Spectral representations of the electron energy loss in composite media

Using a recently developed theory for the inverse longitudinal nonlocal dielectric response for a granular composite [R. G. Barrera and R. Fuchs, Phys. Rev. B 52, 3256 (1995)], we find an effective local dielectric response from the condition that it must give the same electron energy-loss probability density as the actual nonlocal response. This local response is expressed as a spectral representation:a sum of terms with simple poles corresponding to the excitation of bulk and interfacial modes. Taking as a guideline the condition that the strength and location of the poles must satisfy sum rules, a single-surface-mode approximation is proposed. This single-mode approximation is tested for a random system of aluminum spheres in vacuum.