Calculation of total cross sections of multiple-sphere clusters

Abstract

A method for calculating the extinction, absorption, and scattering cross sections of clusters of neighboring spheres for both fixed and random orientations is developed. The analysis employs the superposition formulation for radiative interactions among spheres, in which the total field from the cluster is expressed as a superposition of vector spherical harmonic expansions about each of the spheres in the cluster. Through the use of addition theorems a matrix equation for the expansion coefficients is obtained. Further application of addition theorems on the inverse of the coefficient matrix is shown to yield analytical expressions for the orientation-averaged total cross sections of the sphere cluster. Calculations of the cross sections of pairs of spheres and fractal aggregates of several spheres are presented. It is found that a dipole representation of the field in each sphere does not adequately predict the absorption cross section of clusters of small-size-parameter spheres when the spheres are highly conducting. For this situation several multipole orders are required for an accurate calculation of the absorption cross section. In addition, the predicted absorption of sphere clusters can be significantly greater than that estimated from the sum of the isolated-sphere cross sections.

Effects of aggregation on the permittivity of random media containing monodisperse spheres

“…The vector addition theorems and corresponding translation coefficients can be calculated using recurrence formulas that significantly reduce the number of operations for their computation. 54,55 A more concise form for Eq. (11) is found in Fuller and Mackowski.…”

Complete field descriptions in three-dimensional multiple scattering problems: a transfer-matrix approach

“…Exact solutions to finite multiple scattering systems can provide important insights into the numerous approximate methods that are typically used to describe multiple scattering. Perhaps even more importantly, exact solutions are a good means of studying field fluctuations in multiple scattering systems.Invoking the addition theorem [1, 2], one can rather readily describe a multiple scattering system via coupled matrix equations acting on a spherical wave development of the incident field [3][4][5][6][7]. After a suitable truncation, the complete solution to this problem can then, in principle, be obtained by the inversion of an entire 'system' matrix [3,[5][6][7].…”

“…Analogous problems for two-dimensional systems of infinite cylinders have been treated with success in this manner [8,9]. The enormous number of unknowns and the sparse nature of the matrix, however, often prevents a direct matrix inversion solution for three-dimensional systems of spheres [5,6,[10][11][12].Several authors have proposed the calculation of cluster transfer matrices, T cl N , via recursive approaches which build the transfer matrix one particle at a time [11,13]. A major obstacle in these recursive approaches is that they suffer from numerical instabilities associated with partial-wave space truncations as recently discussed by Siqueira and Sarabandi [12], and by us [5].…”

“…After a suitable truncation, the complete solution to this problem can then, in principle, be obtained by the inversion of an entire 'system' matrix [3,[5][6][7]. Analogous problems for two-dimensional systems of infinite cylinders have been treated with success in this manner [8,9].…”

“…Analogous problems for two-dimensional systems of infinite cylinders have been treated with success in this manner [8,9]. The enormous number of unknowns and the sparse nature of the matrix, however, often prevents a direct matrix inversion solution for three-dimensional systems of spheres [5,6,[10][11][12].…”