Multiple Scattering of Waves. II. The Effective Field in Dense Systems

TLDR

Multiple scattering of waves in particle systems is modeled self‑consistently, allowing for random, partially, or fully ordered arrangements, with the coherent wave’s propagation constant differing from vacuum by a term involving scatterer density and an anisotropic scattering operator. Scattering is described by anisotropic, absorptive, and creation operators; the effective field is related to the average field via a parameter \(c\) defined through an integral over the field, leading to an integral equation under a quasi‑crystalline approximation and a variational expression exact for point scatterers, with comparisons between finite and infinite systems. The extinction theorem is proven, and the macroscopic approach is shown to apply to small systems whose size exceeds the scatterer potential range and correlation range.

Abstract

The multiple scattering of waves interacting with a system of particles is treated by a self-consistent approach. Scattering processes are described by operators that permit anisotropy, absorption, and creation. The scattering system may be randomly, partially, or completely ordered.The propagation constant ${k}^{\ensuremath{'}}$ of the coherent wave in the scatterer medium differs from the vacuum constant $k$ by ${({k}^{\ensuremath{'}})}^{2}={k}^{2}+4\ensuremath{\pi}ncf({\mathbf{k}}^{\ensuremath{'}},{\mathbf{k}}^{\ensuremath{'}})$, where $n$ is the scatterer density and $f$ is an operator whose matrix elements $f(\mathbf{b},\mathbf{a})$ represent the scattering amplitude in direction b for a wave incident in direction a on a single scatterer bound by the forces of its neighbors. The parameter $c$, defined by $cf({\mathbf{k}}^{\ensuremath{'}},{\mathbf{k}}^{\ensuremath{'}})=\ensuremath{\int}\mathrm{exp}(\ensuremath{-}i{\mathbf{k}}^{\ensuremath{'}}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{r})f\ifmmode\times\else\texttimes\fi{}{\ensuremath{\psi}}_{e}(\mathbf{r})d\mathbf{r}$, is a measure of the ratio of the effective field ${\ensuremath{\psi}}_{e}(\mathbf{r})$ to the average field.An integral equation is found for ${\ensuremath{\psi}}_{e}(\mathbf{r})$ with the help of a "quasi-crystalline" approximation. A variational expression is then found for $c$ that becomes exact for point scatterers.A comparison is made of finite and infinite scattering systems. The extinction theorem is proven. The macroscopic viewpoint is found to be applicable to small systems whose size is large compared to the scatterer potential range, and the range of scatterer position correlations.