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