Abstract

We have quantitatively studied the energy-transfer processes between trivalent thulium ions in aluminate glass with different Tm3+ concentrations. Our emphasis has been placed on the determination of the microscopic and macroscopic parameters—the critical radius of these energy transfer processes, i.e., cross relaxation (H43+H63F43+F43) and donor–donor energy migration (H43+H63H43+H63). For the 1.8μm emission in aluminate glass, only a slightly slower increase rather than quenching, even at a high concentration (higher than 15wt%), was observed. Quantitative evidences and explanations indicated that high-order multipolar coupling mechanisms played an important role in energy transfer processes, and it would be helpful to predict efficient host materials and impurity concentrations according to these results to prevent the depopulation of the F43 energy level.

© 2010 Optical Society of America

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  1. Z. S. Xiao, R. Serna, C. N. Afonso, and I. Vickridge, “Broadband infrared emission from Er-Tm:Al2O3 thin films,” Appl. Phys. Lett. 87, 111103 (2005).
    [CrossRef]
  2. D. Q. Chen, Y. S. Wang, F. Bao, and Y. L. Yu, “Broadband near-infrared emission from Tm3+/Er3+ codoped nanostructured glass ceramics,” J. Appl. Phys. 101, 113511 (2007).
    [CrossRef]
  3. A. Polman and F. van Veggel, “Broadband sensitizers for erbium-doped planar optical amplifiers: review,” J. Opt. Soc. Am. B 21, 871-892 (2004).
    [CrossRef]
  4. J. F. Wu, Z. Yao, J. Zong, and S. B. Jiang, “Highly efficient high-power thulium-doped germanate glass fiber laser,” Opt. Lett. 32, 638-640 (2007).
    [CrossRef] [PubMed]
  5. B. R. Judd, “Optical absorption intensities of rare-earth ions,” Phys. Rev. 127, 750-761 (1962).
    [CrossRef]
  6. G. S. Ofelt, “Intensities of crystal spectra of rare-earth ions,” J. Chem. Phys. 37, 511-520 (1962).
    [CrossRef]
  7. D. L. Dexter, “A theory of sensitized luminescence in solids,” J. Chem. Phys. 21, 836-850 (1953).
    [CrossRef]
  8. 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]
  9. D. F. de Sousa, R. Lebullenger, A. C. Hernandes, and L. A. O. Nunes, “Evidence of higher-order mechanisms than dipole-dipole interaction in Tm3+-Tm3+ energy transfer in fluoroindogallate glasses,” Phys. Rev. B 65, 094204 (6pp) (2002).
    [CrossRef]
  10. T. Kushida, “Energy transfer and cooperative optical transitions in rare-earth doped inorganic materials. I. Transition probability calculation,” J. Phys. Soc. Jpn. 34, 1318-1326 (1973).
    [CrossRef]
  11. K. Rajnak, “Approximate excited eigenfunctions for Pr3+ and Tm3+,” J. Chem. Phys. 37, 2440-2444 (1962).
    [CrossRef]
  12. A. I. Burshtein, “Hopping mechanism of energy transfer,” Sov. Phys. JETP 35, 882-885 (1972).
  13. L. V. G. Tarelho, L. Gomes, and I. M. Ranieri, “Determination of microscopic parameters for nonresonant energy-transfer processes in rare-earth-doped crystals,” Phys. Rev. B 56, 14344 (1997).
    [CrossRef]
  14. A. S. S. De Camargo, S. L. De Oliveira, D. F. De Sousa, L. A. O. Nunes, and D. W. Hewak, “Spectroscopic properties and energy transfer parameters of Tm 3 ions in gallium lanthanum sulfide glass,” J. Phys. Condens. Matter 14, 9495-9505 (2002).
    [CrossRef]
  15. D. F. de Sousa and L. A. O. Nunes, “Microscopic and macroscopic parameters of energy transfer between Tm3+ ions in fluoroindogallate glasses,” Phys. Rev. B 66, 024207 (2002).
    [CrossRef]
  16. Z. S. Xiao, R. Serna, F. Xu, and C. N. Afonso, “Critical separation for efficient Tm3+-Tm3+ energy transfer evidenced in nanostructured Tm3+:Al2O3 thin films,” Opt. Lett. 33, 608-610 (2008).
    [CrossRef] [PubMed]
  17. H. Kalaycioglu, A. Sennaroglu, A. Kurt, and G. Ozen, “Spectroscopic analysis of Tm3+:LuAG,” J. Phys. Condens. Matter 19, 036208 (2007).
    [CrossRef]

2008 (1)

2007 (3)

J. F. Wu, Z. Yao, J. Zong, and S. B. Jiang, “Highly efficient high-power thulium-doped germanate glass fiber laser,” Opt. Lett. 32, 638-640 (2007).
[CrossRef] [PubMed]

D. Q. Chen, Y. S. Wang, F. Bao, and Y. L. Yu, “Broadband near-infrared emission from Tm3+/Er3+ codoped nanostructured glass ceramics,” J. Appl. Phys. 101, 113511 (2007).
[CrossRef]

H. Kalaycioglu, A. Sennaroglu, A. Kurt, and G. Ozen, “Spectroscopic analysis of Tm3+:LuAG,” J. Phys. Condens. Matter 19, 036208 (2007).
[CrossRef]

2005 (1)

Z. S. Xiao, R. Serna, C. N. Afonso, and I. Vickridge, “Broadband infrared emission from Er-Tm:Al2O3 thin films,” Appl. Phys. Lett. 87, 111103 (2005).
[CrossRef]

2004 (1)

2002 (3)

D. F. de Sousa, R. Lebullenger, A. C. Hernandes, and L. A. O. Nunes, “Evidence of higher-order mechanisms than dipole-dipole interaction in Tm3+-Tm3+ energy transfer in fluoroindogallate glasses,” Phys. Rev. B 65, 094204 (6pp) (2002).
[CrossRef]

A. S. S. De Camargo, S. L. De Oliveira, D. F. De Sousa, L. A. O. Nunes, and D. W. Hewak, “Spectroscopic properties and energy transfer parameters of Tm 3 ions in gallium lanthanum sulfide glass,” J. Phys. Condens. Matter 14, 9495-9505 (2002).
[CrossRef]

D. F. de Sousa and L. A. O. Nunes, “Microscopic and macroscopic parameters of energy transfer between Tm3+ ions in fluoroindogallate glasses,” Phys. Rev. B 66, 024207 (2002).
[CrossRef]

1997 (1)

L. V. G. Tarelho, L. Gomes, and I. M. Ranieri, “Determination of microscopic parameters for nonresonant energy-transfer processes in rare-earth-doped crystals,” Phys. Rev. B 56, 14344 (1997).
[CrossRef]

1973 (1)

T. Kushida, “Energy transfer and cooperative optical transitions in rare-earth doped inorganic materials. I. Transition probability calculation,” J. Phys. Soc. Jpn. 34, 1318-1326 (1973).
[CrossRef]

1972 (1)

A. I. Burshtein, “Hopping mechanism of energy transfer,” Sov. Phys. JETP 35, 882-885 (1972).

1970 (1)

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]

1962 (3)

K. Rajnak, “Approximate excited eigenfunctions for Pr3+ and Tm3+,” J. Chem. Phys. 37, 2440-2444 (1962).
[CrossRef]

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]

1953 (1)

D. L. Dexter, “A theory of sensitized luminescence in solids,” J. Chem. Phys. 21, 836-850 (1953).
[CrossRef]

Afonso, C. N.

Z. S. Xiao, R. Serna, F. Xu, and C. N. Afonso, “Critical separation for efficient Tm3+-Tm3+ energy transfer evidenced in nanostructured Tm3+:Al2O3 thin films,” Opt. Lett. 33, 608-610 (2008).
[CrossRef] [PubMed]

Z. S. Xiao, R. Serna, C. N. Afonso, and I. Vickridge, “Broadband infrared emission from Er-Tm:Al2O3 thin films,” Appl. Phys. Lett. 87, 111103 (2005).
[CrossRef]

Bao, F.

D. Q. Chen, Y. S. Wang, F. Bao, and Y. L. Yu, “Broadband near-infrared emission from Tm3+/Er3+ codoped nanostructured glass ceramics,” J. Appl. Phys. 101, 113511 (2007).
[CrossRef]

Burshtein, A. I.

A. I. Burshtein, “Hopping mechanism of energy transfer,” Sov. Phys. JETP 35, 882-885 (1972).

Chen, D. Q.

D. Q. Chen, Y. S. Wang, F. Bao, and Y. L. Yu, “Broadband near-infrared emission from Tm3+/Er3+ codoped nanostructured glass ceramics,” J. Appl. Phys. 101, 113511 (2007).
[CrossRef]

De Camargo, A. S. S.

A. S. S. De Camargo, S. L. De Oliveira, D. F. De Sousa, L. A. O. Nunes, and D. W. Hewak, “Spectroscopic properties and energy transfer parameters of Tm 3 ions in gallium lanthanum sulfide glass,” J. Phys. Condens. Matter 14, 9495-9505 (2002).
[CrossRef]

De Oliveira, S. L.

A. S. S. De Camargo, S. L. De Oliveira, D. F. De Sousa, L. A. O. Nunes, and D. W. Hewak, “Spectroscopic properties and energy transfer parameters of Tm 3 ions in gallium lanthanum sulfide glass,” J. Phys. Condens. Matter 14, 9495-9505 (2002).
[CrossRef]

De Sousa, D. F.

A. S. S. De Camargo, S. L. De Oliveira, D. F. De Sousa, L. A. O. Nunes, and D. W. Hewak, “Spectroscopic properties and energy transfer parameters of Tm 3 ions in gallium lanthanum sulfide glass,” J. Phys. Condens. Matter 14, 9495-9505 (2002).
[CrossRef]

D. F. de Sousa and L. A. O. Nunes, “Microscopic and macroscopic parameters of energy transfer between Tm3+ ions in fluoroindogallate glasses,” Phys. Rev. B 66, 024207 (2002).
[CrossRef]

D. F. de Sousa, R. Lebullenger, A. C. Hernandes, and L. A. O. Nunes, “Evidence of higher-order mechanisms than dipole-dipole interaction in Tm3+-Tm3+ energy transfer in fluoroindogallate glasses,” Phys. Rev. B 65, 094204 (6pp) (2002).
[CrossRef]

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]

D. L. Dexter, “A theory of sensitized luminescence in solids,” J. Chem. Phys. 21, 836-850 (1953).
[CrossRef]

Gomes, L.

L. V. G. Tarelho, L. Gomes, and I. M. Ranieri, “Determination of microscopic parameters for nonresonant energy-transfer processes in rare-earth-doped crystals,” Phys. Rev. B 56, 14344 (1997).
[CrossRef]

Hernandes, A. C.

D. F. de Sousa, R. Lebullenger, A. C. Hernandes, and L. A. O. Nunes, “Evidence of higher-order mechanisms than dipole-dipole interaction in Tm3+-Tm3+ energy transfer in fluoroindogallate glasses,” Phys. Rev. B 65, 094204 (6pp) (2002).
[CrossRef]

Hewak, D. W.

A. S. S. De Camargo, S. L. De Oliveira, D. F. De Sousa, L. A. O. Nunes, and D. W. Hewak, “Spectroscopic properties and energy transfer parameters of Tm 3 ions in gallium lanthanum sulfide glass,” J. Phys. Condens. Matter 14, 9495-9505 (2002).
[CrossRef]

Jiang, S. B.

Judd, B. R.

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

Kalaycioglu, H.

H. Kalaycioglu, A. Sennaroglu, A. Kurt, and G. Ozen, “Spectroscopic analysis of Tm3+:LuAG,” J. Phys. Condens. Matter 19, 036208 (2007).
[CrossRef]

Kurt, A.

H. Kalaycioglu, A. Sennaroglu, A. Kurt, and G. Ozen, “Spectroscopic analysis of Tm3+:LuAG,” J. Phys. Condens. Matter 19, 036208 (2007).
[CrossRef]

Kushida, T.

T. Kushida, “Energy transfer and cooperative optical transitions in rare-earth doped inorganic materials. I. Transition probability calculation,” J. Phys. Soc. Jpn. 34, 1318-1326 (1973).
[CrossRef]

Lebullenger, R.

D. F. de Sousa, R. Lebullenger, A. C. Hernandes, and L. A. O. Nunes, “Evidence of higher-order mechanisms than dipole-dipole interaction in Tm3+-Tm3+ energy transfer in fluoroindogallate glasses,” Phys. Rev. B 65, 094204 (6pp) (2002).
[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]

Nunes, L. A. O.

D. F. de Sousa and L. A. O. Nunes, “Microscopic and macroscopic parameters of energy transfer between Tm3+ ions in fluoroindogallate glasses,” Phys. Rev. B 66, 024207 (2002).
[CrossRef]

A. S. S. De Camargo, S. L. De Oliveira, D. F. De Sousa, L. A. O. Nunes, and D. W. Hewak, “Spectroscopic properties and energy transfer parameters of Tm 3 ions in gallium lanthanum sulfide glass,” J. Phys. Condens. Matter 14, 9495-9505 (2002).
[CrossRef]

D. F. de Sousa, R. Lebullenger, A. C. Hernandes, and L. A. O. Nunes, “Evidence of higher-order mechanisms than dipole-dipole interaction in Tm3+-Tm3+ energy transfer in fluoroindogallate glasses,” Phys. Rev. B 65, 094204 (6pp) (2002).
[CrossRef]

Ofelt, G. S.

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

Ozen, G.

H. Kalaycioglu, A. Sennaroglu, A. Kurt, and G. Ozen, “Spectroscopic analysis of Tm3+:LuAG,” J. Phys. Condens. Matter 19, 036208 (2007).
[CrossRef]

Polman, A.

Rajnak, K.

K. Rajnak, “Approximate excited eigenfunctions for Pr3+ and Tm3+,” J. Chem. Phys. 37, 2440-2444 (1962).
[CrossRef]

Ranieri, I. M.

L. V. G. Tarelho, L. Gomes, and I. M. Ranieri, “Determination of microscopic parameters for nonresonant energy-transfer processes in rare-earth-doped crystals,” Phys. Rev. B 56, 14344 (1997).
[CrossRef]

Sennaroglu, A.

H. Kalaycioglu, A. Sennaroglu, A. Kurt, and G. Ozen, “Spectroscopic analysis of Tm3+:LuAG,” J. Phys. Condens. Matter 19, 036208 (2007).
[CrossRef]

Serna, R.

Z. S. Xiao, R. Serna, F. Xu, and C. N. Afonso, “Critical separation for efficient Tm3+-Tm3+ energy transfer evidenced in nanostructured Tm3+:Al2O3 thin films,” Opt. Lett. 33, 608-610 (2008).
[CrossRef] [PubMed]

Z. S. Xiao, R. Serna, C. N. Afonso, and I. Vickridge, “Broadband infrared emission from Er-Tm:Al2O3 thin films,” Appl. Phys. Lett. 87, 111103 (2005).
[CrossRef]

Tarelho, L. V. G.

L. V. G. Tarelho, L. Gomes, and I. M. Ranieri, “Determination of microscopic parameters for nonresonant energy-transfer processes in rare-earth-doped crystals,” Phys. Rev. B 56, 14344 (1997).
[CrossRef]

van Veggel, F.

Vickridge, I.

Z. S. Xiao, R. Serna, C. N. Afonso, and I. Vickridge, “Broadband infrared emission from Er-Tm:Al2O3 thin films,” Appl. Phys. Lett. 87, 111103 (2005).
[CrossRef]

Wang, Y. S.

D. Q. Chen, Y. S. Wang, F. Bao, and Y. L. Yu, “Broadband near-infrared emission from Tm3+/Er3+ codoped nanostructured glass ceramics,” J. Appl. Phys. 101, 113511 (2007).
[CrossRef]

Wu, J. F.

Xiao, Z. S.

Z. S. Xiao, R. Serna, F. Xu, and C. N. Afonso, “Critical separation for efficient Tm3+-Tm3+ energy transfer evidenced in nanostructured Tm3+:Al2O3 thin films,” Opt. Lett. 33, 608-610 (2008).
[CrossRef] [PubMed]

Z. S. Xiao, R. Serna, C. N. Afonso, and I. Vickridge, “Broadband infrared emission from Er-Tm:Al2O3 thin films,” Appl. Phys. Lett. 87, 111103 (2005).
[CrossRef]

Xu, F.

Yao, Z.

Yu, Y. L.

D. Q. Chen, Y. S. Wang, F. Bao, and Y. L. Yu, “Broadband near-infrared emission from Tm3+/Er3+ codoped nanostructured glass ceramics,” J. Appl. Phys. 101, 113511 (2007).
[CrossRef]

Zong, J.

Appl. Phys. Lett. (1)

Z. S. Xiao, R. Serna, C. N. Afonso, and I. Vickridge, “Broadband infrared emission from Er-Tm:Al2O3 thin films,” Appl. Phys. Lett. 87, 111103 (2005).
[CrossRef]

J. Appl. Phys. (1)

D. Q. Chen, Y. S. Wang, F. Bao, and Y. L. Yu, “Broadband near-infrared emission from Tm3+/Er3+ codoped nanostructured glass ceramics,” J. Appl. Phys. 101, 113511 (2007).
[CrossRef]

J. Chem. Phys. (3)

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

D. L. Dexter, “A theory of sensitized luminescence in solids,” J. Chem. Phys. 21, 836-850 (1953).
[CrossRef]

K. Rajnak, “Approximate excited eigenfunctions for Pr3+ and Tm3+,” J. Chem. Phys. 37, 2440-2444 (1962).
[CrossRef]

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

J. Phys. Condens. Matter (2)

H. Kalaycioglu, A. Sennaroglu, A. Kurt, and G. Ozen, “Spectroscopic analysis of Tm3+:LuAG,” J. Phys. Condens. Matter 19, 036208 (2007).
[CrossRef]

A. S. S. De Camargo, S. L. De Oliveira, D. F. De Sousa, L. A. O. Nunes, and D. W. Hewak, “Spectroscopic properties and energy transfer parameters of Tm 3 ions in gallium lanthanum sulfide glass,” J. Phys. Condens. Matter 14, 9495-9505 (2002).
[CrossRef]

J. Phys. Soc. Jpn. (1)

T. Kushida, “Energy transfer and cooperative optical transitions in rare-earth doped inorganic materials. I. Transition probability calculation,” J. Phys. Soc. Jpn. 34, 1318-1326 (1973).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. (1)

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

Phys. Rev. B (4)

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]

D. F. de Sousa, R. Lebullenger, A. C. Hernandes, and L. A. O. Nunes, “Evidence of higher-order mechanisms than dipole-dipole interaction in Tm3+-Tm3+ energy transfer in fluoroindogallate glasses,” Phys. Rev. B 65, 094204 (6pp) (2002).
[CrossRef]

L. V. G. Tarelho, L. Gomes, and I. M. Ranieri, “Determination of microscopic parameters for nonresonant energy-transfer processes in rare-earth-doped crystals,” Phys. Rev. B 56, 14344 (1997).
[CrossRef]

D. F. de Sousa and L. A. O. Nunes, “Microscopic and macroscopic parameters of energy transfer between Tm3+ ions in fluoroindogallate glasses,” Phys. Rev. B 66, 024207 (2002).
[CrossRef]

Sov. Phys. JETP (1)

A. I. Burshtein, “Hopping mechanism of energy transfer,” Sov. Phys. JETP 35, 882-885 (1972).

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Figures (4)

Fig. 1
Fig. 1

Schematic energy diagram of Tm 3 + showing relevant energy transfer processes. PA-CR (D–A), Donor–Acceptor phonon-assisted cross-relaxation; EM (D–D), Donor–Donor energy migration; MPR, Multiphonon relaxation; R DA , average distance between two Tm 3 + ions.

Fig. 2
Fig. 2

Absorption spectra of aluminate glasses with different Tm 3 + concentrations. Inset: Overlap of acceptor’s absorption ( H 6 3 F 4 3 ) and donor’s emission ( H 5 3 F 4 3 ) and the energy mismatch between them.

Fig. 3
Fig. 3

Photoluminescence spectra of Tm 3 + aluminate glasses pump at 791 nm laser in the infrared region. Inset: Normalized PL intensities of 1.4 and 1.8 μ m as a function of Tm 3 + concentration.

Fig. 4
Fig. 4

(a) Intensity of 1.4 μ m . (b) Average distance of ions as a function of Tm 3 + concentration. (b) Critical radius lines.

Tables (3)

Tables Icon

Table 1 Judd–Ofelt Intensity Parameters for Different Tm 3 + Concentrations

Tables Icon

Table 2 Transition Probabilities A J J and Branching Ratios β J J Between Multiplets J and J and Radiative Lifetime τ rad for Different Excited States of Tm 3 + in Aluminate Glass Calculated by Judd–Ofelt Method

Tables Icon

Table 3 Microscopic and Macroscopic Parameters, Critical Radius of Tm 3 + Ion Energy Transfer for Different Interaction Mechanisms in Aluminate Glass

Equations (12)

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

S J J = λ Ω λ | J J U ( λ ) J J | 2 ,
A J J = 64 π 4 e 2 ν ¯ 3 3 h ( 2 J + 1 ) n ( n 2 + 2 ) 2 9 λ Ω λ J J U ( λ ) J J 2 ,
P D - A dd = ( 3 4 c 4 4 π ) ( 1 τ D ) Q A ( 1 R D A ) 1 n 4 f D ems ( E ) f A abs ( E ) E 4 d E
W D A = ( 2 π ) | H DA | 2 S D A ,
P D - A dd = 1 ( 2 J i + 1 ) ( 2 J l + 1 ) ( 2 3 ) ( 2 π ) ( e 2 R D A 3 ) 2 χ dd [ λ Ω λ J i U ( λ ) J j 2 λ Ω λ J l U ( λ ) J k 2 ] S ,
P D - A dq = 1 ( 2 J i + 1 ) ( 2 J l + 1 ) ( 2 π ) ( e 2 R D A 4 ) 2 χ dq [ λ Ω λ J i U ( λ ) J j 2 ] [ 4 f | r A 2 | 4 f 4 f C ( 2 ) 4 f J l U ( 2 ) J k ] 2 S ,
P D - A qq = 1 ( 2 J i + 1 ) ( 2 J l + 1 ) ( 14 5 ) ( 2 π ) ( e 2 R D A 5 ) 2 χ qq [ 4 f | r A 2 | 4 f 4 f | r D 2 | 4 f 4 f C ( 2 ) 4 f 2 J i U ( 2 ) J j J l U ( 2 ) J k ] 2 S ,
C D A ( D ) ( s ) = P D A ( D ) ( s ) R n ,
R C n = C D A ( D ) ( s ) τ D ,
W ET dd = 13 ( C D A dd ) 1 2 ( C D D dd ) 1 2 n d ,
W ET dq = 21 ( C D A dq ) 3 8 ( C D D dq ) 5 8 n d 5 3 ,
W ET qq = 42 ( C D A qq ) 3 10 ( C D D dq ) 7 10 n d 7 3 ,

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