Abstract

Total scattering coefficients for concentric spheres with inner sphere m1 = 2.1050 and concentric spherical shell m2 = 1.4821 have been computed for ν = 2πb/λ over the interval 0.1 (.1) 23.0 (2) 53.0 and for α/ν values of 0, 0.2, 0.4, 0.6, 0.8, 0.9, 0.95, 0.98, 0.99, and 1.00, where α = 2πa/λ, a and b are the radii of the inner and total spheres, and λ is the wavelength. The results are compared with those obtained by a small particle approximation, the approximation suggested by Ryde, the Rayleigh-Gans method and an approximation based on using the single sphere method with a volume averaged refractive index. The Rayleigh-Gans equations for concentric spheres are derived. The small particle approximation permits accurate estimation of the total scattering coefficient for any combination of α and ν up to ν = 1.4.

© 1962 Optical Society of America

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References

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  1. A. L. Aden and M. Kerker, J. Appl. Phys. 22, 1242 (1951).
    [Crossref]
  2. M. Kerker, P. Langleben, and K. L. S. Gunn, J. Meteorol. 8, 424 (1951).
    [Crossref]
  3. D. Atlas, M. Kerker, and W. Hitschfeld, J. Atm. Terrest. Phys. 3, 108 (1953).
    [Crossref]
  4. B. M. Herman and L. J. Battan, Institute of Atmospheric Physics, University of Arizona, Tucson, Arizona, 1960, Scientific Report No. 15 (unpublished).
  5. D. Atlas, W. G. Harper, F. H. Ludlam, and W. C. Macklin, Quart. J. Roy. Meteorol. Soc. 86, 468 (1960).
    [Crossref]
  6. P. Debye and L. K. H. van Beek, J. Chem. Phys. 31, 1595 (1959).
    [Crossref]
  7. H. Oser (private communication).
  8. E. Matijević, M. Kerker, and K. F. Schulz, Discussions Faraday Soc. 30, 178, 223 (1960).
    [Crossref]
  9. E. Matijević, K. F. Schulz, and M. Kerker, J. Colloid Sci. 17, 26 (1962).
    [Crossref]
  10. J. L. Greenstein, “Polarization of Starlight by the Interstellar Medium,” Lowell Obs. Bull. No. 5,  IV, No. 17 (1960).
  11. A. N. Lowan, “Tables of Scattering Functions for Spherical Particles,” National Bureau of Standards, Applied Mathematics Series 4, U. S. Government Printing Office, Washington (1948).
  12. R. O. Gumprecht and C. M. Sliepcevich, Tables of Light Scattering Functions for Spherical Particles (University of MichiganAnn Arbor, Michigan, 1951).
  13. H. C. Van de Hulst, Recherches Astron. Obs. d’Utrecht 11, (1946).
  14. H. C. Van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, Inc., New York, 1957).
  15. P. G. W. Hawksley, Bull. Brit. Coal Utilization Research Assoc. 16, 117, 181 (1952).
  16. R. B. Penndorf, “New Tables of Mie Scattering Functions for Spherical Particles. Part 6: Total Mie Scattering Coefficients for Real Refractive Indices.” Geophysical Research Papers No. 45, A. F. Cambridge Research Center, Bedford, Massachusetts, March, 1956. Parts of this appear in J. Meteorol. 13, 219 (1956); J. Opt. Soc. Am. 46, 1001 (1956); ibid. 47, 603, 1010 (1957); J. Phys. Chem. 62, 1537 (1958).
    [Crossref]
  17. M. Kerker, J. P. Kratohvil, and E. Matijević, J. Phys. Chem. 65, 1713 (1961).
    [Crossref]
  18. A. Güttler, Ann. Physik 11, 65 (1952).
    [Crossref]
  19. J. W. Ryde, Meteorological Factors in Radio-Wave Propagation (The Physical Society, London, 1946), pp. 169–188.
  20. Rayleigh, Proc. Roy. Soc. (London) A90, 219 (1914).
    [Crossref]
  21. J. W. Ryde and B. S. Cooper, Proc. Roy. Soc. (London) A131, 461 (1931).
  22. Reference 14, p. 90.

1962 (1)

E. Matijević, K. F. Schulz, and M. Kerker, J. Colloid Sci. 17, 26 (1962).
[Crossref]

1961 (1)

M. Kerker, J. P. Kratohvil, and E. Matijević, J. Phys. Chem. 65, 1713 (1961).
[Crossref]

1960 (3)

J. L. Greenstein, “Polarization of Starlight by the Interstellar Medium,” Lowell Obs. Bull. No. 5,  IV, No. 17 (1960).

E. Matijević, M. Kerker, and K. F. Schulz, Discussions Faraday Soc. 30, 178, 223 (1960).
[Crossref]

D. Atlas, W. G. Harper, F. H. Ludlam, and W. C. Macklin, Quart. J. Roy. Meteorol. Soc. 86, 468 (1960).
[Crossref]

1959 (1)

P. Debye and L. K. H. van Beek, J. Chem. Phys. 31, 1595 (1959).
[Crossref]

1953 (1)

D. Atlas, M. Kerker, and W. Hitschfeld, J. Atm. Terrest. Phys. 3, 108 (1953).
[Crossref]

1952 (2)

A. Güttler, Ann. Physik 11, 65 (1952).
[Crossref]

P. G. W. Hawksley, Bull. Brit. Coal Utilization Research Assoc. 16, 117, 181 (1952).

1951 (2)

A. L. Aden and M. Kerker, J. Appl. Phys. 22, 1242 (1951).
[Crossref]

M. Kerker, P. Langleben, and K. L. S. Gunn, J. Meteorol. 8, 424 (1951).
[Crossref]

1946 (1)

H. C. Van de Hulst, Recherches Astron. Obs. d’Utrecht 11, (1946).

1931 (1)

J. W. Ryde and B. S. Cooper, Proc. Roy. Soc. (London) A131, 461 (1931).

1914 (1)

Rayleigh, Proc. Roy. Soc. (London) A90, 219 (1914).
[Crossref]

Aden, A. L.

A. L. Aden and M. Kerker, J. Appl. Phys. 22, 1242 (1951).
[Crossref]

Atlas, D.

D. Atlas, W. G. Harper, F. H. Ludlam, and W. C. Macklin, Quart. J. Roy. Meteorol. Soc. 86, 468 (1960).
[Crossref]

D. Atlas, M. Kerker, and W. Hitschfeld, J. Atm. Terrest. Phys. 3, 108 (1953).
[Crossref]

Battan, L. J.

B. M. Herman and L. J. Battan, Institute of Atmospheric Physics, University of Arizona, Tucson, Arizona, 1960, Scientific Report No. 15 (unpublished).

Cooper, B. S.

J. W. Ryde and B. S. Cooper, Proc. Roy. Soc. (London) A131, 461 (1931).

Debye, P.

P. Debye and L. K. H. van Beek, J. Chem. Phys. 31, 1595 (1959).
[Crossref]

Greenstein, J. L.

J. L. Greenstein, “Polarization of Starlight by the Interstellar Medium,” Lowell Obs. Bull. No. 5,  IV, No. 17 (1960).

Gumprecht, R. O.

R. O. Gumprecht and C. M. Sliepcevich, Tables of Light Scattering Functions for Spherical Particles (University of MichiganAnn Arbor, Michigan, 1951).

Gunn, K. L. S.

M. Kerker, P. Langleben, and K. L. S. Gunn, J. Meteorol. 8, 424 (1951).
[Crossref]

Güttler, A.

A. Güttler, Ann. Physik 11, 65 (1952).
[Crossref]

Harper, W. G.

D. Atlas, W. G. Harper, F. H. Ludlam, and W. C. Macklin, Quart. J. Roy. Meteorol. Soc. 86, 468 (1960).
[Crossref]

Hawksley, P. G. W.

P. G. W. Hawksley, Bull. Brit. Coal Utilization Research Assoc. 16, 117, 181 (1952).

Herman, B. M.

B. M. Herman and L. J. Battan, Institute of Atmospheric Physics, University of Arizona, Tucson, Arizona, 1960, Scientific Report No. 15 (unpublished).

Hitschfeld, W.

D. Atlas, M. Kerker, and W. Hitschfeld, J. Atm. Terrest. Phys. 3, 108 (1953).
[Crossref]

Kerker, M.

E. Matijević, K. F. Schulz, and M. Kerker, J. Colloid Sci. 17, 26 (1962).
[Crossref]

M. Kerker, J. P. Kratohvil, and E. Matijević, J. Phys. Chem. 65, 1713 (1961).
[Crossref]

E. Matijević, M. Kerker, and K. F. Schulz, Discussions Faraday Soc. 30, 178, 223 (1960).
[Crossref]

D. Atlas, M. Kerker, and W. Hitschfeld, J. Atm. Terrest. Phys. 3, 108 (1953).
[Crossref]

M. Kerker, P. Langleben, and K. L. S. Gunn, J. Meteorol. 8, 424 (1951).
[Crossref]

A. L. Aden and M. Kerker, J. Appl. Phys. 22, 1242 (1951).
[Crossref]

Kratohvil, J. P.

M. Kerker, J. P. Kratohvil, and E. Matijević, J. Phys. Chem. 65, 1713 (1961).
[Crossref]

Langleben, P.

M. Kerker, P. Langleben, and K. L. S. Gunn, J. Meteorol. 8, 424 (1951).
[Crossref]

Lowan, A. N.

A. N. Lowan, “Tables of Scattering Functions for Spherical Particles,” National Bureau of Standards, Applied Mathematics Series 4, U. S. Government Printing Office, Washington (1948).

Ludlam, F. H.

D. Atlas, W. G. Harper, F. H. Ludlam, and W. C. Macklin, Quart. J. Roy. Meteorol. Soc. 86, 468 (1960).
[Crossref]

Macklin, W. C.

D. Atlas, W. G. Harper, F. H. Ludlam, and W. C. Macklin, Quart. J. Roy. Meteorol. Soc. 86, 468 (1960).
[Crossref]

Matijevic, E.

E. Matijević, K. F. Schulz, and M. Kerker, J. Colloid Sci. 17, 26 (1962).
[Crossref]

M. Kerker, J. P. Kratohvil, and E. Matijević, J. Phys. Chem. 65, 1713 (1961).
[Crossref]

E. Matijević, M. Kerker, and K. F. Schulz, Discussions Faraday Soc. 30, 178, 223 (1960).
[Crossref]

Oser, H.

H. Oser (private communication).

Penndorf, R. B.

R. B. Penndorf, “New Tables of Mie Scattering Functions for Spherical Particles. Part 6: Total Mie Scattering Coefficients for Real Refractive Indices.” Geophysical Research Papers No. 45, A. F. Cambridge Research Center, Bedford, Massachusetts, March, 1956. Parts of this appear in J. Meteorol. 13, 219 (1956); J. Opt. Soc. Am. 46, 1001 (1956); ibid. 47, 603, 1010 (1957); J. Phys. Chem. 62, 1537 (1958).
[Crossref]

Rayleigh,

Rayleigh, Proc. Roy. Soc. (London) A90, 219 (1914).
[Crossref]

Ryde, J. W.

J. W. Ryde and B. S. Cooper, Proc. Roy. Soc. (London) A131, 461 (1931).

J. W. Ryde, Meteorological Factors in Radio-Wave Propagation (The Physical Society, London, 1946), pp. 169–188.

Schulz, K. F.

E. Matijević, K. F. Schulz, and M. Kerker, J. Colloid Sci. 17, 26 (1962).
[Crossref]

E. Matijević, M. Kerker, and K. F. Schulz, Discussions Faraday Soc. 30, 178, 223 (1960).
[Crossref]

Sliepcevich, C. M.

R. O. Gumprecht and C. M. Sliepcevich, Tables of Light Scattering Functions for Spherical Particles (University of MichiganAnn Arbor, Michigan, 1951).

van Beek, L. K. H.

P. Debye and L. K. H. van Beek, J. Chem. Phys. 31, 1595 (1959).
[Crossref]

Van de Hulst, H. C.

H. C. Van de Hulst, Recherches Astron. Obs. d’Utrecht 11, (1946).

H. C. Van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, Inc., New York, 1957).

Ann. Physik (1)

A. Güttler, Ann. Physik 11, 65 (1952).
[Crossref]

Bull. Brit. Coal Utilization Research Assoc. (1)

P. G. W. Hawksley, Bull. Brit. Coal Utilization Research Assoc. 16, 117, 181 (1952).

Discussions Faraday Soc. (1)

E. Matijević, M. Kerker, and K. F. Schulz, Discussions Faraday Soc. 30, 178, 223 (1960).
[Crossref]

J. Appl. Phys. (1)

A. L. Aden and M. Kerker, J. Appl. Phys. 22, 1242 (1951).
[Crossref]

J. Atm. Terrest. Phys. (1)

D. Atlas, M. Kerker, and W. Hitschfeld, J. Atm. Terrest. Phys. 3, 108 (1953).
[Crossref]

J. Chem. Phys. (1)

P. Debye and L. K. H. van Beek, J. Chem. Phys. 31, 1595 (1959).
[Crossref]

J. Colloid Sci. (1)

E. Matijević, K. F. Schulz, and M. Kerker, J. Colloid Sci. 17, 26 (1962).
[Crossref]

J. Meteorol. (1)

M. Kerker, P. Langleben, and K. L. S. Gunn, J. Meteorol. 8, 424 (1951).
[Crossref]

J. Phys. Chem. (1)

M. Kerker, J. P. Kratohvil, and E. Matijević, J. Phys. Chem. 65, 1713 (1961).
[Crossref]

Lowell Obs. Bull. No. 5 (1)

J. L. Greenstein, “Polarization of Starlight by the Interstellar Medium,” Lowell Obs. Bull. No. 5,  IV, No. 17 (1960).

Proc. Roy. Soc. (London) (2)

Rayleigh, Proc. Roy. Soc. (London) A90, 219 (1914).
[Crossref]

J. W. Ryde and B. S. Cooper, Proc. Roy. Soc. (London) A131, 461 (1931).

Quart. J. Roy. Meteorol. Soc. (1)

D. Atlas, W. G. Harper, F. H. Ludlam, and W. C. Macklin, Quart. J. Roy. Meteorol. Soc. 86, 468 (1960).
[Crossref]

Recherches Astron. Obs. d’Utrecht (1)

H. C. Van de Hulst, Recherches Astron. Obs. d’Utrecht 11, (1946).

Other (8)

H. C. Van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, Inc., New York, 1957).

H. Oser (private communication).

R. B. Penndorf, “New Tables of Mie Scattering Functions for Spherical Particles. Part 6: Total Mie Scattering Coefficients for Real Refractive Indices.” Geophysical Research Papers No. 45, A. F. Cambridge Research Center, Bedford, Massachusetts, March, 1956. Parts of this appear in J. Meteorol. 13, 219 (1956); J. Opt. Soc. Am. 46, 1001 (1956); ibid. 47, 603, 1010 (1957); J. Phys. Chem. 62, 1537 (1958).
[Crossref]

A. N. Lowan, “Tables of Scattering Functions for Spherical Particles,” National Bureau of Standards, Applied Mathematics Series 4, U. S. Government Printing Office, Washington (1948).

R. O. Gumprecht and C. M. Sliepcevich, Tables of Light Scattering Functions for Spherical Particles (University of MichiganAnn Arbor, Michigan, 1951).

B. M. Herman and L. J. Battan, Institute of Atmospheric Physics, University of Arizona, Tucson, Arizona, 1960, Scientific Report No. 15 (unpublished).

Reference 14, p. 90.

J. W. Ryde, Meteorological Factors in Radio-Wave Propagation (The Physical Society, London, 1946), pp. 169–188.

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Figures (6)

Fig. 1
Fig. 1

Total scattering coefficient K vs α/ν for values of ν from 0.6 to 1.8.

Fig. 2
Fig. 2

Total scattering coefficient K vs α/ν for values of ν from 2.0 to 3.0.

Fig. 3
Fig. 3

Total scattering coefficient K vs α/ν for ν = 10.0 and 11.0.

Fig. 4
Fig. 4

Total scattering coefficient K vs ρ12 for α/ν = 0, 0.6, and 1.0.

Fig. 5
Fig. 5

Construction for Rayleigh-Gans scattering by two concentric spheres. Inner radius a, total radius b, bisectrix of incident and scattered directions h, plane perpendicular to bisectrix H, refractive indices m1 and m2 and angle between incident and scattered directions θ.

Fig. 6
Fig. 6

Average refractive index approximation for total scattering coefficient K(m) for α/ν = 0.9, total scattering coefficient K, and total scattering coefficient for α/ν = 1.0, K1 vs ν.

Tables (5)

Tables Icon

Table I Total scattering functions K for inner sphere, m1 = 2.1050, spherical shell, m2 = 1.482.

Tables Icon

Table II Additional values for K for m1 = 2.1050, m2 = 1.4821.

Tables Icon

Table III Comparison of total scattering coefficient calculated from small particle approximations, Ks with the exact value K.

Tables Icon

Table IV Comparison of total scattering coefficient calculated from Ryde’s mixture rule KR, with the exact value K.

Tables Icon

Table V Comparison of total scattering coefficient calculated from average refractive index approximation KA, with the exact value K.

Equations (33)

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T = e K b 2 n l ,
ρ 12 = 2 ν [ ( m 1 1 ) + ( m 2 m 1 ) α / ν ] .
K s = ( 8 / 3 ) ν 4 A 2 = ( 8 / 3 ) ν 4 [ ( m 2 2 1 ) ( m 1 2 + 2 m 2 2 ) + ( α / ν 3 ) ( 2 m 2 2 + 1 ) ( m 1 2 m 2 2 ) ( m 2 2 + 2 ) ( m 1 2 + 2 m 2 2 ) + ( α / ν 3 ) ( 2 m 2 2 2 ) ( m 1 2 m 2 2 ) ] 2 .
K s = K 1 ( A / A 1 ) 2 ,
K s = K 0 ( A / A 0 ) 2 ,
K 1 = 0.758 ν 4 + 0.21 ν 5.5
K 1 = 0.7625 ν 4 .
K s = ( 0.758 ν 4 + 0.21 ν 5.5 ) ( A / A 1 ) 2 .
K R = ( 8 / 3 ) ν 4 F 2 ,
F = [ m 1 2 1 m 1 2 + 2 ] [ 1 ( α / ν ) 3 ] + [ m 2 2 1 m 2 2 + 2 ] ( α / ν ) 3 .
K R = K 1 ( F / F 1 ) 2
K R = K 0 ( F / F 0 2 )
i 1 * ( θ ) = V i k 3 ( m 1 ) 2 π H e i k h 2 sin ( θ / 2 ) d h .
u = 2 k a sin ( θ / 2 )
υ = 2 k h sin ( θ / 2 )
h = z b
y = z ( b / a ) .
i 1 * ( θ ) = i k 3 ( m 2 1 ) / 2 π × [ 1 a / b e i z υ π b 2 ( 1 z 2 ) b d z + a / b 1 e i z υ π b 2 ( 1 z 2 ) b d z + a / b a / b e i z υ { π b 2 ( 1 z 2 ) π ( a 2 z 2 b 2 ) } b d z ] + i k 3 ( m 1 1 ) 2 π a / b a / b e i z υ π ( a 2 z 2 b 2 ) b d z
i 1 * ( θ ) = i k 3 ( m 2 1 ) 2 π 1 + 1 e i z υ π b 2 ( 1 z 2 ) b d z + i k 2 ( m 1 m 2 ) 2 π a / b a / b e i z υ π ( a 2 z 2 b 2 ) b d z .
i 1 * ( θ ) = i k 3 ( m 2 1 ) 2 π 1 + 1 e i z υ π b 2 ( 1 z 2 ) b d z + i k 3 ( m 1 m 2 ) 2 π 1 + 1 e i y u π a 2 ( 1 y 2 ) a d y .
i 1 * ( θ ) = i k 3 ( m 2 1 ) 2 π V G ( υ ) + i k 3 ( m 1 m 2 ) 2 π U G ( u )
i 1 * ( θ ) = 2 3 i [ ( m 2 1 ) ν 3 G ( υ ) + ( m 1 m 2 ) α 3 G ( u ) ]
G ( υ ) = 3 υ 3 ( sin υ υ cos υ ) = ( 9 π 2 υ 2 ) 1 2 J 3 / 2 ( υ )
V = 4 3 π b 3 ; U = 4 3 π a 3 .
m 1 = m 2 ; a = b ; a = 0.
i 2 * ( θ ) = i 1 * ( θ ) cos θ .
I ( θ ) = | i 1 * ( θ ) | 2 + | i 2 * ( θ ) | 2 2 k 2 r 2 I 0 ( θ ) ,
K R G = 1 π b 2 0 π I ( θ ) 2 π r 2 sin θ d θ
K R G = 1 ν 2 0 π | i 1 * ( θ ) | 2 ( 1 + cos 2 θ ) sin θ d θ .
K = ( m 2 1 ) 2 ϕ ( ν , ν ) + ( m 1 m 2 ) 2 ( α / ν ) 2 ϕ ( α , α ) + 2 ( m 2 1 ) ( m 1 m 2 ) ( α / ν ) ϕ ( α , ν ) ,
ϕ ( x , y ) = 4 9 x 2 y 2 0 π G ( 2 x sin θ / 2 ) G ( 2 y sin θ / 2 ) × ( 1 + cos 2 θ ) sin θ d θ .
K A = K ( m ) ,
m = m 1 [ 1 ( α / ν ) 3 + m 2 ( a / ν ) 3 ] .