Monte Carlo Trajectory Calculations of Atomic Excitation and Ionization by Thermal Electrons

The excitation and de-excitation of atoms by electron impact has been investigated for states near the ionization limit using classical Monte Carlo trajectory calculations. The trajectories were sampled within the reaction zone with a weight proportional to the equilibrium reaction rate and integrated backward and forward in time to obtain complete histories for the collisions. This method is very much more efficient than the usual technique of sampling outside the reaction zone, and makes an otherwise extremely expensive calculation feasible. The results indicate that for energy transfers less than a few $\mathrm{kT}$ the reaction cross-section is determined by adiabatic collisions, while for transfers greater than a few $\mathrm{kT}$ the impulse approximation is valid. The data have been used to obtain the following convenient approximation for the equilibrium transition kernel valid near the ionization limit: $R({E}_{f}, {E}_{i})=7.8\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}26}{[{A}^{+}]}_{e}{{[e]}_{e}}^{2}{[kT(\mathrm{eV})]}^{\ensuremath{-}\frac{13}{2}}{(\ensuremath{-}\frac{{E}_{<}}{\mathrm{kT}})}^{\ensuremath{-}4.8}\mathrm{exp}(\ensuremath{-}\frac{{E}_{>}}{\mathrm{kT}})$ where ${E}_{<}=min({E}_{i}, {E}_{f})$, and ${E}_{>}=max({E}_{i}, {E}_{f})$. This has been used in conjunction with a conventional master equation to obtain an exact expression for the steady state collisional recombination rate constant $a=2.0\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}27}[e]{[kT(\mathrm{eV})]}^{\ensuremath{-}\frac{9}{2}}$ ${\mathrm{cm}}^{3}$/sec. Comparison with the available experimental data is reasonably good, but there are indications that both radiative cascading and collisions with neutrals may be important under some experimental conditions.