Abstract

The optical transmission spectra of several samples of gold nanoparticle layers were examined using a modified Drude model proposed with a novel elastic scattering parameter, γ′. Although the measured transmission spectra deviated from the simple calculation from Mie scattering, it was explained well by the modified model assuming elastic and inelastic scattering in the form of the collision frequency of free electrons within a metal particle due to the particle boundary. The particle-size and inter-particle-distance dependences of γ′ were extracted within the framework of the proposed model from the curve of best fit of the transmittance spectra.

© 2010 OSA

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    [CrossRef]
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    [CrossRef]
  6. E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
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    [CrossRef]
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2009 (2)

J. H. Sung, B. S. Kim, C. H. Choi, M. W. Lee, S. G. Lee, S. G. Park, E. H. Lee, and B. H. O, “Enhanced luminescence of GaN-based light-emitting diode with a localized surface plasmon resonance,” Microelectron. Eng. 86(4-6), 1120–1123 (2009).
[CrossRef]

J. S. Yang, S. G. Lee, S. G. Park, E. H. Lee, and B. H. O, “Drude Model for the Optical Properties of a Nano-Scale Thin Metal film Revisited,” J. Korean Phys. Soc. 55(6), 2552–2555 (2009).
[CrossRef]

2007 (1)

E. R. Encina and E. A. Coronado, “Resonance conditions for Multipole Plasmon Excitations in Noble Metal Nanorods,” J. Phys. Chem. C 111(45), 16796–16801 (2007).
[CrossRef]

2006 (3)

L. B. Scaffardi and J. O. Tocho, “Size dependence of refractive index of gold nanoparticles,” Nanotechnology 17(5), 1309–1315 (2006).
[CrossRef]

B. Khlebtsov, A. Melnikov, V. Zharov, and N. Khlebtsov, “Absorption and scattering of light by a dimer of metal nanospheres: comparison of dipole and multipole approaches,” Nanotechnology 17(5), 1437–1445 (2006).
[CrossRef]

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311(5758), 189–193 (2006).
[CrossRef] [PubMed]

2005 (1)

S. Berciaud, L. Cognet, P. Tamarat, and B. Lounis, “Observation of intrinsic size effects in the optical response of individual gold nanoparticles,” Nano Lett. 5(3), 515–518 (2005).
[CrossRef] [PubMed]

2004 (1)

N. K. Grady, N. J. Halas, and P. Nordlander, “Influence of dielectric function properties on the optical response of plasmon resonant metallic nanoparticles,” Chem. Phys. Lett. 399(1-3), 167–171 (2004).
[CrossRef]

2003 (3)

G. Xu, M. Tazawa, P. Jin, S. Nakao, and K. Yoshimura, “Wavelength tuning of surface plasmon resonance using dielectric layers on silver island films,” Appl. Phys. Lett. 82(22), 3811–3813 (2003).
[CrossRef]

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
[CrossRef] [PubMed]

G. Raschke, S. Kowarik, T. Franzl,, C. Sönnichsen, T. A. Klar, J. Feldmann, A. Nichtl, and K. Kürzinger “Biomolecular Recognition Based on Single Gold Nanoparticle Light Scattering,” Nano Lett. 3(7), 935–938 (2003).
[CrossRef]

2001 (1)

2000 (2)

J. Vučković, M. Lončar, and A. Scherer, “Surface Plasmon Enhanced Light-Emitting Diode,” IEEE J. Quantum Electron. 36(10), 1131–1144 (2000).
[CrossRef]

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanopartice chain arrays below the diffraction limit,” Phys. Rev. B 62(24), R16356–R16359 (2000).
[CrossRef]

1999 (1)

S. Link and M. A. El-Sayed, “Size and Temperature Dependence of the Plasmon Absorption of Colloidal Nanoparticles,” J. Phys. Chem. B 103(21), 4212–4217 (1999).
[CrossRef]

1998 (1)

H. Inouye, K. Tanaka, I. Tanahashi, and K. Hirao, “Ultrafast dynamics of nonequilibrium electrons in a gold nanoparticle system,” Phys. Rev. B 57(18), 11334–11340 (1998).
[CrossRef]

1992 (1)

M. Xu and J. Dignam, “A new approach to the surface plasmon resonance of small metal particles,” J. Chem. Phys. 96(5), 3370–3378 (1992).
[CrossRef]

1980 (1)

D. E. Aspnes, E. Kinsbron, and D. D. Bacon, “Optical properties of Au: Sample effects,” Phys. Rev. B 21(8), 3290–3299 (1980).
[CrossRef]

1977 (1)

C. G. Granqvist and O. Hunderi, “Optical properties of ultrafine gold particles,” Phys. Rev. B 16(8), 3513–3534 (1977).
[CrossRef]

1975 (1)

R. E. Hetrick and J. Lambe, “Optical properties of small In particles in thin-film form,” Phys. Rev. B 11(4), 1273–1278 (1975).
[CrossRef]

1974 (1)

U. Kreibig, “Electronic properties of small silver particles: the optical constants and their temperature dependence,” J. Phys. F Met. Phys. 4(7), 999–1014 (1974).
[CrossRef]

1972 (1)

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

1970 (1)

M. Théye, “Investigation of the Optical Properties of Au by Means of Thin Semitransparent Films,” Phys. Rev. B 2(8), 3060–3078 (1970).
[CrossRef]

1962 (1)

H. Ehrenreich and H. R. Philipp, “Optical properties of Ag and Cu,” Phys. Rev. 128(4), 1622–1629 (1962).
[CrossRef]

1938 (1)

K. Fuchs and N. F. Mott, “The conductivity of Thin Metallic Films according to the Electron Theory of Metals,” Proc. Camb. Philos. Soc. 34(01), 100 (1938); E. H. Sondheimer, “The Mean Free Path of Electrons in Metals,” Adv. Phys. 1(1), 1–42 (1952).
[CrossRef]

K. Fuchs and N. F. Mott, “The conductivity of Thin Metallic Films according to the Electron Theory of Metals,” Proc. Camb. Philos. Soc. 34(01), 100 (1938); E. H. Sondheimer, “The Mean Free Path of Electrons in Metals,” Adv. Phys. 1(1), 1–42 (1952).
[CrossRef]

Aspnes, D. E.

D. E. Aspnes, E. Kinsbron, and D. D. Bacon, “Optical properties of Au: Sample effects,” Phys. Rev. B 21(8), 3290–3299 (1980).
[CrossRef]

Atwater, H. A.

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanopartice chain arrays below the diffraction limit,” Phys. Rev. B 62(24), R16356–R16359 (2000).
[CrossRef]

Bacon, D. D.

D. E. Aspnes, E. Kinsbron, and D. D. Bacon, “Optical properties of Au: Sample effects,” Phys. Rev. B 21(8), 3290–3299 (1980).
[CrossRef]

Berciaud, S.

S. Berciaud, L. Cognet, P. Tamarat, and B. Lounis, “Observation of intrinsic size effects in the optical response of individual gold nanoparticles,” Nano Lett. 5(3), 515–518 (2005).
[CrossRef] [PubMed]

Brongersma, M. L.

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanopartice chain arrays below the diffraction limit,” Phys. Rev. B 62(24), R16356–R16359 (2000).
[CrossRef]

Choi, C. H.

J. H. Sung, B. S. Kim, C. H. Choi, M. W. Lee, S. G. Lee, S. G. Park, E. H. Lee, and B. H. O, “Enhanced luminescence of GaN-based light-emitting diode with a localized surface plasmon resonance,” Microelectron. Eng. 86(4-6), 1120–1123 (2009).
[CrossRef]

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]

Cognet, L.

S. Berciaud, L. Cognet, P. Tamarat, and B. Lounis, “Observation of intrinsic size effects in the optical response of individual gold nanoparticles,” Nano Lett. 5(3), 515–518 (2005).
[CrossRef] [PubMed]

Coronado, E. A.

E. R. Encina and E. A. Coronado, “Resonance conditions for Multipole Plasmon Excitations in Noble Metal Nanorods,” J. Phys. Chem. C 111(45), 16796–16801 (2007).
[CrossRef]

Dalacu, D.

Dignam, J.

M. Xu and J. Dignam, “A new approach to the surface plasmon resonance of small metal particles,” J. Chem. Phys. 96(5), 3370–3378 (1992).
[CrossRef]

Ehrenreich, H.

H. Ehrenreich and H. R. Philipp, “Optical properties of Ag and Cu,” Phys. Rev. 128(4), 1622–1629 (1962).
[CrossRef]

El-Sayed, M. A.

S. Link and M. A. El-Sayed, “Size and Temperature Dependence of the Plasmon Absorption of Colloidal Nanoparticles,” J. Phys. Chem. B 103(21), 4212–4217 (1999).
[CrossRef]

Encina, E. R.

E. R. Encina and E. A. Coronado, “Resonance conditions for Multipole Plasmon Excitations in Noble Metal Nanorods,” J. Phys. Chem. C 111(45), 16796–16801 (2007).
[CrossRef]

Feldmann, J.

G. Raschke, S. Kowarik, T. Franzl,, C. Sönnichsen, T. A. Klar, J. Feldmann, A. Nichtl, and K. Kürzinger “Biomolecular Recognition Based on Single Gold Nanoparticle Light Scattering,” Nano Lett. 3(7), 935–938 (2003).
[CrossRef]

Franzl,, T.

G. Raschke, S. Kowarik, T. Franzl,, C. Sönnichsen, T. A. Klar, J. Feldmann, A. Nichtl, and K. Kürzinger “Biomolecular Recognition Based on Single Gold Nanoparticle Light Scattering,” Nano Lett. 3(7), 935–938 (2003).
[CrossRef]

Fuchs, K.

K. Fuchs and N. F. Mott, “The conductivity of Thin Metallic Films according to the Electron Theory of Metals,” Proc. Camb. Philos. Soc. 34(01), 100 (1938); E. H. Sondheimer, “The Mean Free Path of Electrons in Metals,” Adv. Phys. 1(1), 1–42 (1952).
[CrossRef]

Grady, N. K.

N. K. Grady, N. J. Halas, and P. Nordlander, “Influence of dielectric function properties on the optical response of plasmon resonant metallic nanoparticles,” Chem. Phys. Lett. 399(1-3), 167–171 (2004).
[CrossRef]

Granqvist, C. G.

C. G. Granqvist and O. Hunderi, “Optical properties of ultrafine gold particles,” Phys. Rev. B 16(8), 3513–3534 (1977).
[CrossRef]

Halas, N. J.

N. K. Grady, N. J. Halas, and P. Nordlander, “Influence of dielectric function properties on the optical response of plasmon resonant metallic nanoparticles,” Chem. Phys. Lett. 399(1-3), 167–171 (2004).
[CrossRef]

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
[CrossRef] [PubMed]

Hartman, J. W.

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanopartice chain arrays below the diffraction limit,” Phys. Rev. B 62(24), R16356–R16359 (2000).
[CrossRef]

Hetrick, R. E.

R. E. Hetrick and J. Lambe, “Optical properties of small In particles in thin-film form,” Phys. Rev. B 11(4), 1273–1278 (1975).
[CrossRef]

Hirao, K.

H. Inouye, K. Tanaka, I. Tanahashi, and K. Hirao, “Ultrafast dynamics of nonequilibrium electrons in a gold nanoparticle system,” Phys. Rev. B 57(18), 11334–11340 (1998).
[CrossRef]

Hunderi, O.

C. G. Granqvist and O. Hunderi, “Optical properties of ultrafine gold particles,” Phys. Rev. B 16(8), 3513–3534 (1977).
[CrossRef]

Inouye, H.

H. Inouye, K. Tanaka, I. Tanahashi, and K. Hirao, “Ultrafast dynamics of nonequilibrium electrons in a gold nanoparticle system,” Phys. Rev. B 57(18), 11334–11340 (1998).
[CrossRef]

Jin, P.

G. Xu, M. Tazawa, P. Jin, S. Nakao, and K. Yoshimura, “Wavelength tuning of surface plasmon resonance using dielectric layers on silver island films,” Appl. Phys. Lett. 82(22), 3811–3813 (2003).
[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]

Khlebtsov, B.

B. Khlebtsov, A. Melnikov, V. Zharov, and N. Khlebtsov, “Absorption and scattering of light by a dimer of metal nanospheres: comparison of dipole and multipole approaches,” Nanotechnology 17(5), 1437–1445 (2006).
[CrossRef]

Khlebtsov, N.

B. Khlebtsov, A. Melnikov, V. Zharov, and N. Khlebtsov, “Absorption and scattering of light by a dimer of metal nanospheres: comparison of dipole and multipole approaches,” Nanotechnology 17(5), 1437–1445 (2006).
[CrossRef]

Kim, B. S.

J. H. Sung, B. S. Kim, C. H. Choi, M. W. Lee, S. G. Lee, S. G. Park, E. H. Lee, and B. H. O, “Enhanced luminescence of GaN-based light-emitting diode with a localized surface plasmon resonance,” Microelectron. Eng. 86(4-6), 1120–1123 (2009).
[CrossRef]

Kinsbron, E.

D. E. Aspnes, E. Kinsbron, and D. D. Bacon, “Optical properties of Au: Sample effects,” Phys. Rev. B 21(8), 3290–3299 (1980).
[CrossRef]

Klar, T. A.

G. Raschke, S. Kowarik, T. Franzl,, C. Sönnichsen, T. A. Klar, J. Feldmann, A. Nichtl, and K. Kürzinger “Biomolecular Recognition Based on Single Gold Nanoparticle Light Scattering,” Nano Lett. 3(7), 935–938 (2003).
[CrossRef]

Kowarik, S.

G. Raschke, S. Kowarik, T. Franzl,, C. Sönnichsen, T. A. Klar, J. Feldmann, A. Nichtl, and K. Kürzinger “Biomolecular Recognition Based on Single Gold Nanoparticle Light Scattering,” Nano Lett. 3(7), 935–938 (2003).
[CrossRef]

Kreibig, U.

U. Kreibig, “Electronic properties of small silver particles: the optical constants and their temperature dependence,” J. Phys. F Met. Phys. 4(7), 999–1014 (1974).
[CrossRef]

Kürzinger, K.

G. Raschke, S. Kowarik, T. Franzl,, C. Sönnichsen, T. A. Klar, J. Feldmann, A. Nichtl, and K. Kürzinger “Biomolecular Recognition Based on Single Gold Nanoparticle Light Scattering,” Nano Lett. 3(7), 935–938 (2003).
[CrossRef]

Lambe, J.

R. E. Hetrick and J. Lambe, “Optical properties of small In particles in thin-film form,” Phys. Rev. B 11(4), 1273–1278 (1975).
[CrossRef]

Lee, E. H.

J. S. Yang, S. G. Lee, S. G. Park, E. H. Lee, and B. H. O, “Drude Model for the Optical Properties of a Nano-Scale Thin Metal film Revisited,” J. Korean Phys. Soc. 55(6), 2552–2555 (2009).
[CrossRef]

J. H. Sung, B. S. Kim, C. H. Choi, M. W. Lee, S. G. Lee, S. G. Park, E. H. Lee, and B. H. O, “Enhanced luminescence of GaN-based light-emitting diode with a localized surface plasmon resonance,” Microelectron. Eng. 86(4-6), 1120–1123 (2009).
[CrossRef]

Lee, M. W.

J. H. Sung, B. S. Kim, C. H. Choi, M. W. Lee, S. G. Lee, S. G. Park, E. H. Lee, and B. H. O, “Enhanced luminescence of GaN-based light-emitting diode with a localized surface plasmon resonance,” Microelectron. Eng. 86(4-6), 1120–1123 (2009).
[CrossRef]

Lee, S. G.

J. H. Sung, B. S. Kim, C. H. Choi, M. W. Lee, S. G. Lee, S. G. Park, E. H. Lee, and B. H. O, “Enhanced luminescence of GaN-based light-emitting diode with a localized surface plasmon resonance,” Microelectron. Eng. 86(4-6), 1120–1123 (2009).
[CrossRef]

J. S. Yang, S. G. Lee, S. G. Park, E. H. Lee, and B. H. O, “Drude Model for the Optical Properties of a Nano-Scale Thin Metal film Revisited,” J. Korean Phys. Soc. 55(6), 2552–2555 (2009).
[CrossRef]

Link, S.

S. Link and M. A. El-Sayed, “Size and Temperature Dependence of the Plasmon Absorption of Colloidal Nanoparticles,” J. Phys. Chem. B 103(21), 4212–4217 (1999).
[CrossRef]

Loncar, M.

J. Vučković, M. Lončar, and A. Scherer, “Surface Plasmon Enhanced Light-Emitting Diode,” IEEE J. Quantum Electron. 36(10), 1131–1144 (2000).
[CrossRef]

Lounis, B.

S. Berciaud, L. Cognet, P. Tamarat, and B. Lounis, “Observation of intrinsic size effects in the optical response of individual gold nanoparticles,” Nano Lett. 5(3), 515–518 (2005).
[CrossRef] [PubMed]

Martinu, L.

Melnikov, A.

B. Khlebtsov, A. Melnikov, V. Zharov, and N. Khlebtsov, “Absorption and scattering of light by a dimer of metal nanospheres: comparison of dipole and multipole approaches,” Nanotechnology 17(5), 1437–1445 (2006).
[CrossRef]

Mott, N. F.

K. Fuchs and N. F. Mott, “The conductivity of Thin Metallic Films according to the Electron Theory of Metals,” Proc. Camb. Philos. Soc. 34(01), 100 (1938); E. H. Sondheimer, “The Mean Free Path of Electrons in Metals,” Adv. Phys. 1(1), 1–42 (1952).
[CrossRef]

Nakao, S.

G. Xu, M. Tazawa, P. Jin, S. Nakao, and K. Yoshimura, “Wavelength tuning of surface plasmon resonance using dielectric layers on silver island films,” Appl. Phys. Lett. 82(22), 3811–3813 (2003).
[CrossRef]

Nichtl, A.

G. Raschke, S. Kowarik, T. Franzl,, C. Sönnichsen, T. A. Klar, J. Feldmann, A. Nichtl, and K. Kürzinger “Biomolecular Recognition Based on Single Gold Nanoparticle Light Scattering,” Nano Lett. 3(7), 935–938 (2003).
[CrossRef]

Nordlander, P.

N. K. Grady, N. J. Halas, and P. Nordlander, “Influence of dielectric function properties on the optical response of plasmon resonant metallic nanoparticles,” Chem. Phys. Lett. 399(1-3), 167–171 (2004).
[CrossRef]

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
[CrossRef] [PubMed]

O, B. H.

J. H. Sung, B. S. Kim, C. H. Choi, M. W. Lee, S. G. Lee, S. G. Park, E. H. Lee, and B. H. O, “Enhanced luminescence of GaN-based light-emitting diode with a localized surface plasmon resonance,” Microelectron. Eng. 86(4-6), 1120–1123 (2009).
[CrossRef]

J. S. Yang, S. G. Lee, S. G. Park, E. H. Lee, and B. H. O, “Drude Model for the Optical Properties of a Nano-Scale Thin Metal film Revisited,” J. Korean Phys. Soc. 55(6), 2552–2555 (2009).
[CrossRef]

Ozbay, E.

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311(5758), 189–193 (2006).
[CrossRef] [PubMed]

Park, S. G.

J. H. Sung, B. S. Kim, C. H. Choi, M. W. Lee, S. G. Lee, S. G. Park, E. H. Lee, and B. H. O, “Enhanced luminescence of GaN-based light-emitting diode with a localized surface plasmon resonance,” Microelectron. Eng. 86(4-6), 1120–1123 (2009).
[CrossRef]

J. S. Yang, S. G. Lee, S. G. Park, E. H. Lee, and B. H. O, “Drude Model for the Optical Properties of a Nano-Scale Thin Metal film Revisited,” J. Korean Phys. Soc. 55(6), 2552–2555 (2009).
[CrossRef]

Philipp, H. R.

H. Ehrenreich and H. R. Philipp, “Optical properties of Ag and Cu,” Phys. Rev. 128(4), 1622–1629 (1962).
[CrossRef]

Prodan, E.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
[CrossRef] [PubMed]

Radloff, C.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
[CrossRef] [PubMed]

Raschke, G.

G. Raschke, S. Kowarik, T. Franzl,, C. Sönnichsen, T. A. Klar, J. Feldmann, A. Nichtl, and K. Kürzinger “Biomolecular Recognition Based on Single Gold Nanoparticle Light Scattering,” Nano Lett. 3(7), 935–938 (2003).
[CrossRef]

Scaffardi, L. B.

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S. Berciaud, L. Cognet, P. Tamarat, and B. Lounis, “Observation of intrinsic size effects in the optical response of individual gold nanoparticles,” Nano Lett. 5(3), 515–518 (2005).
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Figures (3)

Fig. 1
Fig. 1

The size distribution of Au particles for three samples as a function of the effective radius. Scanning electron microscopy (SEM) images for each distribution are shown in the inset. A schematic of the cross-sectional view for the particle layer surrounded by the dielectric medium is also shown in the inset.

Fig. 2
Fig. 2

The optical transmittance simulated with this model and Mie theory are compared with the optical spectra measured for normal incident light as a function of the wavelength.

Fig. 3
Fig. 3

The values of γ′ and Qs extracted from the inverse curve fit are plotted as a function of the effective radius of each metal particle. The solid line is a simple functional curve fit with γ ' = ( B ε s m ) r o 2 r 2 + r o 2 . The extracted Qs(r) values are shown in the inset. Note that Qs(r) lie on an almost straight line.

Equations (9)

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ε = ε b + ε f = ε b + 1 + i ( σ ε o ω ) ,
σ = ( n e 2 m ) ( 1 γ i ω ) ,
ε b ( ω ) = Q ω g x ω g x × [ 1 F ( x , E F , T ) ] × ( x 2 ω 2 + γ b 2 + i 2 ω γ b ) ( x 2 ω 2 + γ b 2 ) 2 + 4 ω 2 γ b 2 d x ,
F = m d v ( t ) d t = e E ( t ) m ( γ + i γ ' ) v ( t ) ,
v = e / m γ i ( ω γ ' ) E σ = ( e 2 m ) ( n p γ i ( ω γ ' ) ) ,
ε e f f = ( 1 V T ) [ i = 1 N ( ε m i V m i ) + ε s m ( V T i = 1 N V m i ) ] ,
i = 1 N ( ε m i V m i ) = ( 1 + ε m , b ) ( i = 1 N V m i ) + i ( 1 ε o ω ) i = 1 N ( σ m i V m i ) ,
i = 1 N σ m i V m i = V M ( n e 2 m ) i Q s i ρ i γ i ( ω γ i ' ) ,
ε e f f = f f [ ( 1 + ε m , b ) + i ( ω p 2 ω )     i Q s i ρ i γ i ( ω γ i ' ) ] + ε s m ( 1 f f ) ,

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