deletions | additions       diff --git a/A_log_log_plot_of__.tex b/A_log_log_plot_of__.tex  index 0a8fc27..174ef75 100644  --- a/A_log_log_plot_of__.tex  +++ b/A_log_log_plot_of__.tex 
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A log--log plot 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}  \begin{equation}  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)  \end{equation}  (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, our new results for $v \leq 0.07~\mathrm{a.u.}$, are primarily due to $\mathrm{s}$-band electrons within linear response.