Abstract

Due to relatively recent advances in theory, light transmission apparatus need not be calibrated against monodisperse standards in order to provide accurate size information for transparent spherical particles. However, the precision of construction necessary for accurate results has not heretofore been evaluated. In this paper the effects of unavoidable approximations in the optical construction of light transmission apparatus are evaluated quantitatively. Tables and graphs are presented which will enable the apparatus designer and user to avoid, or at least account for, errors in interpretation of light transmission data, without resort to empirical calibration.

© 1956 Optical Society of America

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References

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  1. R. O. Gumprecht and C. M. Sliepcevich, J. Phys. Chem. 57, 95 (1953).
    [Crossref]
  2. H. E. Rose, The Measurement of Particle Size in Very Fine Powders (Chemical Publishing Company, New York, 1954).
  3. P. G. W. Hawksley, Brit. Coal Util. Res. Association Bull. 16, 117, 181 (1952).
  4. R. O. Gumprecht and C. M. Sliepcevich, J. Phys. Chem. 57, 90 (1953).
    [Crossref]
  5. R. O. Gumprecht and C. M. Sliepcevich, Tables of Light Scattering Functions for Spherical Particles (University of Michigan, Engineering Research Institute, Special Publications: Tables, Ann Arbor, Michigan, 1951).
  6. Tables of Scattering Functions for Spherical Particles (National Bureau of Standards Applied Mathematics Series-4, U. S. Government Printing Office, Washington, D. C., 1948).
  7. Boll, Gumprecht, and Sliepcevich, J. Opt. Soc. Am. 44, 18 (1954).
    [Crossref]
  8. D. G. Skinner and S. Boas-Traube, Symposium on Particle Size Analysis, p. 57, Suppl. to Trans. Inst. Chem. Eng. 25(1947).
  9. D. Sinclair, Handbook on Aerosols (U. S. Atomic Energy Commission, Washington, D. C., 1950), p. 81.
  10. W. H. Walton, Symposium on Particle Size Analysis, p. 141, Suppl. to Trans. Inst. Chem. Eng. 25(1947).
  11. F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw-Hill Book Company, Inc., New York, 1950).
  12. R. O. Gumprecht, “Particle Size Measurements by Light Scattering,” Ph.D. Dissertation, University of Michigan, Ann Arbor, Michigan, 1952.

1954 (1)

1953 (2)

R. O. Gumprecht and C. M. Sliepcevich, J. Phys. Chem. 57, 95 (1953).
[Crossref]

R. O. Gumprecht and C. M. Sliepcevich, J. Phys. Chem. 57, 90 (1953).
[Crossref]

1952 (1)

P. G. W. Hawksley, Brit. Coal Util. Res. Association Bull. 16, 117, 181 (1952).

1947 (2)

D. G. Skinner and S. Boas-Traube, Symposium on Particle Size Analysis, p. 57, Suppl. to Trans. Inst. Chem. Eng. 25(1947).

W. H. Walton, Symposium on Particle Size Analysis, p. 141, Suppl. to Trans. Inst. Chem. Eng. 25(1947).

Boas-Traube, S.

D. G. Skinner and S. Boas-Traube, Symposium on Particle Size Analysis, p. 57, Suppl. to Trans. Inst. Chem. Eng. 25(1947).

Boll,

Gumprecht,

Gumprecht, R. O.

R. O. Gumprecht and C. M. Sliepcevich, J. Phys. Chem. 57, 95 (1953).
[Crossref]

R. O. Gumprecht and C. M. Sliepcevich, J. Phys. Chem. 57, 90 (1953).
[Crossref]

R. O. Gumprecht and C. M. Sliepcevich, Tables of Light Scattering Functions for Spherical Particles (University of Michigan, Engineering Research Institute, Special Publications: Tables, Ann Arbor, Michigan, 1951).

R. O. Gumprecht, “Particle Size Measurements by Light Scattering,” Ph.D. Dissertation, University of Michigan, Ann Arbor, Michigan, 1952.

Hawksley, P. G. W.

P. G. W. Hawksley, Brit. Coal Util. Res. Association Bull. 16, 117, 181 (1952).

Jenkins, F. A.

F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw-Hill Book Company, Inc., New York, 1950).

Rose, H. E.

H. E. Rose, The Measurement of Particle Size in Very Fine Powders (Chemical Publishing Company, New York, 1954).

Sinclair, D.

D. Sinclair, Handbook on Aerosols (U. S. Atomic Energy Commission, Washington, D. C., 1950), p. 81.

Skinner, D. G.

D. G. Skinner and S. Boas-Traube, Symposium on Particle Size Analysis, p. 57, Suppl. to Trans. Inst. Chem. Eng. 25(1947).

Sliepcevich,

Sliepcevich, C. M.

R. O. Gumprecht and C. M. Sliepcevich, J. Phys. Chem. 57, 95 (1953).
[Crossref]

R. O. Gumprecht and C. M. Sliepcevich, J. Phys. Chem. 57, 90 (1953).
[Crossref]

R. O. Gumprecht and C. M. Sliepcevich, Tables of Light Scattering Functions for Spherical Particles (University of Michigan, Engineering Research Institute, Special Publications: Tables, Ann Arbor, Michigan, 1951).

Walton, W. H.

W. H. Walton, Symposium on Particle Size Analysis, p. 141, Suppl. to Trans. Inst. Chem. Eng. 25(1947).

White, H. E.

F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw-Hill Book Company, Inc., New York, 1950).

Brit. Coal Util. Res. Association Bull. (1)

P. G. W. Hawksley, Brit. Coal Util. Res. Association Bull. 16, 117, 181 (1952).

J. Opt. Soc. Am. (1)

J. Phys. Chem. (2)

R. O. Gumprecht and C. M. Sliepcevich, J. Phys. Chem. 57, 90 (1953).
[Crossref]

R. O. Gumprecht and C. M. Sliepcevich, J. Phys. Chem. 57, 95 (1953).
[Crossref]

Symposium on Particle Size Analysis (2)

D. G. Skinner and S. Boas-Traube, Symposium on Particle Size Analysis, p. 57, Suppl. to Trans. Inst. Chem. Eng. 25(1947).

W. H. Walton, Symposium on Particle Size Analysis, p. 141, Suppl. to Trans. Inst. Chem. Eng. 25(1947).

Other (6)

F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw-Hill Book Company, Inc., New York, 1950).

R. O. Gumprecht, “Particle Size Measurements by Light Scattering,” Ph.D. Dissertation, University of Michigan, Ann Arbor, Michigan, 1952.

D. Sinclair, Handbook on Aerosols (U. S. Atomic Energy Commission, Washington, D. C., 1950), p. 81.

H. E. Rose, The Measurement of Particle Size in Very Fine Powders (Chemical Publishing Company, New York, 1954).

R. O. Gumprecht and C. M. Sliepcevich, Tables of Light Scattering Functions for Spherical Particles (University of Michigan, Engineering Research Institute, Special Publications: Tables, Ann Arbor, Michigan, 1951).

Tables of Scattering Functions for Spherical Particles (National Bureau of Standards Applied Mathematics Series-4, U. S. Government Printing Office, Washington, D. C., 1948).

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Figures (10)

Fig. 1
Fig. 1

Refraction diagram for pinhole within the focal length.

Fig. 2
Fig. 2

Geometry for the light beam within the cone of reception of the receiver when the pinhole is within the focal length.

Fig. 3
Fig. 3

Scattering diagram with symmetrical and dissymmetrical cones of reception.

Fig. 4
Fig. 4

Relationship between g1 and γ, θ, and μ0.

Fig. 5
Fig. 5

Error in R vs α with varying misalignment.

Fig. 6
Fig. 6

Error in R vs μ0 for varying α.

Fig. 7
Fig. 7

Sketch of geometry with combined divergence and misalignment.

Fig. 8
Fig. 8

Error in R due to divergence, ψ0, and misalignment, μ0.

Fig. 9
Fig. 9

Appearance of image of lamp filament in pinhole.

Fig. 10
Fig. 10

R vs drop diameter: corrected and uncorrected curves for θ=0.92°, λ=4800 A.

Tables (4)

Tables Icon

Table I Variation in θ due to pinhole misplacement. F=3.875 inches.

Tables Icon

Table II Variation in θ due to spherical aberration. f=3.875 inches.

Tables Icon

Table III Summary of calculated errors in R due to misalignment. θ=1.00°.

Tables Icon

Table IV Average error in R due to combined divergence and misalignment. α = 150. θ = 1.000.

Equations (30)

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ln I 0 I = π l 4 0 D R K t N D 2 d D
R = 1 + J 0 2 ( α θ ) + J 1 2 ( α θ ) 2
θ = 1 2 D P F
θ = 1 2 D S h + S
θ = 1 2 D P h δ + F ( 1 - δ ) ,
D L - D B h max D P - δ D B δ h max + F ( 1 - δ ) ,
δ = - S S r r 2 f 3 Q ( n , p , q ) ,
Q ( n , p , q ) = 1 8 n ( n - 1 ) [ n + 2 n - 1 q 2 + 4 ( n + 1 ) p q + ( 3 n + 2 ) ( n - 1 ) p 2 + n 3 n - 1 ] ,
δ - r 2 f Q ( n , p , q ) .
F = f + δ ,
θ 1 2 D P f - h r 2 Q f 2 - r 2 Q .
θ θ n 1 1 - h r 2 Q f 3 .
r ¯ = 1 2 D B 2 .
( θ θ n ) Av 1 1 - ( h 2 ) ( D B 2 2 ) 2 1.5 f 3 1 + 1.5 16 ( D B f ) 2 ( h f ) .
( 1 - R ) γ d γ = J 1 2 ( α γ ) γ d γ .
( 1 - R ) = ( 1 - R ) + H 1 - H 2 ,
H 1 = θ θ + μ 0 g 1 π J 1 2 ( α γ ) γ d γ ,
H 2 = θ - μ 0 θ g 2 π J 1 2 ( α γ ) γ d γ .
cos g 1 = b + μ 0 γ ,             γ θ ,
a 2 + b 2 = θ 2 ,
a 2 + ( b + μ 0 ) 2 = γ 2 ,
cos g 1 = γ 2 - θ 2 - μ 0 2 2 γ μ 0 ,             γ θ .
cos g 2 = θ 2 - μ 0 2 - γ 2 2 γ μ 0 ,             γ θ .
R = R + H 2 - H 1 .
R - R R = E ( μ ) ,
μ = μ 0 + ψ .
Ē = μ 0 - ψ 0 μ 0 + ψ 0 I ( μ - μ 0 ) E ( μ ) d μ - ψ 0 + ψ 0 I ( ψ ) d ψ ,
Ē = 1 2 ψ 0 μ 0 - ψ 0 μ 0 + ψ 0 E ( μ ) d μ .
D L - D B h max = 3.0 - 0.5 12 = 0.208.
δ = ( - 1 32 ) ÷ 3.875 = - 0.00806 , D P - δ D B F ( 1 - δ ) + δ h max = 0.125 + 0.00806 ( 0.5 ) 3.875 ( 1.008 ) - 0.00806 ( 12 ) = 0.0339 ,             0.208 > 0.0339.