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 $\overrightarrow{\mathrm{k}}$- and $\omega$-dependent dielectric function ${\epsilon }_M(\overrightarrow{\mathrm{k}},\omega )$ for $\overrightarrow{\mathrm{k}}$ 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 $\mathrm{Im}\left[\frac{-1\mathrm{}}{\epsilon }\right]$ 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 ${\epsilon }_M(\overrightarrow{\mathrm{k}},\omega )$ by Singhal for some discrete $\overrightarrow{\mathrm{k}}$ values. The general importance of the ZBCS for the understanding of the energy-loss spectrum and for more complicated systems is pointed out.