Cell Sizing: a Small-Angle Light-Scattering Method for Sizing Particles of Low Relative Refractive Index P. F. Mullaneyand P. N. Dean

Biomedical Research Group, Los Alamos Scientific Laboratory, University of California, Los Alamos, New Mexico 87544. Received 23 June 1969.

Edwards and Brinkworth1 recently reported a technique for the control of particle size in a La Mer aerosol generator in which the input signal to the generator control system was derived from the light-scattering signal due to product particles. The light-scattering method consisted of measuring the ratio of intensity of scattered light, [I(θ2)/I(θ1)], at two angles, θ2 and θ1 (θ2 > θ1), which lie within the main forward scattering lobe for the range of particle size parameters, a, of interest (α = πd/λ, where λ = wavelength of the incident light and d = particle diameter). The possibility of extending this method to large a and low values of m (the refractive index of the particle, with respect to the surrounding medium) is discussed in this communication. This is the situation when biological cells are suspended in normal saline, buffer, or other aqueous media.

Hodkinson and Greenleaves2 have shown, for particles with diameters larger than three or four times the wavelength of the incident light (α ≥ 9), that scattering can be considered as com­posed of contributions from Fraunhofer diffraction, transmission with refraction, and external reflection. For large a and small 0, the intensity distribution within the main forward scattering lobe is essentially that of Fraunhofer diffraction and, hence, is inde­pendent of m. The intensity, I (θ), is then proportional to

Edwards and Brinkworth calculated the ratio Z(20°)/7(ll°) for 0 ≤ α ≤ 10 using Eq. (1) and found it to be a function of α, exhibiting a nearly linear region for values of a between 3 and 7. In their apparatus, θ2 = 20°, θ1 = 11°, thus setting the upper limit on α at about 10. This method can be extended to larger α if an optical system is employed which permits scattering measure­ments to be made at very small angles. The first minimum in the Fraunhofer diffraction pattern occurs at an angle, θmin, given by α sinømin = 3.83 so that θmin = 2.0° for α = 100.

We are currently conducting light-scattering experiments with biological cells suspended in normal saline using a helium-neon laser.3 The small cross section diameter and narrow divergence of the beam allow routine scattering measurements to be made at angles as small as 0.5°. We have calculated I(θ2)/I(θ1) as a func­tion of α for several values of θ2 and θ1 using Eq. (1). Plots of these functions are shown in Fig. 1. Notice that, in each case considered, there is a region where the ratio of intensities is nearly a linear function of α, extending from α = 25 to α = 80 for θ2 = 2.0°, θ1 = 1.0°; from α = 30 to α = 90 for θ2 = 1.5°, θ1 = 0.5°; and from α = 40 to α = 160 for θ2 = 1.0°, θ1 = 0.5°.

Kattawar and Plass4 have published results of Mie5 theory cal­culations for absorbing and nonabsorbing spheres. In particular, they have investigated the relationship between the half-width of the main scattering lobe and α for nonabsorbing spheres with values of m between 1.01 and 2.00. The half-width of this lobe is proportional to 1/α for α » l. This is interpreted as indicating that diffraction effects are always the principal mechanism for forward scattering, even for small m.

Since there appears to be some question as to the validity of the diffraction theory for opaque, circular disks when applied to the case of transparent spheres,6,7 the intensity ratio was calculated according to the Mie theory computer codes DAMIE and DBMIE8 for the following three values of m: 1.2, 1.1, 1.04, and θ1 = 0.5°, θ2 =

Fig. 1. Plot of I(θ2)/I(θ1) as a function of a for several values of θ1 and θ2. Calculations based on Fraunhofer diffraction as the only scattering mechanism. (O) θ2 = 1.5°, θ1 = 0.5°; (× ) θ2 =

1.0°, θ1 = 0.5°; and (Δ)θ2 = 2.0°, θ1 = 1.0°.

Fig. 2. Comparison of 7(2.0°)/I(0.5°) calculated on the basis of diffraction alone (—) and Mie theory for the relative refractive indices 1.04 (•); 1.1 (■); and 1.2 (A). The spheres were as­

sumed nonabsorbing for the Mie calculations.

2.0°. These values of m are close to those for plastic micro­spheres, several spores,9 and biological cells,10 respectively, im­mersed in water.

Results of the Mie and diffraction calculations are compared in Fig. 2. The agreement is quite good; the region of near linearity

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extends from α = 20 to α = 85. The greatest deviations of ƒ(2.0°) /I (0.5°), calculated by Mie theory from the diffraction approximation, occur at α = 40 for m = 1.1 ( ~ 1 5 % low) and at α = 20 for m = 1.2 ( ~ 7 % low). Use of the diffraction curve to estimate α would result in values 10-12% too large in these two cases.

I t is well known that the minima in the extinction curve are the result of destructive interference between diffracted and refracted light. The first minimum occurs when p = 2α(m – 1) has a value of 7.63 for m → l.11 The quantity p represents the phase difference between a ray traversing the particle along a diameter and a ray traversing the same distance in the surrounding medium. For a = 20, m = 1.2, and a = 40, m = 1.1, p = 8. Thus, the small reduction of signal calculated by Mie theory is the result of destructive interference effects for these values of a.

Since the diffraction and Mie methods agree quite well even for small m, it suggests that this type of scattering measurement could yield information on particle size and be relatively independent of the effects of transmitted light. The apparent insensitivity to m would allow calibration with any convenient particles, opaque or transparent, without knowledge of the refractive index. Of particular interest is the application of this method to particles of very low refractive index (i.e., biological cells immersed in buffer or other appropriate medium). We have investigated the rela­tionship between the total light scattered between 0.5° and 2.0° and α for biological cells using the Hodkinson model3 and found the predicted signal nearly proportional to α3, suggesting the possibility of obtaining absolute volume and volume distribution information. Experiments to date on cells have yielded correct relative volume, but absolute volume is too large—probably due to the additive effect of refracted light present at small angles as m → 1. The present analysis suggests the ratio of intensities method as an attractive means of obtaining absolute size when this is required. Experimental investigation of this method is planned for the near future.

This work was performed under the auspices of the U.S. Atomic Energy Commission.

References

1. J. Edwards and B. J. Brinkworth, J. Sci. Instrum. 45, 636 (1968).

2. J. R. Hodkinson and I. Greenleaves, J. Opt. Soc. Amer. 53 , 577 (1963).

3. P . F . Mullaney, M. A. Van Dilla, J. R. Coulter, and P. N. Dean, Rev. Sci. Instrum. 40, 1029 (1969).

4. G. W. Kat tawar and G. N. Plass, Appl. Opt. 6, 1377 (1967). 5. G. Mie, Ann. Phys. 25, 377 (1908). 6. W. PL Walton, Trans. Inst. Chem. Eng. Suppl. 25, 141

(1947). 7. G. F . Lothian and P . F . Chappel, J. Appl. Chem. 1, 475

(1951). 8. J. V. Dave, Subroutines for Computing the Parameters of the

Electromagnetic Radiation Scattered by a Sphere (IBM Scien­tific Center, Palo Alto, California, 1968), Rept. No. 320-3237.

9. K. F . Ross and E. Billings, J. Gen. Microbiol. 16, 418 (1958). 10. J. Barer, in Physical Techniques in Biological Research, G.

Oster and A. Pollister, Eds. (Academic Press Inc., New York, 1956), Vol. 3, p. 29.

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

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