Abstract

The mean scattering cross section of a polydispersion upon which electromagnetic radiation is incident is found to depend primarily upon the ratio of the third to second moment of the size distribution function, i.e., the volume–surface mean diameter, and is only weakly dependent on the shape of the size distribution function. This result is applicable to mean diameters ranging from very large to very small compared with the wavelength of incident light. Mean scattering coefficients are presented for polydispersions with refractive indices varying from m−1≪1 to m=1.85. The results of this investigation can be used to measure mean particle size, as represented by the ratio of third to second moment of size distribution function, to an accuracy comparable to experimental accuracy even though the size distribution function is not known.

© 1966 Optical Society of America

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References

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  1. D. Sinclair, J. Opt. Soc. Am. 37, 475 (1947).
    [Crossref] [PubMed]
  2. R. M. Tabibian, W. Heller, and J. M. Epel, J. Colloid. Sci. 11, 195 (1956).
    [Crossref]
  3. W. Heller, Interdisciplinary Conference on Electromagnetic Scattering (Pergamon Press, New York, N. Y., 1963), p. 101.
  4. H. C. van de Hulst, Light Scattering By Small Particles (John Wiley & Sons, New York, N. Y., 1956), p. 114.
  5. M. L. Wallach and W. Heller, J. Phys. Chem. 68, 924 (1964).
    [Crossref]
  6. R. A. Dobbins, AIAA J. 1, 1940 (1963).
    [Crossref]
  7. Ref. 4, p. 176.
  8. R. A. Mugele and H. D. Evans, Ind. Eng. Chem. 43, 1317 (1951).
    [Crossref]
  9. J. H. Roberts and M. J. WebbAIAA J. 2, 3583 (1964).
  10. Ref. 4, p. 177.
  11. R. B. Penndorf, J. Opt. Soc. Am. 47, 1010 (1957).
    [Crossref]
  12. See Ref. 4, p. 70.
  13. Our attention has been directed by a reviewer to the extensive tabulation of scattering coefficients due to Kerker et al.14 deposited with the American Documentation Institute. We have checked our calculations with the results of a calculation using the tabulation of Kerker et al. and find that we obtain for ρ32=6.0, m=1.76, K¯=3.104. From Table II for the same ρ32 and m=1.75 we find K¯=3.233.
  14. M. Kerker and et al., J. Phys. Chem. 65, 1713 (1961).
    [Crossref]
  15. E. J. Meehan and Z. Z. Hugus, J. Opt. Soc. Am. 51, 260 (1961).
    [Crossref]
  16. R. B. Penndorf, J. Phys. Chem. 62, 1537 (1958).
    [Crossref]

1964 (2)

M. L. Wallach and W. Heller, J. Phys. Chem. 68, 924 (1964).
[Crossref]

J. H. Roberts and M. J. WebbAIAA J. 2, 3583 (1964).

1963 (1)

R. A. Dobbins, AIAA J. 1, 1940 (1963).
[Crossref]

1961 (2)

M. Kerker and et al., J. Phys. Chem. 65, 1713 (1961).
[Crossref]

E. J. Meehan and Z. Z. Hugus, J. Opt. Soc. Am. 51, 260 (1961).
[Crossref]

1958 (1)

R. B. Penndorf, J. Phys. Chem. 62, 1537 (1958).
[Crossref]

1957 (1)

1956 (1)

R. M. Tabibian, W. Heller, and J. M. Epel, J. Colloid. Sci. 11, 195 (1956).
[Crossref]

1951 (1)

R. A. Mugele and H. D. Evans, Ind. Eng. Chem. 43, 1317 (1951).
[Crossref]

1947 (1)

Dobbins, R. A.

R. A. Dobbins, AIAA J. 1, 1940 (1963).
[Crossref]

Epel, J. M.

R. M. Tabibian, W. Heller, and J. M. Epel, J. Colloid. Sci. 11, 195 (1956).
[Crossref]

Evans, H. D.

R. A. Mugele and H. D. Evans, Ind. Eng. Chem. 43, 1317 (1951).
[Crossref]

Heller, W.

M. L. Wallach and W. Heller, J. Phys. Chem. 68, 924 (1964).
[Crossref]

R. M. Tabibian, W. Heller, and J. M. Epel, J. Colloid. Sci. 11, 195 (1956).
[Crossref]

W. Heller, Interdisciplinary Conference on Electromagnetic Scattering (Pergamon Press, New York, N. Y., 1963), p. 101.

Hugus, Z. Z.

Kerker, M.

M. Kerker and et al., J. Phys. Chem. 65, 1713 (1961).
[Crossref]

Meehan, E. J.

Mugele, R. A.

R. A. Mugele and H. D. Evans, Ind. Eng. Chem. 43, 1317 (1951).
[Crossref]

Penndorf, R. B.

R. B. Penndorf, J. Phys. Chem. 62, 1537 (1958).
[Crossref]

R. B. Penndorf, J. Opt. Soc. Am. 47, 1010 (1957).
[Crossref]

Roberts, J. H.

J. H. Roberts and M. J. WebbAIAA J. 2, 3583 (1964).

Sinclair, D.

Tabibian, R. M.

R. M. Tabibian, W. Heller, and J. M. Epel, J. Colloid. Sci. 11, 195 (1956).
[Crossref]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering By Small Particles (John Wiley & Sons, New York, N. Y., 1956), p. 114.

Wallach, M. L.

M. L. Wallach and W. Heller, J. Phys. Chem. 68, 924 (1964).
[Crossref]

Webb, M. J.

J. H. Roberts and M. J. WebbAIAA J. 2, 3583 (1964).

AIAA J. (2)

R. A. Dobbins, AIAA J. 1, 1940 (1963).
[Crossref]

J. H. Roberts and M. J. WebbAIAA J. 2, 3583 (1964).

Ind. Eng. Chem. (1)

R. A. Mugele and H. D. Evans, Ind. Eng. Chem. 43, 1317 (1951).
[Crossref]

J. Colloid. Sci. (1)

R. M. Tabibian, W. Heller, and J. M. Epel, J. Colloid. Sci. 11, 195 (1956).
[Crossref]

J. Opt. Soc. Am. (3)

J. Phys. Chem. (3)

R. B. Penndorf, J. Phys. Chem. 62, 1537 (1958).
[Crossref]

M. Kerker and et al., J. Phys. Chem. 65, 1713 (1961).
[Crossref]

M. L. Wallach and W. Heller, J. Phys. Chem. 68, 924 (1964).
[Crossref]

Other (6)

W. Heller, Interdisciplinary Conference on Electromagnetic Scattering (Pergamon Press, New York, N. Y., 1963), p. 101.

H. C. van de Hulst, Light Scattering By Small Particles (John Wiley & Sons, New York, N. Y., 1956), p. 114.

Ref. 4, p. 177.

Ref. 4, p. 176.

See Ref. 4, p. 70.

Our attention has been directed by a reviewer to the extensive tabulation of scattering coefficients due to Kerker et al.14 deposited with the American Documentation Institute. We have checked our calculations with the results of a calculation using the tabulation of Kerker et al. and find that we obtain for ρ32=6.0, m=1.76, K¯=3.104. From Table II for the same ρ32 and m=1.75 we find K¯=3.233.

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Figures (5)

Fig. 1
Fig. 1

Mean specific scattering cross section vs mean size for certain distribution functions [—— rectangular, - - - - parabolic, — – — Eq. (11), a=1.13, δ=1.26].

Fig. 2
Fig. 2

Average values of K ¯/ρ32 for 20 upper-limit distribution functions. (Standard deviation, S.D., given as a percentage of the mean.)

Fig. 3
Fig. 3

Upper-limit function with a=1.13 and δ=1.26. ( D ¯/D=0.281, D32/ D ¯ =0.431, w=1.13).

Fig. 4
Fig. 4

Scattering coefficient vs phase shift for selected values of refractive index. (— – — m−1≪1, —— m=1.33, – – – – m=1.40, ⋯ m=1.50).

Fig. 5
Fig. 5

Mean scattering coefficient vs mean size for selected of half-width w, D ¯/D=0.281. (—— w=1.07, —— · —— w=0.51, – – – – w=0.20.)

Tables (2)

Tables Icon

Table I Comparison of mean cross section for large and small values of size number (m−1≪1, α≫1)

Tables Icon

Table II K ¯ vs ρ32 for upper limit function (a=1.130, δ=1.260) selected values of m.

Equations (17)

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T = exp [ - K ( α , m ) ( π / 4 ) D 2 C n l ] .
T = exp [ - π 4 C n l 0 D K ( D , m ) N r ( D ) D 2 d D ] ,
K ¯ 0 D K ( D , m ) N r ( D ) D 2 d D / 0 D N r ( D ) D 2 d D .
C v = C n π 6 0 D N r ( D ) D 3 d D .
D 32 0 D N r ( D ) D 3 d D / 0 D N r ( D ) D 2 d D .
T = exp [ - 3 2 ( K ¯ / D 32 ) C v l ] .
C n π 6 0 N r ( D ) D 3 d D / C n π 0 N r ( D ) D 2 d D = D 32 / 6 .
K = 2 - ( 4 sin ρ / ρ ) + 4 ( 1 - cos ρ ) / ρ 2 ,
N r ( D ) { 1 , D D = 0 , D > D ,
N r ( D ) D / D - ( D / D ) 2 ,
N r ( D ) exp - [ δ ln ( a D / D - D ) ] 2 D 4 ( D - D ) .
K ¯ = 0 K ( ζ , m ) exp { - δ ln [ a ζ / ( ζ - ζ ) ] } 2 ζ 2 ( ζ - ζ ) d ζ 0 exp { - δ ln [ a ζ / ( ζ - ζ ) ] } 2 ζ 2 ( ζ - ζ ) d ζ .
K = ( 8 / 3 ) α 4 [ ( m 2 - 1 ) / ( m 2 + 2 ) ] 2 .
K ¯ = ( 8 / 3 ) [ m 2 - 1 ) / ( m 2 + 2 ) ] α 62 4 ,
α 62 4 = ( π λ ) 4 0 N r ( D ) D 6 d D / 0 N r ( D ) D 2 d D .
K ¯ = 0.271 [ ( m 2 - 1 ) / ( m - 1 ) 2 ( m 2 + 2 ) ] 2 ρ 32 4 .
w = ( D + - D - ) / D ¯ .