deletions | additions       diff --git a/section_Discussion_A_logarithmic_version__.tex b/section_Discussion_A_logarithmic_version__.tex  index 2d52e69..7469593 100644  --- a/section_Discussion_A_logarithmic_version__.tex  +++ b/section_Discussion_A_logarithmic_version__.tex 
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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, according to this analysis of  our new results 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 $1.48 \pm 0.02$.