Abstract

The absorption, scattering, and differential scattering cross sections are presented for polydisperse aggregates of prescribed fractal dimension and uniform primary particle size. These optical properties are formulated for polydisperse aggregates in terms of the primary particle diameter, the appropriate moments of the discrete size distribution function, and the mean-square radius of gyration. The absorption and scattering cross sections are compared with Rayleigh theory in the small size limit and with the results of the computational simulations of Mountain and Mulholland [ Langmuir 4, 1321 ( 1988)] for intermediate and large aggregates. The differential scattering cross sections are well correlated by the law of Guinier together with a power-law expression for the larger sizes. The cross sections that are described herein apply in particular to polydisperse fractallike aggregates that are formed by cluster–cluster aggregation and possess a size scale that is pertinent to laboratory experiments and industrial processes.

© 1991 Optical Society of America

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References

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  1. A. I. Medalia, “Morphology of aggregates I. Calculation of shape and bulkiness factors: application to computer-simulated Random Flocs,” J. Colloid Interface Sci. 24, 393–404 (1967).
    [Crossref]
  2. A. I. Medalia, “Morphology of Aggregates VI. Effective volume of aggregates of carbon black from electron microscopy; application to vehicle absorption and to die swell of filled rubber,” J. Colloid Interface Sci. 32, 115–131 (1970).
    [Crossref]
  3. R. A. Dobbins, C. M. Megaridis, “Morphology of flame-generated soot as determined by thermophoretic sampling,” Langmuir 3, 254–259 (1987).
    [Crossref]
  4. C. M. Megaridis, R. A. Dobbins, “Morphological description of flame-generated materials,” Combust. Sci. Tecnol. 71, 95–109 (1990).
    [Crossref]
  5. J. de Ris, “Fire radiation—a review,” presented at Seventeenth Symposium (International) on Combustion (The Combustion Institute, Pittsburgh, PA, 1979), p. 1003.
    [Crossref]
  6. C. L. Tien, S. C. Lee, “Flame radiation,” Prog. Energy Combust. Sci. 8, 41–59 (1982).
    [Crossref]
  7. S. C. Graham, “The refractive indices of isolated and of aggregated soot particles,” Combust. Sci. Technol. 9, 159–163 (1974).
    [Crossref]
  8. J. D. Felske, T. T. Charalampopoulos, H. S. Hura, “Determination of the refractive indices of soot particles from the reflectivities of compressed soot pellets,” Combust. Sci. Technol. 37, 263–284 (1984).
    [Crossref]
  9. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  10. B. L. Drolen, C. L. Tien, “Absorption and scattering of agglomerated soot particles,” J. Quant. Spectrosc. Radiat. Transfer 37, 433–448 (1987).
    [Crossref]
  11. A. R. Jones, “Scattering efficiency factors for agglomerates of small spheres,” J. Phys. D 12, 1661–1672 (1979).
    [Crossref]
  12. J. D. Felske, P. F. Hsu, J. C. Ku, “The effect of soot particle optical inhomogeneity and agglomeration on the analysis of light scattering measurements in flames,” J. Quant. Spectrosc. Radiat. Transfer 35, 447–465 (1986).
    [Crossref]
  13. M. V. Berry, I. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 571–591 (1986).
    [Crossref]
  14. R. A. Dobbins, R. J. Santoro, H. G. Semerjian, “Interpretation of optical measurement of soot in flames,” Prog. Astronaut. Aeronaut. 92, 208–237 (1984).
  15. R. D. Mountain, G. W. Mulholland, “Light scattering from simulated smoke agglomerates,” Langmuir 4, 1321–1326 (1988).
    [Crossref]
  16. A. Guinier, G. Fournet, Small-Angle Scattering of X-Rays, translated by C. B. Walker (Wiley, New York, 1955).
  17. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957);republished by Dover, New York, 1981.
    [PubMed]
  18. R. Jullien, R. Botet, Aggregation and Fractal Aggregates (World Scientific, Singapore, 1987).
  19. D. W. Schaefer, J. E. Martin, P. Wiltzius, D. S. Cannell, “Aggregation of colloidal silica,” in Kinetics of Aggregation and Gelation, F. Family, D. P. Landau, eds. (Elsevier, New York, 1984).
  20. R. Botet, R. Jullien, “A theory of aggregating systems of particles: the clustering of clusters process,” Ann. Phys. Paris 13, 153–220 (1988).
  21. J. E. Martin, A. J. Hurd, “Scattering from fractals,” J. Appl. Crystallogr. 20, 61–78 (1987).
    [Crossref]
  22. F. R. Faxvog, D. M. Roessler, “Carbon aerosol visibility vs particle size distribution,” Appl. Opt. 17, 2612–2616 (1988).
  23. D. M. Roessler, “Diesel particle mass concentration by optical techniques,” Appl. Opt. 21, 4077–4086 (1982).
    [Crossref] [PubMed]
  24. D. M. Roessler, D.-S. Y. Wang, M. Kerker, “Optical absorption by randomly oriented carbon spheroids,” Appl. Opt. 22, 3648–3651 (1983).
    [Crossref] [PubMed]
  25. J. R. Ouimette, R. C. Flagan, “The extinction coefficient of multicomponent aerosols,” Atmos. Environ. 16, 2405–2419 (1982).
    [Crossref]

1990 (1)

C. M. Megaridis, R. A. Dobbins, “Morphological description of flame-generated materials,” Combust. Sci. Tecnol. 71, 95–109 (1990).
[Crossref]

1988 (3)

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

R. Botet, R. Jullien, “A theory of aggregating systems of particles: the clustering of clusters process,” Ann. Phys. Paris 13, 153–220 (1988).

F. R. Faxvog, D. M. Roessler, “Carbon aerosol visibility vs particle size distribution,” Appl. Opt. 17, 2612–2616 (1988).

1987 (3)

J. E. Martin, A. J. Hurd, “Scattering from fractals,” J. Appl. Crystallogr. 20, 61–78 (1987).
[Crossref]

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

B. L. Drolen, C. L. Tien, “Absorption and scattering of agglomerated soot particles,” J. Quant. Spectrosc. Radiat. Transfer 37, 433–448 (1987).
[Crossref]

1986 (2)

J. D. Felske, P. F. Hsu, J. C. Ku, “The effect of soot particle optical inhomogeneity and agglomeration on the analysis of light scattering measurements in flames,” J. Quant. Spectrosc. Radiat. Transfer 35, 447–465 (1986).
[Crossref]

M. V. Berry, I. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 571–591 (1986).
[Crossref]

1984 (2)

R. A. Dobbins, R. J. Santoro, H. G. Semerjian, “Interpretation of optical measurement of soot in flames,” Prog. Astronaut. Aeronaut. 92, 208–237 (1984).

J. D. Felske, T. T. Charalampopoulos, H. S. Hura, “Determination of the refractive indices of soot particles from the reflectivities of compressed soot pellets,” Combust. Sci. Technol. 37, 263–284 (1984).
[Crossref]

1983 (1)

1982 (3)

J. R. Ouimette, R. C. Flagan, “The extinction coefficient of multicomponent aerosols,” Atmos. Environ. 16, 2405–2419 (1982).
[Crossref]

D. M. Roessler, “Diesel particle mass concentration by optical techniques,” Appl. Opt. 21, 4077–4086 (1982).
[Crossref] [PubMed]

C. L. Tien, S. C. Lee, “Flame radiation,” Prog. Energy Combust. Sci. 8, 41–59 (1982).
[Crossref]

1979 (1)

A. R. Jones, “Scattering efficiency factors for agglomerates of small spheres,” J. Phys. D 12, 1661–1672 (1979).
[Crossref]

1974 (1)

S. C. Graham, “The refractive indices of isolated and of aggregated soot particles,” Combust. Sci. Technol. 9, 159–163 (1974).
[Crossref]

1970 (1)

A. I. Medalia, “Morphology of Aggregates VI. Effective volume of aggregates of carbon black from electron microscopy; application to vehicle absorption and to die swell of filled rubber,” J. Colloid Interface Sci. 32, 115–131 (1970).
[Crossref]

1967 (1)

A. I. Medalia, “Morphology of aggregates I. Calculation of shape and bulkiness factors: application to computer-simulated Random Flocs,” J. Colloid Interface Sci. 24, 393–404 (1967).
[Crossref]

Berry, M. V.

M. V. Berry, I. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 571–591 (1986).
[Crossref]

Bohren, C. F.

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

Botet, R.

R. Botet, R. Jullien, “A theory of aggregating systems of particles: the clustering of clusters process,” Ann. Phys. Paris 13, 153–220 (1988).

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

Cannell, D. S.

D. W. Schaefer, J. E. Martin, P. Wiltzius, D. S. Cannell, “Aggregation of colloidal silica,” in Kinetics of Aggregation and Gelation, F. Family, D. P. Landau, eds. (Elsevier, New York, 1984).

Charalampopoulos, T. T.

J. D. Felske, T. T. Charalampopoulos, H. S. Hura, “Determination of the refractive indices of soot particles from the reflectivities of compressed soot pellets,” Combust. Sci. Technol. 37, 263–284 (1984).
[Crossref]

de Ris, J.

J. de Ris, “Fire radiation—a review,” presented at Seventeenth Symposium (International) on Combustion (The Combustion Institute, Pittsburgh, PA, 1979), p. 1003.
[Crossref]

Dobbins, R. A.

C. M. Megaridis, R. A. Dobbins, “Morphological description of flame-generated materials,” Combust. Sci. Tecnol. 71, 95–109 (1990).
[Crossref]

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

R. A. Dobbins, R. J. Santoro, H. G. Semerjian, “Interpretation of optical measurement of soot in flames,” Prog. Astronaut. Aeronaut. 92, 208–237 (1984).

Drolen, B. L.

B. L. Drolen, C. L. Tien, “Absorption and scattering of agglomerated soot particles,” J. Quant. Spectrosc. Radiat. Transfer 37, 433–448 (1987).
[Crossref]

Faxvog, F. R.

Felske, J. D.

J. D. Felske, P. F. Hsu, J. C. Ku, “The effect of soot particle optical inhomogeneity and agglomeration on the analysis of light scattering measurements in flames,” J. Quant. Spectrosc. Radiat. Transfer 35, 447–465 (1986).
[Crossref]

J. D. Felske, T. T. Charalampopoulos, H. S. Hura, “Determination of the refractive indices of soot particles from the reflectivities of compressed soot pellets,” Combust. Sci. Technol. 37, 263–284 (1984).
[Crossref]

Flagan, R. C.

J. R. Ouimette, R. C. Flagan, “The extinction coefficient of multicomponent aerosols,” Atmos. Environ. 16, 2405–2419 (1982).
[Crossref]

Fournet, G.

A. Guinier, G. Fournet, Small-Angle Scattering of X-Rays, translated by C. B. Walker (Wiley, New York, 1955).

Graham, S. C.

S. C. Graham, “The refractive indices of isolated and of aggregated soot particles,” Combust. Sci. Technol. 9, 159–163 (1974).
[Crossref]

Guinier, A.

A. Guinier, G. Fournet, Small-Angle Scattering of X-Rays, translated by C. B. Walker (Wiley, New York, 1955).

Hsu, P. F.

J. D. Felske, P. F. Hsu, J. C. Ku, “The effect of soot particle optical inhomogeneity and agglomeration on the analysis of light scattering measurements in flames,” J. Quant. Spectrosc. Radiat. Transfer 35, 447–465 (1986).
[Crossref]

Huffman, D. R.

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

Hura, H. S.

J. D. Felske, T. T. Charalampopoulos, H. S. Hura, “Determination of the refractive indices of soot particles from the reflectivities of compressed soot pellets,” Combust. Sci. Technol. 37, 263–284 (1984).
[Crossref]

Hurd, A. J.

J. E. Martin, A. J. Hurd, “Scattering from fractals,” J. Appl. Crystallogr. 20, 61–78 (1987).
[Crossref]

Jones, A. R.

A. R. Jones, “Scattering efficiency factors for agglomerates of small spheres,” J. Phys. D 12, 1661–1672 (1979).
[Crossref]

Jullien, R.

R. Botet, R. Jullien, “A theory of aggregating systems of particles: the clustering of clusters process,” Ann. Phys. Paris 13, 153–220 (1988).

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

Kerker, M.

Ku, J. C.

J. D. Felske, P. F. Hsu, J. C. Ku, “The effect of soot particle optical inhomogeneity and agglomeration on the analysis of light scattering measurements in flames,” J. Quant. Spectrosc. Radiat. Transfer 35, 447–465 (1986).
[Crossref]

Lee, S. C.

C. L. Tien, S. C. Lee, “Flame radiation,” Prog. Energy Combust. Sci. 8, 41–59 (1982).
[Crossref]

Martin, J. E.

J. E. Martin, A. J. Hurd, “Scattering from fractals,” J. Appl. Crystallogr. 20, 61–78 (1987).
[Crossref]

D. W. Schaefer, J. E. Martin, P. Wiltzius, D. S. Cannell, “Aggregation of colloidal silica,” in Kinetics of Aggregation and Gelation, F. Family, D. P. Landau, eds. (Elsevier, New York, 1984).

Medalia, A. I.

A. I. Medalia, “Morphology of Aggregates VI. Effective volume of aggregates of carbon black from electron microscopy; application to vehicle absorption and to die swell of filled rubber,” J. Colloid Interface Sci. 32, 115–131 (1970).
[Crossref]

A. I. Medalia, “Morphology of aggregates I. Calculation of shape and bulkiness factors: application to computer-simulated Random Flocs,” J. Colloid Interface Sci. 24, 393–404 (1967).
[Crossref]

Megaridis, C. M.

C. M. Megaridis, R. A. Dobbins, “Morphological description of flame-generated materials,” Combust. Sci. Tecnol. 71, 95–109 (1990).
[Crossref]

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

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]

Ouimette, J. R.

J. R. Ouimette, R. C. Flagan, “The extinction coefficient of multicomponent aerosols,” Atmos. Environ. 16, 2405–2419 (1982).
[Crossref]

Percival, I.

M. V. Berry, I. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 571–591 (1986).
[Crossref]

Roessler, D. M.

Santoro, R. J.

R. A. Dobbins, R. J. Santoro, H. G. Semerjian, “Interpretation of optical measurement of soot in flames,” Prog. Astronaut. Aeronaut. 92, 208–237 (1984).

Schaefer, D. W.

D. W. Schaefer, J. E. Martin, P. Wiltzius, D. S. Cannell, “Aggregation of colloidal silica,” in Kinetics of Aggregation and Gelation, F. Family, D. P. Landau, eds. (Elsevier, New York, 1984).

Semerjian, H. G.

R. A. Dobbins, R. J. Santoro, H. G. Semerjian, “Interpretation of optical measurement of soot in flames,” Prog. Astronaut. Aeronaut. 92, 208–237 (1984).

Tien, C. L.

B. L. Drolen, C. L. Tien, “Absorption and scattering of agglomerated soot particles,” J. Quant. Spectrosc. Radiat. Transfer 37, 433–448 (1987).
[Crossref]

C. L. Tien, S. C. Lee, “Flame radiation,” Prog. Energy Combust. Sci. 8, 41–59 (1982).
[Crossref]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957);republished by Dover, New York, 1981.
[PubMed]

Wang, D.-S. Y.

Wiltzius, P.

D. W. Schaefer, J. E. Martin, P. Wiltzius, D. S. Cannell, “Aggregation of colloidal silica,” in Kinetics of Aggregation and Gelation, F. Family, D. P. Landau, eds. (Elsevier, New York, 1984).

Ann. Phys. Paris (1)

R. Botet, R. Jullien, “A theory of aggregating systems of particles: the clustering of clusters process,” Ann. Phys. Paris 13, 153–220 (1988).

Appl. Opt. (3)

Atmos. Environ. (1)

J. R. Ouimette, R. C. Flagan, “The extinction coefficient of multicomponent aerosols,” Atmos. Environ. 16, 2405–2419 (1982).
[Crossref]

Combust. Sci. Technol. (2)

S. C. Graham, “The refractive indices of isolated and of aggregated soot particles,” Combust. Sci. Technol. 9, 159–163 (1974).
[Crossref]

J. D. Felske, T. T. Charalampopoulos, H. S. Hura, “Determination of the refractive indices of soot particles from the reflectivities of compressed soot pellets,” Combust. Sci. Technol. 37, 263–284 (1984).
[Crossref]

Combust. Sci. Tecnol. (1)

C. M. Megaridis, R. A. Dobbins, “Morphological description of flame-generated materials,” Combust. Sci. Tecnol. 71, 95–109 (1990).
[Crossref]

J. Appl. Crystallogr. (1)

J. E. Martin, A. J. Hurd, “Scattering from fractals,” J. Appl. Crystallogr. 20, 61–78 (1987).
[Crossref]

J. Colloid Interface Sci. (2)

A. I. Medalia, “Morphology of aggregates I. Calculation of shape and bulkiness factors: application to computer-simulated Random Flocs,” J. Colloid Interface Sci. 24, 393–404 (1967).
[Crossref]

A. I. Medalia, “Morphology of Aggregates VI. Effective volume of aggregates of carbon black from electron microscopy; application to vehicle absorption and to die swell of filled rubber,” J. Colloid Interface Sci. 32, 115–131 (1970).
[Crossref]

J. Phys. D (1)

A. R. Jones, “Scattering efficiency factors for agglomerates of small spheres,” J. Phys. D 12, 1661–1672 (1979).
[Crossref]

J. Quant. Spectrosc. Radiat. Transfer (2)

J. D. Felske, P. F. Hsu, J. C. Ku, “The effect of soot particle optical inhomogeneity and agglomeration on the analysis of light scattering measurements in flames,” J. Quant. Spectrosc. Radiat. Transfer 35, 447–465 (1986).
[Crossref]

B. L. Drolen, C. L. Tien, “Absorption and scattering of agglomerated soot particles,” J. Quant. Spectrosc. Radiat. Transfer 37, 433–448 (1987).
[Crossref]

Langmuir (2)

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 (1)

M. V. Berry, I. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 571–591 (1986).
[Crossref]

Prog. Astronaut. Aeronaut. (1)

R. A. Dobbins, R. J. Santoro, H. G. Semerjian, “Interpretation of optical measurement of soot in flames,” Prog. Astronaut. Aeronaut. 92, 208–237 (1984).

Prog. Energy Combust. Sci. (1)

C. L. Tien, S. C. Lee, “Flame radiation,” Prog. Energy Combust. Sci. 8, 41–59 (1982).
[Crossref]

Other (6)

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

J. de Ris, “Fire radiation—a review,” presented at Seventeenth Symposium (International) on Combustion (The Combustion Institute, Pittsburgh, PA, 1979), p. 1003.
[Crossref]

A. Guinier, G. Fournet, Small-Angle Scattering of X-Rays, translated by C. B. Walker (Wiley, New York, 1955).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957);republished by Dover, New York, 1981.
[PubMed]

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

D. W. Schaefer, J. E. Martin, P. Wiltzius, D. S. Cannell, “Aggregation of colloidal silica,” in Kinetics of Aggregation and Gelation, F. Family, D. P. Landau, eds. (Elsevier, New York, 1984).

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Figures (5)

Fig. 1
Fig. 1

Comparison of orientation averaged structure function IA for vv differential scattering vs qdp for randomly oriented, monodisperse aggregates with n = 12, 32, 119 primary particles and xp = 0.005, 0.01, 0.02, 0.05, 0.10, 0.157. Dots are from the computer simulation,15 while the open symbols are from calculations by the porous sphere model.

Fig. 2
Fig. 2

Comparison of IA/n2 for vv differential scattering vs q 2 R g 2 for randomly oriented, monodisperse aggregates. The dots are for forty-eight aggregates with n = 10−700 from the computer simulation.15 The squares display data calculated by the porous sphere model with n = 12, 32, 119 for values of xp given in Fig. 1.

Fig. 3
Fig. 3

I A ¯ / n 2 ¯ for vv differential scattering vs q 2 R g 2 ¯ for forty-five sets randomly oriented, polydisperse aggregates. Data points displayed by open symbols were calculated using the porous sphere model for lognormal distributions with n0/ng/n, = 1/5/30, 1/15/100, 1/30/ 150; σg = 0.1, 0.3, 0.5; xp = 0.01, 0.02, 0.05, 0.10, 0.157. The smooth curve is given by the Guinier plus power laws [Eq. (26) with c1 = c2 = 1 and Eq. (28)]. The dots are for monodisperse aggregates from the computer simulation15 with n = 10−700.

Fig. 4
Fig. 4

Comparison of J A ¯ / n 2 ¯ vs ( k 2 R g 2 ¯ ) 1 / 2 for total scattering by randomly oriented polydisperse aggregates. The dots represent results of a computational simulation15; the open symbols represent the porous sphere model predictions with xp = 0.02, 0.05, 0.10, 0.157, 0.25; the smooth curve is given by Eq. (30).

Fig. 5
Fig. 5

Comparison of specific extinction Pext vs πD/λ for a sphere using the Lorenz–Mie theory and an aggregate using Eq. (40). For the aggregate curve the volume-equivalent diameter D = n1/3dp. The refractive index is m = 2 − i1, the primary particle diameter dp is 40 nm, and the wavelength of light is 555 nm.

Tables (2)

Tables Icon

Table I Values of E(m) and F(m) for Selected Values of Refractive Index m

Tables Icon

Table II Values of c1 and c2 Appearing In Eq. (26)a

Equations (54)

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

C j A
C j A ¯
D 30 3
D 60 6
I A ¯
J A ¯
n q ¯
R g 2 ¯
ρ s a A ¯
ω A ¯
p ( n ) = exp [ ( ln n n g 3 2 σ g ) 2 ] 3 2 π σ g S 0 n ,
n q ¯ = n n q p ( n ) .
D 30 3 = n 1 ¯ d p 3 ,
D 60 6 = n 2 ¯ d p 6 .
C j A ¯ = n C j A ( n , π d p λ , m ) p ( n ) ,
G ( m p s ) = η υ G ( m ) ,
G ( m ) = m 2 1 m 2 + 2 ,
η υ = n d p 3 D p s 3 .
m p s = ( 2 η υ G ( m ) + 1 1 η υ G ( m ) ) 1 / 2 .
E ( m ) = Im [ G ( m ) ] , F ( m ) = | G ( m ) | 2 ,
E ( m p s ) = η υ E ( m ) , F ( m p s ) = η υ 2 F ( m ) .
C abs = 4 π x p 3 E ( m ) k 2 ,
C sca = 8 π x p 6 F ( m ) 3 k 2 ,
C vv = x p 6 F ( m ) k 2 ,
C abs A ¯ = 4 π n 1 ¯ x p 3 E ( m ) k 2 ,
C sca A ¯ = 8 π n 2 ¯ x p 6 F ( m ) 3 k 2 ,
C vv A ( θ ) ¯ = n 2 ¯ x p 6 F ( m ) k 2 .
D p s = 2 C r R g ,
R g 2 = 1 n i r i 2 ,
n = K f ( R g d p ) D f ,
I A = k 2 C vv A ( θ ) x p 6 F ( m ) .
I A n 2 1 q 2 R g 2 3 ; q 2 R g 2 1 ,
I A n 2 = G 2 ( u ) ,
G ( u ) = 3 u 3 ( sin u u cos u ) , u = 2 π D p s λ sin ( θ / 2 ) .
G ( u ) 1 u 2 10 .
R g 2 ¯ n R g 2 ( n ) n 2 p ( n ) n n 2 p ( n ) .
I A ¯ n 2 ¯ = c 1 exp ( c 2 q 2 R g 2 ¯ 3 ) .
( ln I A ¯ ) ( ln k ) = D f .
I A ¯ n 2 ¯ = ( 3 D f 2 e q 2 R g 2 ¯ ) D f / 2 for q 2 R g 2 ¯ 1.5 D f ,
J A ¯ n 2 ¯ = 8 π 3 ( 1 2 3 k 2 R g 2 ¯ ) .
J A ¯ n 2 ¯ = 8 π 3 g ( k 2 R g 2 ¯ ) ,
g ( k 2 R g 2 ¯ ) = ( 1 + 4 3 D f k 2 R g 2 ¯ ) D f / 2 .
g ( k 2 R g 2 ¯ ) K f n ( 3 D f 4 k 2 d p 2 ) D f / 2 ,
C abs A ¯ = 4 π n 1 ¯ x p 3 E ( m ) k 2 ,
C sca A ¯ = 8 π n 2 ¯ x p 6 F ( m ) 3 k 2 g ( k 2 R g 2 ¯ ) ,
g ( k 2 R g 2 ¯ ) = ( 1 + 4 3 D f k 2 R g 2 ¯ ) D f / 2 .
C vv A ¯ ( θ ) = n 2 ¯ x p 6 F ( m ) k 2 f ( q 2 R g 2 ¯ ) ,
f ( q 2 R g 2 ¯ ) = { exp ( q 2 R g 2 ¯ 3 ) ; q 2 R g 2 ¯ 1.5 D f , ( 3 D f 2 e q 2 R g 2 ¯ ) D f / 2 ; q 2 R g 2 ¯ > 1.5 D f .
ω p = 2 3 x p 3 F ( m ) E ( m ) .
ω A ¯ = ρ sa A ¯ 1 + ρ sa A ¯ ,
ρ sa A ¯ = f n n 1 ¯ ω p g ( k 2 R g 2 ¯ ) .
ρ sa A = K f ω p ( 3 D f 16 x p 2 ) D f / 2 .
P ext = λ C ext A ¯ π 6 D 30 3 .
P ext = 6 π E ( m ) ( 1 + ρ sa A ¯ ) .

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