deletions | additions       diff --git a/The_energy_transfered_to_the__.tex b/The_energy_transfered_to_the__.tex  index 75ced3d..c942901 100644  --- a/The_energy_transfered_to_the__.tex  +++ b/The_energy_transfered_to_the__.tex 
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The energy transfered to the electrons of the target atom due to a constant velocity moving proton is monitored.   At the time scales of the simulations, the large mass of the proton guarantees a the change in its velocity is relatively small.  For simplicity the proton is forced to move at constant velocity, hence total energy of the system is not conserved.   The excess in total energy is instead used as a measure of the stopping power as a function of the proton velocity.  As the proton moves, the time-dependent Kohn-Sham (TDKS) equation \cite{Runge_1984} describes electronic density and energy of the system due to the dynamics of effective single particle states under the external potential generated by the proton and the crystal of Cu nuclei. The TDKS equation can be written as:  \begin{equation}  \mathrm i\hbar\frac\partial{\partial t}\psi_i(\textbf r, t) = \left\{-\frac{\hbar^2\nabla^2}{2m} + V_\text{KS}[n(t)](\{\mathbf R_i(t)\}_i, \mathbf r, t)\right\}\psi_i(\textbf r, t)  \label{eq:tdks1}   \end{equation}  The Kohn-Sham (KS) effective potential $V_\text{KS}$ is given as  \begin{equation}  V_\text{KS}[n](\{\textbf R_i(t)\}_i, \textbf r, t) = \textit{V}_\text{ext}(\{\textbf R_i(t)\}_i, \textbf r, t) + \textit{V}_\text{H}[n](\textbf r, t) + \textit{V}_\text{XC}[n](\textbf r, t)  \label{eq:tdks3}  \end{equation}  where the external potential is $V_\text{ext}(\{\textbf R_i(t)\}_i, \textbf r, t)$ due to ionic core potential (with ions at positions $\mathbf R_i(t)$), $V_{H}(\textbf r, t)$ is the Hartree potential comprising the classical electrostatic interactions between electrons and $\textit{V}_{xc}(\textbf r, t)$ denotes the exchange-correlation (XC) potential. The spatial and time coordinates are represented by $\mathbf r$ and $t$ respectively.   At time $t$ the instantaneous density is given by $n(t)$.  The exchange-correlation potential used in this study is due to Perdew-Burke-Ernzerhof (PBE) ~\cite{Perdew_1992,Perdew_1996}, using a norm-conserving Troullier-Martins pseudopotential, with $17$ explicit electrons per Cu atom in the valence band, the coulomb potential is generated.  The simulations of the collisions consist of a well-defined trajectories of the projectile (proton) in the metallic bulk.   The calculations were done using the code \textsc{Qbox} \cite{Gygi_2008} with time-dependent modifications \cite{Schleife_2012}.