deletions | additions       diff --git a/We_observe_in_Fig_ref__.tex b/We_observe_in_Fig_ref__.tex  index 7178338..911aa1f 100644  --- a/We_observe_in_Fig_ref__.tex  +++ b/We_observe_in_Fig_ref__.tex 
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As shown in Fig.~\ref{fig:log_stopping_power}, for $v < 0.1~\mathrm{a.u.}$ the results mimics the response of an effective electron gas with one electron per $\mathrm{Cu}$ atom.  The resulting curves shown in Fig.~\ref{fig:log_stopping_power} agree with the shows that for $v > 0.3~\mathrm{a.u.}$ at least the $11$ electrons per atom (full valence) behaves as the stopping electron gas.  We observe $S_\text{e}$ kink around $v \sim 0.1 ~\mathrm{a.u.}$ due to a mixture of $d$-band in the electronic density of states.   For energy loss our new results for $v \leq 0.06~\mathrm{a.u.}$, 0.07~\mathrm{a.u.}$,  are primarily due to $s$-band electrons. In the simulation we directly show a crossover region between the two linear regimes, and we find that the friction in direct proportion to the velocity with a power law of with exponent $1.455$.  The kink we found at $v = 0.06~\mathrm{a.u.}$ can be explained by conservation laws in the effective homogeneous electron gas and general properties of electronic density of states in crystalline $\mathrm{Cu}$.