deletions | additions       diff --git a/We_observe_in_Fig_ref__.tex b/We_observe_in_Fig_ref__.tex  index 0900b77..175e7e8 100644  --- a/We_observe_in_Fig_ref__.tex  +++ b/We_observe_in_Fig_ref__.tex 
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We observe in Fig.~\ref{fig:log_stopping_power} 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 the Linhard 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} \mathrm{d}\omega\omega \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 $v < 0.1~\mathrm{a.u.}$ the results mimic the response of an effective electron gas with one electron per $\mathrm{Cu}$ atom.