Sensitivity of Surface-Plasmon Dispersion and Damping to Potential Barrier Shape

I have evaluated exactly the random-phase-approximation surface-plasmon linear dispersion coefficient, $\alpha ={\alpha }_1+i{\alpha }_2\equiv limit\; of{q}^{-1\mathrm{}}[\frac{{\omega }_s\mathrm{}\left(q\right)\mathrm{}}{{\omega }_s\mathrm{}\left(0\right)-1\mathrm{}}\text{as}q\to 0]$ for $2<~r_s<~6$, using self-consistent (Lang-Kohn) potential barriers, and also using model barriers of adjustable surface diffuseness, $a$. I find $\alpha$ to be markedly sensitive to $a$, casting doubt on the reliability of using non-self-consistent model barriers to determine surface-plasmon properties. At Al density ($r_s\simeq 2$), for example, I find $\alpha =-(0.24+0.57i)\mathrm{}$ Å using the Lang-Kohn potential, in contrast to Beck and Celli's step-function ($a=0\mathrm{}$) barrier result, $\alpha =-(0.07+0.2i)\mathrm{}$ Å.