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 $Q\approx 1.5k_F$. 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.

Figure 1

Electron DOS for aluminum and sodium as calculated within the LDA and compared to the corresponding DOS for HEG at equal densities. The Fermi energies are here set to ${\varepsilon }_F=0$.

Figure 2

Experimental $S(Q,\omega )$ at $T=300$ K (circles) compared to calculations (lines) for the HEG within RPA, and for real sodium metal within RPA, TDLDA, and TDLDA+LT for selected momentum transfers $Q$. The spectra are shifted vertically in steps of 0.1 1/eV (lower panel), and 0.05 1/eV (upper panel). The momentum-transfer values for each spectrum are indicated on the right both in units of $a_0^{-1}$ and in units of the Fermi wave vector $k_F$ (latter in brackets).

Figure 3

(a) and (b) The dynamic structure factor for Na and the dispersion of the energy of its peak from experimental data in different temperatures. The particle-hole continuum is marked with lines and the single-particle excitation dispersion $Q^2/2m$ with the dashed line, as annotated in (b). (c) Both experimental dispersions overlaid with results from electron-energy-loss spectroscopy.[58] Note the different ordinate axis in (c). Arrows mark the kinks in the dispersion; first a flattening at $Q\approx k_F$ and an acceleration at $Q\approx 1.6k_F$. (d)–(f) Theoretical dispersion contours. We assume here $m^{*}=m=1$.