Normalized scattering diagram for atmosplieric aerosols with power law for particle size distribution Vadim Raskin

Elizabeth City State University, Box 801, Elizabeth City, North Carolina 27909. Received 25 April 1983. 0003-6935/83/233670-02$01.00/0. © 1983 Optical Society of America. In two previous publications1,2 normalized scattering di­

agrams and total scattering coefficients were obtained in an­alytic form for a polydisperse aerosol systems of spherical particles with gamma-particle size distribution and Junge particle size distribution. The calculations were based on the theory of light scattering by tenuous particles developed by Shifrin.3 In this Letter the theory of light scattering by ten­uous particles is used to obtain the normalized scattering di­agram and total coefficient of scattering for the power law model of particle size distribution.

Junge indicated that in many cases the microstructure of the atmospheric aerosol can be described by the following function ƒ(a):

3670 APPLIED OPTICS / Vol. 22, No. 23 / 1 December 1983

where a is the radius of the aerosol particle, and amin is the minimal radius of the aerosol particle. The latest observa­tional data obtained by the Air Force Geophysical Observa­tory4 confirm the existence of aerosol boundary layer char­acterized by the Junge slopes from - 3 to - 6 . According to this information we can describe the aerosol particle size distribution for this layer by the following expression:

In this Letter we shall consider n = 5 and n = 6. First let us look at n = 5:

The parameters B and amin are5

where N is the number of particles in the unit volume, and a is the mean radius of the particles. Following the same ap­proach as before1,2 we shall use the following expression for the scattering diagram of the single particle:

where

Here I{β,a) is the scattering diagram of a single particle; β is the angle of scattering.

where α is the polarizability given by the Lorentz-Lorenz formula, n is the relative refractive index, and I0 is the in­tensity of the incident light.

where λ is the wavelength of the incident light. For the scattering diagram of the polydisperse aerosol

system Ī(β) we obtain the following expression:

where

These integrals are easy to calculate, and after introducing the variable u = 2mam i n = 3/2mā we obtain the following re­sult:

The normahzed scattering diagram Īn(β) we define (as in Refs. 1 and 2) by

It is convenient to present the end result in the following form:

(We set I0 = 1.) Let us now consider the case when n = 6:

where IR (ā) is Rayleigh's scattering diagram for monodisperse aerosol with particles having a radius of ā and with concen­tration N; ζ - 5 (u) is a function of u:

Using the same calculational procedure we obtain the fol­lowing expression for normalized scattering diagram corre­sponding to polydisperse aerosol system with f(a) = c/a6:

We will obtain the scattering coefficient of the polydisperse aerosol with f(a) = B/a5 by integrating the scattering coeffi­cient of a single particle on a particle size distribution. Fol­lowing Refs. 1 and 2 we describe the scattering coefficient of a single particle Kp by the following expression6:

where

(we set I0 = 1), where

where Κ(λ) is the scattering coefficient of the polydisperse aerosol.

where

The results presented here can be useful for calculating the optical properties of atmospheric aerosols, hydrosols, and other colloidal systems.

References 1. V. F. Raskin, Appl. Opt. 20, 3290 (1981). 2. V. F. Raskin, Appl. Opt. 21, 3808 (1982). 3. K. S. Shifrin, Scattering of Light in a Turbid Medium (Nauka,

Moscow, 1951; NASA Technical Translation TTF477, 1968). 4. T. S. Cress, Altitudinal Variations of Aerosol Size Distributions

over Northern Europe.in Atmospheric aerosols, A. Deepak, Ed. (Spectrum Press, Hampton, Va., 1982).

5. K. S. Shifrin and V. F. Raskin, "The Atmospheric Indicatrice, Corresponding to the Generalized Junge Distribution," in Pro­ceedings, Voyekoυ Main Geophysical Observatory (Leningrad, 1961), Vol. 105.

6. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

As the end result of the calculation we obtain

1 December 1983 / Vol. 22, No. 23 / APPLIED OPTICS 3671