Abstract

The general belief that the sphere of equal volume provides a better approximation for the extinction cross section of a nonspherical particle than a sphere of equal surface area at small values of the size parameters is not correct. At some values of x, the equal volume sphere is a better approximation; at others, the equal surface area sphere is better. Details depend on the shape, size, and refractive index. For strongly absorbing particles at x > π, the extinction cross section of an equal volume sphere σEV provides the lower bound, and σEVSN/SEV (where SN is the surface area of considered nonspherical particle, and SEV is the surface area of equal volume sphere) provides the upper bound on the extinction cross section of an arbitrarily shaped nonspherical particle.

© 1982 Optical Society of America

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References

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  1. A. B. Pluchino, Appl. Opt. 20, 531 (1981).
    [CrossRef] [PubMed]
  2. A. B. Pluchino, Appl. Opt. 20, 2986 (1981).
    [CrossRef] [PubMed]
  3. D. M. Roessler, F. R. Faxvog, Appl. Opt. 18, 1399 (1979).
    [CrossRef] [PubMed]
  4. S. G. Jennings, R. G. Pinnick, Atmos. Environ. 14, 1123 (1980).
    [CrossRef]
  5. P. Chýlek, V. Ramaswamy, R. Cheng, R. G. Pinnick, Appl. Opt. 20, 2980 (1981).
    [CrossRef] [PubMed]
  6. A. I. Medalia, L. W. Richards, J. Colloid Interface Sci. 40, 233 (1972).
    [CrossRef]
  7. J. Janzen, J. Colloid Interface Sci. 69, 436 (1979).
    [CrossRef]
  8. J. Janzen, Appl. Opt. 19, 2977 (1980).
    [CrossRef] [PubMed]
  9. P. Chýlek, J. Opt Soc. Am. 67, 1348 (1977).
    [CrossRef]
  10. P. Barber, C. Yeh, Appl. Opt. 14, 2866 (1975).
    [CrossRef]
  11. P. Chýlek, V. Ramaswamy, W. Wiscombe, J. Atmos Sci. 39, 1886 (1982).
    [CrossRef]
  12. R. A. Dobbins, G. S. Jizmagian, J. Opt, Soc. Am. 56, 1345 (1966).
    [CrossRef]
  13. S. G. Warren, Rev. Geophys. Space Phys. 20, 67 (1982).
    [CrossRef]

1982 (2)

P. Chýlek, V. Ramaswamy, W. Wiscombe, J. Atmos Sci. 39, 1886 (1982).
[CrossRef]

S. G. Warren, Rev. Geophys. Space Phys. 20, 67 (1982).
[CrossRef]

1981 (3)

1980 (2)

J. Janzen, Appl. Opt. 19, 2977 (1980).
[CrossRef] [PubMed]

S. G. Jennings, R. G. Pinnick, Atmos. Environ. 14, 1123 (1980).
[CrossRef]

1979 (2)

1977 (1)

P. Chýlek, J. Opt Soc. Am. 67, 1348 (1977).
[CrossRef]

1975 (1)

P. Barber, C. Yeh, Appl. Opt. 14, 2866 (1975).
[CrossRef]

1972 (1)

A. I. Medalia, L. W. Richards, J. Colloid Interface Sci. 40, 233 (1972).
[CrossRef]

1966 (1)

R. A. Dobbins, G. S. Jizmagian, J. Opt, Soc. Am. 56, 1345 (1966).
[CrossRef]

Barber, P.

P. Barber, C. Yeh, Appl. Opt. 14, 2866 (1975).
[CrossRef]

Cheng, R.

Chýlek, P.

P. Chýlek, V. Ramaswamy, W. Wiscombe, J. Atmos Sci. 39, 1886 (1982).
[CrossRef]

P. Chýlek, V. Ramaswamy, R. Cheng, R. G. Pinnick, Appl. Opt. 20, 2980 (1981).
[CrossRef] [PubMed]

P. Chýlek, J. Opt Soc. Am. 67, 1348 (1977).
[CrossRef]

Dobbins, R. A.

R. A. Dobbins, G. S. Jizmagian, J. Opt, Soc. Am. 56, 1345 (1966).
[CrossRef]

Faxvog, F. R.

Janzen, J.

J. Janzen, Appl. Opt. 19, 2977 (1980).
[CrossRef] [PubMed]

J. Janzen, J. Colloid Interface Sci. 69, 436 (1979).
[CrossRef]

Jennings, S. G.

S. G. Jennings, R. G. Pinnick, Atmos. Environ. 14, 1123 (1980).
[CrossRef]

Jizmagian, G. S.

R. A. Dobbins, G. S. Jizmagian, J. Opt, Soc. Am. 56, 1345 (1966).
[CrossRef]

Medalia, A. I.

A. I. Medalia, L. W. Richards, J. Colloid Interface Sci. 40, 233 (1972).
[CrossRef]

Pinnick, R. G.

Pluchino, A. B.

Ramaswamy, V.

Richards, L. W.

A. I. Medalia, L. W. Richards, J. Colloid Interface Sci. 40, 233 (1972).
[CrossRef]

Roessler, D. M.

Warren, S. G.

S. G. Warren, Rev. Geophys. Space Phys. 20, 67 (1982).
[CrossRef]

Wiscombe, W.

P. Chýlek, V. Ramaswamy, W. Wiscombe, J. Atmos Sci. 39, 1886 (1982).
[CrossRef]

Yeh, C.

P. Barber, C. Yeh, Appl. Opt. 14, 2866 (1975).
[CrossRef]

Appl. Opt. (6)

Atmos. Environ. (1)

S. G. Jennings, R. G. Pinnick, Atmos. Environ. 14, 1123 (1980).
[CrossRef]

J. Atmos Sci. (1)

P. Chýlek, V. Ramaswamy, W. Wiscombe, J. Atmos Sci. 39, 1886 (1982).
[CrossRef]

J. Colloid Interface Sci. (2)

A. I. Medalia, L. W. Richards, J. Colloid Interface Sci. 40, 233 (1972).
[CrossRef]

J. Janzen, J. Colloid Interface Sci. 69, 436 (1979).
[CrossRef]

J. Opt Soc. Am. (1)

P. Chýlek, J. Opt Soc. Am. 67, 1348 (1977).
[CrossRef]

J. Opt, Soc. Am. (1)

R. A. Dobbins, G. S. Jizmagian, J. Opt, Soc. Am. 56, 1345 (1966).
[CrossRef]

Rev. Geophys. Space Phys. (1)

S. G. Warren, Rev. Geophys. Space Phys. 20, 67 (1982).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Extinction cross-section σEXT of a spherical strongly absorbing particle is a monotonically increasing function of the size parameter x.

Fig. 2
Fig. 2

Normalized extinction cross-section QEXT = σEXT/S of a strongly absorbing or reflecting particle is a monotonically decreasing function of size parameter x for ~x < π.

Fig. 3
Fig. 3

Shapes of the four nonspherical particles used for numerical calculations.

Fig. 4
Fig. 4

Error functions ɛ1, ɛ2, ɛ3, and ɛ4 for the case of nonspherical T 2 + particle as a function of size parameter x. Notice that for x < 0.6, the equal surface area sphere (ɛ4) provides a more accurate approximation than the equal volume sphere (ɛ1).

Fig. 5
Fig. 5

Same as Fig. 4, however, for T 2 particle.

Fig. 6
Fig. 6

Same as Fig. 4, however, for T 4 + particle.

Fig. 7
Fig. 7

Same as Fig. 4, however, for T 4 particle.

Tables (3)

Tables Icon

Table I According to the Weaker Form of Asymptotic Behavior As Suggested by Eq. (30), the Extinction Cross Section/Unit Projected Area Should Be a Function of the Size Parameter and Refractive Index, and It Should Have Only a Weak Dependence on the Shape of Nonspherical Particles. Shown Results Are for Spherical Particle and for T 2 ± Particles With = 0.2 and for T 4 ± With = 0.1. While σ/P Varies Significantly With the Size Parameter x, It Is Almost the Same for All Five Considered Shapes for the Size Parameter x > 1.

Tables Icon

Table II Extinction Cross-Section σAPPROX Given By Eq. (25) Is Compared With Exact Values σEBCM Obtained Using the Extended Boundary Condition Method. Shape of Particles is Described By Eq. (19) and Refractive Index m = 2 – l. In This Case of Moderately Nonspherical Particles, the Predicted Accuracy of Approximation Is ~2%. All Exact Values σEBCM Are Within 0.6% From σAPPROX. Extinction Cross Section is in (μm)2

Tables Icon

Table III Extinction Cross Sections of Nonspherical Particles Calculated From the Proposed Approximation (25) for the Refractive Index m = 2 – l. The Extinction of 3-D Particles (All Three Dimensions Are of the Same Magnitude) Can Be Calculated With the Accuracy of about ±10%. In the Case of Plates (2-D) and Columns (1-D Particles), the Accuracy Decreases to ±30−40%. Extinction Cross Section is in (μm)2

Equations (33)

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x = ( 6 π 2 V N ) 1 / 3 / λ .
S E V S 1 < S N .
V 2 > V N .
r E S > r E V .
σ 1 = σ E V V 1 V N = σ E V ,
σ 2 = σ E V S E V S N ,
σ 3 = σ E S V E S V N ,
σ 4 = σ E S S 2 S N = σ E S .
ε 1 = ( σ 1 σ N ) / σ N ,
ε 2 = ( σ 2 σ N ) / σ N ,
ε 3 = ( σ 3 σ N ) / σ N ,
ε 4 = ( σ 4 σ N ) / σ N .
ε 1 < ε 2 .
ε 3 < ε 4 .
σ E V / σ E S < 1.
ε 1 < ε 4 .
σ E V S E V > σ E S S E S .
ε 3 < ε 1 < ε 4 < ε 2 .
r = r 0 [ 1 ± ε T n ( θ ) ] ,
σ / P = 2 ,
P N 1 4 S N ,
σ E V P E V σ N P N σ E S P E S ,
σ E V < σ N = σ E S ,
σ E V < σ N < σ E V S N / S E V ,
σ = σ E V S N + S E V 2 S E V ± σ E V S N S E V 2 S E V ,
σ E V < σ N < σ E S ,
σ = 1 2 ( σ E S + σ E V ) ± 1 2 ( σ E S σ E V ) .
σ E X T / P = C ,
C = 2 ,
σ / P = C < 2 ,
σ E V P E V σ N P N  at  x = 2 π r E V / λ ,
σ E S P E S σ N P N  at  x = 2 π r E S / λ .
ε 3 < ε 1 < 0 < ε 4 < ε 2 ,

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