Measurement of the Dispersion of Electronic Surface Excitations via Analysis of Inelastic Low-Energy-Electron Diffraction

A quantum-field theory of inelastic low-energy-electron diffraction (ILEED) is applied to examine four significant limitations on the uniqueness and precision with which the dispersion relation of electronic surface excitations can be determined from a model analysis of experimental ILEED intensities. The example of surface plasmons on Al (111) is described in detail. A grid size of $\delta w\cong 0.050$ eV in the loss energy and $\delta {\theta }^{\prime }\cong 0.5°$ in the final-state polar angle is established as being sufficiently small to provide a precise distinction between differing surface-plasmon dispersion relations, yet sufficiently large to permit the use of a two-step model of the diffraction process rather than a complete dynamical calculation in the analysis. Comparison of two-step with dynamical isotropic-scatterer inelastic-collision model calculations reveals that plots of the scattered intensity in a given exit direction as a function of the loss energy (i.e., "loss profiles") provide the most appropriate method of data presentation for theoretical analysis via a two-step model. The loss-energy ($\Delta w\sim 1$ eV) and angular ($\Delta {\theta }^{\prime }\sim 2°$) resolution of present ILEED spectrometers is shown not to be a limiting factor of the precision of our analysis. Finally, we demonstrate that within the context of both the two-step and dynamical model calculations, the conservation laws fail to provide a sufficiently detailed description of ILEED intensities to permit a determination of the surface-plasmon dispersion from kinematical considerations alone. This dispersion can be extracted accurately from observed ILEED intensities only via a complete analysis of the loss profiles for various values of the incident-beam parameters using a two-step (or more complicated) model. These four results are combined to propose an analytical procedure which provides an accurate yet economical determination of the surface branches of the electronic excitation spectra of the valence-electron fluid in a solid.