Microscopic calculation of surface-plasmon dispersion and damping

Exact calculations of the linear coefficient of surface-plasmon dispersion and damping $\alpha ={\alpha }_1+i{\alpha }_2\equiv {\lim }_{q\to 0}{q}^{-1\mathrm{}}[\frac{{\omega }_s\mathrm{}\left(q\right)\mathrm{}}{{\omega }_s\mathrm{}\left(0\right)\mathrm{}}-1]$, using the random-phase-approximation dynamical equation, for a jellium-vacuum interface characterized by a smooth, finite, surface potential barrier are described in detail. $\alpha$ is shown to be markedly sensitive to the shape of this barrier, casting doubt on the reliability of using non-self-consistent potential barriers to determine surface plasmon properties. For the case of an Al-density ($r_s=2\mathrm{}$) substrate, the sensitivity of $\alpha$ to barrier shape should manifest itself through a rather strong increase of ${\alpha }_1$ with alkali-impurity adsorption.