deletions | additions       diff --git a/For_the_off_channeling_trajectory__.tex b/For_the_off_channeling_trajectory__.tex  index e13cf1a..eb70fb6 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}. The direction of the velocities are 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").  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" and therefore densities.  The viability along with the necessity of using this trick 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.   To obtain the $S_\text{e}$ we compute the slopes of the curves by a linear fit of the form $y = a + bx$ (black solid lines) using our data from $x > 5~a_0$ (to eliminate the transient region) to a given maximum position of $x$ determined by minimizing reentrancy in the periodic supercell into the initial position.