deletions | additions       diff --git a/The_kink_we_found_at__.tex b/The_kink_we_found_at__.tex  index 19357a5..e3abcdd 100644  --- a/The_kink_we_found_at__.tex  +++ b/The_kink_we_found_at__.tex 
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The kink we found at $v = 0.07~\mathrm{a.u.}$ 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 $2\hbar k_\text{F}$ k_\text{F}v$  and $2\hbar k_\text{F} v$ k_\text{F}$  (plus small  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 close to $1$ and that $k_\text{F} = 0.72/a_0$ for the effective homogeneous electron gas of $\mathrm{Cu}$ $\mathrm{s}$-electrons \cite{Ashcroft_2003}, we can derive an approximate value of $v_\text{kink}$ caused by the participation of $\mathrm{d}$-electrons.  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 near  agreement with the our  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}, which means that both the DFT-based estimate and the full  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) becomes 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.