deletions | additions       diff --git a/We_observe_in_Fig_ref__.tex b/We_observe_in_Fig_ref__.tex  index 5c8c631..c6217dd 100644  --- a/We_observe_in_Fig_ref__.tex  +++ b/We_observe_in_Fig_ref__.tex 
...
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 $\rho$ of the homogeneous electron gas~\cite{Giuliani_2005}. 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 \Im\left(\frac{1}{\varepsilon_\text{RPA}(\rho, 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 mimics mimic  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 participates in  the stopping electron gas. gas within linear response.  We observe a  $S_\text{e}$ kink around $v \sim 0.1 0.07  ~\mathrm{a.u.}$ due to a mixture of $d$-band in the electronic density of states. For energy loss Similarly,  our new results for $v \leq 0.07~\mathrm{a.u.}$, are primarily due to $s$-band electrons. 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 proportion relation  to the velocity with a power lawof  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 $\mathrm{Cu}$.  The minimum energy loss with maximum momentum transfer from an electron to an ion moving with velocity $v$ are respectively $2\hbar k_\text{F}$ and $2\hbar k_\text{F} v$ (plus corrections of order $m_\text{e}/m_\text{p}$).   Due to Pauli exclusion only electrons in the energy range $E_\text{F} \pm 2\hbar k_\text{F} v$ can participate in the stopping process.   Taking into account that DFT band structure predicts that the $\mathrm{d}$-band edge is $\Delta_\text{DFT} = 1.6~\mathrm{eV}$ below the Fermi energy (see for example, Fig.~3(a) in Ref.~\cite{Lin_2008}),   that electron (band) effective mass are close to $1$ and $k_\text{F} = 0.72$ for the effective homogeneous electron gas of $\mathrm{Cu}$ $\mathrm{s}$-electrons \cite{Ashcroft_2003}, we can derive a value of th $v_\text{kink}$.  Based in this DFT ground state density of states plus conservation laws we obtain an estimate of $v_\text{kink} = \Delta/(2\hbar k_\text{F}) = 0.41~\mathrm{a.u.}$ in qualitative agreement with the TDDFT prediction.  In reality, the $\mathrm{d}$-band is about $\Delta_\text{exp} = 2~\mathrm{eV}$ below the Fermi energy as indicated by ARPES \cite{Knapp_1979}, that means that both the DFT-based estimate and the TDDFT result should be giving an underestimation of 25\% of the kink location.  The second (negative) kink at $v = 0.3~\mathrm{a.u.}$ is more difficult to explain precisely as the qualitative description in terms of $k_\text{F}$ (as in the homogeneous electron gas) become more ambiguous, but it is related to the point at which the whole conduction band (11 $\mathrm{s} + \mathrm{d}$ electrons) starts participating in the process.