Abstract

Two dimensional arrays of monodispersed Ag-nanoparticles separated by different gaps with sub-10 nm precision are fabricated on anodic alumina substrates with self-organized pores. Light scattering spectra from the arrays evolve with the gaps, revealing plasmonic coupling among the nanoparticles, which can be satisfactorily interpreted by analytical formulae derived from generic dipolar approximation. The general formulism lays down a foundation for predicting the Q factor of an array of metallic nano-particles and its geometric characteristics.

© 2008 Optical Society of America

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    [CrossRef]
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  14. W.  Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, "Optical properties of two interacting gold nanoparticles," Opt. Commun. 220, 137-141 (2003).
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    [CrossRef]
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    [CrossRef]

2007

P. K. Jain, W. Huang, and M. A. El-Sayed, "On the universal scaling behavior of the distance decay of plasmon coupling in metal nanoparticle pairs: A Plasmon Ruler Equation," Nano Lett. 7, 2080-2088 (2007).
[CrossRef]

2006

H. H. Wang,  et al., "Highly Raman Enhancing-Substrates Based on Silver nanoparticle Arrays with Tunable Sub-10 nm Gaps," Adv. Mater. 18, 491-495 (2006).
[CrossRef]

B. Khlebtsov, A. Melnikov, B. Zharov, and N. Khlebtsov, "Absorption and scattering of light by a dimmer of metal nanospheres: Comparison of dipole and multipole approaches," Nanotech. 17, 1437-1445 (2006).
[CrossRef]

2005

S. Zou and G. C. Schatz, Response to comment on "Silver nanoparticle array structures that produce remarkable narrow plasmon line shapes,"J. Chem. Phys. 102, 122 (2005).

D. W. Thompson, "Optical characterization of porous alumina from vacuum ultraviolet to midinfrared," J. Appl. Phys. 97, 113511 (2005).
[CrossRef]

L.  Gunnarsson, T.  Rindzevicious, J. Prikulis, B. Kasemo, M. Käll, S. Zou, and G. C. Schatz, "Confined Plasmons in Nanofabricated Single Silver Particle Pairs: Experimental Observations of Strong Interparticle Interactions," J. Phys. Chem. B 109, 1079-1087 (2005).

2004

2003

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424,824-830 (2003).
[CrossRef] [PubMed]

K. H. Su, Q. H. Wei, X. Zhang, J. J. Mock, D. R. Smith, and S. Schultz, "Interparticle coupling effects on plasmon resonances of nanogold particles," Nano Lett. 3, 1087-1090 (2003).
[CrossRef]

C. L. Haynes, A. D. McFarland, L. L. Zao, R. P. Von Duyne, G. C. Schatz, L. Gunnarsson, J. Prikulis, B. Kasemo, and M. Käll, "Nanoparticle Optics: The Importance of Radiative Dipole Coupling in Two-Dimensional Nanoparticle Arrays," J. Phys. Chem. B 107, 7337-7342 (2003).

W.  Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, "Optical properties of two interacting gold nanoparticles," Opt. Commun. 220, 137-141 (2003).
[CrossRef]

2001

1997

S. Nie and S. R. Emory, "Probing Single Molecules and Single Nanoparticles by Surface-Enhanced Raman Scattering," Science 275, 1102-1106 (1997).
[CrossRef] [PubMed]

1995

J. C. Hulteen and R. P. Van Duyne, "Nanosphere lithography: A materials general fabrication process for periodic particle array surfaces," J. Vac. Sci. Technol. A 13, 1553-1558 (1995).

X. -D. Xiang,  et al., "A Combinatorial Approach to Materials Discovery," Science 268, 1738-1740 (1995).
[CrossRef] [PubMed]

1993

V. A. Markel, "Coupled Dipole approach to Scattering of Light from a One-Diemnsional Periodic Dipole Structure," J. Mod. Opt. 40, 2281-2291 (1993).
[CrossRef]

1988

B. T. Draine, "The Discrete-Dipole Approxiamtion and its Application to Interstellar graphite Grains," Astrophys. J. 333, 848-872 (1988).
[CrossRef]

1985

M. Moskovits, "Surface enhanced spectroscopy," Rev. Mod. Phys. 57, 783-826 (1985).
[CrossRef]

1983

B. N. J. Persson and A. Liebsch, "Optical properties of two-dimensional systems of randomly distributed particles," Phys. Rev. B 28, 4247-4254 (1983).

1977

J. J. Olivero and R. L. Longbothum, "Empirical fits to the Voigt linewidth: A brief review" J. Quant. Spectrosc. Radiat. Transfer 17, 233-236 (1977).
[CrossRef]

1908

G. Mie, "Beitrage zur optik truber medien speziel kolloidaler metallosungen," Ann. Phys. 25, 377-445 (1908).
[CrossRef]

1857

M. Faraday, "On the color of colloidal gold," Phil. Trans. R. Soc. London 147, 145-181 (1857).
[CrossRef]

Aussenegg, F. R.

W.  Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, "Optical properties of two interacting gold nanoparticles," Opt. Commun. 220, 137-141 (2003).
[CrossRef]

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424,824-830 (2003).
[CrossRef] [PubMed]

Christofilos, D.

C. Voisin, N. D. Fatti, D. Christofilos, and F. Vallee, "Ultrafast Electron Dynamics and Optical Nonlinearities in Metal Nanoparticles," J. Phys. Chem. B,  105, 2264-2280 (2001).

Collinge, M. J.

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424,824-830 (2003).
[CrossRef] [PubMed]

Draine, B. T.

M. J. Collinge and B. T. Draine, "Discrete-dipole approximation with polarizabilities that account for both finite wavelength and target geometry," J. Opt. Soc. Am. A 21, 2023-2028 (2004).
[CrossRef]

B. T. Draine, "The Discrete-Dipole Approxiamtion and its Application to Interstellar graphite Grains," Astrophys. J. 333, 848-872 (1988).
[CrossRef]

Duan, C.

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424,824-830 (2003).
[CrossRef] [PubMed]

El-Sayed, M. A.

P. K. Jain, W. Huang, and M. A. El-Sayed, "On the universal scaling behavior of the distance decay of plasmon coupling in metal nanoparticle pairs: A Plasmon Ruler Equation," Nano Lett. 7, 2080-2088 (2007).
[CrossRef]

Emory, S. R.

S. Nie and S. R. Emory, "Probing Single Molecules and Single Nanoparticles by Surface-Enhanced Raman Scattering," Science 275, 1102-1106 (1997).
[CrossRef] [PubMed]

Faraday, M.

M. Faraday, "On the color of colloidal gold," Phil. Trans. R. Soc. London 147, 145-181 (1857).
[CrossRef]

Fatti, N. D.

C. Voisin, N. D. Fatti, D. Christofilos, and F. Vallee, "Ultrafast Electron Dynamics and Optical Nonlinearities in Metal Nanoparticles," J. Phys. Chem. B,  105, 2264-2280 (2001).

Geissler, M.

M. Geissler and Y. Xia, "Patterning: Principles and some new developments," Adv. Mater. 16, 1249-1269 (2004).
[CrossRef]

Gunnarsson, L.

L.  Gunnarsson, T.  Rindzevicious, J. Prikulis, B. Kasemo, M. Käll, S. Zou, and G. C. Schatz, "Confined Plasmons in Nanofabricated Single Silver Particle Pairs: Experimental Observations of Strong Interparticle Interactions," J. Phys. Chem. B 109, 1079-1087 (2005).

C. L. Haynes, A. D. McFarland, L. L. Zao, R. P. Von Duyne, G. C. Schatz, L. Gunnarsson, J. Prikulis, B. Kasemo, and M. Käll, "Nanoparticle Optics: The Importance of Radiative Dipole Coupling in Two-Dimensional Nanoparticle Arrays," J. Phys. Chem. B 107, 7337-7342 (2003).

Guo, Y.

Haynes, C. L.

C. L. Haynes, A. D. McFarland, L. L. Zao, R. P. Von Duyne, G. C. Schatz, L. Gunnarsson, J. Prikulis, B. Kasemo, and M. Käll, "Nanoparticle Optics: The Importance of Radiative Dipole Coupling in Two-Dimensional Nanoparticle Arrays," J. Phys. Chem. B 107, 7337-7342 (2003).

Hohenau, A.

W.  Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, "Optical properties of two interacting gold nanoparticles," Opt. Commun. 220, 137-141 (2003).
[CrossRef]

Huang, G.

Huang, W.

P. K. Jain, W. Huang, and M. A. El-Sayed, "On the universal scaling behavior of the distance decay of plasmon coupling in metal nanoparticle pairs: A Plasmon Ruler Equation," Nano Lett. 7, 2080-2088 (2007).
[CrossRef]

Hulteen, J. C.

J. C. Hulteen and R. P. Van Duyne, "Nanosphere lithography: A materials general fabrication process for periodic particle array surfaces," J. Vac. Sci. Technol. A 13, 1553-1558 (1995).

Jain, P. K.

P. K. Jain, W. Huang, and M. A. El-Sayed, "On the universal scaling behavior of the distance decay of plasmon coupling in metal nanoparticle pairs: A Plasmon Ruler Equation," Nano Lett. 7, 2080-2088 (2007).
[CrossRef]

Käll, M.

L.  Gunnarsson, T.  Rindzevicious, J. Prikulis, B. Kasemo, M. Käll, S. Zou, and G. C. Schatz, "Confined Plasmons in Nanofabricated Single Silver Particle Pairs: Experimental Observations of Strong Interparticle Interactions," J. Phys. Chem. B 109, 1079-1087 (2005).

C. L. Haynes, A. D. McFarland, L. L. Zao, R. P. Von Duyne, G. C. Schatz, L. Gunnarsson, J. Prikulis, B. Kasemo, and M. Käll, "Nanoparticle Optics: The Importance of Radiative Dipole Coupling in Two-Dimensional Nanoparticle Arrays," J. Phys. Chem. B 107, 7337-7342 (2003).

Kasemo, B.

L.  Gunnarsson, T.  Rindzevicious, J. Prikulis, B. Kasemo, M. Käll, S. Zou, and G. C. Schatz, "Confined Plasmons in Nanofabricated Single Silver Particle Pairs: Experimental Observations of Strong Interparticle Interactions," J. Phys. Chem. B 109, 1079-1087 (2005).

C. L. Haynes, A. D. McFarland, L. L. Zao, R. P. Von Duyne, G. C. Schatz, L. Gunnarsson, J. Prikulis, B. Kasemo, and M. Käll, "Nanoparticle Optics: The Importance of Radiative Dipole Coupling in Two-Dimensional Nanoparticle Arrays," J. Phys. Chem. B 107, 7337-7342 (2003).

Khlebtsov, B.

B. Khlebtsov, A. Melnikov, B. Zharov, and N. Khlebtsov, "Absorption and scattering of light by a dimmer of metal nanospheres: Comparison of dipole and multipole approaches," Nanotech. 17, 1437-1445 (2006).
[CrossRef]

Khlebtsov, N.

B. Khlebtsov, A. Melnikov, B. Zharov, and N. Khlebtsov, "Absorption and scattering of light by a dimmer of metal nanospheres: Comparison of dipole and multipole approaches," Nanotech. 17, 1437-1445 (2006).
[CrossRef]

Kotmann, J. P.

Krenn, J. R.

W.  Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, "Optical properties of two interacting gold nanoparticles," Opt. Commun. 220, 137-141 (2003).
[CrossRef]

Lamprecht, B.

W.  Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, "Optical properties of two interacting gold nanoparticles," Opt. Commun. 220, 137-141 (2003).
[CrossRef]

Leitner, A.

W.  Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, "Optical properties of two interacting gold nanoparticles," Opt. Commun. 220, 137-141 (2003).
[CrossRef]

Liebsch, A.

B. N. J. Persson and A. Liebsch, "Optical properties of two-dimensional systems of randomly distributed particles," Phys. Rev. B 28, 4247-4254 (1983).

Lin, J.

Liu, Y.

Longbothum, R. L.

J. J. Olivero and R. L. Longbothum, "Empirical fits to the Voigt linewidth: A brief review" J. Quant. Spectrosc. Radiat. Transfer 17, 233-236 (1977).
[CrossRef]

Markel, V. A.

V. A. Markel, "Coupled Dipole approach to Scattering of Light from a One-Diemnsional Periodic Dipole Structure," J. Mod. Opt. 40, 2281-2291 (1993).
[CrossRef]

Martin, O. J. F.

McFarland, A. D.

C. L. Haynes, A. D. McFarland, L. L. Zao, R. P. Von Duyne, G. C. Schatz, L. Gunnarsson, J. Prikulis, B. Kasemo, and M. Käll, "Nanoparticle Optics: The Importance of Radiative Dipole Coupling in Two-Dimensional Nanoparticle Arrays," J. Phys. Chem. B 107, 7337-7342 (2003).

Melnikov, A.

B. Khlebtsov, A. Melnikov, B. Zharov, and N. Khlebtsov, "Absorption and scattering of light by a dimmer of metal nanospheres: Comparison of dipole and multipole approaches," Nanotech. 17, 1437-1445 (2006).
[CrossRef]

Mie, G.

G. Mie, "Beitrage zur optik truber medien speziel kolloidaler metallosungen," Ann. Phys. 25, 377-445 (1908).
[CrossRef]

Mock, J. J.

K. H. Su, Q. H. Wei, X. Zhang, J. J. Mock, D. R. Smith, and S. Schultz, "Interparticle coupling effects on plasmon resonances of nanogold particles," Nano Lett. 3, 1087-1090 (2003).
[CrossRef]

Moskovits, M.

M. Moskovits, "Surface enhanced spectroscopy," Rev. Mod. Phys. 57, 783-826 (1985).
[CrossRef]

Nie, S.

S. Nie and S. R. Emory, "Probing Single Molecules and Single Nanoparticles by Surface-Enhanced Raman Scattering," Science 275, 1102-1106 (1997).
[CrossRef] [PubMed]

Olivero, J. J.

J. J. Olivero and R. L. Longbothum, "Empirical fits to the Voigt linewidth: A brief review" J. Quant. Spectrosc. Radiat. Transfer 17, 233-236 (1977).
[CrossRef]

Persson, B. N. J.

B. N. J. Persson and A. Liebsch, "Optical properties of two-dimensional systems of randomly distributed particles," Phys. Rev. B 28, 4247-4254 (1983).

Prikulis, J.

L.  Gunnarsson, T.  Rindzevicious, J. Prikulis, B. Kasemo, M. Käll, S. Zou, and G. C. Schatz, "Confined Plasmons in Nanofabricated Single Silver Particle Pairs: Experimental Observations of Strong Interparticle Interactions," J. Phys. Chem. B 109, 1079-1087 (2005).

C. L. Haynes, A. D. McFarland, L. L. Zao, R. P. Von Duyne, G. C. Schatz, L. Gunnarsson, J. Prikulis, B. Kasemo, and M. Käll, "Nanoparticle Optics: The Importance of Radiative Dipole Coupling in Two-Dimensional Nanoparticle Arrays," J. Phys. Chem. B 107, 7337-7342 (2003).

Rechberger, W.

W.  Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, "Optical properties of two interacting gold nanoparticles," Opt. Commun. 220, 137-141 (2003).
[CrossRef]

Rindzevicious, T.

L.  Gunnarsson, T.  Rindzevicious, J. Prikulis, B. Kasemo, M. Käll, S. Zou, and G. C. Schatz, "Confined Plasmons in Nanofabricated Single Silver Particle Pairs: Experimental Observations of Strong Interparticle Interactions," J. Phys. Chem. B 109, 1079-1087 (2005).

Schatz, G. C.

L.  Gunnarsson, T.  Rindzevicious, J. Prikulis, B. Kasemo, M. Käll, S. Zou, and G. C. Schatz, "Confined Plasmons in Nanofabricated Single Silver Particle Pairs: Experimental Observations of Strong Interparticle Interactions," J. Phys. Chem. B 109, 1079-1087 (2005).

S. Zou and G. C. Schatz, Response to comment on "Silver nanoparticle array structures that produce remarkable narrow plasmon line shapes,"J. Chem. Phys. 102, 122 (2005).

C. L. Haynes, A. D. McFarland, L. L. Zao, R. P. Von Duyne, G. C. Schatz, L. Gunnarsson, J. Prikulis, B. Kasemo, and M. Käll, "Nanoparticle Optics: The Importance of Radiative Dipole Coupling in Two-Dimensional Nanoparticle Arrays," J. Phys. Chem. B 107, 7337-7342 (2003).

Schultz, S.

K. H. Su, Q. H. Wei, X. Zhang, J. J. Mock, D. R. Smith, and S. Schultz, "Interparticle coupling effects on plasmon resonances of nanogold particles," Nano Lett. 3, 1087-1090 (2003).
[CrossRef]

Smith, D. R.

K. H. Su, Q. H. Wei, X. Zhang, J. J. Mock, D. R. Smith, and S. Schultz, "Interparticle coupling effects on plasmon resonances of nanogold particles," Nano Lett. 3, 1087-1090 (2003).
[CrossRef]

Su, K. H.

K. H. Su, Q. H. Wei, X. Zhang, J. J. Mock, D. R. Smith, and S. Schultz, "Interparticle coupling effects on plasmon resonances of nanogold particles," Nano Lett. 3, 1087-1090 (2003).
[CrossRef]

Thompson, D. W.

D. W. Thompson, "Optical characterization of porous alumina from vacuum ultraviolet to midinfrared," J. Appl. Phys. 97, 113511 (2005).
[CrossRef]

Vallee, F.

C. Voisin, N. D. Fatti, D. Christofilos, and F. Vallee, "Ultrafast Electron Dynamics and Optical Nonlinearities in Metal Nanoparticles," J. Phys. Chem. B,  105, 2264-2280 (2001).

Van Duyne, R. P.

J. C. Hulteen and R. P. Van Duyne, "Nanosphere lithography: A materials general fabrication process for periodic particle array surfaces," J. Vac. Sci. Technol. A 13, 1553-1558 (1995).

Voisin, C.

C. Voisin, N. D. Fatti, D. Christofilos, and F. Vallee, "Ultrafast Electron Dynamics and Optical Nonlinearities in Metal Nanoparticles," J. Phys. Chem. B,  105, 2264-2280 (2001).

Von Duyne, R. P.

C. L. Haynes, A. D. McFarland, L. L. Zao, R. P. Von Duyne, G. C. Schatz, L. Gunnarsson, J. Prikulis, B. Kasemo, and M. Käll, "Nanoparticle Optics: The Importance of Radiative Dipole Coupling in Two-Dimensional Nanoparticle Arrays," J. Phys. Chem. B 107, 7337-7342 (2003).

Wang, H. H.

H. H. Wang,  et al., "Highly Raman Enhancing-Substrates Based on Silver nanoparticle Arrays with Tunable Sub-10 nm Gaps," Adv. Mater. 18, 491-495 (2006).
[CrossRef]

Wei, Q. H.

K. H. Su, Q. H. Wei, X. Zhang, J. J. Mock, D. R. Smith, and S. Schultz, "Interparticle coupling effects on plasmon resonances of nanogold particles," Nano Lett. 3, 1087-1090 (2003).
[CrossRef]

Xia, Y.

M. Geissler and Y. Xia, "Patterning: Principles and some new developments," Adv. Mater. 16, 1249-1269 (2004).
[CrossRef]

Xiang, X. -D.

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Figures (6)

Fig. 1.
Fig. 1.

Variation in interparticle spacing (Δr) vs. mean interparticle spacing (r̄). Inset shows the histogram of interparticle spacing (r) for a sample with r̄=30 nm.

Fig. 2.
Fig. 2.

Normalized scattering spectra from Ag nanoparticle arrays with mean interparticle spacings: 50 nm (violet line), 45 nm (blue line), 40 nm (green line), 35 nm (orange line), and 30 nm (red line).

Fig. 3.
Fig. 3.

Transverse-mode resonance peak (Ω T ) and Lorentzian width (Γ L ) vs. mean interparticle spacing (r̄). Red solid lines are theoretical fittings.

Fig. 4.
Fig. 4.

Scattering intensity at the peak of transverse-mode resonance, I * T ), (filled circles) and Q factor (open squares) vs. mean interparticle spacing (r̄). Red solid line is a theoretical fitting.

Appendix Fig. 1.
Appendix Fig. 1.

Top view scanning electron microscopy image of Ag/AAO substrate with a mean diameter of 25 nm and a mean interparticle spacing of 30 nm. Inset shows the corresponding cross-section transmission electron microscopy image.

Appendix Fig. 2.
Appendix Fig. 2.

Voigt line width (ΓV) vs. mean interparticle spacing (r̄).

Tables (1)

Tables Icon

Table 1. Comparison of the fitted plasma frequency, ωp , and relaxation time, τ, of the Drude model from the optical constant, ε, of Ag (Ref. 22) and those from the variation of the spectral peak shifting, Ω T , the spectral width broadening, ΓL , and the peak intensity, I T ), based on Eqs. (4)-(6). The errors represent the fitting error. The analysis of the fitting results is given in Appendix.

Equations (21)

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α eff = α T 1 α T U ,
U = Σ j i [ k 2 sin 2 θ ij r i j i k ( 3 cos 2 θ i j 1 ) r i j 2 + ( 3 cos 2 θ i j 1 ) r i j 3 ] e i k r i j ,
U F · r 3 + i B
Ω T 2 = ω p 2 [ 1 F ( S r ) 3 ] ( 1 + C ε m ) + ( ε m 1 ) F ( S r ) 3 ,
Γ = 1 τ + S 3 B ω p 2 ( 1 + C ε m ) Ω T + ε m 1 1 + C ε m S 3 B Ω T ,
I ( Ω T ) N ( r ) Ω T 4 α eff ( Ω T ) 2 = N ( r ) { Ω T S 3 [ ( ε m 1 ) Ω T 2 + ω p 2 ] Γ ( 1 + C ε m ) } 2 ,
α T ( ω ) = R 2 h ε ( ω ) ε m 3 ε m + 3 L [ ε ( ω ) ε m ]
ε ( ω ) = 1 ω p 2 ω ( ω + i τ )
α T ( ω ) ( S 3 1 + C ε m ) [ ( ε m 1 ) ω 2 + ω p 2 ] ω p 2 1 + C ε m ω 2 i ω τ
E loc ( r i ) = E inc ( r i ) + Σ j i e ikr ij r ij 3 [ k 2 r ij × ( r ij × P j ) + 1 i k r ij r ij 2 × { r ij 2 P j 3 r ij ( r ij · P j ) } ]
α eff = α T 1 α T U
U = Σ j i [ k 2 sin 2 θ ij r ij i k ( 3 cos 2 θ ij 1 ) r ij 2 + ( 3 cos 2 θ ij 1 ) r ij 3 ] e ikr ij
α eff = S 3 [ ( ε m 1 ) ω 2 + ω p 2 ] [ ( 1 S 3 A ) ω p 2 { ( 1 + C ε m ) + ( ε m 1 ) S 3 A } ω 2 ] i [ ( ε m 1 ) S 3 B ω 2 + ( 1 + C ε m ) ( ω τ ) + S 3 B ω p 2 ]
U F · r 3 + i B
Ω T 2 = ω p 2 [ 1 F ( S r ) 3 ] ( 1 + C ε m ) + ( ε m 1 ) F ( S r ) 3 .
Γ = 1 τ + S 3 B ω p 2 ( 1 + C ε m ) Ω T + ε m 1 1 + C ε m S 3 B Ω T .
I ( Ω T ) N ( r ) Ω T 4 α eff ( Ω T ) 2 = N ( r ) { Ω T S 3 [ ( ε m 1 ) Ω T 2 + ω p 2 ] Γ ( 1 + C ε m ) } 2 ,
Ω T = ( ω p 3 ) 1 F ( R r ) 3 ,
Γ = 1 τ + R 3 B ω p 2 3 Ω T ,
I ( Ω T ) N ( r ) R 6 ω p 4 ( Ω T Γ ) 2 .
Δ Ω T = 3 2 F 2 S 3 ε m ( C + 1 ) Ω T r 6 [ 1 + C ε m + ( ε m 1 ) F ( S r ) 3 ] 2 Δ r

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