Shape of plasmon bands at the Brillouin-zone boundary

Recently an off-diagonal element of the density response function was measured by inelastic scattering of x rays from standing wave fields (coherent inelastic x-ray scattering). The observed peak-valley structure demonstrated the existence of plasmon bands separated by a plasmon band gap for scattering vectors k close to the L point of the Brillouin zone in Si. In standard inelastic x-ray scattering spectroscopy which yields the diagonal density response the plasmon band gap is obscured by strong line broadening of the two plasmons resulting in a single broad peak. Here we explain that by measuring both the diagonal and the off-diagonal response for k=G111/2 the line shapes of the two plasmons at the Brillouin-zone boundary can be reconstructed. Theoretical results obtained from a 2×2 effective dielectric matrix evaluated to second order in the local pseudopotential are in very good qualitative agreement with experiment