Determination of the Surface-Plasmon Dispersion Relation in Aluminum by Inelastic Electron Diffraction

Abstract

Inelastic differential cross sections for low-energy ($10\lesssim E\lesssim {10}^3$ eV) electrons scattered from the (111) and (100) surfaces of aluminum are analyzed using a quantum field theory of inelastic processes. The analysis permits the determination of the dispersion relation and damping of excitations created by the electrons. For nominally clean Al(111), the surface-plasmon dispersion relation is found to be $\hslash {\omega }_s\left(p_{\parallel }\right)=10.1-0.7\mathrm{}{p}_{\parallel }+10\mathrm{}{p}_{\parallel }^2$, ${10}^{-2\mathrm{}}\lesssim p_{\parallel }\lesssim 1$ ${\mathrm{\AA }}^{-1\mathrm{}}$, for momenta measured in reciprocal angstroms and energies measured in electron volts. With this choice of the dispersion relation, the predictions of the theory also are in fairly good agreement with experimental inelastic scattering data from Al(100) surfaces. The accuracy of the method of analysis and possible improvements on it are discussed.