Laser diffraction particle sizing : sampling and inversion

Abstract

The inverse problem of obtaining particle size distributions from observations of the angular distribution of near forward scattered light is reexamined. Asymptotic analysis of the forward problem reveals the information content of the observations, and the sources of non-uniqueness and instability in inverting them. A sampling criterion, such that the observations uniquely specify the size distribution is derived, in terms of the largest particle size, and an angle above which the intensity is indistinguishable from an asymptote. The instability of inverting unevenly spaced data is compared to that of super-resolving Fourier spectra. Resolution is shown to be inversely proportional to the angular range of observations. The problem is rephrased so that the size weighted number density is sought from the intensity weighted by the scattering angle cubed. Algorithms which impose positivity and bounds on particle size improve the stability of inversions. The forward problem can be represented by an over-determined matrix equation by choosing a large integration increment in size dependent on the frequency content of the angular intensity, further improving stability. Experimental data obtained using a linear CCD array illustrates the theory, with standard polystyrene spheres as scatterers. The scattering from single and tri-modal distributions is successfully inverted.