Energy-apportionment techniques based upon detailed atomic cross sections

We briefly discuss four of the principal energy-apportionment techniques currently used in energy-deposition calculations—the Fowler equation, the Monte Carlo approach, the discrete-energy-bin method, and the Peterson-Green integral equation in the context of the continuous-slowing-down approximation (CSDA). Using a complete set of analytic inelastic cross sections for molecular hydrogen, we calculate, by each of these four methods, the energy per ion pair $W$ associated with the degradation in ${\mathrm{H}}_2$ of incident electrons with initial energies ranging between the ionization threshold and 1000 eV. We find that the results obtained from the first three of these methods are in very good agreement with one another, while the values of $W$ obtained by the last method in the CSDA are consistently larger than the corresponding values obtained from any of the discrete-energy-loss approaches. Lastly, we isolate the cause of the CSDA discrepancies and propose two techniques for modifying the low-energy predictions of the CSDA so that they are in much better agreement with those of the three discrete energy-apportionment methods for all incident-electron energies.