deletions | additions       diff --git a/The_simulations_of_the_collisions__.tex b/The_simulations_of_the_collisions__.tex  index 5f12e33..c97f3b3 100644  --- a/The_simulations_of_the_collisions__.tex  +++ b/The_simulations_of_the_collisions__.tex 
...
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.   The need to take into account off-channeling trajectories is described in \cite{Schleife_2015}.  Following the scheme of Ref.~\cite{Schleife_2012} the TDKS equation (see Eq. \ref{eq:tdks1}) was solved numerically   with a time step of, at most, $0.121~\mathrm{attoseconds}$ was used (which is 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 wavefunctions were then propagated for up to tens of 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.   The increase of $E$ as a function of projectile displacement $x$ enables us to extract the electronic stopping power as a time-averaged quantity.  \begin{equation}  S_\text{e} = \overline{\mathrm{d}E(x)/\mathrm{d}x}  \label{eq:stopping}   \end{equation}  $S_\text{e}$ has the dimension of a force (e.g. $E_\text{h}/a_0$) and it is the drag force acting on the projectile.