Effective-medium theory of a heterogeneous medium with individual grains having a nonlocal dielectric function

A formulation is given to obtain the effective dielectric function ${\epsilon }^{*}\left(\omega \right)$ of a heterogeneous medium in which nonlocality due to the spatial dispersion of the individual grains is important. The formulation is then applied to the calculation of the ${\epsilon }^{*}\left(\omega \right)$ of a medium consisting of spherical metallic grains. A very general method of solving the boundary-value problems involving a nonlocal $\epsilon$ is presented. The results are presented in both the average—$T$-matrix approximation and the coherent-potential approximation. Numerical results are obtained with a hydrodynamic model of the metallic dielectric function although in principle other nonlocal dielectric functions could also be used. Finally the optical absorption by dye molecules adsorbed on metallic spheres is calculated, and the results are compared to those obtained by using the local-dielectric-function model for the metal. The important effects of the nonlocality manifest themselves in the characteristic shifts of the resonances and in the decrease in the peak heights.