopic properties of lead fluoroborate and heavy metal oxide glasses doped with Yb3+," J. Non-Cryst. Solids 304, 233-237 (2002).
[CrossRef]

Heo, J.

Y. B. Shin, H. T. Lim, Y. G. Choi, Y. S. Kim, and J. Heo, "2.0 µm emission properties and energy transfer between Ho3+ and Tm3+ in PbO-Bi2O3-Ga2O3 glasses," J. Am. Ceram. Soc. 83, 787-791 (2000).
[CrossRef]

Y. G. Choi and J. Heo, "1.3 µm emission and multiphonon relaxation phenomena in PbO-Bi2O3-Ga2O3 glasses doped with rare-earths," J. Non-Cryst. Solids 217, 199-207 (1997).
[CrossRef]

J. Heo, Y. B. Shin, and J. N. Jang, "Spectroscopic analysis of Tm3+ in PbO-Bi2O3-Ga2O3 glass," Appl. Opt. 34, 4284-4289 (1995).
[CrossRef] [PubMed]

Jackson, S. D.

Jagosich, F. H.

F. H. Jagosich, L. Gomes, L. V. G. Tarelho, L. C. Courrol, I. M. Ranieri, "Deactivation effects of the lowest excited states of Er3+ and Ho3+ introduced by Nd3+ ions in LiYF crystals," J. Appl. Phys. 91, 624-632 (2002).
[CrossRef]

Jang, J. N.

Kassab, L. R. P.

L. R. P. Kassab, L. C. Courrol, A. S. Morais, C. M. S. P. Mendes, S. H. Tatumi, N. U. Wetter, and L. Gomes, "Spectroscopic properties of lead fluoroborate and heavy metal oxide glasses doped with Yb3+," J. Non-Cryst. Solids 304, 233-237 (2002).
[CrossRef]

L. R. P. Kassab, S. H. Tatumi, A. S. Morais, L. C. Courrol, N. U. Wetter, and V. L. R. Salvador, "Optical properties of Nd doped Bi2O3-PbO-Ga2O3 glasses," Opt. Express 6, 104-108 (2000).
[CrossRef] [PubMed]

Kim, Y. S.

Y. B. Shin, H. T. Lim, Y. G. Choi, Y. S. Kim, and J. Heo, "2.0 µm emission properties and energy transfer between Ho3+ and Tm3+ in PbO-Bi2O3-Ga2O3 glasses," J. Am. Ceram. Soc. 83, 787-791 (2000).
[CrossRef]

King, T.

Kostka, P.

D. Lezal, J. Pedlikova, P. Kostka, J. Bludska, M. Poulain, and J. Zavadil, "Heavy metal oxide glasses: preparation and physical properties," J. Non-Cryst. Solids 284, 288-295 (2001).
[CrossRef]

Lezal, D.

D. Lezal, J. Pedlikova, P. Kostka, J. Bludska, M. Poulain, and J. Zavadil, "Heavy metal oxide glasses: preparation and physical properties," J. Non-Cryst. Solids 284, 288-295 (2001).
[CrossRef]

Lim, H. T.

Y. B. Shin, H. T. Lim, Y. G. Choi, Y. S. Kim, and J. Heo, "2.0 µm emission properties and energy transfer between Ho3+ and Tm3+ in PbO-Bi2O3-Ga2O3 glasses," J. Am. Ceram. Soc. 83, 787-791 (2000).
[CrossRef]

McCumber, D. E.

D. E. McCumber, "Einstein relations connection broadband emission and absorption spectra," Phys. Rev. 136, 954-957 (1964).
[CrossRef]

Mendes, C. M. S. P.

L. R. P. Kassab, L. C. Courrol, A. S. Morais, C. M. S. P. Mendes, S. H. Tatumi, N. U. Wetter, and L. Gomes, "Spectroscopic properties of lead fluoroborate and heavy metal oxide glasses doped with Yb3+," J. Non-Cryst. Solids 304, 233-237 (2002).
[CrossRef]

Messadeq, Y.

L. D. da Vila, L. Gomes, L. V. G. Tarelho, S. J. L. Ribeiro, and Y. Messadeq, "Mechanism of the Yb-Er energy transfer in fluorozirconate glass," J. Appl. Phys. 93, 1-8 (2003).
[CrossRef]

Morais, A. S.

L. R. P. Kassab, L. C. Courrol, A. S. Morais, C. M. S. P. Mendes, S. H. Tatumi, N. U. Wetter, and L. Gomes, "Spectroscopic properties of lead fluoroborate and heavy metal oxide glasses doped with Yb3+," J. Non-Cryst. Solids 304, 233-237 (2002).
[CrossRef]

L. R. P. Kassab, S. H. Tatumi, A. S. Morais, L. C. Courrol, N. U. Wetter, and V. L. R. Salvador, "Optical properties of Nd doped Bi2O3-PbO-Ga2O3 glasses," Opt. Express 6, 104-108 (2000).
[CrossRef] [PubMed]

Pedlikova, J.

D. Lezal, J. Pedlikova, P. Kostka, J. Bludska, M. Poulain, and J. Zavadil, "Heavy metal oxide glasses: preparation and physical properties," J. Non-Cryst. Solids 284, 288-295 (2001).
[CrossRef]

Poulain, M.

D. Lezal, J. Pedlikova, P. Kostka, J. Bludska, M. Poulain, and J. Zavadil, "Heavy metal oxide glasses: preparation and physical properties," J. Non-Cryst. Solids 284, 288-295 (2001).
[CrossRef]

Powell, R. C.

R. C. Powell, Physics of Solid State Laser Material (Springer, New York, 1998), Chap. 5.
[CrossRef]

Ranieri, I. M.

F. H. Jagosich, L. Gomes, L. V. G. Tarelho, L. C. Courrol, I. M. Ranieri, "Deactivation effects of the lowest excited states of Er3+ and Ho3+ introduced by Nd3+ ions in LiYF crystals," J. Appl. Phys. 91, 624-632 (2002).
[CrossRef]

Ribeiro, S. J. L.

L. D. da Vila, L. Gomes, L. V. G. Tarelho, S. J. L. Ribeiro, and Y. Messadeq, "Mechanism of the Yb-Er energy transfer in fluorozirconate glass," J. Appl. Phys. 93, 1-8 (2003).
[CrossRef]

Salvador, V. L. R.

Shin, Y. B.

Y. B. Shin, H. T. Lim, Y. G. Choi, Y. S. Kim, and J. Heo, "2.0 µm emission properties and energy transfer between Ho3+ and Tm3+ in PbO-Bi2O3-Ga2O3 glasses," J. Am. Ceram. Soc. 83, 787-791 (2000).
[CrossRef]

J. Heo, Y. B. Shin, and J. N. Jang, "Spectroscopic analysis of Tm3+ in PbO-Bi2O3-Ga2O3 glass," Appl. Opt. 34, 4284-4289 (1995).
[CrossRef] [PubMed]

Tarelho, L. V. G.

L. D. da Vila, L. Gomes, L. V. G. Tarelho, S. J. L. Ribeiro, and Y. Messadeq, "Mechanism of the Yb-Er energy transfer in fluorozirconate glass," J. Appl. Phys. 93, 1-8 (2003).
[CrossRef]

F. H. Jagosich, L. Gomes, L. V. G. Tarelho, L. C. Courrol, I. M. Ranieri, "Deactivation effects of the lowest excited states of Er3+ and Ho3+ introduced by Nd3+ ions in LiYF crystals," J. Appl. Phys. 91, 624-632 (2002).
[CrossRef]

Tatumi, S. H.

L. R. P. Kassab, L. C. Courrol, A. S. Morais, C. M. S. P. Mendes, S. H. Tatumi, N. U. Wetter, and L. Gomes, "Spectroscopic properties of lead fluoroborate and heavy metal oxide glasses doped with Yb3+," J. Non-Cryst. Solids 304, 233-237 (2002).
[CrossRef]

L. R. P. Kassab, S. H. Tatumi, A. S. Morais, L. C. Courrol, N. U. Wetter, and V. L. R. Salvador, "Optical properties of Nd doped Bi2O3-PbO-Ga2O3 glasses," Opt. Express 6, 104-108 (2000).
[CrossRef] [PubMed]

Watts, R. K.

R. K. Watts, Optical Properties of Ions in Solids (Plenum, New York, 1975), pp. 307.
[CrossRef]

Wetter, N. U.

L. R. P. Kassab, L. C. Courrol, A. S. Morais, C. M. S. P. Mendes, S. H. Tatumi, N. U. Wetter, and L. Gomes, "Spectroscopic properties of lead fluoroborate and heavy metal oxide glasses doped with Yb3+," J. Non-Cryst. Solids 304, 233-237 (2002).
[CrossRef]

L. R. P. Kassab, S. H. Tatumi, A. S. Morais, L. C. Courrol, N. U. Wetter, and V. L. R. Salvador, "Optical properties of Nd doped Bi2O3-PbO-Ga2O3 glasses," Opt. Express 6, 104-108 (2000).
[CrossRef] [PubMed]

Zavadil, J.

D. Lezal, J. Pedlikova, P. Kostka, J. Bludska, M. Poulain, and J. Zavadil, "Heavy metal oxide glasses: preparation and physical properties," J. Non-Cryst. Solids 284, 288-295 (2001).
[CrossRef]

Appl. Opt. (1)

J. Am. Ceram. Soc. (1)

Y. B. Shin, H. T. Lim, Y. G. Choi, Y. S. Kim, and J. Heo, "2.0 µm emission properties and energy transfer between Ho3+ and Tm3+ in PbO-Bi2O3-Ga2O3 glasses," J. Am. Ceram. Soc. 83, 787-791 (2000).
[CrossRef]

J. Appl. Phys. (2)

L. D. da Vila, L. Gomes, L. V. G. Tarelho, S. J. L. Ribeiro, and Y. Messadeq, "Mechanism of the Yb-Er energy transfer in fluorozirconate glass," J. Appl. Phys. 93, 1-8 (2003).
[CrossRef]

F. H. Jagosich, L. Gomes, L. V. G. Tarelho, L. C. Courrol, I. M. Ranieri, "Deactivation effects of the lowest excited states of Er3+ and Ho3+ introduced by Nd3+ ions in LiYF crystals," J. Appl. Phys. 91, 624-632 (2002).
[CrossRef]

J. Non-Cryst. Solids (3)

L. R. P. Kassab, L. C. Courrol, A. S. Morais, C. M. S. P. Mendes, S. H. Tatumi, N. U. Wetter, and L. Gomes, "Spectroscopic properties of lead fluoroborate and heavy metal oxide glasses doped with Yb3+," J. Non-Cryst. Solids 304, 233-237 (2002).
[CrossRef]

D. Lezal, J. Pedlikova, P. Kostka, J. Bludska, M. Poulain, and J. Zavadil, "Heavy metal oxide glasses: preparation and physical properties," J. Non-Cryst. Solids 284, 288-295 (2001).
[CrossRef]

Y. G. Choi and J. Heo, "1.3 µm emission and multiphonon relaxation phenomena in PbO-Bi2O3-Ga2O3 glasses doped with rare-earths," J. Non-Cryst. Solids 217, 199-207 (1997).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Express (1)

Phys. Chem. Glasses (1)

W. H. Dumbaugh, "Lead bismuthate glasses," Phys. Chem. Glasses 19, 121-125 (1978).

Phys. Rev. (1)

D. E. McCumber, "Einstein relations connection broadband emission and absorption spectra," Phys. Rev. 136, 954-957 (1964).
[CrossRef]

Other (3)

R. C. Powell, Physics of Solid State Laser Material (Springer, New York, 1998), Chap. 5.
[CrossRef]

W. H. Dumbaugh, "Heavy metal oxide glasses containing Bi2O3," Phys. Chem. 27, 119-122 (1986).

R. K. Watts, Optical Properties of Ions in Solids (Plenum, New York, 1975), pp. 307.
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Simplified energy level diagram of the ytterbium-erbium system showing radiative (solid line) and nonradiative transitions (dashed curve).

Fig. 2
Fig. 2

Fluorescence decay of Yb 3 + ( F 5 2 2 F 7 2 2 ) in Yb:Er:PBG glass ( 0.1 mol % of Yb 3 + and 0.01 mol % of Er 3 + ) and in the Yb:PBG glass ( 0.1 mol % of Yb 3 + ); theoretical fit and experimental results are indicated.

Fig. 3
Fig. 3

Spectra superposition between the Yb 3 + emission ( F 5 2 2 F 7 2 2 ) and the Er 3 + absorption ( I 15 2 4 I 11 2 4 ) cross sections in Yb:Er:PBG glass ( 0.1 mol % of Yb 3 + and 0.01 mol % of Er 3 + ); absorption cross section of Yb 3 + in PBG glass is also presented ( 0.1 mol % of Yb 3 + ).

Tables (3)

Tables Icon

Table 1 Total Lifetime of the Yb 3 + Donor in Single ( τ D ) and Codoped (τ samples) a

Tables Icon

Table 2 Theoretical ( k d ) and Experimental ( k ) Transfer Parameters for Yb:Er:PBG Glasses a

Tables Icon

Table 3 Calculated Efficiency of the ET1 Process (eff1) in Yb:Er:PBG Glasses

Equations (15)

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

C D A = R D A 6 τ D ,
C D D = R D D 6 τ D .
R D D 6 = 6 c τ D ( 2 π ) 4 n 2 g D low g D up σ emis D ( λ ) σ abs D ( λ ) d λ ,
R D A 6 = 6 c τ D ( 2 π ) 4 n 2 g D low g D up σ emis D ( λ ) σ abs A ( λ ) d λ ,
C A D = R A D 6 τ A ,
σ emi ( λ ) = σ abs ( λ ) Z l Z u exp ( E z l h c λ 1 k T ) ,
Yb ( F 5 2 2 ) Yb ( F 5 2 2 ) ,
C D D = ( 108 ± 12 ) × 10 40 cm 6 s 1 , R D D = 12.92 Å ,
Yb ( F 5 2 2 ) Er ( I 11 2 4 ) ,
C D A = ( 18 ± 2 ) × 10 40 cm 6 s 1 , R D A = 9.60 Å .
k d = 21 c A c D ( C D D 3 C D A ) 1 4 ,
k h = 20 c A c D ( C D D C D A ) 1 2 .
k = 1 A τ 1 A τ D ,
eff ( 1 ) = k k + γ ( theor ) 2 + k d ,
γ ( theor ) = [ 4 3 π ( 3 2 ) ] c A ( C D A ) ( 1 2 ) .

Metrics