deletions | additions       diff --git a/For_the_off_channeling_trajectory__.tex b/For_the_off_channeling_trajectory__.tex  index a606b01..f2be53e 100644  --- a/For_the_off_channeling_trajectory__.tex  +++ b/For_the_off_channeling_trajectory__.tex 
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For the off-channeling trajectory, the procedure for computing the $S_\text{e}$ is depicted in Fig. \ref{fig:fit_off_channel}.   We tried two directions, [$0.704657, 0.609709, 0.362924$] [$0.705, 0.610, 0.363$]  and [$0.309017, [$0.309,  0.5, 0.809017$] 0.809$]  (given normalized here).   The first was choosen chosen  by visual inspection of the supercell in order to \emph{not} match any simple channel and avoid also an immediate head on collision. The second in is  the normalized version of $[1, \phi, \phi^2]$, where $\phi$ is the Golden Ratio, Ratio ($\sim 1.618$),  which is a number know for its mathematical properties as an irrational number. It is important to note here an interesting geometrical fact that if the direction is incommensurate with the crystal direction, all available densities are probed 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 this geometrical averaging method was shown earlier in \cite{Schleife_2015}. 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.