deletions | additions       diff --git a/For_the_off_channeling_trajectory__.tex b/For_the_off_channeling_trajectory__.tex  index 208337b..5600d35 100644  --- a/For_the_off_channeling_trajectory__.tex  +++ b/For_the_off_channeling_trajectory__.tex 
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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, for a direction incommensurate with the crystal 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 and therefore densities.  The viability along with and  the necessity of 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, atom during an off-channeling trajectory,  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.   The slope ($b$) gives the electronic stopping for this off-channeling case.