Quantum-Mechanical Random-Phase-Approximation Calculation of the Surface-Plasmon Dispersion Relation for a Semi-Infinite Electron Gas

The surface-plasmon (SPO) dispersion relation for a semi-infinite electron gas is computed for the quantum-mechanical random-phase approximation. The surface is assumed to be perfectly reflecting and the boundary-value problem is solved by a symmetric continuation of the electron gas. The linear response of the electron gas to a perturbing charge is described by a function $\nu$ which satisfies an integral equation. The integral equation for $\nu$ is solved numerically and the SPO dispersion relation is obtained by finding the pole in the density response of the electron gas to this perturbation. Graphs of the real and imaginary part of the SPO dispersion relation are given.