deletions | additions       diff --git a/We_observe_in_Fig_ref__.tex b/We_observe_in_Fig_ref__.tex  index 175e7e8..813fd7c 100644  --- a/We_observe_in_Fig_ref__.tex  +++ b/We_observe_in_Fig_ref__.tex 
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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 $v < 0.1~\mathrm{a.u.}$ the results mimic 0.07~\mathrm{a.u.}$  the response of an the  effective electron gas with one electron per $\mathrm{Cu}$ atom. mimics the TDDFT results.  The resulting curves shown 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 $v \sim 0.07 ~\mathrm{a.u.}$ $v\sim 0.07~\mathrm{a.u.}$  due to a mixture of $d$-band $\mathrm{d}$-band  in the electronic density of states. Similarly, our new results for $v \leq 0.07~\mathrm{a.u.}$, are primarily due to $s$-band $\mathrm{s}$-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 $1.455$.