Abstract

Using a technique based on Mie theory, we calculate the scattering intensity of light from particles of arbitrary size that differ slightly from spherical shape. The boundary conditions are approximated linearly in the deviations r from a spherical shape, which are expanded in spherical harmonics. The validity of the approach is restricted to rRo and r ≪ λ/2π, Ro and λ being the radius of the sphere and the wavelength of the light in the outer medium, respectively. The scattering cross section is calculated for a liquid drop fluctuating around the spherical equilibrium shape and for rigid nonspherical homogeneous bodies with specific and random orientation with respect to the scattering geometry. We find that only moments of the deviations in the range 2 ≤ l ≤ 4πRo/λ contribute to the scattering. Only those moments with azimuthal symmetry with respect to the ingoing wave vector contribute to the scattering. For small spheres, the depolarization ratio is significantly influenced by ellipsoidal deviations.

© 1982 Optical Society of America

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References

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  1. G. Duckwitz-Peterlein, G. Eilenberger, and P. Overath, “Phospholipid exchange between bilayer membranes,” Biochim. Biophys. Acta 469, 311–325 (1977).
    [Crossref] [PubMed]
  2. J. P. Reeves and R. M. Dowben, “Water permeability of phospholipid vesicles,” J. Membr. Biol. 3, 123–141 (1970).
    [Crossref]
  3. P. J. Wyatt, “Differential light scattering techniques for microbiology,” in Methods in Microbiology, J. R. Norris and D. W. Ribbons, eds. (Academic, New York, 1973), Vol. 8.
    [Crossref]
  4. H. C. van de Hulst, “Light scattering by small particles,” (Wiley, New York, 1957).
  5. W. Helfrich, “The size of bilayer vesicles generated by sonication,” Phys. Lett. A 50, 115–116 (1974).
    [Crossref]
  6. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
    [Crossref]
  7. P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. 30, 57–136 (1909).
    [Crossref]
  8. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  9. L. Mandelstam, “Über die Rauhigkeit freier Flüssigkeits-oberflächen,” Ann. Phys. 41, 609–624 (1913).
    [Crossref]
  10. M. A. Bonchiat and D. Langevin, “Relation between molecular properties and the intensity scattered by a liquid interface,” J. Colloid Interface Sci. 63, 193–211 (1978); see also L. Kramer, “Theory of light scattering from fluctuations of membranes and monolayers,” J. Chem. Phys. 55, 2097–2105 (1971).
    [Crossref]
  11. E. F. Grabowski and I. A. Cowen, “Thermal excitations of a bilipid membrane,” Biophys. J. 18, 23–28 (1977).
  12. P. L. Marston, “Rainbow phenomena and the detection of non sphericity in drops,” Appl. Opt. 19, 680–685 (1980).
    [Crossref] [PubMed]
  13. K. Sassen, “Angular scattering and rainbow formation in pendant drops,” J. Opt. Soc. Am. 69, 1083–1089 (1979).
    [Crossref]
  14. H. H. Blau, D. J. McCleese, and D. Watson, “Scattering by individual transparent spheres,” Appl. Opt. 9, 2522–2528 (1970).
    [Crossref] [PubMed]
  15. C. Yeh, “Perturbation approach to the diffraction of electromagnetic waves by arbitrarily shaped dielectric obstacles,” Phys. Rev. A 135, 1193–1201 (1964).
  16. U. A. Erna, “Exact solution for the scattering of electromagnetic waves from bodies of arbitrary shape III. Obstacles with arbitrary electromagnetic properties,” Phys. Rev. 179, 1238–1246 (1968).
    [Crossref]
  17. P. Barber and C. Yeh, “Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies,” Appl. Opt. 14, 2864–2872 (1975).
    [Crossref] [PubMed]
  18. P. Latimer, A. Brunsting, B. E. Pyle, and C. Moore, “Effects of asphericity on single particle scattering,” Appl. Opt. 17, 3152–3158 (1978).
    [Crossref] [PubMed]
  19. A. Mugnai and W. J. Wiscombe, “Scattering of radiation by moderately non spherical particles,” J. Atmos. Sci. 37, 1291–1307 (1980).
    [Crossref]
  20. J. W. S. Rayleigh, The Theory of Sound (Dover, New York, 1945), Vol. II, pp. 371–375.
  21. L. D. Landau and E. M. Lifshitz, Lehrbuch der theoretischen Physik. Bd. VI, Hydrodynamik (Akademie Verlag, Berlin, 1974), pp. 274–277.
  22. J. R. Henderson and J. Lekner, “Surface oscillations and the surface thickness of classical and quantum droplets,” Mol. Phys. 36, 781–789 (1978).
    [Crossref]
  23. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), pp. 563–568.
  24. M. Kerker uses, in fact, Riccati–Hankel functions of the second kind. However, the asymptotic behavior of those is such that they represent ingoing spherical waves. Equations (3.3.56) and (3.3.57) for the scattering amplitudes in Kerker’s book (Ref. 8) consequently contain a factor (−)n+1 that conflicts with the results of other authors where (−)n+1 is absent. See, e.g., Ref. 4, p. 125.
  25. M. E. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957), p. 60, Eq. (4.28a).
  26. The prime denoting differentiation is used with some inconsistency. In connection with the Riccati–Bessel function, the prime means differentiation with respect to the argument throughout: ψ′(Nα) = d/d(Nα)ψ(Nα) or ζ(α) = d/dαζ(α). In connection with the radial functions fni and gni we always mean differentiation with respect to R at the point R= Ro, e.g., f′ni= d/dRf(kR)|R=Ro.
  27. M. Abramovitz and J. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), p. 439, Eqs. (10.1.19)–(10.1.22), p. 445, Eq. (10.3.1).
  28. Ref. 27, p. 365, Eq. (9.3.1).
  29. J. D. Harvey and N. W. Woolford, “Laser light scattering studies of bull spermatozoae, I. Orientational effects,” Biophys. J. 31, 147–156 (1980).
  30. W. Helfrich, “Elastic properties of lipid bilayers: theory and possible experiments,” Z. Naturforsch. 28c, 693–703 (1973).
  31. R. M. Servuss, W. Harbich, and W. Helfrich, “Measurement of the curvature-elastic modulus of egg lecithin bilayers,” Biochim. Biophys. Acta 436, 900–903 (1976).
    [Crossref] [PubMed]
  32. W. Harbich, Institut für Atom-und Festkörperphysik, WE1b, Freie Universität, Königin Luisestrasse 28–30, 1000 Berlin 33, West Germany (personal communication).
  33. E. Boroske, Institut für Atom-und Festkörperphysik, WE1b, Freie Universität, Königin Luisestrasse 28–30, 1000 Berlin 33, West Germany (personal communication).
  34. S. Aragón and M. Elwenspoek (to be published).
  35. Ref. 25, p. 62, Eq. (4.34).

1980 (3)

P. L. Marston, “Rainbow phenomena and the detection of non sphericity in drops,” Appl. Opt. 19, 680–685 (1980).
[Crossref] [PubMed]

A. Mugnai and W. J. Wiscombe, “Scattering of radiation by moderately non spherical particles,” J. Atmos. Sci. 37, 1291–1307 (1980).
[Crossref]

J. D. Harvey and N. W. Woolford, “Laser light scattering studies of bull spermatozoae, I. Orientational effects,” Biophys. J. 31, 147–156 (1980).

1979 (1)

1978 (3)

J. R. Henderson and J. Lekner, “Surface oscillations and the surface thickness of classical and quantum droplets,” Mol. Phys. 36, 781–789 (1978).
[Crossref]

M. A. Bonchiat and D. Langevin, “Relation between molecular properties and the intensity scattered by a liquid interface,” J. Colloid Interface Sci. 63, 193–211 (1978); see also L. Kramer, “Theory of light scattering from fluctuations of membranes and monolayers,” J. Chem. Phys. 55, 2097–2105 (1971).
[Crossref]

P. Latimer, A. Brunsting, B. E. Pyle, and C. Moore, “Effects of asphericity on single particle scattering,” Appl. Opt. 17, 3152–3158 (1978).
[Crossref] [PubMed]

1977 (2)

E. F. Grabowski and I. A. Cowen, “Thermal excitations of a bilipid membrane,” Biophys. J. 18, 23–28 (1977).

G. Duckwitz-Peterlein, G. Eilenberger, and P. Overath, “Phospholipid exchange between bilayer membranes,” Biochim. Biophys. Acta 469, 311–325 (1977).
[Crossref] [PubMed]

1976 (1)

R. M. Servuss, W. Harbich, and W. Helfrich, “Measurement of the curvature-elastic modulus of egg lecithin bilayers,” Biochim. Biophys. Acta 436, 900–903 (1976).
[Crossref] [PubMed]

1975 (1)

1974 (1)

W. Helfrich, “The size of bilayer vesicles generated by sonication,” Phys. Lett. A 50, 115–116 (1974).
[Crossref]

1973 (1)

W. Helfrich, “Elastic properties of lipid bilayers: theory and possible experiments,” Z. Naturforsch. 28c, 693–703 (1973).

1970 (2)

J. P. Reeves and R. M. Dowben, “Water permeability of phospholipid vesicles,” J. Membr. Biol. 3, 123–141 (1970).
[Crossref]

H. H. Blau, D. J. McCleese, and D. Watson, “Scattering by individual transparent spheres,” Appl. Opt. 9, 2522–2528 (1970).
[Crossref] [PubMed]

1968 (1)

U. A. Erna, “Exact solution for the scattering of electromagnetic waves from bodies of arbitrary shape III. Obstacles with arbitrary electromagnetic properties,” Phys. Rev. 179, 1238–1246 (1968).
[Crossref]

1964 (1)

C. Yeh, “Perturbation approach to the diffraction of electromagnetic waves by arbitrarily shaped dielectric obstacles,” Phys. Rev. A 135, 1193–1201 (1964).

1913 (1)

L. Mandelstam, “Über die Rauhigkeit freier Flüssigkeits-oberflächen,” Ann. Phys. 41, 609–624 (1913).
[Crossref]

1909 (1)

P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. 30, 57–136 (1909).
[Crossref]

1908 (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
[Crossref]

Aragón, S.

S. Aragón and M. Elwenspoek (to be published).

Barber, P.

Blau, H. H.

Bonchiat, M. A.

M. A. Bonchiat and D. Langevin, “Relation between molecular properties and the intensity scattered by a liquid interface,” J. Colloid Interface Sci. 63, 193–211 (1978); see also L. Kramer, “Theory of light scattering from fluctuations of membranes and monolayers,” J. Chem. Phys. 55, 2097–2105 (1971).
[Crossref]

Boroske, E.

E. Boroske, Institut für Atom-und Festkörperphysik, WE1b, Freie Universität, Königin Luisestrasse 28–30, 1000 Berlin 33, West Germany (personal communication).

Brunsting, A.

Cowen, I. A.

E. F. Grabowski and I. A. Cowen, “Thermal excitations of a bilipid membrane,” Biophys. J. 18, 23–28 (1977).

Debye, P.

P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. 30, 57–136 (1909).
[Crossref]

Dowben, R. M.

J. P. Reeves and R. M. Dowben, “Water permeability of phospholipid vesicles,” J. Membr. Biol. 3, 123–141 (1970).
[Crossref]

Duckwitz-Peterlein, G.

G. Duckwitz-Peterlein, G. Eilenberger, and P. Overath, “Phospholipid exchange between bilayer membranes,” Biochim. Biophys. Acta 469, 311–325 (1977).
[Crossref] [PubMed]

Eilenberger, G.

G. Duckwitz-Peterlein, G. Eilenberger, and P. Overath, “Phospholipid exchange between bilayer membranes,” Biochim. Biophys. Acta 469, 311–325 (1977).
[Crossref] [PubMed]

Elwenspoek, M.

S. Aragón and M. Elwenspoek (to be published).

Erna, U. A.

U. A. Erna, “Exact solution for the scattering of electromagnetic waves from bodies of arbitrary shape III. Obstacles with arbitrary electromagnetic properties,” Phys. Rev. 179, 1238–1246 (1968).
[Crossref]

Grabowski, E. F.

E. F. Grabowski and I. A. Cowen, “Thermal excitations of a bilipid membrane,” Biophys. J. 18, 23–28 (1977).

Harbich, W.

R. M. Servuss, W. Harbich, and W. Helfrich, “Measurement of the curvature-elastic modulus of egg lecithin bilayers,” Biochim. Biophys. Acta 436, 900–903 (1976).
[Crossref] [PubMed]

W. Harbich, Institut für Atom-und Festkörperphysik, WE1b, Freie Universität, Königin Luisestrasse 28–30, 1000 Berlin 33, West Germany (personal communication).

Harvey, J. D.

J. D. Harvey and N. W. Woolford, “Laser light scattering studies of bull spermatozoae, I. Orientational effects,” Biophys. J. 31, 147–156 (1980).

Helfrich, W.

R. M. Servuss, W. Harbich, and W. Helfrich, “Measurement of the curvature-elastic modulus of egg lecithin bilayers,” Biochim. Biophys. Acta 436, 900–903 (1976).
[Crossref] [PubMed]

W. Helfrich, “The size of bilayer vesicles generated by sonication,” Phys. Lett. A 50, 115–116 (1974).
[Crossref]

W. Helfrich, “Elastic properties of lipid bilayers: theory and possible experiments,” Z. Naturforsch. 28c, 693–703 (1973).

Henderson, J. R.

J. R. Henderson and J. Lekner, “Surface oscillations and the surface thickness of classical and quantum droplets,” Mol. Phys. 36, 781–789 (1978).
[Crossref]

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Lehrbuch der theoretischen Physik. Bd. VI, Hydrodynamik (Akademie Verlag, Berlin, 1974), pp. 274–277.

Langevin, D.

M. A. Bonchiat and D. Langevin, “Relation between molecular properties and the intensity scattered by a liquid interface,” J. Colloid Interface Sci. 63, 193–211 (1978); see also L. Kramer, “Theory of light scattering from fluctuations of membranes and monolayers,” J. Chem. Phys. 55, 2097–2105 (1971).
[Crossref]

Latimer, P.

Lekner, J.

J. R. Henderson and J. Lekner, “Surface oscillations and the surface thickness of classical and quantum droplets,” Mol. Phys. 36, 781–789 (1978).
[Crossref]

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Lehrbuch der theoretischen Physik. Bd. VI, Hydrodynamik (Akademie Verlag, Berlin, 1974), pp. 274–277.

Mandelstam, L.

L. Mandelstam, “Über die Rauhigkeit freier Flüssigkeits-oberflächen,” Ann. Phys. 41, 609–624 (1913).
[Crossref]

Marston, P. L.

McCleese, D. J.

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
[Crossref]

Moore, C.

Mugnai, A.

A. Mugnai and W. J. Wiscombe, “Scattering of radiation by moderately non spherical particles,” J. Atmos. Sci. 37, 1291–1307 (1980).
[Crossref]

Overath, P.

G. Duckwitz-Peterlein, G. Eilenberger, and P. Overath, “Phospholipid exchange between bilayer membranes,” Biochim. Biophys. Acta 469, 311–325 (1977).
[Crossref] [PubMed]

Pyle, B. E.

Rayleigh, J. W. S.

J. W. S. Rayleigh, The Theory of Sound (Dover, New York, 1945), Vol. II, pp. 371–375.

Reeves, J. P.

J. P. Reeves and R. M. Dowben, “Water permeability of phospholipid vesicles,” J. Membr. Biol. 3, 123–141 (1970).
[Crossref]

Rose, M. E.

M. E. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957), p. 60, Eq. (4.28a).

Sassen, K.

Servuss, R. M.

R. M. Servuss, W. Harbich, and W. Helfrich, “Measurement of the curvature-elastic modulus of egg lecithin bilayers,” Biochim. Biophys. Acta 436, 900–903 (1976).
[Crossref] [PubMed]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), pp. 563–568.

van de Hulst, H. C.

H. C. van de Hulst, “Light scattering by small particles,” (Wiley, New York, 1957).

Watson, D.

Wiscombe, W. J.

A. Mugnai and W. J. Wiscombe, “Scattering of radiation by moderately non spherical particles,” J. Atmos. Sci. 37, 1291–1307 (1980).
[Crossref]

Woolford, N. W.

J. D. Harvey and N. W. Woolford, “Laser light scattering studies of bull spermatozoae, I. Orientational effects,” Biophys. J. 31, 147–156 (1980).

Wyatt, P. J.

P. J. Wyatt, “Differential light scattering techniques for microbiology,” in Methods in Microbiology, J. R. Norris and D. W. Ribbons, eds. (Academic, New York, 1973), Vol. 8.
[Crossref]

Yeh, C.

P. Barber and C. Yeh, “Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies,” Appl. Opt. 14, 2864–2872 (1975).
[Crossref] [PubMed]

C. Yeh, “Perturbation approach to the diffraction of electromagnetic waves by arbitrarily shaped dielectric obstacles,” Phys. Rev. A 135, 1193–1201 (1964).

Ann. Phys. (3)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
[Crossref]

P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. 30, 57–136 (1909).
[Crossref]

L. Mandelstam, “Über die Rauhigkeit freier Flüssigkeits-oberflächen,” Ann. Phys. 41, 609–624 (1913).
[Crossref]

Appl. Opt. (4)

Biochim. Biophys. Acta (2)

R. M. Servuss, W. Harbich, and W. Helfrich, “Measurement of the curvature-elastic modulus of egg lecithin bilayers,” Biochim. Biophys. Acta 436, 900–903 (1976).
[Crossref] [PubMed]

G. Duckwitz-Peterlein, G. Eilenberger, and P. Overath, “Phospholipid exchange between bilayer membranes,” Biochim. Biophys. Acta 469, 311–325 (1977).
[Crossref] [PubMed]

Biophys. J. (2)

J. D. Harvey and N. W. Woolford, “Laser light scattering studies of bull spermatozoae, I. Orientational effects,” Biophys. J. 31, 147–156 (1980).

E. F. Grabowski and I. A. Cowen, “Thermal excitations of a bilipid membrane,” Biophys. J. 18, 23–28 (1977).

J. Atmos. Sci. (1)

A. Mugnai and W. J. Wiscombe, “Scattering of radiation by moderately non spherical particles,” J. Atmos. Sci. 37, 1291–1307 (1980).
[Crossref]

J. Colloid Interface Sci. (1)

M. A. Bonchiat and D. Langevin, “Relation between molecular properties and the intensity scattered by a liquid interface,” J. Colloid Interface Sci. 63, 193–211 (1978); see also L. Kramer, “Theory of light scattering from fluctuations of membranes and monolayers,” J. Chem. Phys. 55, 2097–2105 (1971).
[Crossref]

J. Membr. Biol. (1)

J. P. Reeves and R. M. Dowben, “Water permeability of phospholipid vesicles,” J. Membr. Biol. 3, 123–141 (1970).
[Crossref]

J. Opt. Soc. Am. (1)

Mol. Phys. (1)

J. R. Henderson and J. Lekner, “Surface oscillations and the surface thickness of classical and quantum droplets,” Mol. Phys. 36, 781–789 (1978).
[Crossref]

Phys. Lett. A (1)

W. Helfrich, “The size of bilayer vesicles generated by sonication,” Phys. Lett. A 50, 115–116 (1974).
[Crossref]

Phys. Rev. (1)

U. A. Erna, “Exact solution for the scattering of electromagnetic waves from bodies of arbitrary shape III. Obstacles with arbitrary electromagnetic properties,” Phys. Rev. 179, 1238–1246 (1968).
[Crossref]

Phys. Rev. A (1)

C. Yeh, “Perturbation approach to the diffraction of electromagnetic waves by arbitrarily shaped dielectric obstacles,” Phys. Rev. A 135, 1193–1201 (1964).

Z. Naturforsch. (1)

W. Helfrich, “Elastic properties of lipid bilayers: theory and possible experiments,” Z. Naturforsch. 28c, 693–703 (1973).

Other (15)

J. W. S. Rayleigh, The Theory of Sound (Dover, New York, 1945), Vol. II, pp. 371–375.

L. D. Landau and E. M. Lifshitz, Lehrbuch der theoretischen Physik. Bd. VI, Hydrodynamik (Akademie Verlag, Berlin, 1974), pp. 274–277.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), pp. 563–568.

M. Kerker uses, in fact, Riccati–Hankel functions of the second kind. However, the asymptotic behavior of those is such that they represent ingoing spherical waves. Equations (3.3.56) and (3.3.57) for the scattering amplitudes in Kerker’s book (Ref. 8) consequently contain a factor (−)n+1 that conflicts with the results of other authors where (−)n+1 is absent. See, e.g., Ref. 4, p. 125.

M. E. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957), p. 60, Eq. (4.28a).

The prime denoting differentiation is used with some inconsistency. In connection with the Riccati–Bessel function, the prime means differentiation with respect to the argument throughout: ψ′(Nα) = d/d(Nα)ψ(Nα) or ζ(α) = d/dαζ(α). In connection with the radial functions fni and gni we always mean differentiation with respect to R at the point R= Ro, e.g., f′ni= d/dRf(kR)|R=Ro.

M. Abramovitz and J. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), p. 439, Eqs. (10.1.19)–(10.1.22), p. 445, Eq. (10.3.1).

Ref. 27, p. 365, Eq. (9.3.1).

P. J. Wyatt, “Differential light scattering techniques for microbiology,” in Methods in Microbiology, J. R. Norris and D. W. Ribbons, eds. (Academic, New York, 1973), Vol. 8.
[Crossref]

H. C. van de Hulst, “Light scattering by small particles,” (Wiley, New York, 1957).

W. Harbich, Institut für Atom-und Festkörperphysik, WE1b, Freie Universität, Königin Luisestrasse 28–30, 1000 Berlin 33, West Germany (personal communication).

E. Boroske, Institut für Atom-und Festkörperphysik, WE1b, Freie Universität, Königin Luisestrasse 28–30, 1000 Berlin 33, West Germany (personal communication).

S. Aragón and M. Elwenspoek (to be published).

Ref. 25, p. 62, Eq. (4.34).

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

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Figures (3)

Fig. 1
Fig. 1

Scattering geometry. The ingoing wave vector (ki) is parallel to z; the scattered light (kf) is observed in the zy plane. pi is the polarization of the incident light and χ is the angle between pi and x, the latter being perpendicular to the scattering plane. The scattering amplitudes S1 and S2 [Eqs. (18) and (19)] refer to observation of the scattered light polarized parallel and perpendicular to x, respectively.

Fig. 2
Fig. 2

Relation of the scattering geometry and the rotated coordinate system used to expand the surface of the scattering center in spherical coordinates. Only β of the set of Eulerian angles Ω = (αβγ) is indicated.

Fig. 3
Fig. 3

A scattering center with specific orientation with respect to the scattering geometry. We have indicated the cone around which the figure axis may rotate without changing the scattering intensity.

Equations (102)

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E 0 = X ^ e i k z = n = 1 2 n + 1 n ( n + 1 ) i n [ m o 1 n ( 1 ) - i n e 1 n ( 1 ) ] ,
m 1 e o n = ± ψ n ( k R ) k R sin θ P n 1 ( cos θ ) cos sin ϕ ê θ - ψ n ( k R ) k R d P n 1 ( cos θ ) d θ sin cos ϕ ê ϕ
n 1 e o n = n ( n + 1 ) ψ n ( k R ) ( k R ) 2 P n 1 ( cos θ ) sin cos ϕ ê R + 1 k R d ψ n ( k R ) d ( k R ) d P n 1 ( cos θ ) d θ sin cos ϕ ê θ ± d ψ n ( k R ) d ( k R ) P n 1 ( cos θ ) k R sin θ cos sin ϕ ê ϕ .
E s = n = 1 2 n + 1 n ( n + 1 ) i n [ a n m o 1 n ( 3 ) - i b n n e 1 n ( 3 ) ]
E t = n = 1 2 n + 1 n ( n + 1 ) i n [ c n m o 1 n ( 1 ) - i d n n e 1 n ( 1 ) ] .
curl E = i ω B .
n ^ × C * = 0 ,
E * ( R ) = n = 1 i n 2 n + 1 n ( n + 1 ) { - i n ( n + 1 ) f n o ( α ) P n 1 cos ϕ ê R + [ P n 1 sin θ f n 1 ( α ) - i d P n 1 d θ f n 2 ( α ) ] cos ϕ ê θ + [ - d P n 1 d θ f n 1 ( α ) + i P n 1 sin θ f n 2 ( α ) ] sin ϕ ê ϕ }
H * ( R ) = 1 μ o ω n = 1 i n 2 n + 1 n ( n + 1 ) × { - i n ( n + 1 ) g n 0 ( α ) P n 1 sin ϕ ê R + [ P n 1 sin θ g n 1 ( α ) - i d P n 1 d θ g n 2 ( α ) ] sin ϕ ê θ + [ d P n 1 d θ g n 1 ( α ) - i P n 1 sin θ g n 2 ( α ) ] cos ϕ ê ϕ } .
f n 0 = α - 2 [ ψ n ( α ) + b n ζ n ( α ) - d n ψ n ( N α ) / N 2 ] ,
f n 1 = α - 1 [ ψ n ( α ) + a n ζ n ( α ) - c n ψ n ( N α ) / N ] ,
f n 2 = α - 1 [ ψ n ( α ) + b n ζ n ( α ) - d n ψ n ( N α ) / N ] ,
g n 0 = α - 2 [ ψ n ( α ) + a n ζ n ( α ) - c n ψ n ( N α ) / N ] ,
g n 1 = α - 1 [ ψ n ( α ) + b n ζ n ( α ) - d n ψ n ( N α ) ] ,
g n 2 = α - 1 [ ψ n ( α ) + a n ζ n ( α ) - c n ψ n ( N α ) ] .
d σ d Ω = k - 2 S 1 2 sin 2 χ
d σ d Ω = k - 2 S 2 2 cos 2 χ .
S 1 = n = 1 2 n + 1 n ( n + 1 ) ( b n P n 1 sin θ + a n d P n 1 d θ )
S 2 = n = 1 2 n + 1 n ( n + 1 ) ( a n P n 1 sin θ + b n d P n 1 d θ ) .
σ SC = 2 π k - 2 n = 1 ( 2 n + 1 ) ( a n 2 + b n 2 ) .
r ˜ ( θ , ϕ ) = R o + r ( θ , ϕ ) ,
r ( θ , ϕ ) = l = 2 m = - l + l ρ l m Y l m ( θ , ϕ ) .
ρ l m ( Ω ) = m = - l l ρ l m D m m l ( Ω ) .
r ( θ , ϕ ) = l = 2 m = - l l ρ l m ( Ω ) Y l m ( θ , ϕ ) .
ρ l m = r ( θ , ϕ ) Y l m * ( θ , ϕ ) sin θ d θ d ϕ ,
ρ l m * = ( - ) m ρ l - m .
n ^ = ( 1 , - 1 R o r θ , - 1 R o sin θ r ϕ ) .
ρ l m 2 = 2 k B T K 1 ( l + 1 ) ( l - 1 ) ,
C θ * ( R o ) + R C θ * R = R o · r + 1 R o r θ C R * ( R o ) = 0
C ϕ * ( R o ) + R C ϕ * R = R o · r + 1 R o sin θ r ϕ C R * ( R o ) = 0.
f n 0 0 n ( n + 1 ) / R o = f n 0 0 = k / α 3 ( 1 - N 2 ) / N 2 × d 0 ψ n ( N α ) n ( n + 1 ) , f n 1 0 = 0 , g n 0 0 = 0 , g n 1 0 = k / α ( 1 - N 2 ) / N c n 0 ψ n ( N α ) , g n 2 0 = k / α ( 1 - N 2 ) / N c n 0 ψ n ( N α ) .
n = 1 i n 2 n + 1 n ( n + 1 ) [ ( P n 1 sin θ g n 1 - i d P n 1 d θ g n 2 ) sin ϕ + ( P n 1 sin θ g n 1 0 - i d P n 1 d θ g n 2 0 ) sin ϕ l m ρ l m ( Ω ) Y l m ] = 0 ,
n = 1 i n 2 n + 1 n ( n + 1 ) [ ( d P n 1 d θ g n 1 - i P n 1 sin θ g n 2 ) cos ϕ + ( d P n 1 d θ g n 1 0 - i P n 1 sin θ g n 2 0 ) cos ϕ l m ρ l m ( Ω ) Y l m ] = 0 ,
n = 1 i n 2 n + 1 n ( n + 1 ) [ ( P n 1 sin θ f n 1 - i d P n 1 d θ f n 2 ) cos ϕ - i f n 0 0 / R 0 n ( n + 1 ) cos ϕ l m ρ l m ( Ω ) d d θ ( P n 1 Y l m ) ] = 0 ,
n = 1 i n 2 n + 1 n ( n + 1 ) [ ( d P n 1 d θ f n 1 - i P n 1 sin θ f n 2 ) sin ϕ - i f n 0 0 / R 0 n ( n + 1 ) l m ρ l m ( Ω ) × ( sin ϕ - i m cos ϕ ) P n 1 Y l m sin θ ] = 0.
0 π sin θ d θ ( P n 1 sin θ d P n 1 d θ + d P n 1 d θ P n 1 sin θ ) = 0
0 π sin θ d θ ( d P n m d θ d P n m d θ + m 2 sin 2 θ P n m P n m ) = 2 n ( n + 1 ) 2 n + 1 ( n + m ) ! ( n - m ) ! δ n n .
g q 1 + [ 2 i q q ( q + 1 ) ] - 1 l ρ l ( 2 l + 1 2 π ) 1 / 2 × n [ g n 1 o 0 π sin θ d θ ( P n 1 P q 1 sin 2 θ + d P n 1 d θ d P q 1 d θ ) P l o - i g n 2 o o π sin θ d θ ( d P n 1 d θ P q 1 + P n 1 d P q 1 d θ ) P l 0 sin θ ] = 0 ,
g q 2 + i [ 2 i q q ( q + 1 ) ] - 1 l ρ l ( 2 l + 1 2 π ) 1 / 2 × n [ g n 1 o o π sin θ d θ ( d P q 1 d θ P n 1 + P q 1 d P n 1 d θ ) P l o sin θ - i g n 2 o o π sin θ d θ ( P n 1 P q 1 sin 2 θ + d P n 1 d θ d P q 1 d θ ) P l o ] = 0 ,
f q 1 - n i n ( n + 1 ) f n o o 2 R 0 i q q ( q + 1 ) l ρ l ( 2 l + 1 2 π ) 1 / 2 o π sin θ d θ × [ d d θ ( P n 1 P l 0 ) P q 1 sin θ + P n 1 P l 0 sin θ d P q 1 d θ ] = 0 ,
f q 2 + n n ( n + 1 ) f n o o 2 R 0 i q q ( q + 1 ) l ρ l ( 2 l + 1 2 π ) 1 / 2 o π sin θ d θ × [ d d θ ( P n 1 P l 0 ) d P q 1 d θ + P n 1 P l 0 P q 1 sin 2 θ ] = 0.
f n 1 0 = f n 2 0 = g n 1 0 = g n 2 0 = 0 ,
a n 0 = - ψ n ( α ) ψ n ( N α ) - N ψ n ( N α ) ψ n ( α ) ζ n ( α ) ψ n ( N α ) - N ζ n ( α ) ψ n ( N α ) ,
b n 0 = - N ψ n ( α ) ψ n ( N α ) - ψ n ( N α ) ψ n ( α ) N ζ n ( α ) ψ n ( N α ) - ζ n ( α ) ψ n ( N α ) ,
c n 0 = i N / [ ψ n ( N α ) ζ n ( α ) - N ψ n ( N α ) ζ n ( α ) ] ,
d n 0 = i N / [ N ψ n ( N α ) ζ n ( α ) - ψ n ( N α ) ζ n ( α ) ] .
ζ n ( α ) ψ n ( α ) - ζ n ( α ) ψ n ( α ) = i .
a q = a q 0 + l k ρ l ( Ω ) a q l 1
b q = b q 0 + l k ρ l ( Ω ) b q l 1 ,
a q l 1 = ( 1 - N ) i - q 2 q ( q + 1 ) N ( 2 l + 1 2 π ) 1 / 2 × n ψ q ( N α ) Δ l n q ( N α ) ζ q ( α ) ψ q ( N α ) - N ζ q ( α ) ψ q ( N α )
b q l 1 = ( 1 - N ) i - q 2 q ( q + 1 ) N ( 2 l + 1 2 π ) 1 / 2 × n ψ q ( N α ) χ l n q ( N α ) - ψ q ( N α ) Γ l n q ( N α ) ζ q ( α ) ψ q ( N α ) - N ζ q ( α ) ψ q ( N α ) .
Δ l n q ( N α ) = i n 2 n + 1 n ( n + 1 ) c n 0 [ ψ n ( N α ) D ( l n q ) - i ψ n ( N α ) B ( l n q ) ] ,
Γ l n q ( N α ) = i n 2 n + 1 n ( n + 1 ) c n 0 [ i ψ n ( N α ) B ( l n q ) - ψ n ( N α ) D ( l n q ) ] ,
χ l n q ( N α ) = i n 2 n + 1 n ( n + 1 ) d n 0 α 2 q ( q + 1 ) ψ n ( N α ) A ( l n q ) .
A ( l n q ) = 2 [ q ( q + 1 ) n ( n + 1 ) 2 q + 1 ] 1 / 2 C ( l n q , 01 ) C ( l n q , 00 ) , D ( l n q ) = ½ [ q ( q + 1 ) + n ( n + 1 ) - l ( l + 1 ) ] A ( l n q ) ,
B ( l n q ) = - k = 0 1 / 2 ( l - 1 - q - n ) ( 2 l - 1 - 4 k ) A ( l - 1 - 2 k , n , q ) .
S 1 l 1 = n = 1 2 n + 1 n ( n + 1 ) ( b n l 1 P n 1 sin θ + a n l 1 d P n 1 d θ )
S 2 l 1 = n = 1 2 n + 1 n ( n + 1 ) ( a n l 1 P n 1 sin θ + b n l 1 d P n 1 d θ ) ,
I 1 , 2 1 = l 1 2 l + 1 ρ l 2 S 1 , 2 l 1 2 cos 2 sin 2 χ
ρ l 2 = 2 l + 1 8 π 3 d Ω ρ l 2 .
I 1 , 2 1 = 2 k B T K l 1 2 + l ( l + 1 ) S 1 , 2 l 1 2 cos 2 sin 2 χ .
ψ ν ( z ) ½ z ν + 1 ( ν + 1 / 2 ) 1 / 2 ( e 2 ν + 1 ) ν + 1 / 2
ϕ ν ( z ) - ½ z - ν ( ν + 1 / 2 ) 1 / 2 ( e 2 ν + 1 ) - ν - 1 / 2
b 1 o 2 3 i N 2 - 1 N 2 + 2 α 3 ,
b 2 o 1 15 i N 2 - 1 2 N 2 + 3 α 5 ,
a 1 o 1 45 i ( N 2 - 1 ) α 5 .
a 12 1 i N - 1 18 N ( 5 2 π ) 1 / 2 D ( 211 ) α 3 ,
a 22 1 - i N - 1 360 ( 5 2 π ) 1 / 2 B ( 212 ) α 5 ,
a 23 1 N - 1 180 ( 7 2 π ) 1 / 2 D ( 312 ) α 4 ,
b 12 1 - i N - 1 12 ( 5 2 π ) 1 / 2 [ 2 N 2 A ( 211 ) + α D ( 211 ) ] α 2 ,
b 22 1 i N - 1 15 N ( 5 2 π ) 1 / 2 B ( 212 ) α 3 ,
b 23 1 - N - 1 15 ( 7 2 π ) 1 / 2 [ N 2 A ( 312 ) + α 6 D ( 312 ) ] α 3 .
A ( 211 ) = - 4 / 15 ,             D ( 211 ) = 8 / 15 ,             A ( 312 ) = - 12 / 35 , D ( 312 ) = 24 / 35 ,             B ( 212 ) = 12 / 5.
S 2 ( 90 ° ) = 3 2 a 1 - 5 2 b 2 .
S 2 0 ( 90 ° ) = 1 15 i ( N 2 - 1 ) 2 2 N 2 + 3 α 5 .
S 2 , 2 1 ( 90 ° ) = - 16 45 5 2 π i N - 1 N α 3 .
I 2 ( 90 ° ) = k - 2 1 225 [ ( N 2 - 1 ) 2 2 N 2 + 3 ] 2 α 10 + ρ 2 2 5 2 π ( 16 45 ) 2 ( N - 1 N ) 2 α 6 ,
σ ( 90 ° ) = 1 625 ( N 2 - 1 ) 2 α 4 + k 2 ρ 2 2 1 25 N 2 64 45 .
ρ l o ( Ω ) = ρ l o 2 π 2 l + 1 P l ( cos Ω ) ,
S 1 , 2 = S 1 , 2 0 + k ρ l ( Ω ) S 1 , 2 1 ,
I 1 , 2 = [ I 1 , 2 0 + k - 2 2 k ρ l ( Ω ) ] Re ( S 1 , 2 0 S 1 , 2 1 ) cos 2 sin 2 χ ,
- 1 + 1 d x [ d d x ( P n 1 P l 0 ) P q 1 + P n 1 P l 0 d P q 1 d x ] = - 1 + 1 d d x ( P n 1 P q 1 P l 0 ) d x = 0.
- 1 + 1 d x [ ( 1 - x 2 ) d d x ( P n 1 P l 0 ) d P q 1 d x + P n 1 P l 0 1 - x 2 P q 1 ] .
- - 1 + 1 d x P n 1 P l 0 ( - 2 x d P n 1 d x + ( 1 - x 2 ) d 2 P q 1 d x 2 ) = - 1 + 1 d x P q 1 P l 0 [ q ( q + 1 ) - 1 1 - x 2 ] P n 1
q ( q + 1 ) - 1 + 1 d x P n 1 P q 1 P l 0 .
A ( l n q ) = - 1 + 1 d x P n 1 P q 1 P l 0 .
- 1 + 1 d x [ ( 1 - x 2 ) d P n 1 d x d P q 1 d x + P n 1 P q 1 ( 1 - x 2 ) ] P l 0 .
- 1 + 1 d x [ q ( q + 1 ) - P n 1 P q 1 P l 0 1 - x 2 - ( 1 - x 2 ) P n 1 d P q 1 d x d P l 0 d x ] ,
- 1 + 1 d x [ q ( q + 1 ) - ( 1 - x 2 ) P n 1 d P q 1 d x d P l 0 d x ] .
- 1 + 1 d x ( 1 - x 2 ) d P n 1 d x P q 1 d P l 0 d x - l ( l + 1 ) - 1 + 1 d x P n 1 P q 1 P l 0 .
- 1 + 1 d x [ n ( n + 1 ) - 1 1 - x 2 ] P l 0 P n 1 P q 1 - - 1 + 1 d x ( 1 - x 2 ) P l 0 d P n 1 d x d P q 1 d x ,
[ q ( q + 1 ) + n ( n + 1 ) - l ( l + 1 ) ] - 1 + 1 P n 1 P q 1 P l 0 d x - - 1 + 1 d x [ P l 0 P n 1 P q 1 1 - x 2 + ( 1 - x 2 ) P l 0 d P n 1 d x d P q 1 d x ] ;
- 1 + 1 d x [ P l 0 P n 1 P q 1 1 - x 2 + ( 1 + x 2 ) P l 0 d P n 1 d x d P q 1 d x ] = ½ [ q ( q + 1 ) + n ( n + 1 ) - l ( l + 1 ) ] A ( l n q ) .
P q 1 ( x ) = - q ( q + 1 ) P q - 1 ( x ) , P n 1 ( x ) = [ 4 π 2 n + 1 n ( n + 1 ) ] 1 / 2 Y n 1 e - i ϕ ,
P q 1 ( x ) = [ 4 π 2 q + 1 q ( q + 1 ) ] 1 / 2 Y q 1 * e i ϕ .
A ( l n q ) = 1 2 π [ ( 4 π ) 3 q ( q + 1 ) n ( n + 1 ) ( 2 l + 1 ) ( 2 q + 1 ) ( 2 n + 1 ) ] 1 / 2 × o 2 π d ϕ o π sin θ d θ Y n 1 Y q 1 * Y l o ,
A ( l n q ) = 2 [ q ( q + 1 ) n ( n + 1 ) ] 1 / 2 2 q + 1 C ( l n q , 01 ) C ( l n q , 00 ) .
B ( l n q ) - 1 + 1 d x d d x ( P n 1 P q 1 ) P l 0 = - - 1 + 1 d x P n 1 P q 1 P l 1 ( 1 - x 2 ) 1 / 2 .
P j - 1 1 - P j + 1 1 = ( 2 j + 1 ) ( 1 - x 2 ) 1 / 2 P j 0 .
B ( l n q ) = k = 0 ( 2 l - 1 - 4 k ) A ( l - 1 - 2 k , n , q ) .
B ( l n q ) = - k = 0 ( 2 l + 3 + 4 k ) A ( l + 1 + 2 k , n q ) .
l = n + q - 1 ,             n + q - 2 , ,             n - q + 1.