Abstract

Light scattered by single particles is frequently measured to determine particle volume. The particle is illuminated by a light beam; it scatters to one or more photodetectors. Usually no consideration has been given to effects of particle shape. This study applies recently developed theoretical techniques for predicting scattering by spheroids in order to compare representative scattered fluxes for several particle shapes and orientations. It is found that shape and orientation can strongly influence the measurement of whole particle size. The effects of refractive index are also found to be significant but smaller.

© 1978 Optical Society of America

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  1. P. J. Wyatt, Appl. Opt. 7, 1879 (1968).
    [Crossref] [PubMed]
  2. P. J. Wyatt, D. T. Phillips, J. Theor. Biol. 37, 493 (1972).
    [Crossref] [PubMed]
  3. P. F. Mullaney, M. A. Van Dilla, J. R. Coulter, D. N. Dean, Rev. Sci. Instrum. 40, 1029 (1969).
    [Crossref] [PubMed]
  4. P. F. Mullaney, J. M. Crowell, G. C. Saltman, J. C. Martin, R. D. Hiebert, C. A. Goad, J. Histochem. Cytochem. 24, 298 (1976).
    [Crossref] [PubMed]
  5. T. M. Jovin, S. J. Morris, G. Striker, H. A. Schultens, M. Digweed, D. G. Arndt-Jovin, J. Histochem. Cytochem. 24, 269 (1976).
    [Crossref] [PubMed]
  6. C. C. Gravatt, Aerosol Measurements; Proceedings of a Seminar on Aerosol Measurements, Gaithersburg, MD, 7 May 1974, NBS Special Publication 412 (U.S. Government Printing Office, Washington, D.C., 1974).
  7. C. C. Gravatt, J. Air Pollut. Control Assoc. 23, 1035 (1973).
    [Crossref] [PubMed]
  8. L. W. Casperson, C. Yeh, W. F. Yeung, Appl. Opt. 16, 1104 (1977).
    [Crossref] [PubMed]
  9. D. D. Cooke, M. Kerker, Appl. Opt. 14, 734 (1975).
    [Crossref] [PubMed]
  10. S. Asano, G. Yamamoto, Appl. Opt. 14, 29 (1975).
    [PubMed]
  11. P. Barber, C. Yeh, Appl. Opt. 14, 2864 (1975).
    [Crossref] [PubMed]
  12. P. Latimer, J. Colloid Interface Sci. 53, 102 (1975).
    [Crossref]
  13. P. Latimer, P. Barber, J. Colloid Interface Sci. 63, 310 (1978).
    [Crossref]
  14. H. Chew, M. Kerker, D. D. Cooke, Opt. Lett. 1, 138 (1977).
    [Crossref] [PubMed]
  15. J. R. Hodkinson, Br. J. Appl. Phys. 14, 1431 (1963).
    [Crossref]
  16. D. A. Cross, P. Latimer, Appl. Opt. 11, 1225 (1972).
    [Crossref] [PubMed]
  17. P. Latimer, Plant Physiol. 34, 193 (1959).
    [Crossref] [PubMed]
  18. M. M. Frojmovic, A. O. Kagawa, S. G. Mason, Biochem. Biophys. Res. Commun. 62, 17 (1975).
    [Crossref] [PubMed]
  19. P. Latimer, G. V. R. Born, F. Michal, Arch. Biochem. Biophys. 180, 151 (1977).
    [Crossref] [PubMed]
  20. H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1959), p. 256.

1978 (1)

P. Latimer, P. Barber, J. Colloid Interface Sci. 63, 310 (1978).
[Crossref]

1977 (3)

1976 (2)

P. F. Mullaney, J. M. Crowell, G. C. Saltman, J. C. Martin, R. D. Hiebert, C. A. Goad, J. Histochem. Cytochem. 24, 298 (1976).
[Crossref] [PubMed]

T. M. Jovin, S. J. Morris, G. Striker, H. A. Schultens, M. Digweed, D. G. Arndt-Jovin, J. Histochem. Cytochem. 24, 269 (1976).
[Crossref] [PubMed]

1975 (5)

1973 (1)

C. C. Gravatt, J. Air Pollut. Control Assoc. 23, 1035 (1973).
[Crossref] [PubMed]

1972 (2)

P. J. Wyatt, D. T. Phillips, J. Theor. Biol. 37, 493 (1972).
[Crossref] [PubMed]

D. A. Cross, P. Latimer, Appl. Opt. 11, 1225 (1972).
[Crossref] [PubMed]

1969 (1)

P. F. Mullaney, M. A. Van Dilla, J. R. Coulter, D. N. Dean, Rev. Sci. Instrum. 40, 1029 (1969).
[Crossref] [PubMed]

1968 (1)

1963 (1)

J. R. Hodkinson, Br. J. Appl. Phys. 14, 1431 (1963).
[Crossref]

1959 (1)

P. Latimer, Plant Physiol. 34, 193 (1959).
[Crossref] [PubMed]

Arndt-Jovin, D. G.

T. M. Jovin, S. J. Morris, G. Striker, H. A. Schultens, M. Digweed, D. G. Arndt-Jovin, J. Histochem. Cytochem. 24, 269 (1976).
[Crossref] [PubMed]

Asano, S.

Barber, P.

P. Latimer, P. Barber, J. Colloid Interface Sci. 63, 310 (1978).
[Crossref]

P. Barber, C. Yeh, Appl. Opt. 14, 2864 (1975).
[Crossref] [PubMed]

Born, G. V. R.

P. Latimer, G. V. R. Born, F. Michal, Arch. Biochem. Biophys. 180, 151 (1977).
[Crossref] [PubMed]

Casperson, L. W.

Chew, H.

Cooke, D. D.

Coulter, J. R.

P. F. Mullaney, M. A. Van Dilla, J. R. Coulter, D. N. Dean, Rev. Sci. Instrum. 40, 1029 (1969).
[Crossref] [PubMed]

Cross, D. A.

Crowell, J. M.

P. F. Mullaney, J. M. Crowell, G. C. Saltman, J. C. Martin, R. D. Hiebert, C. A. Goad, J. Histochem. Cytochem. 24, 298 (1976).
[Crossref] [PubMed]

Dean, D. N.

P. F. Mullaney, M. A. Van Dilla, J. R. Coulter, D. N. Dean, Rev. Sci. Instrum. 40, 1029 (1969).
[Crossref] [PubMed]

Digweed, M.

T. M. Jovin, S. J. Morris, G. Striker, H. A. Schultens, M. Digweed, D. G. Arndt-Jovin, J. Histochem. Cytochem. 24, 269 (1976).
[Crossref] [PubMed]

Frojmovic, M. M.

M. M. Frojmovic, A. O. Kagawa, S. G. Mason, Biochem. Biophys. Res. Commun. 62, 17 (1975).
[Crossref] [PubMed]

Goad, C. A.

P. F. Mullaney, J. M. Crowell, G. C. Saltman, J. C. Martin, R. D. Hiebert, C. A. Goad, J. Histochem. Cytochem. 24, 298 (1976).
[Crossref] [PubMed]

Goldstein, H.

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1959), p. 256.

Gravatt, C. C.

C. C. Gravatt, J. Air Pollut. Control Assoc. 23, 1035 (1973).
[Crossref] [PubMed]

C. C. Gravatt, Aerosol Measurements; Proceedings of a Seminar on Aerosol Measurements, Gaithersburg, MD, 7 May 1974, NBS Special Publication 412 (U.S. Government Printing Office, Washington, D.C., 1974).

Hiebert, R. D.

P. F. Mullaney, J. M. Crowell, G. C. Saltman, J. C. Martin, R. D. Hiebert, C. A. Goad, J. Histochem. Cytochem. 24, 298 (1976).
[Crossref] [PubMed]

Hodkinson, J. R.

J. R. Hodkinson, Br. J. Appl. Phys. 14, 1431 (1963).
[Crossref]

Jovin, T. M.

T. M. Jovin, S. J. Morris, G. Striker, H. A. Schultens, M. Digweed, D. G. Arndt-Jovin, J. Histochem. Cytochem. 24, 269 (1976).
[Crossref] [PubMed]

Kagawa, A. O.

M. M. Frojmovic, A. O. Kagawa, S. G. Mason, Biochem. Biophys. Res. Commun. 62, 17 (1975).
[Crossref] [PubMed]

Kerker, M.

Latimer, P.

P. Latimer, P. Barber, J. Colloid Interface Sci. 63, 310 (1978).
[Crossref]

P. Latimer, G. V. R. Born, F. Michal, Arch. Biochem. Biophys. 180, 151 (1977).
[Crossref] [PubMed]

P. Latimer, J. Colloid Interface Sci. 53, 102 (1975).
[Crossref]

D. A. Cross, P. Latimer, Appl. Opt. 11, 1225 (1972).
[Crossref] [PubMed]

P. Latimer, Plant Physiol. 34, 193 (1959).
[Crossref] [PubMed]

Martin, J. C.

P. F. Mullaney, J. M. Crowell, G. C. Saltman, J. C. Martin, R. D. Hiebert, C. A. Goad, J. Histochem. Cytochem. 24, 298 (1976).
[Crossref] [PubMed]

Mason, S. G.

M. M. Frojmovic, A. O. Kagawa, S. G. Mason, Biochem. Biophys. Res. Commun. 62, 17 (1975).
[Crossref] [PubMed]

Michal, F.

P. Latimer, G. V. R. Born, F. Michal, Arch. Biochem. Biophys. 180, 151 (1977).
[Crossref] [PubMed]

Morris, S. J.

T. M. Jovin, S. J. Morris, G. Striker, H. A. Schultens, M. Digweed, D. G. Arndt-Jovin, J. Histochem. Cytochem. 24, 269 (1976).
[Crossref] [PubMed]

Mullaney, P. F.

P. F. Mullaney, J. M. Crowell, G. C. Saltman, J. C. Martin, R. D. Hiebert, C. A. Goad, J. Histochem. Cytochem. 24, 298 (1976).
[Crossref] [PubMed]

P. F. Mullaney, M. A. Van Dilla, J. R. Coulter, D. N. Dean, Rev. Sci. Instrum. 40, 1029 (1969).
[Crossref] [PubMed]

Phillips, D. T.

P. J. Wyatt, D. T. Phillips, J. Theor. Biol. 37, 493 (1972).
[Crossref] [PubMed]

Saltman, G. C.

P. F. Mullaney, J. M. Crowell, G. C. Saltman, J. C. Martin, R. D. Hiebert, C. A. Goad, J. Histochem. Cytochem. 24, 298 (1976).
[Crossref] [PubMed]

Schultens, H. A.

T. M. Jovin, S. J. Morris, G. Striker, H. A. Schultens, M. Digweed, D. G. Arndt-Jovin, J. Histochem. Cytochem. 24, 269 (1976).
[Crossref] [PubMed]

Striker, G.

T. M. Jovin, S. J. Morris, G. Striker, H. A. Schultens, M. Digweed, D. G. Arndt-Jovin, J. Histochem. Cytochem. 24, 269 (1976).
[Crossref] [PubMed]

Van Dilla, M. A.

P. F. Mullaney, M. A. Van Dilla, J. R. Coulter, D. N. Dean, Rev. Sci. Instrum. 40, 1029 (1969).
[Crossref] [PubMed]

Wyatt, P. J.

P. J. Wyatt, D. T. Phillips, J. Theor. Biol. 37, 493 (1972).
[Crossref] [PubMed]

P. J. Wyatt, Appl. Opt. 7, 1879 (1968).
[Crossref] [PubMed]

Yamamoto, G.

Yeh, C.

Yeung, W. F.

Appl. Opt. (6)

Arch. Biochem. Biophys. (1)

P. Latimer, G. V. R. Born, F. Michal, Arch. Biochem. Biophys. 180, 151 (1977).
[Crossref] [PubMed]

Biochem. Biophys. Res. Commun. (1)

M. M. Frojmovic, A. O. Kagawa, S. G. Mason, Biochem. Biophys. Res. Commun. 62, 17 (1975).
[Crossref] [PubMed]

Br. J. Appl. Phys. (1)

J. R. Hodkinson, Br. J. Appl. Phys. 14, 1431 (1963).
[Crossref]

J. Air Pollut. Control Assoc. (1)

C. C. Gravatt, J. Air Pollut. Control Assoc. 23, 1035 (1973).
[Crossref] [PubMed]

J. Colloid Interface Sci. (2)

P. Latimer, J. Colloid Interface Sci. 53, 102 (1975).
[Crossref]

P. Latimer, P. Barber, J. Colloid Interface Sci. 63, 310 (1978).
[Crossref]

J. Histochem. Cytochem. (2)

P. F. Mullaney, J. M. Crowell, G. C. Saltman, J. C. Martin, R. D. Hiebert, C. A. Goad, J. Histochem. Cytochem. 24, 298 (1976).
[Crossref] [PubMed]

T. M. Jovin, S. J. Morris, G. Striker, H. A. Schultens, M. Digweed, D. G. Arndt-Jovin, J. Histochem. Cytochem. 24, 269 (1976).
[Crossref] [PubMed]

J. Theor. Biol. (1)

P. J. Wyatt, D. T. Phillips, J. Theor. Biol. 37, 493 (1972).
[Crossref] [PubMed]

Opt. Lett. (1)

Plant Physiol. (1)

P. Latimer, Plant Physiol. 34, 193 (1959).
[Crossref] [PubMed]

Rev. Sci. Instrum. (1)

P. F. Mullaney, M. A. Van Dilla, J. R. Coulter, D. N. Dean, Rev. Sci. Instrum. 40, 1029 (1969).
[Crossref] [PubMed]

Other (2)

C. C. Gravatt, Aerosol Measurements; Proceedings of a Seminar on Aerosol Measurements, Gaithersburg, MD, 7 May 1974, NBS Special Publication 412 (U.S. Government Printing Office, Washington, D.C., 1974).

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1959), p. 256.

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Figures (14)

Fig. 1
Fig. 1

Schematic diagram showing the incident beam of intensity I0 and the measured scattered light Is from a spheroidal particle at the origin. The axis of symmetry of the particle (arrow) makes a polar angle ψ with the beam and y axis and an azimuthal angle ω. This coordinate system is chosen so that the photocell which views light scattered at angle θ always lies in the xy plane.

Fig. 2
Fig. 2

The angular (θ) dependence of scattering of a sphere of volume 1 μm3, λ = 0.5 μm, and m = 1.05. These parameter values approximate those of a small bacterial cell in water when illuminated with red light. Curves for σ(θ,ψ,ω) to the left and right of the particle are shown in this and subsequent figures as dashed and solid lines, respectively, both in a conventional semilog Cartesian plot (left) and in a semilog polar plot (right). In this latter, ρ = log10σ(θ,ψ,ω) + 7.0. On the right is also shown a top view of the scattering particle. If the incident beam is of intensity I0, the scattered beam will be of intensity Is which is proportional to σ(θ,ψ,ω). Arrows indicating the incident beam and light scattered at a convenient angle, 74° in this case, are shown to facilitate comparisons. Figures 25 are drawn to the same scale.

Fig. 3
Fig. 3

The angular scattering pattern of a prolate spheroid of volume 1 μm3, υ = 2.0, m = 1.05, λ = 0.5 μm, for ψ = 0°. The axis of symmetry of the particle is in the plane of the paper and parallel to the beam.

Fig. 4
Fig. 4

The angular scattering pattern of the prolate spheroid of Fig. 3 in another orientation: ψ = 40°, ω = 0°. The particle axis lies in the plane of the paper and points at 40° to the right of the beam. Note the asymmetry of the patterns produced by this asymmetrically oriented particle as compared with the symmetry of the patterns and particle orientations of the earlier figures.

Fig. 5
Fig. 5

The angular scattering pattern of an oblate spheroid of volume 1 μm3, υ = 0.5, m = 1.05, λ = 0.5 μm, ψ = 40°, and ω = 0°. The particle axis lies in the plane of the paper and points at 40° to the right of the beam. Note that this oblate ellipsoid points in the same direction as did the prolate ellipsoid of Fig. 4.

Fig. 6
Fig. 6

Small angle scattering by spheres and spheroids of specified orientations [left: prolate (υ = 2.0); right: oblate (υ = 0.5)]. In each case, m = 1.05, volume = 10 μm3, λ = 0.5 μm. Point code: ○, sphere; ×, spheroid, ψ = 0, ω = 0; *, ψ = π/2, ω = 0; and Δ, ψ = π/2, ω = π/2. These orientations generally lead to extreme values of σ(θ,ψ,ω). When ψ = 0, the axis of symmetry is parallel to the beam. When ψ = π/2, the axis is perpendicular to the beam. Then, when ω = 0, the axis lies in the scattering plane; when ω = π/2, the axis is perpendicular to both the beam and the scattering plane.

Fig. 7
Fig. 7

Small angle scattering by spheres and spheroids of the size, shape, orientations, etc., of Fig. 6, but of larger refractive index: m = 1.20. The point code is that of Fig. 6.

Fig. 8
Fig. 8

The size dependence of small angle scattering by spheres and spheroids: left, prolate (υ = 2) and right, oblate (υ = 0.5), m = 1.05, and λ = 0.5 μm. The point code is that of Fig. 6.

Fig. 9
Fig. 9

The size dependence of small angle scattering by particles like those in Fig. 8 but of refractive index m = 1.20. The point code is that of Fig. 6.

Fig. 10
Fig. 10

The dependence of 90° scattering by spheres and prolate spheroids (υ = 2) on particle size. In all cases, the particle refractive index is m = 1.20. Left: the photodetector collects only light scattered at θ = 90°, and the incident beam is monochromatic (λ = 0.5 μm). Right: a wide photodetector collects equal amounts of light scattered at all θ values between 80° and 100° in the xy plane. In addition, the incident beam contains equal amounts of light of all wavelengths in the 0.4–0.7-μm range. The point code is that of Figs. 69. The curves for ψ = π/2, ω = 0 for 90° scattering were found to fall on those for ψ = π/2 and ω = π/2 and were omitted for clarity.

Fig. 11
Fig. 11

Randomly oriented populations of spheroids. Probability distributions of the theoretical photocell response ratio σ(10°, ψ,ω)/σ(5°, ψ,ω): left, prolate (υ = 2) and, right, oblate (υ = 0.5) spheroids. Particle volume is 10 μm3, m = 1.05, and λ = 0.5 μm. Each photocell is assumed to collect all light at θ ± 0.5° within the xy plane. Also indicated are the response ratios for spheres of the indicated sizes. For randomly oriented particles, it is assumed that all values of ω are equally probable, and the probability of a given ψ valve is proportional to sinψ.

Fig. 12
Fig. 12

Randomly oriented populations of spheroids: probability distributions of the predicted photodetector response ratio, σav(10°, ψ)/σav(5°, ψ), for the randomly oriented spheroids of Fig. 11: left, prolate; right, oblate. Each ringlike photodetector is assumed to collect light scattered at θ ±0.5° and at all azimuthal angles.

Fig. 13
Fig. 13

Predicted response of a photodetector which collects light scattered in the range θ = 1–20° and all azimuthal angles ω: R(1 − 20°, ψ), as a function of particle size. Responses for prolate spheroids (υ = 2) are shown on the left, those for oblate ones (υ = 0.5) on the right. R(1 − 20°, ψ) values for spheres are shown as solid lines without points. Unconnected points denote responses for the following orientations: Δ, ψ = 0°; ×, ψ = 45°; ⊙, ψ = 90°. In each case, the volume = 10 μm3, λ = 0.5 μm, and m = 1.05.

Fig. 14
Fig. 14

The angular dependence of scattering by the spheroids of Fig. 4 as predicted by the three hybrid methods of Table I. The solid curves represent scattering to the right, the dashed curves that to the left.

Tables (1)

Tables Icon

Table I Methods for Calculating the Differential Scattering Cross Section σell(θ,ψ,ω) of a Spheroid of Refractive Index mell Oriented with its Polar Axis Making an Angle ψ with the Beam and at an Azimuthal Angle ω from that of an Equivalent Sphere σsp(θ); this Sphere is of Radius ac or ag′ and Relative Refractive Index msp

Equations (6)

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σ av ( θ , ψ ) = 1 2 π 0 2 π σ ( θ , ψ , ω ) d ω .
R ( θ 1 θ 2 , ψ ) = θ 1 θ 2 σ av ( θ , ψ ) sin θ d θ .
σ ell ( θ , ψ , ω ) = F ( θ , ψ , ω ) σ sp ( θ ) .
δ = arccos [ sin ( θ / 2 ) sin ψ cos ω cos ( θ / 2 ) cos ψ ] ,
ψ * = arccos [ cos ψ cos ( θ / 2 ) + sin ψ sin ( θ / 2 ) cos ψ ] ,
ω * = arcsin ( sin ψ sin ω / sin ψ * ) .

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