Analysis of the modified point-matching method in the electrostatic problem for axisymmetric particles

摘要

An integral modification of the generalized point-matching method (GPMMi) in the electrostatic problem for axisymmetric particles is developed. Scalar potentials that determine electric fields are represented as expansions in terms of eigenfunctions of the Laplace operator in the spherical coordinate system. Unknown expansion coefficients are determined from infinite systems of linear algebraic equations (ISLAEs), which are obtained from the requirement of a minimum of the integrated residual in the boundary conditions on the particle surface. Matrix elements of ISLAEs and expansion coefficients of the "scattered" field at large index values are analyzed analytically and numerically. It is shown analytically that the applicability condition of the GPMMi coincides with that for the extended boundary conditions method (D center dot D'DDe). As model particles, oblate pseudospheroids with semiaxes a = 1 and b aecurrency sign 1 are considered, which are obtained as a result of the inversion of prolate spheroids with the same semiaxes with respect to the coordinate origin. For pseudospheroids, the range of applicability of the considered methods is determined by the condition . Numerical calculations show that, as a rule, the D center dot D'DDe yields considerably more accurate results in this range, with the time consumption being substantially shorter. Beyond the D center dot D'DDe range of applicability, the GPMMi approach can yield reasonable results for the calculation of the polarizability, which should be considered as approximate and which should be verified with other approaches. For oblate nonconvex pseudospheroids (i.e., at ), it is shown that the spheroidal model works well if pseudospheroids are replaced with ordinary "effective" oblate spheroids. Semiaxes a (ef) and b (ef) of the effective spheroids are determined from the requirement of the particle volumes, as well as from the equality of the maximal longitudinal and transverse dimensions of particles or their lengths. As a result, the polarizability of pseudospheroids can be calculated by simple explicit formulas with an error of about 0.5-2%.