A logarithmic version of the findings of Fig. \ref{fig:stopping_power} is depicted in Fig. \ref{fig:log_stopping_power}, where we have observed that the resulting curve is not as particularly simple. In order to interpret the results we also calculated the linear response stopping \(S_\text{L}(n, v)\) \cite{Lindhard_1964_book} based on a simple Lindhard RPA dielectric function \(\varepsilon_\text{RPA}\) for different effective densities \(n\) of the homogeneous electron gas \cite{Giuliani_2005} \[S_\text{L}(n, v) = \frac{2 e^2}{\pi v^2} \int_0^\infty \frac{\mathrm{d}k}{k} \int_0^{k v} \omega\mathrm{d}\omega \Im\left(\frac{1}{\varepsilon_\text{RPA}(n, k, \omega)}\right)\] (which assumes a proton effective charge of \(Z_1 = 1\)). As shown in Fig. \ref{fig:log_stopping_power}, for \(v < 0.07~\mathrm{a.u.}\) the response of the effective electron gas with one electron per \(\mathrm{Cu}\) mimics the TDDFT results. While more sophisticated dielectric models can be used \cite{Morawetz_1996}, we use the minimal model that can explain the simulation in the different regimes.
The resulting curves in Fig. \ref{fig:log_stopping_power} shows that for \(v > 0.3~\mathrm{a.u.}\) at least the \(11\) electrons per atom (full valence) participates in the stopping electron gas within linear response. We observe a \(S_\text{e}\) kink around \(v\sim 0.07~\mathrm{a.u.}\) due to a mixture of \(\mathrm{d}\)-band in the electronic density of states. Similarly, according to this analysis of our new results, for \(v \leq 0.07~\mathrm{a.u.}\), are primarily due to \(\mathrm{s}\)-band electrons within linear response. In the simulation we directly show a crossover region between the two linear regimes, and we find that the friction is in direct relation to the velocity with a power law with exponent \(1.48 \pm 0.02\). (In linear scale kinks are represented by changes of curvature, here logarithmic scale is more appropriate to discuss the physics.)