Density response function and the dynamic structure factor of thin metal films: Nonlocal effects

We use the hydrodynamic model of the bounded electron gas to evaluate the density response function of a thin film, including the effects of electron-gas dispersion (nonlocal effects). We obtain the contribution from the complete spectrum of plasma modes to the inelastic differential scattering cross section for keV electrons. Our results are for a sharp electron-density profile at the slab surfaces. Because of nonlocal effects, the spectrum is composed of a series of distinct bulk plasmons, in addition to the two surface plasmons of a thin film. We present a detailed analysis of the dynamic structure factor of a thin film in the small-wave-vector limit. We show that for sufficiently thin films, the limits $\beta \to 0$ (which defines the local limit) and $q_{\parallel }L\to 0$ (where $q_{\parallel }$ is the wave vector of the plasmon and $L$ is half the film thickness) are not interchangable in our expression for the differential cross section. Thus the local-approximation result of Ritchie for the transmission probability of a fast electron through a thin film is recovered in the $\beta \to 0$ limit only for $q_{\parallel }L>\mathrm{}1\mathrm{}$. We also show the close relationship between the hydrodynamic density response function and the density response function obtained in the semiclassical random-phase approximation with classical specular scattering at the boundaries.