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Alfredo A. Correa added In_order_to_interpret_the__.tex
over 8 years ago
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In order to interpret the results we calculated the linear response stopping $S_\text{L}(n, v)$ \cite{Lindhard_1964} based on the Linhard RPA dielectric function $\epsilon_\text{RPA}$ for different effective densities $n$ \cite{Giuliani_2005}.
\begin{equation}
S_\text{L}(n, v) = \frac{2 e^2}{\pi v^2} \int_0^\infty \mathrm{d}k k^{-1} \int_0^{k v} \mathrm{d}\omega\omega \varepsilon_{
\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 the response of an electron gas with one electron per $\mathrm{Cu}$ atom.
The resulting curves shown in Fig.~\label{fig:} 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 $v \leq 0.06~\mathrm{a.u.}$, 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 $v = 0.1~\mathrm{a.u.}$ can be explained by conservation laws in the 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 $m_\text{e}/m_\text{p}$).
Therefore electrons in the 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 is $\Delta = 1.5~\mathrm{eV}$ below the Fermi energy,
that electron (band) effective mass are close to $1$ for and $k_\text{F} = 0.72$ for $\mathrm{s}$-electrons.
Based in this DFT ground state plus conservation laws we obtain an estimate of $v_\text{kink} = \Delta/\hbar/k_\text{F} = 0.081~\mathrm{a.u.}$ in qualitative agreement with the TDDFT prediction.
The second (negative) kink at $v = 0.3~\mathrm{a.u.}$ is more difficult to explain as the qualitative description in terms of $k_\text{F}$ become more ambiguous, but it is related to the point at which the whole conduction band (11 electrons) starts participating in the process.
