# Collision Frequency and Energy Transfer Rate in e-He Scattering
## 3 Results and Discussion
### Effective Collision Frequency and Energy Transfer Rate  
The results of our effective collision frequency calculations are shown in Figure 4, compared with the results from other authors. The use of the momentum transfer cross-section calculated by the optical potential method led to an interesting result. The resulting effective frequency shows a maximum at an energy of approximately 5 eV, which has not been found in other theoretical works that mostly use constant values for the momentum transfer cross-section. Experimental data from [PERSON] et al. [45] are consistent with this maximum, and at energies lower than the maximum, our results match well with other theoretical and experimental works [46]. The effective collision frequency given by expression (16) exhibits a maximum with increasing electron temperature. This behavior can be explained by a complex interplay between the electron velocity distribution and the momentum-transfer cross-section. With increasing electron temperature \(T_{e}\), the fraction of fast electrons in the Maxwell distribution also increases. At the same time, the momentum transfer cross-section does not depend on the electron temperature but on the energy of the incident electron. If the Maxwell distribution has a maximum at electron energies corresponding to the region where the cross section increases, a rise in the collision frequency can be observed. However, a further increase in the fraction of fast electrons leads to a decrease in the collision frequency. Figure 5 illustrates the energy transfer rate between electrons and helium atoms, with the rate governed by the relation (17). The energy transfer rate behavior reflects a competition between the electron and helium temperatures. The transfer rate peaks at lower electron temperatures when helium atoms have higher temperatures (which is rather impossible). Figure 3: Momentum-transfer cross-section of electron scattering on a helium atom. The results are compared with experimental data from [PERSON] et al. [42], [PERSON] et al. [41], and [PERSON] et al. [43], as well as theoretical results from [PERSON] [44], [PERSON] et al. [35]. As the electron temperature increases beyond the helium temperature, the energy transfer rate diminishes, eventually becoming negative. The negative values indicate that electrons begin to transfer energy to the helium atoms, reversing the direction of energy flow. This crossover occurs because the temperature difference (\(T_{e}-T_{He}\)) changes sign, causing a change in the direction of the energy flow. Figure 6 shows the dependence of the energy transfer rate on the electron temperature at the helium gas temperature of \(T_{He}=5\) eV. Figures 7 and 8 illustrate the energy transfer over time for electron temperatures of \(T_{e}=5\) eV and \(T_{e}=20\) eV, respectively, with \(T_{He}=5\) eV. Figure 4: Effective collision frequency between an electron and a helium atom. The results of this study are compared with data from [PERSON] et al. [45], [PERSON] et al. [46], [PERSON] et al. [22], [PERSON] et al. [24], and [PERSON] [26]. A distinct maximum at approximately 5 eV is observed, consistent with the experimental data from [PERSON] et al. [45], which has not been reported in previous theoretical studies. Figure 5: Energy transfer rate between electrons and helium atoms as a function of electron and helium temperatures. In Figure 6, the energy transfer rate shows a gradual increase with electron energy until it reaches a maximum near 5 eV. This figure shows that both [PERSON] data and ours exhibit a peak, as explained above. However, our data lie above the [PERSON] curve in the region of negative energy transfer rates. This discrepancy is explained by the fact that in [PERSON] model, the magnitude of the energy transfer rate increases more rapidly with rising electron temperature, as the collision frequency in his model increases with electron temperature (see Figure 4).