deletions | additions       diff --git a/At_low_velocity_our_results__.tex b/At_low_velocity_our_results__.tex  index be4deeb..e90fac1 100644  --- a/At_low_velocity_our_results__.tex  +++ b/At_low_velocity_our_results__.tex 
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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 $v = 0.6~\mathrm{a.u.}$ 0.06~\mathrm{a.u.}$  can be explained by conservation laws in the homogeneous electron gas. 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 $\sim $\Delta =  1.5~\mathrm{eV}$ below the Fermi energy,and we use  that the values of the electron (band)  effective mass are close to $1$ for and $k_\text{F}$ $\mathrm{s}$-electrons $k_\text{F} = 0.72$ for $\mathrm{s}$-electrons.   We obtain an estimate of $v_\text{kink} = \Delta/\hbar/k_\text{F} = 0.076~\mathrm{a.u.}$.  Below $0.02~\mathrm{a.u.}$, the lack of experimental points preclude a direct comparison, but we find a large deviation from linear behavior, one possible explanation is that bound effects break down the linear response (Linhard) approximation that was useful to interpret the different regimes and crossover at $v > 0.02~\mathrm{a.u.}$