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Method of Computing Cross sections

Our cross section calculations proceed in three steps: (i) smoothing, (ii) interpolation/extrapolation and (iii) computation of the cross section.

Especially at low $J$, the reaction probabilities as a function of energy are dominated by a series of overlapping resonances. These resonance features are not conserved as a function of $J$, so they will be largely averaged out in any calculation of the total cross section. Because we do not compute cross sections at all relevant values of $J$, at each energy of interest, we must interpolate to obtain reaction probabilities $J < J_{\max}$, and extrapolate for $J>J_{\max}$, where $J_{\max}=35$ is the largest $J$ value we computed. In order to reliably interpolate, however, we must smooth out the reaction probabilities curves. We employed two types of smoothing filters : moving window averaging(mw), and Savitzy-Golay(sg).[83] For the mw calculations, we included 35 points to the right and 35 points to the left of each point in the average value. This simple form of smoothing conserves the area and the average of the distribution (the zeroth and first moment). Savitzy-Golay smoothing is a more sophisticated method of smoothing designed to preserve higher moments. We applied the sg filters to the computed reaction probabilities, using 25 points to the right and left of each point, conserving through the second moment.

The smoothed reaction probability curves were then used as input to the interpolation/extrapolation routines. Interpolation was done separately for each of the three initial projections: $\Omega_i = 0,1^+$ and $
1^-$, where $1^+$ and $
1^-$ refer to even or odd spectroscopic parity, respectively. For both the sg and the mw filtered curves, we used second order (quadratic) and third order (cubic) polynomial interpolation. These higher order polynomials are more reliable for interpolation than for extrapolation since for $J>J_{\max}$ the higher order extrapolations may begin to increase and never reach the abscissa. We found that high order extrapolation led to spurious oscillations in the cross sections that were basically artifacts; we therefore employed linear extrapolation. With linear extrapolation, there is negligible difference between the curves obtained with quadratic and cubic interpolation.

The cross section is computed from

\begin{displaymath}
\sigma_{(\nu_i=0,j_i=1)}(E_{tr}) = \frac{1}{3} \frac{\pi}{k^...
...{\Omega_i ,p} (2J + 1)
P_{\nu_i,j_i,\Omega_i}^{J,p} (E_{tr}),
\end{displaymath} (6)

where $k = \sqrt{2 \mu_R (E-\epsilon_{\nu_i j_i})}= \sqrt{2 \mu_R
E_{tr}}$. $E_{tr}$ is the translational energy in the entrance channel and $\epsilon_{\nu_i j_i}$ is the rovibrational energy of the initial state. The $1/3$ factor arises from the electronic degeneracy; the reactants can couple to form both doublet (ground state) and quartet states.
next up previous
Next: Results and Discussion Up: Time-dependent quantum mechanical calculations Previous: Initial Conditions
Anthony J. H. M. Meijer 2000-10-05