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Edwin E. Quashie edited Figure_ref_fig_fit_graph_shows__.tex
over 8 years ago
Commit id: f1faa1cf79e407de1b08ecb4c42e4b1ad5aaa71e
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Figure \ref{fig:fit_graph}
shows explains how
we obtained the slopes of Figure
\ref{fig:energy_distance}. In Figure \ref{fig:fit_graph}, \ref{fig:energy_distance} are evaluated. Here the
magenta dot-dash (magenta) curve
shows represents the adiabatic potential energy surface (APES) curve at $v = 0 ~\mathrm{a.u.}$;
which is obtained due to the Born-Oppenheimer approximation. The oscillations in the curves reflect the periodicity of the $\mathrm{Cu}$ lattice. At $v = 0 ~\mathrm{a.u.}$, there is no transfer of energy, just oscillations of the total energy with a zero slope. To obtain the slope for $v = 0.06 ~\mathrm{a.u.}$ which represents the $S_\text{e}$ at this velocity,
we subtract the adiabatic
curve results ($v = 0 ~\mathrm{a.u.}$)
are subtracted from
those of $v = 0.06$
curve due to the non-adiabatic effect at velocities greater than $0 ~\mathrm{a.u.}$ to obtain
only the
red non-adiabatic contributions; it is shown as a solid line
curve (red) with little oscillations. This subtraction is done to remove the oscillations that result from the periodic lattice. A linear fit , $y = a +
bx$ (blue line) gives bx$, dash line (blue) yields a slope of $0.00698913~E_\text{h}/a_0$ with an error of $\pm 6.936 \times 10^{-5}~E_\text{h}/a_0$. We then proceed to do a linear fit in addition to an oscillatory function $y = a + bx + A\cos(k x +
\phi)$ (black line) \phi)$, dot line (black) to capture any remnant oscillation. This oscillation fit
gives generates a slope of $0.00743456~E_\text{h}/a_0$ with an error of $\pm 8.52 \times 10^{-7}~E_\text{h}/a_0$. For higher velocities, a linear fit alone is enough to obtain reasonable error values.%