Light scattering cross section as a function of pair distribution density

This paper presents a simple method of approximation for calculating the scattering cross section for a random orientated convex particle illuminated by a non polarized electromagnetic wave. This method is proved efficient for sphere and spheroids as the scattering efficiency is smaller than one and as the material is optically either soft or hard. Introduction Since the nineteenth century, physicists have tried to calculate the optical properties of macroscopic bodies, particularly their scattering cross sections. This has led to several theories, known as approximations of the exact theory [1]: / 1 d λ << : Rayleigh (R) approximation / d 1 λ < et 1 m 1 − << : Rayleigh-Debye-Gans (RDG) approximation / d 1 λ > et 1 m 1 − << : Anomalous Diffraction (AD) approximation / 1 d λ >> et 1 m 0 − >> : Fraunhofer Diffraction (FD) approximation d and m are respectively the object size (for instance, the sphere diameter) and the ratio of the refractive indices for object and suspending medium. The exact theory is due to Mie [2], who has solved the Maxwell’s equations for the propagation of an electromagnetic wave around an object. Solution needs appropriate boundary conditions at the interface particle-medium. Since the Mie’s ground-breaking paper, numerous works have been achieved in order to extend the method to bodies with more complex shape than the sphere one. Among these, one may emphasize investigations about the optical properties of sphere clusters. The latter require the use of sophisticated algorithms and the corresponding programming on computers. However, computations have been performed as the spheres, i.e the primary particles in the sphere cluster, are Rayleigh [3], Rayleigh-Debye-Gans [4] or Mie [5] scatterers. As an obvious fact, the optical properties of a particle are related to its geometrical shape or its morphology. Several morphological functions have been proposed or appear in the calculations: the surface function for a convex particle, as dealt by Mie for a sphere the local pair (inter-particle distance) distribution density function for a fractal cluster consisting in RDG scatterers Journal of Quantitative Spectroscopy & Radiative Transfer, 2009, 110(3), 240-246 doi:10.1016/j.jqsrt.2008.10.003 2 the chord length distribution density for particle studied in the framework of AD approximation [6] The different morphological functions are studied by means of integral geometry, which searches the relations between them [7]. The purpose of this paper is to search an approximate expression relating the light scattering cross section to a morphological function for a convex particle. Thus, one aims to calculate the light scattering cross section of simply shaped bodies by proposing a new approximation involving more recent results obtained for sphere clusters. The section 2 reviews the main results concerning the sphere clusters. The section 3 develops the new approximation. The section 4 shows the results of the comparison of the new approximation and exact theories when the body is a sphere or a spheroid. Scattering cross section for sphere clusters Let us consider a sphere cluster, consisting in N spherical primary particles with the same radius a. Its scattering cross section is denoted CN. One will begin by treating the case of a cluster of RDG scatterers, itself being a RDG scatterer. Rayleigh-Debye-Gans approximation Ω The volume element is the primary particle. Each volume element gives RDG scattering and does so independently of the other volume elements. The waves scattered in a given direction by all these elements interfere because of the different positions of the volume elements in space. The scattering cross section (averaged over all the possible orientations) of the cluster obeys the relation ( ) ( ) 2 1 ( 4 ) N C N F S q d