deletions | additions       diff --git a/figures/radial_force/caption.tex b/figures/radial_force/caption.tex  index 25c26e4..9d4edac 100644  --- a/figures/radial_force/caption.tex  +++ b/figures/radial_force/caption.tex 
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\label{fig:force_on_neighbor} (color online). $\mathrm{H^+}$ in Copper: Radial force exerted on host atom as a function of parallel distance to projectile at different projectile velocities $v$. The force is “negative” radial. For visualization purposes the non-adiabatic curves have been shifted vertically upwards.  Figure \ref{fig:force_on_neighbor} shows the radial force exerted on a neighboring Cu atom closest to $\mathrm{H^+}$ trajectory as a function of parallel distance to the projectile at different projectile velocities along the [100] channeling trajectory. The forces on the nuclei are evaluated using the time-dependent electron density, $n(\mathbf{r}, t)$.  The force is obtained by the application of the Hellmann-Feynman theorem \cite{Hellmann_2015,1941,Feynman_1939}.% but not applied to the ground state in this case. \cite{Hellmann_2015,1941,Feynman_1939}.  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 $\mathrm{H^+}$ and neighbor $\mathrm{Cu}$ atom.  As the proton moves away 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 becomes persistent as the velocity of the proton increases.%