Dynamical response function in sodium studied by inelastic x-ray scattering spectroscopy

We present the dynamic structure factor S(Q, omega) of Na valence electrons in the range of momentum transfer 0.5k(F) < Q < 2.4k(F) and energy transfer 3 eV < omega < 30 eV determined by inelastic x-ray scattering spectroscopy. In this range, we observe how the collective plasmon excitations decay into the single-particle excitation continuum. We compare the results to calculations using time-dependent density-functional theory with different approximations. The failure of both random-phase approximation and time-dependent local-density approximation (TDLDA) is shown to become important at k(F) < Q < 2.4k(F), while TDLDA with an additional inclusion of quasiparticle lifetime effects reproduces the experimental spectra well. The experimental valence-electron response reaches the single-particle spectrum surprisingly early, at Q approximate to 1.5k(F). This is manifested both in the spectral shape and the peak dispersion. The experimental spectra are nearly free of any fine structure, confirming that the peak-shoulder structure observed in many other materials is due to band-structure effects, which turn out to be negligible in Na.