Light scattering theory from monodisperse spheroidal particles in the Rayleigh–Debye–Gans regime

Abstract

A general expression for the scattered intensity by populations of Rayleigh–Debye–Gans (RDG) spheroids has been derived for single and independent scattering from nonabsorbing monodisperse particles. The form factor—which contains information on the particle size, shape, and orientation—and the orientation distribution function are expanded in terms of a complete set of spherical harmonics for the polar and azimuthal angles, θ and φ. These expansions lead to a systematic solution procedure for the inverse problem of determining the orientation distribution function for populations of RDG spheroids from scattered intensities in two dimensions. With Fourier sine and cosine transforms of the form factor with respect to φ, the two-dimensional problem (θ, φ) is reduced to several linear one-dimensional problems at various scattering angles, θ. A quadratic optimization routine is used to solve these problems for obtaining the most pertinent orientation distribution coefficients. As an example, the form factors have been calculated for populations of RDG spheroids undergoing rotary diffusion. The diffusion time, initial orientation, aspect ratio, and particle volume have been examined in these simulations with two-dimensional contour plots. Changing the incident beam polarization changes the scattered intensity contours but not the form factor contours, hence yielding no additional information. The inverse problem has been solved and tested with simulated data from azimuthally symmetric populations of spheroids. The performance of the inverse algorithm has been examined for various particle aspect ratios, sizes, and angles, θ. With error-free simulated data, the orientation distribution functions can be precisely determined over the complete angular range from data obtainable over only a small angular range. These results show the robustness of the new approach. When up to 1% random error is introduced in the simulated intensity data, the solution to the inverse problem becomes less precise but is still reasonable. The implications of the inverse algorithm for analyzing experimental data are discussed.