Abstract

An effective-medium theory (EMT) is developed to predict the effective permittivity εeff of dense random dispersions of high optical-conductivity metals such as Ag, Au, and Cu. Dependence of εeff on the volume fraction ϕ, a microstructure parameter κ related to the static structure factor and particle radius a, is studied. In the electrostatic limit, the upper and lower bounds of κ correspond to Maxwell–Garnett and Bruggeman EMTs, respectively. Finite size effects are significant when |β2(ka/n)3| becomes O(1), where β, k, and n denote the nanoparticle polarizability, wavenumber, and matrix refractive index, respectively. The coupling between the particle and effective medium results in a red-shift in the resonance peak, a nonlinear dependence of εeff on ϕ, and Fano resonance in εeff.

© 2012 Optical Society of America

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2011 (3)

T. Cong, S. N. Wani, P. A. Paynter, and R. Sureshkumar, “Structure and optical properties of self-assembled multicomponent plasmonic nanogels,” Appl. Phys. Lett. 99, 043112 (2011).
[CrossRef]

D. Erickson, D. Sinton, and D. Psaltis, “Optofluidics for energy applications,” Nat. Photon. 5, 583–590 (2011).
[CrossRef]

N. J. Halas, S. Lal, W. S. Chang, S. Link, and P. Nordlander, “Plasmons in strongly coupled metallic nanostructures,” Chem. Rev. 111, 3913–3961 (2011).
[CrossRef]

2010 (7)

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9, 707–715 (2010).
[CrossRef]

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257–2298 (2010).
[CrossRef]

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[CrossRef]

S. Torkamani, S. N. Wani, Y. J. Tang, and R. Sureshkumar, “Plasmon-enhanced microalgal growth in miniphotobioreactors,” Appl. Phys. Lett. 97, 043703–043703 (2010).
[CrossRef]

J. Yao, A. P. Le, S. K. Gray, J. S. Moore, J. A. Rogers, and R. G. Nuzzo, “Functional nanostructured plasmonic materials,” Adv. Mater. 22, 1102–1110 (2010).
[CrossRef]

P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photon. Rev. 4, 795–808 (2010).
[CrossRef]

V. E. Ferry, J. N. Munday, and H. A. Atwater, “Design considerations for plasmonic photovoltaics,” Adv. Mater. 22, 4794–4808 (2010).
[CrossRef]

2009 (2)

M. G. Blaber, M. D. Arnold, and M. J. Ford, “Search for the ideal plasmonic nanoshell: the effects of surface scattering and alternatives to gold and silver,” J. Phys. Chem. C 113, 3041–3045 (2009).
[CrossRef]

H. Garcia, R. Kalyanaraman, and R. Sureshkumar, “Nonlinear optical properties of multi-metal nanocomposites in a glass matrix,” J. Phys. B 42, 175401 (2009).
[CrossRef]

2008 (1)

J. Biener, G. W. Nyce, A. M. Hodge, M. M. Biener, A. V. Hamza, and S. A. Maier, “Nanoporous plasmonic metamaterials,” Adv. Mater. 20, 1211–1217 (2008).
[CrossRef]

2007 (5)

W. A. Murray and W. L. Barnes, “Plasmonic materials,” Adv. Mater. 19, 3771–3782 (2007).
[CrossRef]

J. Trice, D. Thomas, C. Favazza, R. Sureshkumar, and R. Kalyanaraman, “Pulsed-laser-induced dewetting in nanoscopic metal films: theory and experiments,” Phys. Rev. B 75, 235439 (2007).
[CrossRef]

S. Pillai, K. R. Catchpole, T. Trupke, and M. A. Green, “Surface plasmon enhanced silicon solar cells,” J. Appl. Phys. 101, 093105–093108 (2007).
[CrossRef]

F. Kaminski, V. Sandoghdar, and M. Agio, “Finite-difference time-domain modeling of decay rates in the near field of metal nanostructures,” J. Comp. Theor. Nanosci. 4, 635–643 (2007).

H. Garcia, J. Trice, R. Kalyanaraman, and R. Sureshkumar, “Self-consistent determination of plasmonic resonances in ternary nanocomposites,” Phys. Rev. B 75, 045439 (2007).
[CrossRef]

2006 (4)

V. Yannopapas, “Effective-medium description of disordered photonic alloys,” J. Opt. Soc. Am. B 23, 1414–1419 (2006).
[CrossRef]

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

L. M. Liz-Marzán, “Tailoring surface plasmons through the morphology and assembly of metal nanoparticles,” Langmuir 22, 32–41 (2006).
[CrossRef]

O. Popov, A. Zilbershtein, and D. Davidov, “Random lasing from dye-gold nanoparticles in polymer films: enhanced gain at the surface-plasmon-resonance wavelength,” Appl. Phys. Lett. 89,191116 (2006).
[CrossRef]

2005 (1)

P. Mallet, C. A. Guérin, and A. Sentenac, “Maxwell–Garnett mixing rule in the presence of multiple scattering: derivation and accuracy,” Phys. Rev. B 72, 014205 (2005).
[CrossRef]

2004 (2)

C. Oubre and P. Nordlander, “Optical properties of metallodielectric nanostructures calculated using the finite difference time domain method,” J. Phys. Chem. B 108, 17740–17747 (2004).
[CrossRef]

A. Biswas, O. C. Aktas, U. Schurmann, U. Saeed, V. Zaporojtchenko, F. Faupel, and T. Strunskus, “Tunable multiple plasmon resonance wavelengths response from multicomponent polymer-metal nanocomposite systems,” Appl. Phys. Lett. 84, 2655–2657 (2004).
[CrossRef]

2003 (1)

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles:  the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[CrossRef]

2002 (1)

S. Koo and A. S. Sangani, “Effective-medium theories for predicting hydrodynamic transport properties of bidisperse suspensions,” Phys. Fluids 14, 3522–3533 (2002).
[CrossRef]

2001 (4)

P. D. M. Spelt, M. A. Norato, A. S. Sangani, M. S. Greenwood, and L. L. Tavlarides, “Attenuation of sound in concentrated suspensions: theory and experiments,” J. Fluid Mech. 430, 51–86 (2001).
[CrossRef]

D. D. Smith, L. A. Snow, L. Sibille, and E. Ignont, “Tunable optical properties of metal nanoparticle sol–gel composites,” J. Non-Cryst. Solids 285, 256–263 (2001).
[CrossRef]

A. Dawson and P. V. Kamat, “Semiconductor–metal nanocomposites. Photoinduced fusion and photocatalysis of gold-capped TiO2 (TiO2/Gold) nanoparticles,” J. Phys. Chem. B 105, 960–966 (2001).
[CrossRef]

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, “Plasmonics—a route to nanoscale optical devices,” Adv. Mater. 13, 1501–1505 (2001).
[CrossRef]

2000 (2)

1999 (1)

T. Li, J. Moon, A. A. Morrone, J. J. Mecholsky, D. R. Talham, and J. H. Adair, “Preparation of Ag/SiO2 nanosize composites by a reverse micelle and sol–gel technique,” Langmuir 15, 4328–4334 (1999).
[CrossRef]

1998 (1)

Z. Liu, H. Wang, H. Li, and X. Wang, “Red shift of plasmon resonance frequency due to the interacting Ag nanoparticles embedded in single crystal Sio2 by implantation,” Appl. Phys. Lett. 72, 1823–1825 (1998).
[CrossRef]

1995 (1)

T. L. Dodd, D. A. Hammer, A. S. Sangani, and D. L. Koch, “Numerical simulations of the effect of hydrodynamic interactions on diffusivities of integral membrane proteins,” J. Fluid Mech. 293, 147–180 (1995).
[CrossRef]

1994 (1)

G. Mo and A. S. Sangani, “A method for computing Stokes flow interactions among spherical objects and its application to suspensions of drops and porous particles,” Phys. Fluids 6, 1637–1652 (1994).
[CrossRef]

1991 (1)

A. S. Sangani, “A pairwise interaction theory for determining the linear acoustic properties of dilute bubbly liquids,” J. Fluid Mech. 232, 221–284 (1991).
[CrossRef]

1988 (2)

A. S. Sangani and C. Yao, “Bulk thermal conductivity of composites with spherical inclusions,” J. Appl. Phys. 63, 1334–1341 (1988).
[CrossRef]

R. L. Hightower and C. B. Richardson, “Resonant Mie scattering from a layered sphere,” Appl. Opt. 27, 4850–4855 (1988).
[CrossRef]

1987 (1)

A. S. Sangani and W. Lu, “Elastic coefficients of composites containing spherical inclusions in a periodic array,” J. Mech. Phys. Solids 35, 1–21 (1987).
[CrossRef]

1977 (1)

D. M. Wood and N. W. Ashcroft, “Effective medium theory of the optical properties of small particle composites,” Philos. Mag. 35, 269–280 (1977).
[CrossRef]

1969 (1)

N. F. Carnahan and K. E. Starling, “Equation of state for nonattracting rigid spheres,” J. Chem. Phys. 51, 635–636 (1969).
[CrossRef]

1961 (1)

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124, 1866 (1961).
[CrossRef]

1904 (1)

J. C. M. Garnett, “Colours in metal glasses and in metallic films,” Phil. Trans. R. Soc. A 203, 385–420 (1904).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical tables (Dover, 1970).

Adair, J. H.

T. Li, J. Moon, A. A. Morrone, J. J. Mecholsky, D. R. Talham, and J. H. Adair, “Preparation of Ag/SiO2 nanosize composites by a reverse micelle and sol–gel technique,” Langmuir 15, 4328–4334 (1999).
[CrossRef]

Agio, M.

F. Kaminski, V. Sandoghdar, and M. Agio, “Finite-difference time-domain modeling of decay rates in the near field of metal nanostructures,” J. Comp. Theor. Nanosci. 4, 635–643 (2007).

Aktas, O. C.

A. Biswas, O. C. Aktas, U. Schurmann, U. Saeed, V. Zaporojtchenko, F. Faupel, and T. Strunskus, “Tunable multiple plasmon resonance wavelengths response from multicomponent polymer-metal nanocomposite systems,” Appl. Phys. Lett. 84, 2655–2657 (2004).
[CrossRef]

Arnold, M. D.

M. G. Blaber, M. D. Arnold, and M. J. Ford, “Search for the ideal plasmonic nanoshell: the effects of surface scattering and alternatives to gold and silver,” J. Phys. Chem. C 113, 3041–3045 (2009).
[CrossRef]

Ashcroft, N. W.

D. M. Wood and N. W. Ashcroft, “Effective medium theory of the optical properties of small particle composites,” Philos. Mag. 35, 269–280 (1977).
[CrossRef]

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart, and Winston, 1976).

Atwater, H. A.

V. E. Ferry, J. N. Munday, and H. A. Atwater, “Design considerations for plasmonic photovoltaics,” Adv. Mater. 22, 4794–4808 (2010).
[CrossRef]

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[CrossRef]

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, “Plasmonics—a route to nanoscale optical devices,” Adv. Mater. 13, 1501–1505 (2001).
[CrossRef]

Barnes, W. L.

W. A. Murray and W. L. Barnes, “Plasmonic materials,” Adv. Mater. 19, 3771–3782 (2007).
[CrossRef]

Biener, J.

J. Biener, G. W. Nyce, A. M. Hodge, M. M. Biener, A. V. Hamza, and S. A. Maier, “Nanoporous plasmonic metamaterials,” Adv. Mater. 20, 1211–1217 (2008).
[CrossRef]

Biener, M. M.

J. Biener, G. W. Nyce, A. M. Hodge, M. M. Biener, A. V. Hamza, and S. A. Maier, “Nanoporous plasmonic metamaterials,” Adv. Mater. 20, 1211–1217 (2008).
[CrossRef]

Biswas, A.

A. Biswas, O. C. Aktas, U. Schurmann, U. Saeed, V. Zaporojtchenko, F. Faupel, and T. Strunskus, “Tunable multiple plasmon resonance wavelengths response from multicomponent polymer-metal nanocomposite systems,” Appl. Phys. Lett. 84, 2655–2657 (2004).
[CrossRef]

Blaber, M. G.

M. G. Blaber, M. D. Arnold, and M. J. Ford, “Search for the ideal plasmonic nanoshell: the effects of surface scattering and alternatives to gold and silver,” J. Phys. Chem. C 113, 3041–3045 (2009).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).

Boltasseva, A.

P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photon. Rev. 4, 795–808 (2010).
[CrossRef]

Brongersma, M. L.

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, “Plasmonics—a route to nanoscale optical devices,” Adv. Mater. 13, 1501–1505 (2001).
[CrossRef]

Carnahan, N. F.

N. F. Carnahan and K. E. Starling, “Equation of state for nonattracting rigid spheres,” J. Chem. Phys. 51, 635–636 (1969).
[CrossRef]

Catchpole, K. R.

S. Pillai, K. R. Catchpole, T. Trupke, and M. A. Green, “Surface plasmon enhanced silicon solar cells,” J. Appl. Phys. 101, 093105–093108 (2007).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Clarendon Press, 1961).

Chang, W. S.

N. J. Halas, S. Lal, W. S. Chang, S. Link, and P. Nordlander, “Plasmons in strongly coupled metallic nanostructures,” Chem. Rev. 111, 3913–3961 (2011).
[CrossRef]

Chong, C. T.

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9, 707–715 (2010).
[CrossRef]

Choy, T. C.

T. C. Choy, Effective Medium Theory (Oxford University, 1999).

Cong, T.

T. Cong, S. N. Wani, P. A. Paynter, and R. Sureshkumar, “Structure and optical properties of self-assembled multicomponent plasmonic nanogels,” Appl. Phys. Lett. 99, 043112 (2011).
[CrossRef]

Coronado, E.

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles:  the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[CrossRef]

Davidov, D.

O. Popov, A. Zilbershtein, and D. Davidov, “Random lasing from dye-gold nanoparticles in polymer films: enhanced gain at the surface-plasmon-resonance wavelength,” Appl. Phys. Lett. 89,191116 (2006).
[CrossRef]

Dawson, A.

A. Dawson and P. V. Kamat, “Semiconductor–metal nanocomposites. Photoinduced fusion and photocatalysis of gold-capped TiO2 (TiO2/Gold) nanoparticles,” J. Phys. Chem. B 105, 960–966 (2001).
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Figures (9)

Fig. 1.
Fig. 1.

A schematic of the geometry considered for the EMT. The problem of finding the conditionally averaged field in a random medium was approximated with the problem of calculating the fields in this geometry. As r, E(r)=x^exp(ikeffz). The unconditionally averaged wave is assumed to be X-polarized in the present analysis. The choice of r1 is arbitrary for a given origin O.

Fig. 2.
Fig. 2.

Real and imaginary parts of the permittivity of high optical-conductivity metals Ag, Au, and Cu that are considered in this work. Permittivity data is taken from [47]. Ag has the lowest imaginary permittivity over a broad range of wavelengths.

Fig. 3.
Fig. 3.

β for Ag spheres in a εm=7 medium. Resonance occurs for λ600nm and β23i. (β) is a small number away from resonance, and (β) changes sign from negative to positive upon moving from blue to red regions about resonance.

Fig. 4.
Fig. 4.

(εeff) predicted by the scalar EMT for random and well-separated microstructures. Here, λ680nm and β=4.973+0.973i. β has a resonance peak at λ600nm, as shown in Fig. 3.

Fig. 5.
Fig. 5.

εeff for a composite with Ag NPs in an εm=7 medium calculated with the scalar EMT. Random (a) and (b) and well-separated random composites (c) and (d) for ϕ=2% (solid), 6% (dashed), and 10% (dotted) were considered. The resonance peak is more red-shifted and broad for a random system. A well-separated system shows a more intense resonance with a symmetric peak in comparison. Stronger coupling in a random system leads to a tail in the blue region.

Fig. 6.
Fig. 6.

Linear coefficient A for composites in an εm=7 matrix with Ag NPs with diameters dp=10 (solid), 30 (dashed), 50 (dotted), and 100 nm (inset). Here, k*a0.45dpεpλ<1 only for the blue curve. Quadrupolar and octupolar resonance peaks are present for large particles, as seen in the curves in the insets. Dipole resonance is most prominent and red-shifts as dp is increased. The linear coefficient becomes less significant for large particles as they screen most of the E field from their interior.

Fig. 7.
Fig. 7.

Quadratic coefficient B calculated for random (a) and (b) and well-separated random (c) and (d) composites in an εm=7 matrix containing Ag NPs with diameters dp=10 (solid), 30 (dashed), 50 (dotted), and 100 nm (inset). Here, k*a0.45dpεpλ<1 is less than 1 only for the solid curve that is given by 34(β+4)β2 for (a) and (b); and 3β2 for (c) and (d). Weak coupling in well-separated random systems leads to a smaller B in comparison to random systems.

Fig. 8.
Fig. 8.

Effect of κ on εeff for ϕ=5%. Microstructures with κ=ϕ1/32.71 (solid curve), κ1.75 (dashed curve), and κ1.25 (dotted curve). Small values of κ lead to a stronger coupling that distorts the Lorentzian shape of εeff even for relatively small values of ϕ such as 5%. Calculations were performed with the scalar EMT.

Fig. 9.
Fig. 9.

Characterization of the dielectric response of a random plasmonic composite in the κϕ space. The dashed line represents the upper bound for a random composite with well-separated particles, and the gray dotted line represents a random hard-sphere composite. Locations of Lorentzian and Fano responses are shown in black and gray circles, respectively. Unfilled circles denote locations in which broad lineshapes are observed.

Equations (56)

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×Em=iωμmHm,×Hm=iωεmEm,
×E+gp×(EmEp)=iωμ0H,
×H+gp×(HmHp)=iω[εmE+(εpεm)gpE].
εeffE=εmE+(εpεm)gpE.
gpE(r)|rr1|aE(r)|r1P(r1)dVr1.
gpE(r)=Ω(ϕ,keff)ϕE(r),
keff2=km2(kp2km2)Ωϕ.
(2+keff2)E=0
Λ(keff2)=keff2km2ϕ(kp2km2)Ω(keff2)=0.
Ra=κ=(1S(0)ϕ)13,
S(0)0[P(r|r1)P(r1)]dr.
S(0)=(1ϕ)41+4ϕ+4ϕ24ϕ3+ϕ4.
Θp=Θm
εpΘr|p=εmΘr|m.
E=×(rΨ)+××(rΦ)
H=iωε×(rΦ)+1iωμ0××(rΨ),
Eθ=1sinθΨϕ+1r2(rΦ)rθ,
Eϕ=Ψθ+1r2(rΦ)rϕ,
Hθ=iωεsinθΦϕ+1iωμ0r2(rΨ)rθ,
Hϕ=iωεΦθ+1iωμ0r2(rΨ)rϕ.
Ψp=Ψm,
r(rΨ)|p=r(rΨ)|m,
εpΦp=εmΦm,
r(rΦ)|p=r(rΦ)|m.
=exp(ikeff·r1)3εmεp+2εmz^,r<a
ES(r)|r1=exp(ikeff·r1)9εmεeff(εp+2εm)(2εeff+εm)2κ3(εp+εm)(εeff+εm)z^,r<a
Er(r)|r1=exp(ikeff·r1)isinθcosϕkp2r2n=1dnin(2n+1)πn(θ)ψn(kpr),
Eθ(r)|r1=exp(ikeff·r1)cosϕkpr×n=1in(2n+1)n(n+1)[cnπn(θ)ψn(kpr)idnτn(θ)ψn(kpr)],
Eϕ(r)|r1=exp(ikeff·r1)sinϕkpr×n=1in(2n+1)n(n+1)[cnτn(θ)ψn(kpr)idnπn(θ)ψn(kpr)].
εeff/εm=1+3βϕ,β=(εpεmεp+2εm).
εeff/εm=1+9εeff(2εeff+εm)2κ3β(εeffεm)γ,γβϕ.
εeff/εm=1+2γ1γ=1+3γ+3γ2+O(γ3).
ϕ(εpεeffεp+2εeff)+(1ϕ)(εmεeffεm+2εeff)=0,
εeff/εm=1+3γ+34(β+4)γ2+O(γ3).
|β|,or equivalently,εp=2εm,
2β(εeff12εeff+1)=κ3.
β1ϕ.
β2ϕ.
εeff/εeff=1+Aϕ+Bϕ2+O(ϕ3).
A=3β+i2β2(k*a)3+O[(k*a)6],
βn=εpεmεp+n+1nεm.
A=1ϕ(εeffεm1).
B=1ϕ2(εeffεm1Aϕ).
cn=[kpψn(kma)Bnkpχn(kma)kmψn(kpa)][kmψn(keffR)bnkmζn(keffR)ψn(kmR)Bnχn(kmR)]
dn=[ψn(kma)Anχn(kma)ψn(kpa)][ψn(keffR)anζn(keffR)ψn(kmR)Anχn(kmR)].
ψn(z)zjn(z),
χn(z)zyn(z),
ζn(z)zhn(1)(z).
An=kmψn(kma)ψn(kpa)ψn(kma)ψn(kpa)kmχn(kma)ψn(kpa)χn(kma)ψn(kpa)
Bn=kmψn(kma)ψn(kpa)ψn(kma)ψn(kpa)kmχn(kma)ψn(kpa)χn(kma)ψn(kpa).
an=ψn(keffR)[ψn(kmR)Anχn(kmR)]kmψn(keffR)[ψn(kmR)Anχn(kmR)]ζn(keffR)[ψn(kmR)Anχn(kmR)]kmζn(keffR)[ψn(kmR)Anχn(kmR)]
bn=kmψn(keffR)[ψn(kmR)Bnχn(kmR)]ψn(keffR)[ψn(kmR)Bnχn(kmR)]kmζn(keffR)[ψn(kmR)Bnχn(kmR)]kmζn(keffR)[ψn(kmR)Bnχn(kmR)].
εeff=εm+q=1N[(εqεm)gqE].
Λ(keff2)=keff2km2q=1N[ϕq(kq2km2)Ωq(keff2)],
κqr=(δqrSqr(0)ϕq)13.
Sqr=[P(r,aq|0,ar)P(r,aq)]dVr.

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