Effective electric field in an inhomogeneous medium

Abstract

This paper investigates the effective field acting in an infinite, nonmagnetic, inhomogeneous medium. A self-consistent, integrodifferential equation may be written for the effective field at a point, accounting for scattering at every other site in the medium. One may solve this equation by iteration to write the effective field as a power series of operators acting on the uniform-medium effective field. We rewrite this series so that all local fields are augmented by all orders of the self-field. This means that each order of scattering is represented by a single integral operator. The action of this scattering operator on an arbitrary field may be expressed in terms of integrals over reciprocal space, if the product of the fluctuating function and the field is Fourier transformed. One of the integrals contains a weighting function that discriminates in favor of Fourier wave numbers similar to the uniform medium wave number. This integral gives the Bragg-like contributions to the effective field. The other integral extends over all of reciprocal space and contributes equally to the effective field for all fluctuation wave vectors. This integral determines the background effective field. A general term of the complete multiple-scattering expansion for the effective field is given. If the fluctuation distribution has a single cosine Fourier wave vector, then the effective-field expansion simplifies considerably. The Bragg contributions are examined. Finally, the background effective field that ignores all Bragg-like contributions is given, where all orders of scattering are considered. This field is modulated by the fluctuations in the medium in a straightforward way and it obviously provides a better approximation of the local field than either the incident or uniform-medium fields. This background field is used to find the effective dielectric constant that is correct for media in which the inhomogeneities are uncorrelated over lengths comparable to a wavelength.