Fig. 1: Silicon electron loss function approximations.
The electron loss function, \({{{{\rm{Im}}}}}\left\{-1/\epsilon (\omega ,k)\right\}\), for silicon are plotted as a function of ω and k. a shows the Lindhard approximation to the electron loss function; b shows the Mermin approximation built into the DarkELF program24,25; c shows the GPAW DFT calculation of the dielectric function built into DarkELF; d shows the QCDark calculation of the imaginary part of the dielectric function28 screened by the Lindhard \(1/{\left\vert \epsilon (\omega ,k)\right\vert }^{2}\) as given by Eq. (7). d, due to the low k—grid used in the calculation, is Gaussian smoothed with σω = 0.5 eV and σk = 200 eV. In all plots, the plasmon is visible for low k and ω ~ 16.6 eV. The lines indicate the minimum momentum, \({k}_{\min }\), required to transfer energy ω from an incoming particle with speed β. Note that the plasmon threshold occurs close to β ~ 10−2.
