Abstract

An analysis of radiative absorption and scattering by clusters of spheres in the Rayleigh limit is developed with an electrostatics analysis. This approach assumes that the largest dimension of the cluster is significantly smaller than the wavelength of the radiation. The electric field that is incident upon and scattered by the cluster can then be represented by the gradient of a potential that in turn satisfies Laplace’s equation. An analytical solution for the potential that exactly satisfies the boundary conditions at the surfaces of the spheres is obtained with a coupled spherical harmonics method. The components of the polarizability tensor and the absorption, scattering, and depolarization factors are obtained from the solution. Calculations are performed on fractallike clusters of spheres, with refractive-index values that are characteristic of carbonaceous soot in the visible and the IR wavelengths. Results indicate that the absorption cross sections of fractal soot clusters can be significantly larger in the mid-IR wavelengths than what is predicted for Rayleigh-limit spheres that have the same total volume. The absorption cross section (relative to a sphere of the same volume) is dependent on the number of spheres in the aggregate for aggregates with up to approximately 100 primary spheres, and for larger aggregates the relative absorption becomes constant. The predicted spectral variation of soot absorption in the visible and the mid-IR wavelengths is shown to agree well with experimental measurements.

© 1995 Optical Society of America

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References

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  1. R. Viskanta, M. P. Menguc, “Radiation heat transfer in combustion systems,” Prog. Energy Combust. Sci. 13, 97–160 (1987).
    [CrossRef]
  2. R. A. Dobbins, C. M. Megaridis, “Absorption and scattering of light by polydisperse aggregates,” Appl. Opt. 30, 4747–4754 (1991).
    [CrossRef] [PubMed]
  3. H. Horvath, “Atmospheric light absorption—a review,” Atmos. Environ. 27, 293–317 (1993).
    [CrossRef]
  4. R. A. Dobbins, C. M. Megaridis, “Morphology of flame-generated soot as determined by thermophoretic sampling,” Langmuir 3, 254–259 (1987).
    [CrossRef]
  5. R. Jullien, R. Botet, Aggregation and Fractal Aggregates (World Scientific, Singapore, 1987).
  6. R. D. Mountain, G. W. Mulholland, “Light scattering from simulated smoke agglomerates,” Langmuir 4, 1321–1326 (1988).
    [CrossRef]
  7. A. R. Jones, “Electromagnetic wave scattering by assemblies of particles in the Rayleigh approximation,” Proc. R. Soc. London Ser. A 366, 111–127 (1979).
    [CrossRef]
  8. M. V. Berry, I. C. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 577–591 (1986).
    [CrossRef]
  9. D. W. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A 11, 2851–2861 (1994).
    [CrossRef]
  10. D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London Ser. A 433, 599–614 (1991).
    [CrossRef]
  11. S. C. Lee, C. L. Tien, “Optical constants of soot in hydrocarbon flames,” in Eighteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, PA, 1981), pp. 1159–1166.
    [CrossRef]
  12. J. M. Gerardy, M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations: the long wavelength limit,” Phys. Rev. B 22, 4950–4959 (1980).
    [CrossRef]
  13. D. J. Jeffrey, “Conduction though a random suspension of spheres,” Proc. R. Soc. London Ser. A 335, 355–367 (1973).
    [CrossRef]
  14. J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics (Prentice-Hall, Englewood Cliffs, N.J., 1965).
  15. D. W. Mackowski, “Phoretic behavior of asymmetric particles in thermal nonequilibrium with the gas: two sphere aggregates,” J. Colloid Interface Sci. 140, 138–1157 (1990).
    [CrossRef]
  16. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  17. E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics (Cambridge U. Press, Cambridge, 1931).
  18. S. C. Lee, C. L. Tien, “Effect of soot shape on soot radiation,” J. Quant. Spectrosc. Radiat. Transfer 29, 259–265 (1983).
    [CrossRef]
  19. D. W. Mackowski, R. A. Altenkirch, M. P. Menguc, K. Saito, “Radiative properties of chain-agglomerated soot formed in hydrocarbon diffusion flames,” in Twenty-Second Symposium International on Combustion (The Combustion Institute, Pa., 1989), pp. 1263–1269.
  20. S. B. Singham, C. F. Borhen, “Scattering of unpolarized and polarized light by particle aggregates of different size and fractal dimension,” Langmuir 9, 1431–1435 (1993).
    [CrossRef]
  21. D. M. Roessler, F. R. Faxvog, “Optical properties of agglomerated acetylene smoke particles at 0.5145-μm and 10.6-μm wavelengths,” J. Opt. Soc. Am. 70, 230–235 (1980).
    [CrossRef]

1994

1993

S. B. Singham, C. F. Borhen, “Scattering of unpolarized and polarized light by particle aggregates of different size and fractal dimension,” Langmuir 9, 1431–1435 (1993).
[CrossRef]

H. Horvath, “Atmospheric light absorption—a review,” Atmos. Environ. 27, 293–317 (1993).
[CrossRef]

1991

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London Ser. A 433, 599–614 (1991).
[CrossRef]

R. A. Dobbins, C. M. Megaridis, “Absorption and scattering of light by polydisperse aggregates,” Appl. Opt. 30, 4747–4754 (1991).
[CrossRef] [PubMed]

1990

D. W. Mackowski, “Phoretic behavior of asymmetric particles in thermal nonequilibrium with the gas: two sphere aggregates,” J. Colloid Interface Sci. 140, 138–1157 (1990).
[CrossRef]

1988

R. D. Mountain, G. W. Mulholland, “Light scattering from simulated smoke agglomerates,” Langmuir 4, 1321–1326 (1988).
[CrossRef]

1987

R. A. Dobbins, C. M. Megaridis, “Morphology of flame-generated soot as determined by thermophoretic sampling,” Langmuir 3, 254–259 (1987).
[CrossRef]

R. Viskanta, M. P. Menguc, “Radiation heat transfer in combustion systems,” Prog. Energy Combust. Sci. 13, 97–160 (1987).
[CrossRef]

1986

M. V. Berry, I. C. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 577–591 (1986).
[CrossRef]

1983

S. C. Lee, C. L. Tien, “Effect of soot shape on soot radiation,” J. Quant. Spectrosc. Radiat. Transfer 29, 259–265 (1983).
[CrossRef]

1980

J. M. Gerardy, M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations: the long wavelength limit,” Phys. Rev. B 22, 4950–4959 (1980).
[CrossRef]

D. M. Roessler, F. R. Faxvog, “Optical properties of agglomerated acetylene smoke particles at 0.5145-μm and 10.6-μm wavelengths,” J. Opt. Soc. Am. 70, 230–235 (1980).
[CrossRef]

1979

A. R. Jones, “Electromagnetic wave scattering by assemblies of particles in the Rayleigh approximation,” Proc. R. Soc. London Ser. A 366, 111–127 (1979).
[CrossRef]

1973

D. J. Jeffrey, “Conduction though a random suspension of spheres,” Proc. R. Soc. London Ser. A 335, 355–367 (1973).
[CrossRef]

Altenkirch, R. A.

D. W. Mackowski, R. A. Altenkirch, M. P. Menguc, K. Saito, “Radiative properties of chain-agglomerated soot formed in hydrocarbon diffusion flames,” in Twenty-Second Symposium International on Combustion (The Combustion Institute, Pa., 1989), pp. 1263–1269.

Ausloos, M.

J. M. Gerardy, M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations: the long wavelength limit,” Phys. Rev. B 22, 4950–4959 (1980).
[CrossRef]

Berry, M. V.

M. V. Berry, I. C. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 577–591 (1986).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Borhen, C. F.

S. B. Singham, C. F. Borhen, “Scattering of unpolarized and polarized light by particle aggregates of different size and fractal dimension,” Langmuir 9, 1431–1435 (1993).
[CrossRef]

Botet, R.

R. Jullien, R. Botet, Aggregation and Fractal Aggregates (World Scientific, Singapore, 1987).

Brenner, H.

J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics (Prentice-Hall, Englewood Cliffs, N.J., 1965).

Dobbins, R. A.

R. A. Dobbins, C. M. Megaridis, “Absorption and scattering of light by polydisperse aggregates,” Appl. Opt. 30, 4747–4754 (1991).
[CrossRef] [PubMed]

R. A. Dobbins, C. M. Megaridis, “Morphology of flame-generated soot as determined by thermophoretic sampling,” Langmuir 3, 254–259 (1987).
[CrossRef]

Faxvog, F. R.

Gerardy, J. M.

J. M. Gerardy, M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations: the long wavelength limit,” Phys. Rev. B 22, 4950–4959 (1980).
[CrossRef]

Happel, J.

J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics (Prentice-Hall, Englewood Cliffs, N.J., 1965).

Hobson, E. W.

E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics (Cambridge U. Press, Cambridge, 1931).

Horvath, H.

H. Horvath, “Atmospheric light absorption—a review,” Atmos. Environ. 27, 293–317 (1993).
[CrossRef]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Jeffrey, D. J.

D. J. Jeffrey, “Conduction though a random suspension of spheres,” Proc. R. Soc. London Ser. A 335, 355–367 (1973).
[CrossRef]

Jones, A. R.

A. R. Jones, “Electromagnetic wave scattering by assemblies of particles in the Rayleigh approximation,” Proc. R. Soc. London Ser. A 366, 111–127 (1979).
[CrossRef]

Jullien, R.

R. Jullien, R. Botet, Aggregation and Fractal Aggregates (World Scientific, Singapore, 1987).

Lee, S. C.

S. C. Lee, C. L. Tien, “Effect of soot shape on soot radiation,” J. Quant. Spectrosc. Radiat. Transfer 29, 259–265 (1983).
[CrossRef]

S. C. Lee, C. L. Tien, “Optical constants of soot in hydrocarbon flames,” in Eighteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, PA, 1981), pp. 1159–1166.
[CrossRef]

Mackowski, D. W.

D. W. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A 11, 2851–2861 (1994).
[CrossRef]

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London Ser. A 433, 599–614 (1991).
[CrossRef]

D. W. Mackowski, “Phoretic behavior of asymmetric particles in thermal nonequilibrium with the gas: two sphere aggregates,” J. Colloid Interface Sci. 140, 138–1157 (1990).
[CrossRef]

D. W. Mackowski, R. A. Altenkirch, M. P. Menguc, K. Saito, “Radiative properties of chain-agglomerated soot formed in hydrocarbon diffusion flames,” in Twenty-Second Symposium International on Combustion (The Combustion Institute, Pa., 1989), pp. 1263–1269.

Megaridis, C. M.

R. A. Dobbins, C. M. Megaridis, “Absorption and scattering of light by polydisperse aggregates,” Appl. Opt. 30, 4747–4754 (1991).
[CrossRef] [PubMed]

R. A. Dobbins, C. M. Megaridis, “Morphology of flame-generated soot as determined by thermophoretic sampling,” Langmuir 3, 254–259 (1987).
[CrossRef]

Menguc, M. P.

R. Viskanta, M. P. Menguc, “Radiation heat transfer in combustion systems,” Prog. Energy Combust. Sci. 13, 97–160 (1987).
[CrossRef]

D. W. Mackowski, R. A. Altenkirch, M. P. Menguc, K. Saito, “Radiative properties of chain-agglomerated soot formed in hydrocarbon diffusion flames,” in Twenty-Second Symposium International on Combustion (The Combustion Institute, Pa., 1989), pp. 1263–1269.

Mountain, R. D.

R. D. Mountain, G. W. Mulholland, “Light scattering from simulated smoke agglomerates,” Langmuir 4, 1321–1326 (1988).
[CrossRef]

Mulholland, G. W.

R. D. Mountain, G. W. Mulholland, “Light scattering from simulated smoke agglomerates,” Langmuir 4, 1321–1326 (1988).
[CrossRef]

Percival, I. C.

M. V. Berry, I. C. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 577–591 (1986).
[CrossRef]

Roessler, D. M.

Saito, K.

D. W. Mackowski, R. A. Altenkirch, M. P. Menguc, K. Saito, “Radiative properties of chain-agglomerated soot formed in hydrocarbon diffusion flames,” in Twenty-Second Symposium International on Combustion (The Combustion Institute, Pa., 1989), pp. 1263–1269.

Singham, S. B.

S. B. Singham, C. F. Borhen, “Scattering of unpolarized and polarized light by particle aggregates of different size and fractal dimension,” Langmuir 9, 1431–1435 (1993).
[CrossRef]

Tien, C. L.

S. C. Lee, C. L. Tien, “Effect of soot shape on soot radiation,” J. Quant. Spectrosc. Radiat. Transfer 29, 259–265 (1983).
[CrossRef]

S. C. Lee, C. L. Tien, “Optical constants of soot in hydrocarbon flames,” in Eighteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, PA, 1981), pp. 1159–1166.
[CrossRef]

Viskanta, R.

R. Viskanta, M. P. Menguc, “Radiation heat transfer in combustion systems,” Prog. Energy Combust. Sci. 13, 97–160 (1987).
[CrossRef]

Appl. Opt.

Atmos. Environ.

H. Horvath, “Atmospheric light absorption—a review,” Atmos. Environ. 27, 293–317 (1993).
[CrossRef]

J. Colloid Interface Sci.

D. W. Mackowski, “Phoretic behavior of asymmetric particles in thermal nonequilibrium with the gas: two sphere aggregates,” J. Colloid Interface Sci. 140, 138–1157 (1990).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transfer

S. C. Lee, C. L. Tien, “Effect of soot shape on soot radiation,” J. Quant. Spectrosc. Radiat. Transfer 29, 259–265 (1983).
[CrossRef]

Langmuir

S. B. Singham, C. F. Borhen, “Scattering of unpolarized and polarized light by particle aggregates of different size and fractal dimension,” Langmuir 9, 1431–1435 (1993).
[CrossRef]

R. A. Dobbins, C. M. Megaridis, “Morphology of flame-generated soot as determined by thermophoretic sampling,” Langmuir 3, 254–259 (1987).
[CrossRef]

R. D. Mountain, G. W. Mulholland, “Light scattering from simulated smoke agglomerates,” Langmuir 4, 1321–1326 (1988).
[CrossRef]

Opt. Acta

M. V. Berry, I. C. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 577–591 (1986).
[CrossRef]

Phys. Rev. B

J. M. Gerardy, M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations: the long wavelength limit,” Phys. Rev. B 22, 4950–4959 (1980).
[CrossRef]

Proc. R. Soc. London Ser. A

D. J. Jeffrey, “Conduction though a random suspension of spheres,” Proc. R. Soc. London Ser. A 335, 355–367 (1973).
[CrossRef]

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London Ser. A 433, 599–614 (1991).
[CrossRef]

A. R. Jones, “Electromagnetic wave scattering by assemblies of particles in the Rayleigh approximation,” Proc. R. Soc. London Ser. A 366, 111–127 (1979).
[CrossRef]

Prog. Energy Combust. Sci.

R. Viskanta, M. P. Menguc, “Radiation heat transfer in combustion systems,” Prog. Energy Combust. Sci. 13, 97–160 (1987).
[CrossRef]

Other

R. Jullien, R. Botet, Aggregation and Fractal Aggregates (World Scientific, Singapore, 1987).

S. C. Lee, C. L. Tien, “Optical constants of soot in hydrocarbon flames,” in Eighteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, PA, 1981), pp. 1159–1166.
[CrossRef]

J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics (Prentice-Hall, Englewood Cliffs, N.J., 1965).

D. W. Mackowski, R. A. Altenkirch, M. P. Menguc, K. Saito, “Radiative properties of chain-agglomerated soot formed in hydrocarbon diffusion flames,” in Twenty-Second Symposium International on Combustion (The Combustion Institute, Pa., 1989), pp. 1263–1269.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics (Cambridge U. Press, Cambridge, 1931).

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Figures (10)

Fig. 1
Fig. 1

Sphere cluster coordinate system.

Fig. 2
Fig. 2

Isopotential lines, m = 1.6 + 0.6i, with (a) parallel and (b) perpendicular incident polarization.

Fig. 3
Fig. 3

Same as Fig. 2, but with m = 3 + 2i.

Fig. 4
Fig. 4

Comparison of electrostatics and wave-equation predictions of random-orientation absorption.

Fig. 5
Fig. 5

Comparison of electrostatics and wave-equation predictions of random-orientation scattering.

Fig. 6
Fig. 6

Absorption efficiency of straight chains of spheres and prolate spheroids (normalized by equal-volume sphere absorption efficiency) versus the chain length or the spheroid aspect ratio.

Fig. 7
Fig. 7

40-sphere fractal cluster generated with the sequential algorithm.

Fig. 8
Fig. 8

Absorption, scattering, and 90° polarization behavior of fractal clusters and straight chains versus the number of spheres.

Fig. 9
Fig. 9

Spectral refractive index used in calculations, from Lee and Tien.11

Fig. 10
Fig. 10

Absorption cross section of fractal cluster (relative to a sphere of equal volume) versus the radiation wavelength.

Tables (2)

Tables Icon

Table 1 Results for a Two-Sphere Cluster

Tables Icon

Table 2 Ratios of Spectral Absorption Coefficients at the 0.5145- and 10.6-μm Wavelengths for Soot Aggregates as Predicted by the Electrostatics (ES) Model for Fractal Clusters, Rayleigh-Limit Spherical Particles and Infinite-Length Cylinders, and Experimental Measurement Results from Roessler and Faxvog (Ref. 21)

Equations (51)

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E 2 = Φ 2 ,             E 1 , i = Φ 1 , i .
2 Φ = 0.
Φ 2 = Φ 0 + i = 1 N S Φ s , i .
Φ s , i = n = 1 m = - n n a m n i u m n - ( r i , θ i , ϕ i ) ,
u m n - = r i - ( n + 1 ) P n m ( cos θ i ) exp ( i m ϕ i ) .
Φ 1 , i = d 0 i + n = 1 m = - n n d m n i u m n + ( r i , θ i , ϕ i ) ,
u m n + = r i n P n m ( cos θ i ) exp ( i m ϕ i ) .
E 0 = e x E 0 x + e y E 0 y + e z E 0 z .
Φ 0 = x E 0 x + y E 0 y + z E 0 z ,
E 0 x 2 + E 0 y 2 + E 0 z 2 = 1.
Φ 0 = m = - 1 1 p m u m 1 + ( r , θ , ϕ ) ,
p - 1 = E 0 x + i E 0 y = sin ( β ) exp ( i γ ) , p 0 = E 0 z = cos ( β ) , p 1 = - ½ ( E 0 x - i E 0 y ) = - ½ sin ( β ) exp ( - i γ ) .
Φ 2 ( r = a i ) = Φ 1 , i ( r = a i ) ,
r · Φ 2 ( r = a i ) = i r · Φ 1 , i ( r = a i ) ,
u m n - ( r j , θ j , ϕ j ) = l = 1 k = - l l C m n k l i j u k l + ( r i , θ i , ϕ i ) ,             R i j > r i .
C m n k l i j = ( - 1 ) n + m ( n + l - m + k ) ! ( n - m ) ! ( l + k ) ! × u ( m - k ) ( n + l ) - ( R i j , Θ i j , Φ i j ) .
a m n i + f n i j = 1 j i N S l = 1 N O k = - l l C k l m n i j a k l j = - f 1 i p m δ n 1 ,             m = - n , - n + 1 , n ; n + 1 , 2 , N O .
f n i = a i 2 n + 1 n ( i - 1 ) n ( i + 1 ) + 1 .
Φ s r R C = p · r 4 π r 3 ,
p = α E 0 .
α = [ α x x α x y α x z α x y α y y α y z α x z α y z α z z ] .
Φ s = ( α E 0 ) · r 4 π r 3 .
P 1 - 1 ( cos θ ) = - ½ sin θ ,             P 1 0 ( cos θ ) = cos θ , P 1 1 ( cos θ ) = sin θ ,
Φ s ( r R c ) = [ - 1 2 a - 11 T sin θ exp ( - i ϕ ) + a 01 T cos θ + a 11 T sin θ exp ( i ϕ ) ] 1 r 2 ,
a m n T = i = 1 N s a m n i .
Φ s , x = ( α x x sin θ cos ϕ + α x y sin θ sin ϕ + α x z cos θ ) 1 4 π r 2 .
α x x = 2 π ( 2 a 11 T , x - a - 11 T , x ) , α x y = 2 π i ( 2 a 11 T , x + a - 11 T , x ) , α x z = 4 π a 01 T , x .
α x ν = 2 π ( 2 a 11 T , ν - a - 11 T , ν ) , α y ν = 2 π i ( 2 a 11 T , ν + a - 11 T , ν ) , α z ν = 4 π a 01 T , ν ,
C abs , ν = k Im ( α ν ν ) ,
C sca , ν = 1 6 π k 4 ( α x ν 2 + α y ν 2 + α z ν 2 ) .
C abs = ( C abs , x + C abs , y + C abs , z ) ,
C sca = ( C sca , x + C sca , y + C sca , z ) .
G ( θ ) = 3 C sca 80 π [ 6 - M + ( 2 + 3 M ) cos 2 θ ] .
G H H = 3 C sca 40 π [ 1 - M + ( 2 + 3 M ) cos 2 θ ] , G V H = G H V = 3 C sca 40 π ( 1 - M ) , G V V = 3 C sca 40 π ( 3 + 2 M ) .
M = Re ( α 1 * α 2 + α 1 * α 3 + α 2 * α 3 ) α 1 2 + α 2 2 + α 3 2 .
α 1 2 + α 2 2 + α 3 2 = μ ν α μ ν 2 ,
α 1 + α 2 + α 3 = α x x + α y y + α z z .
M = α x x + α y y + α z z 2 - μ ν α μ ν 2 2 μ ν α μ ν 2 .
C abs = 4 π k a V 3 E , C sca = 8 π 3 k 4 a V 6 F ,
E = 1 12 π N S Im ( α ¯ x x + α ¯ y y + α ¯ z z ) ,
F = 1 48 π 2 N S 2 [ α ¯ x x 2 + α ¯ y y 2 + α ¯ z z 2 + 2 ( α ¯ x y 2 + α ¯ x z 2 + α ¯ y z 2 ) ] .
E = Im - 1 + 2 , F = | - 1 + 2 | 2 .
Q abs = C abs π a V 2 = 4 x V E ,
Q sca = C sca π a V 2 = 8 x V 4 F 3 ,
C abs , indep = 4 π k a V 3 Im - 1 + 2 ,
C sca , indep = 8 π 3 k 4 a V 6 | - 1 + 2 | 2 .
C abs ps = k V 3 Im [ - 1 1 + L 1 ( - 1 ) + 2 - 1 1 + L 2 ( - 1 ) ] ,
L 1 = 1 - e 2 e 2 ( 1 2 e ln 1 + e 1 - e - 1 ) , L 2 = 1 2 ( 1 - L 1 ) .
N s = k f ( R g d p ) D f ,
R g 2 = 1 N s i = 1 N S r i 2 ,
P ( 90 ° ) = S 12 ( 90 ° ) S 11 ( 90 ° ) = 3 + 2 M 6 - M .

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