deletions | additions       diff --git a/We_observe_in_Fig_ref__.tex b/We_observe_in_Fig_ref__.tex  index 813fd7c..2c52aa6 100644  --- a/We_observe_in_Fig_ref__.tex  +++ b/We_observe_in_Fig_ref__.tex 
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\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.  The resulting curvesshown  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.   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 $1.455$.