deletions | additions       diff --git a/For_the_off_channeling_trajectory__.tex b/For_the_off_channeling_trajectory__.tex  index eb70fb6..a9a6146 100644  --- a/For_the_off_channeling_trajectory__.tex  +++ b/For_the_off_channeling_trajectory__.tex 
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We tried two directions, [$0.704657, 0.609709, 0.362924$] and [$0.309017, 0.5, 0.809017$]   (given normalized here).   The first was choosen by visual inspection of the supercell in order to not match any simple channel.  The second in the normalized version of $[1, \phi, \phi^2]$, where $\phi$ is the Golden Ratio, which is a number know for its mathematical properties as an irrational number ("the most irrational number"). 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" impact parameters (distances of closest approach to host atom)  and therefore densities. The viability along with the necessity of using this trick 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,