  A sharp feature in the charge-density excitation spectra of single-crystal
MgB$_{2}$, displaying a remarkable cosine-like, periodic energy dispersion with
momentum transfer ($q$) along the $c^{*}$-axis, has been observed for the first
time by high-resolution non-resonant inelastic x-ray scattering (NIXS).
Time-dependent density-functional theory calculations show that the physics
underlying the NIXS data is strong coupling between single-particle and
collective degrees of freedom, mediated by large crystal local-field effects.
As a result, the small-$q$ collective mode residing in the single-particle
excitation gap of the B $\pi$ bands reappears periodically in higher Brillouin
zones. The NIXS data thus embody a novel signature of the layered electronic
structure of MgB$_{2}$.
