deletions | additions       diff --git a/Fig_ref_fig_force_on_neighbor_shows__.tex b/Fig_ref_fig_force_on_neighbor_shows__.tex  index 6eb1a11..5e3fa7d 100644  --- a/Fig_ref_fig_force_on_neighbor_shows__.tex  +++ b/Fig_ref_fig_force_on_neighbor_shows__.tex 
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Fig. \ref{fig:force_on_neighbor} shows the radial force exerted on a neighboring $\mathrm{Cu}$ atom closest to $\mathrm{H^+}$ trajectory as a function of parallel distance to the projectile at different projectile velocities along the $\langle 100\rangle$ channeling trajectory. The forces on the nuclei are evaluated using the time-dependent electron density, $n(\mathbf{r}, t)$. The force is evaluated by the Hellmann-Feynman theorem \cite{Feynman_1939,1941,Hellmann_2015}.  The adiabatic force is recovered for $v \to 0$ with no oscillations as expected.   The maximum value for the force is obtained at the closest distance between the $\mathrm{H^+}$ and a  neighbor $\mathrm{Cu}$ atom. As the proton moves further from the $\mathrm{Cu}$ atom, the force decreases significantly and eventually reduces to zero. As the velocity increases the position of the maximum value of the force shifts due to the complex shape-structure of the curves and results oscillations. These oscillations remain persistent with the increase of projectile velocity.% of the proton increases.%