deletions | additions       diff --git a/Figure_ref_fig_fit_graph_explains__.tex b/Figure_ref_fig_fit_graph_explains__.tex  index b842190..6bd10ee 100644  --- a/Figure_ref_fig_fit_graph_explains__.tex  +++ b/Figure_ref_fig_fit_graph_explains__.tex 
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Figure \ref{fig:fit_graph} explains how the slopes of Figure \ref{fig:energy_distance} are evaluated.   Here the dot-dash (magenta) curve represents the adiabatic potential energy surface (APES) curve at $v = 0 ~\mathrm{a.u.}$; due to the Born-Oppenheimer approximation. The oscillations in the curves reflect the periodicity of In  the $\mathrm{Cu}$ lattice. At $v = 0 ~\mathrm{a.u.}$, there is no transfer of energy, just oscillations of the total energy with low velocities cases extracting  azero slope. To obtain the  slope for $v = 0.06 ~\mathrm{a.u.}$ which represents the $S_\text{e}$ at this velocity, the adiabatic results ($v = 0 ~\mathrm{a.u.}$) are subtracted from those of $v = 0.06 ~\mathrm{a.u.}$ to obtain only the non-adiabatic contributions; it is shown as becomes more challenging, first because  a solid line (red) with little oscillations. This subtraction longer simulation  is done required  to remove sample  the oscillations that result from crystalline structure and second because  the periodic lattice. A linear fit , $y = a + bx$ (blue line) yields a slope of $6.989 \times 10^{-3}~E_\text{h}/a_0$ natural oscillations associated  with an error the crystal periodicity becomes relatively larger.   Figure \ref{fig:fit_graph} explains how the slopes  of $\pm 6.936 \times 10^{-5}~E_\text{h}/a_0$. We then proceed to do a linear fit Figure \ref{fig:energy_distance} are evaluated  in addition to an oscillatory function $y = a + bx + A\cos(k x + \phi)$ (black line) to capture any remnant oscillation. This oscillation fit generates a slope of $7.435 \times 10^{-3}~E_\text{h}/a_0$ with an error of $\pm 8.52 \times 10^{-7}~E_\text{h}/a_0$. these cases.  For higher velocities, a linear fit alone is enough to obtain reasonable error values.%