deletions | additions       diff --git a/The_simulations_of_the_collisions__.tex b/The_simulations_of_the_collisions__.tex  index c556232..f1fbd19 100644  --- a/The_simulations_of_the_collisions__.tex  +++ b/The_simulations_of_the_collisions__.tex 
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The simulations of the collisions consist of a well-defined trajectories of the projectile (proton) in the FCC  metallic bulk. bulk with experimental density.  The calculations were done using the code \textsc{Qbox} \cite{Gygi_2008} with time-dependent modifications \cite{Schleife_2012}.   The Kohn-Sham (KS) orbitals are expanded in a supercell plane-wave basis.   These KS orbitals are evolved in time with a self-consistent Hamiltonian that is a functional of the density. The fourth-order Runge-Kutta scheme (RK4) \cite{Schleife_2012} is used to propagate these orbitals. orbitals in time.  The advantages of using plane-wave approach is that, it systematically deals with basis-size effects which was a drawback for earlier approaches \cite{Pruneda_2007, Correa_2012} and finite-size error in the simulations are overcomed by considering large simulation cells \cite{Schleife_2015}. Periodic boundary conditions are used throughout this study.   The best supercell size was selected so as to reduce the specious size  effects while maintaining controllable computational demands. The calculations used $(3\times3\times3)$ conventional cells containing 108 host $\mathrm{Cu}$ atoms and $\mathrm{H^+}$.  % also represented by a Troullier-Martins pseudopotential ($17$ valence electrons per copper atom are explicitly considered).   To integrate the Brillouin zone a single $k$-point ($\Gamma$) was used.   The basis set is sampled with a $130~\mathrm{Ry}$ energy cutoff.  The projectile $\mathrm{H^+}$ was initially placed in the crystal and the a  time-independent DFT calculation was completed to obtain the converged ground state results that are required for subsequent evolution. We then perform TDDFT calculations on the electronic system. system with the moving proton.  Following the method introduced by Pruneda {\emph et al.} \cite{Pruneda_2005} the projectile is then allowed to move with a constant velocity subjected to a straight uniform movement along a [100] channeling trajectory (also called hyper-channeling trajectory). This minimizes the collision of the projectile with the host atoms.   In the off-channeling case the projectile takes a random trajectory through the host material yielding a stronger interaction between the projectile and the host atom due large charge density close to the target.   Following the scheme \cite{Schleife_2012} the TDKS equation (see Eq. \ref{eq:tdks1}) was then solved numerically. A time step of, at most, $0.121~\mathrm{attoseconds}$ was used, which is below within  the stability limit for the numerical explicit time-integration scheme for these type of basis set. (High velocity points are simulated with smaller time steps.) The resultant wave functions were then propagated for several femtoseconds.  The total electronic energy ($E$) of the system changes as a function of the projectile position ($x$) since the projectile (forced to maintain its velocity) deposits energy into the electronic system as it moves through the host atoms.