Abstract

We present angular scattering functions for loosely packed aggregates of 250 and 500 identical spheres near the Rayleigh size limit before and after the application of successive layers of an absorbing mantle. All measurements were obtained by using the microwave analog technique. Gross features of the scattering by aggregates without a mantle can be interpreted in terms of coherent scattering from the unit spheres acting independently of each other. The largest deviations from this approximation occur after the first minimum in forward scattering and extend to a scattering angle of 60° or 80° for our models. This intermediate range is also where the largest differences occur in the scattering from one aggregate to another. The angular extent of the range is largest for aggregates with the smallest dimensions. The scattering function is usually flat in the backscattering hemisphere and has little or no backscattering increase. The coherent scattering approximation breaks down when the aggregates are coated, and an equivalent spheres approximation becomes a better representation. The maximum degree of polarization near a scattering angle of 90° first decreases and then increases again as the mantle grows thicker.

© 1993 Optical Society of America

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References

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  1. C. F. Bohren, D. R. Huffmann, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  2. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  3. R. H. Zerull, “Laboratory investigations and optical properties of grains,” in Properties and Interactions of Interplanetary Dust, R. H. Giese, P. Lamy, eds. (Reidel, Dordrecht, The Netherlands, 1985), pp. 197–266.
    [CrossRef]
  4. J. M. Greenberg, J. I. Hage, “From interstellar dust to comets: a unification of observational constraints,” Astrophys. J. 361, 260–274 (1990).
    [CrossRef]
  5. J. M. Greenberg, B. Å. S. Gustafson, “A comet fragment model for zodiacal light particles,” Astron. Astrophys. 93, 35–42 (1981).
  6. A. P. Boss, “Protostellar formation in rotation interstellar clouds. VII. Opacity and fragmentation,” Astrophys. J. 331, 370–376 (1988).
    [CrossRef]
  7. R. A. West, P. H. Smith, “Evidence for aggregate particles in the atmospheres of Titan and Jupiter,” Icarus 90, 330–333 (1991).
    [CrossRef]
  8. P. Latimer, “Experimental tests of a theoretical method for predicting light scattering by aggregates,” Appl. Opt. 24, 3231–3239 (1985).
    [CrossRef] [PubMed]
  9. J. I. Hage, J. M. Greenberg, “A model for the optical properties of porous grains,” Astrophys. J. 361, 251–259 (1990).
    [CrossRef]
  10. E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
    [CrossRef]
  11. K. A. Fuller, G. W. Kattawar, “Consummate solution to the problem of classical electromagnetic scattering by an ensemble of spheres. II: Clusters of arbitrary configuration,” Opt. Lett. 13, 1063–1065 (1988).
    [CrossRef] [PubMed]
  12. R. T. Wang, B. Å. S. Gustafson, “Angular scattering and polarization by randomly oriented dumbbells and chains of spheres,” in Proceedings of the 1983 CSL Scientific Conference on Obscuration and Aerosol Research, J. Farmer, R. H. Kohl, eds. (U. S. Army Chemical Systems Laboratory, Aberdeen, Md., 1984), pp. 237–247.
  13. B. Å. S. Gustafson, Scattering by Ensembles of Small Particles, Experiment, Theory, and Application, Vol. 17 of Reports From the Observatory of Lund, (Lund Observatory, Lund, Sweden, 1980).
  14. B. Å. S. Gustafson, “Dominant particle parameters in side scattering by some aggregates of cylinders,” in Proceedings of the 1982 CSL Scientific Conference on Obscuration and Aerosol Research, R. H. Kohl, ed. (U. S. Army Chemical Systems Laboratory, Aberdeen, Md., 1983), pp. 281–292.
  15. R. H. Giese, K. Weiss, R. H. Zerull, T. Ono, “Large fluffy particles: a possible explanation of the optical properties of interplanetary dust,” Astron. Astrophys. 65, 265–272 (1978).
  16. J. R. Bottiger, E. S. Fry, R. C. Thompson, “Phase matrix measurements for electromagnetic scattering by sphere aggregates,” in Light Scattering by Irregularly Shaped Particles, D. W. Schuerman, ed. (Plenum, New York, 1980), pp. 283–290.
    [CrossRef]
  17. K. Weiss-Wrana, “Optical properties of interplanetary dust: comparison with light scattering by larger meteoritic and terrestrial grains,” Astron. Astrophys. 126, 240–250 (1983).
  18. However, the scattering matrix for a single particle can contain only seven independent elements, and symmetry relations usually reduce this number further. See Sec. 5.2 of Ref. 2.
  19. B. Å. S. Gustafson, “Comet ejection and dynamics of nonspherical dust particles and meteoroids,” Astrophys. J. 337, 945–949 (1989).
    [CrossRef]
  20. The index of refraction was obtained by fitting the scattering from a sphere made from the compound with Mie theory. The procedure and the fit are described by K. Schulz in “Mikrowellenanalogiestreuversuche an Lockeren Materialagglomeraten zur Interpretation der Winkelabhängigkeit von Streuintensität und Polarisation Kometarer Staubpartikel,” M.S. thesis (Ruhr-Universität, Bochum, Germany, 1991), Sec. 4.1.4.
    [PubMed]
  21. D. A. G. Bruggeman, “Berechnung verschiederner physikalischer Konstanten von heterogen Substanzen. I. Dielectrizitätskonstaten und Leitfahigkeiten der Mischkörper aus isotropen Substanzen,” Ann. Phys. (Leipzig) 24, 636–679 (1935).
  22. G. A. Niklasson, C. G. Granqvist, O. Hunderi, “Effective medium models for the optical properties of inhomogeneous materials,” Appl. Opt. 20, 26–30 (1981).
    [CrossRef] [PubMed]
  23. C. F. Bohren, “Applicability of effective-medium theories to problems of scattering and absorption by nonhomogeneous atmospheric particles,” J. Atmos. Sci. 43, 468–475.
  24. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes. The Art of Scientific Computing (fortran Version) (Cambridge U. Press, Cambridge, 1989), Sect. 10.6.
  25. M. Kerker, The Scattering of Light (Academic, New York, 1969), Sec. 8.1.2.

1991 (1)

R. A. West, P. H. Smith, “Evidence for aggregate particles in the atmospheres of Titan and Jupiter,” Icarus 90, 330–333 (1991).
[CrossRef]

1990 (2)

J. M. Greenberg, J. I. Hage, “From interstellar dust to comets: a unification of observational constraints,” Astrophys. J. 361, 260–274 (1990).
[CrossRef]

J. I. Hage, J. M. Greenberg, “A model for the optical properties of porous grains,” Astrophys. J. 361, 251–259 (1990).
[CrossRef]

1989 (1)

B. Å. S. Gustafson, “Comet ejection and dynamics of nonspherical dust particles and meteoroids,” Astrophys. J. 337, 945–949 (1989).
[CrossRef]

1988 (2)

1985 (1)

1983 (1)

K. Weiss-Wrana, “Optical properties of interplanetary dust: comparison with light scattering by larger meteoritic and terrestrial grains,” Astron. Astrophys. 126, 240–250 (1983).

1981 (2)

G. A. Niklasson, C. G. Granqvist, O. Hunderi, “Effective medium models for the optical properties of inhomogeneous materials,” Appl. Opt. 20, 26–30 (1981).
[CrossRef] [PubMed]

J. M. Greenberg, B. Å. S. Gustafson, “A comet fragment model for zodiacal light particles,” Astron. Astrophys. 93, 35–42 (1981).

1978 (1)

R. H. Giese, K. Weiss, R. H. Zerull, T. Ono, “Large fluffy particles: a possible explanation of the optical properties of interplanetary dust,” Astron. Astrophys. 65, 265–272 (1978).

1973 (1)

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

1935 (1)

D. A. G. Bruggeman, “Berechnung verschiederner physikalischer Konstanten von heterogen Substanzen. I. Dielectrizitätskonstaten und Leitfahigkeiten der Mischkörper aus isotropen Substanzen,” Ann. Phys. (Leipzig) 24, 636–679 (1935).

Bohren, C. F.

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

C. F. Bohren, “Applicability of effective-medium theories to problems of scattering and absorption by nonhomogeneous atmospheric particles,” J. Atmos. Sci. 43, 468–475.

Boss, A. P.

A. P. Boss, “Protostellar formation in rotation interstellar clouds. VII. Opacity and fragmentation,” Astrophys. J. 331, 370–376 (1988).
[CrossRef]

Bottiger, J. R.

J. R. Bottiger, E. S. Fry, R. C. Thompson, “Phase matrix measurements for electromagnetic scattering by sphere aggregates,” in Light Scattering by Irregularly Shaped Particles, D. W. Schuerman, ed. (Plenum, New York, 1980), pp. 283–290.
[CrossRef]

Bruggeman, D. A. G.

D. A. G. Bruggeman, “Berechnung verschiederner physikalischer Konstanten von heterogen Substanzen. I. Dielectrizitätskonstaten und Leitfahigkeiten der Mischkörper aus isotropen Substanzen,” Ann. Phys. (Leipzig) 24, 636–679 (1935).

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes. The Art of Scientific Computing (fortran Version) (Cambridge U. Press, Cambridge, 1989), Sect. 10.6.

Fry, E. S.

J. R. Bottiger, E. S. Fry, R. C. Thompson, “Phase matrix measurements for electromagnetic scattering by sphere aggregates,” in Light Scattering by Irregularly Shaped Particles, D. W. Schuerman, ed. (Plenum, New York, 1980), pp. 283–290.
[CrossRef]

Fuller, K. A.

Giese, R. H.

R. H. Giese, K. Weiss, R. H. Zerull, T. Ono, “Large fluffy particles: a possible explanation of the optical properties of interplanetary dust,” Astron. Astrophys. 65, 265–272 (1978).

Granqvist, C. G.

Greenberg, J. M.

J. I. Hage, J. M. Greenberg, “A model for the optical properties of porous grains,” Astrophys. J. 361, 251–259 (1990).
[CrossRef]

J. M. Greenberg, J. I. Hage, “From interstellar dust to comets: a unification of observational constraints,” Astrophys. J. 361, 260–274 (1990).
[CrossRef]

J. M. Greenberg, B. Å. S. Gustafson, “A comet fragment model for zodiacal light particles,” Astron. Astrophys. 93, 35–42 (1981).

Gustafson, B. Å. S.

B. Å. S. Gustafson, “Comet ejection and dynamics of nonspherical dust particles and meteoroids,” Astrophys. J. 337, 945–949 (1989).
[CrossRef]

J. M. Greenberg, B. Å. S. Gustafson, “A comet fragment model for zodiacal light particles,” Astron. Astrophys. 93, 35–42 (1981).

R. T. Wang, B. Å. S. Gustafson, “Angular scattering and polarization by randomly oriented dumbbells and chains of spheres,” in Proceedings of the 1983 CSL Scientific Conference on Obscuration and Aerosol Research, J. Farmer, R. H. Kohl, eds. (U. S. Army Chemical Systems Laboratory, Aberdeen, Md., 1984), pp. 237–247.

B. Å. S. Gustafson, Scattering by Ensembles of Small Particles, Experiment, Theory, and Application, Vol. 17 of Reports From the Observatory of Lund, (Lund Observatory, Lund, Sweden, 1980).

B. Å. S. Gustafson, “Dominant particle parameters in side scattering by some aggregates of cylinders,” in Proceedings of the 1982 CSL Scientific Conference on Obscuration and Aerosol Research, R. H. Kohl, ed. (U. S. Army Chemical Systems Laboratory, Aberdeen, Md., 1983), pp. 281–292.

Hage, J. I.

J. M. Greenberg, J. I. Hage, “From interstellar dust to comets: a unification of observational constraints,” Astrophys. J. 361, 260–274 (1990).
[CrossRef]

J. I. Hage, J. M. Greenberg, “A model for the optical properties of porous grains,” Astrophys. J. 361, 251–259 (1990).
[CrossRef]

Huffmann, D. R.

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

Hunderi, O.

Kattawar, G. W.

Kerker, M.

M. Kerker, The Scattering of Light (Academic, New York, 1969), Sec. 8.1.2.

Latimer, P.

Niklasson, G. A.

Ono, T.

R. H. Giese, K. Weiss, R. H. Zerull, T. Ono, “Large fluffy particles: a possible explanation of the optical properties of interplanetary dust,” Astron. Astrophys. 65, 265–272 (1978).

Pennypacker, C. R.

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes. The Art of Scientific Computing (fortran Version) (Cambridge U. Press, Cambridge, 1989), Sect. 10.6.

Purcell, E. M.

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Schulz, K.

The index of refraction was obtained by fitting the scattering from a sphere made from the compound with Mie theory. The procedure and the fit are described by K. Schulz in “Mikrowellenanalogiestreuversuche an Lockeren Materialagglomeraten zur Interpretation der Winkelabhängigkeit von Streuintensität und Polarisation Kometarer Staubpartikel,” M.S. thesis (Ruhr-Universität, Bochum, Germany, 1991), Sec. 4.1.4.
[PubMed]

Smith, P. H.

R. A. West, P. H. Smith, “Evidence for aggregate particles in the atmospheres of Titan and Jupiter,” Icarus 90, 330–333 (1991).
[CrossRef]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes. The Art of Scientific Computing (fortran Version) (Cambridge U. Press, Cambridge, 1989), Sect. 10.6.

Thompson, R. C.

J. R. Bottiger, E. S. Fry, R. C. Thompson, “Phase matrix measurements for electromagnetic scattering by sphere aggregates,” in Light Scattering by Irregularly Shaped Particles, D. W. Schuerman, ed. (Plenum, New York, 1980), pp. 283–290.
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes. The Art of Scientific Computing (fortran Version) (Cambridge U. Press, Cambridge, 1989), Sect. 10.6.

Wang, R. T.

R. T. Wang, B. Å. S. Gustafson, “Angular scattering and polarization by randomly oriented dumbbells and chains of spheres,” in Proceedings of the 1983 CSL Scientific Conference on Obscuration and Aerosol Research, J. Farmer, R. H. Kohl, eds. (U. S. Army Chemical Systems Laboratory, Aberdeen, Md., 1984), pp. 237–247.

Weiss, K.

R. H. Giese, K. Weiss, R. H. Zerull, T. Ono, “Large fluffy particles: a possible explanation of the optical properties of interplanetary dust,” Astron. Astrophys. 65, 265–272 (1978).

Weiss-Wrana, K.

K. Weiss-Wrana, “Optical properties of interplanetary dust: comparison with light scattering by larger meteoritic and terrestrial grains,” Astron. Astrophys. 126, 240–250 (1983).

West, R. A.

R. A. West, P. H. Smith, “Evidence for aggregate particles in the atmospheres of Titan and Jupiter,” Icarus 90, 330–333 (1991).
[CrossRef]

Zerull, R. H.

R. H. Giese, K. Weiss, R. H. Zerull, T. Ono, “Large fluffy particles: a possible explanation of the optical properties of interplanetary dust,” Astron. Astrophys. 65, 265–272 (1978).

R. H. Zerull, “Laboratory investigations and optical properties of grains,” in Properties and Interactions of Interplanetary Dust, R. H. Giese, P. Lamy, eds. (Reidel, Dordrecht, The Netherlands, 1985), pp. 197–266.
[CrossRef]

Ann. Phys. (Leipzig) (1)

D. A. G. Bruggeman, “Berechnung verschiederner physikalischer Konstanten von heterogen Substanzen. I. Dielectrizitätskonstaten und Leitfahigkeiten der Mischkörper aus isotropen Substanzen,” Ann. Phys. (Leipzig) 24, 636–679 (1935).

Appl. Opt. (2)

Astron. Astrophys. (3)

J. M. Greenberg, B. Å. S. Gustafson, “A comet fragment model for zodiacal light particles,” Astron. Astrophys. 93, 35–42 (1981).

R. H. Giese, K. Weiss, R. H. Zerull, T. Ono, “Large fluffy particles: a possible explanation of the optical properties of interplanetary dust,” Astron. Astrophys. 65, 265–272 (1978).

K. Weiss-Wrana, “Optical properties of interplanetary dust: comparison with light scattering by larger meteoritic and terrestrial grains,” Astron. Astrophys. 126, 240–250 (1983).

Astrophys. J. (5)

A. P. Boss, “Protostellar formation in rotation interstellar clouds. VII. Opacity and fragmentation,” Astrophys. J. 331, 370–376 (1988).
[CrossRef]

J. I. Hage, J. M. Greenberg, “A model for the optical properties of porous grains,” Astrophys. J. 361, 251–259 (1990).
[CrossRef]

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

B. Å. S. Gustafson, “Comet ejection and dynamics of nonspherical dust particles and meteoroids,” Astrophys. J. 337, 945–949 (1989).
[CrossRef]

J. M. Greenberg, J. I. Hage, “From interstellar dust to comets: a unification of observational constraints,” Astrophys. J. 361, 260–274 (1990).
[CrossRef]

Icarus (1)

R. A. West, P. H. Smith, “Evidence for aggregate particles in the atmospheres of Titan and Jupiter,” Icarus 90, 330–333 (1991).
[CrossRef]

J. Atmos. Sci. (1)

C. F. Bohren, “Applicability of effective-medium theories to problems of scattering and absorption by nonhomogeneous atmospheric particles,” J. Atmos. Sci. 43, 468–475.

Opt. Lett. (1)

Other (11)

R. T. Wang, B. Å. S. Gustafson, “Angular scattering and polarization by randomly oriented dumbbells and chains of spheres,” in Proceedings of the 1983 CSL Scientific Conference on Obscuration and Aerosol Research, J. Farmer, R. H. Kohl, eds. (U. S. Army Chemical Systems Laboratory, Aberdeen, Md., 1984), pp. 237–247.

B. Å. S. Gustafson, Scattering by Ensembles of Small Particles, Experiment, Theory, and Application, Vol. 17 of Reports From the Observatory of Lund, (Lund Observatory, Lund, Sweden, 1980).

B. Å. S. Gustafson, “Dominant particle parameters in side scattering by some aggregates of cylinders,” in Proceedings of the 1982 CSL Scientific Conference on Obscuration and Aerosol Research, R. H. Kohl, ed. (U. S. Army Chemical Systems Laboratory, Aberdeen, Md., 1983), pp. 281–292.

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

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

R. H. Zerull, “Laboratory investigations and optical properties of grains,” in Properties and Interactions of Interplanetary Dust, R. H. Giese, P. Lamy, eds. (Reidel, Dordrecht, The Netherlands, 1985), pp. 197–266.
[CrossRef]

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes. The Art of Scientific Computing (fortran Version) (Cambridge U. Press, Cambridge, 1989), Sect. 10.6.

M. Kerker, The Scattering of Light (Academic, New York, 1969), Sec. 8.1.2.

However, the scattering matrix for a single particle can contain only seven independent elements, and symmetry relations usually reduce this number further. See Sec. 5.2 of Ref. 2.

J. R. Bottiger, E. S. Fry, R. C. Thompson, “Phase matrix measurements for electromagnetic scattering by sphere aggregates,” in Light Scattering by Irregularly Shaped Particles, D. W. Schuerman, ed. (Plenum, New York, 1980), pp. 283–290.
[CrossRef]

The index of refraction was obtained by fitting the scattering from a sphere made from the compound with Mie theory. The procedure and the fit are described by K. Schulz in “Mikrowellenanalogiestreuversuche an Lockeren Materialagglomeraten zur Interpretation der Winkelabhängigkeit von Streuintensität und Polarisation Kometarer Staubpartikel,” M.S. thesis (Ruhr-Universität, Bochum, Germany, 1991), Sec. 4.1.4.
[PubMed]

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Figures (11)

Fig. 1
Fig. 1

Block diagram showing the equipment for transmitting microwaves and receiving the scattered signal. A phase reference and a compensation signal used to cancel unwanted scattering are split off directly from the transmitter. Orthomode transducers and waveguide switches (not shown) control the state of polarization. The receiving antenna can be positioned at any scattering angle ϴ = (0, 175°). By including the forward direction, ϴ = 0, one can determine the effective cross sections for extinction, scattering, absorption, and radiation pressure.

Fig. 2
Fig. 2

Schematic illustration of the analog and digital data processing. The currents Idx and IDy are used to derive the complex elements of the amplitude scattering (Jones) matrix. The absolute value of the electric field |E| is derived in real time by analog processing for easier compensation of unwanted signals. All other components are obtained digitally.

Fig. 3
Fig. 3

Example of a set of calibration measurements. The angular scattering pattern of a size parameter x = 10.14 sphere with refractive index 1.526 (crosses) is fully resolved. The calibration factor is unambiguously obtained by matching the scattering function to the exact solution by using Mie theory.

Fig. 4
Fig. 4

(a) Model 4-M2 consists of 500 spheres with two layers of absorbing material. This model was used to represent scattering of visual light by a comet dust particle consisting of aggregated, essentially unaltered, interstellar grains. The dimensions are scaled by the same ratio as the wavelength λ, or approximately 15,000 times. (b) Aggregate 4 before the 500 spheres were coated. At microwave frequencies, sets of two nylon spheres represent the elongated silicate core of an interstellar dust grain so that this model represents 250 aggregated interstellar dust grains stripped of both their outer mantle of complex ices and of their inner organic refractory mantle. Each sphere is 1.588 ± 0.020 mm in diameter and has a circumference to wavelength ratio (size parameter) near 0.58 at the laboratory wavelength λ.

Fig. 5
Fig. 5

(a) Rayleigh solution is a reasonable approximation to the independent scattering by unit spheres without a mantle (Mie curves). The scattering is therefore not strongly dependent on our choice of homogeneous spheres as building units for the aggregates. (b) The angular dependence of the degree of linear polarization for Rayleigh particles with isotropic polarizability (solid curve) is nearly identical with that by independently scattering unit spheres (Mie curves).

Fig. 6
Fig. 6

Aggregate 3: (a) Measured angular dependence of total scattered intensity from the 250 spheres in this aggregate without a mantle (crosses) and averaged over azimuth rotation about three axes. The solid curve represents scattering by a sphere of the same material with a geometric cross section equal to the average cross section of the aggregate. The dashed curve is for a sphere of equal volume to that occupied by matter in the aggregate. Independent and incoherent scattering by unit spheres is given by the dotted curve. None of these curves satisfactorily reproduce the measurements. (b) Measured degree of linear polarization of radiation scattered from this aggregate as a function of scattering angle (crosses) averaged over all available orientations (rotation about three axes). The polarization by independent unit spheres is a good approximation at all angles. This also supports the coherent scattering interpretation. The calculated polarization both from the equal cross-section sphere (solid curve) and the equal volume sphere (dashed curve) are poor approximations. (c) Among the approximations we tried, coherent scattering best reproduces the measured scattering from this aggregate (crosses) at all angles. Coherent scattering was calculated from a computer reconstruction of the coordinates of each sphere from photographs. Coordinates could only be obtained for 249 of the 250 particles. The solid curve is an average over the same axis of rotation as the measurements. The dashed curve is for an average over random orientations and the dotted line is for incoherent scattering.

Fig. 7
Fig. 7

Aggregates 1 and 4: (a) Scattering per sphere from aggregate 1 (diamonds) consisting of 250 spheres and from aggregate 4 with 500 spheres (crosses). The aggregates are made from identical unit spheres and have similar packing density. Apart from the forward peak, the largest differences occur at intermediate angles (zone II). (b) The degree of linear polarization in scattering from aggregates 1 and 4 is quite similar. The only significant differences occur in zone II, between 30° and 70° scattering angle.

Fig. 8
Fig. 8

Measured angular dependence of total scattered intensity from the 500-sphere aggregate 4 (crosses) averaged over azimuth rotation about three axes and from aggregate 5 consisting of 250 spheres (diamonds) averaged over two axes of rotation. Each sphere in aggregate 5 is the double in volume of the unit spheres making up aggregate 4, so the amount of material is the same in both aggregates. The difference has no significant effect on the angular scattering. (b) The measured angular dependence of the degree of polarization from both aggregate 4 and aggregate 5 resembles Rayleigh polarization.

Fig. 9
Fig. 9

Effect of varying packing density on the angular dependence of total scattered intensity is shown after averaging over rotation about one axis. The set of 250 spheres in aggregate 6 is more disperse than those in 2, and the packing is densest in aggregate 3 (see Table 1). The inserted cross sections give a visual impression of the packing. The forward-scattering cone is narrowest for the fluffiest and therefore largest aggregate (dotted curve) and widest for the densest aggregate (solid curve). Backscattering increases with higher packing. This may be because of the larger degree of coherence in the scattering from different parts of the aggregate and partially because of the coupling between unit spheres. (b) The maximum polarization decreases slightly with increased packing. The dotted curve is for aggregate 6 with the lowest packing, the dashed curve is for aggregate 2, and the solid curve is for the densest aggregate (3). All data are averaged over one axis of rotation.

Fig. 10
Fig. 10

Measured total scattering functions and degree of linear polarization for aggregate 3 (crosses) are compared to the equivalent spheres approximation (solid curve) and to coherent scattering (dashed curve). Coherent scattering is a better approximation both in total intensity (a) and in polarization (b) before the aggregate is coated. The set of equivalent spheres is a better approximation after the aggregates are coated with a mantle. The effect of adding one coat (Model 3-M1) on the total scattering function is shown in (c) and the effect on the degree of polarization is shown in (d). Scattering by Model 3-M3 is shown in (e) and (f), whereas that of 4-M4 is in (g) and (h). The inserts show representative geometric cross sections.

Fig. 11
Fig. 11

Effect of coating the 500-sphere aggregate 4 with an absorbing mantle is to emphasize and narrow the forward-scattering peak and to decrease backscattering. Several oscillations with well-defined minima can be seen as the coat grows thicker. The thin mantle (thin solid curve) contains 2.77 times the compound volume of the spheres, so Model 4-M2 in Table 2 is actually quite thick. The thick mantle is Model 4-M4 and has 17.2 times the volume of the spheres. (b) Addition of the first absorbing mantle to the 500-sphere aggregate 4 decreased the maximum degree of polarization. As two additional layers were added to form the thick mantle, the maximum increases again and appears to shift toward the Brewster angle in the ϴ = (60–70°) interval.

Tables (2)

Tables Icon

Table 1 Physical Parameters of Scattering Bodies Without Mantles

Tables Icon

Table 2 Physical Parameters of Scattering Bodies with Mantles

Equations (5)

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

[ E l , s E r , s ] = [ S 2 S 3 S 4 S 1 ] × exp ( ikr + ikz ) ikr × [ E l , o E r , o ] .
[ I l , s I r , s ] = [ i 22 i 21 i 12 i 11 ] × [ I l , o I r , o ] .
i ( ϴ ) = i 11 + i 12 + i 21 + i 22 2
P ( ϴ ) = ( i 11 + i 12 ) ( i 21 + i 22 ) i 11 + i 12 + i 21 + i 22
f m 2 m ¯ 2 m 2 + 2 m ¯ 2 + ( 1 f ) 1 m ¯ 2 1 + 2 m ¯ 2 = 0 .

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