deletions | additions       diff --git a/For_the_off_channeling_trajectory__.tex b/For_the_off_channeling_trajectory__.tex  index 3535011..208337b 100644  --- a/For_the_off_channeling_trajectory__.tex  +++ b/For_the_off_channeling_trajectory__.tex 
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(given normalized here).   The first direction was chosen by visual inspection of the supercell in order to \emph{not} match any simple channel but also avoid an immediate head on collision.  The second direction is the normalized version of $[1, \phi, \phi^2]$, where $\phi$ is the golden ratio ($\sim 1.618$), which guarantees a trajectory most incommensurate with the cell due to its mathematical properties as an irrational number.  It is important to note here an interesting geometrical fact that if the that, for a  directionis  incommensurate with the crystal direction, directions,  all available densities and impact parameters (distances of closest approach to host atom)  are probed (averaged)  eventually for a long enough trajectory. Our simulations are limited in space (and time) but it is clear that the trajectories explore a wide range of impact parameters(distances of closest approach to host atom)  and therefore densities. The viability along with the necessity of using considering  this geometrical averaging method was shown earlier in \cite{Schleife_2015}. In Fig.~\ref{fig:fit_off_channel}, the sharp peaks show when the proton is in the vicinity of a host $\mathrm{Cu}$ atom,   while the smaller peaks and flatter regions indicate that the proton is not very close to any host atom.