Theory of a zone-boundary collective state in A1: A model calculation

A two-band model, which previously was used successfully to evaluate the optical absorption in A1, is applied to derive the $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$- and $\ensuremath{\omega}$-dependent dielectric function ${\ensuremath{\epsilon}}_{M}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},\ensuremath{\omega})$ for $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ parallel to the [100] direction with use of degenerate perturbation theory. Within the nearly-free-electron approximation, it is shown that a pair of (200) Bragg planes gives rise to another pole in the energy-loss function $\mathrm{Im}[\frac{\ensuremath{-}1}{\ensuremath{\epsilon}}]$ and hence to a collective mode. Both the dispersion of the mode throughout the first Brillouin zone and the strength of the mode are evaluated and are found to agree very well with electron-energy-loss spectroscopy data. A detailed discussion of the nature of this mode is given. The mode is of the same origin as the so-called zone-boundary collective state (ZBCS) first proposed by Foo and Hopfield in Na. Comparison is made with a numerical calculation of ${\ensuremath{\epsilon}}_{M}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},\ensuremath{\omega})$ by Singhal for some discrete $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ values. The general importance of the ZBCS for the understanding of the energy-loss spectrum and for more complicated systems is pointed out.