Simple mathematical expressions for spectral extinction and scattering properties of small size parameter particles, including examples for soot and TiO2

Focusing on small size-parameter particles, this paper compares several easily-computable approximations to the Mie equations, including those by Rayleigh, Penndorf and Wiscombe, with only the first term of Mie-coefficient expansion, and the exact solution. A symbolic algebra algorithm was also developed using Mathematica(tm) for solving the exact Mie equations, which is more than an order of magnitude shorter than an available FORTRAN algorithm for the same purpose. The comparisons of the approximations and the errors incurred are evaluated for a wide range of complex refractive indices (1.0 less than or equal to n less than or equal to 5.0 and 0.001 less than or equal to k less than or equal to 50), and for the size-parameter range of 0.0 less than or equal to x less than or equal to 1.0. While the choice of approximation depends on the size parameter and the refractive index, the first-term approximation is the best in most cases, with considerable reduction of cpu time. As a specific example, the extinction and scattering coefficients for soot and TiO2 particle suspensions are computed as a function of the size-parameter by the above-described approximations, by two other available approximations, and by the exact solution.