# Collision Frequency and Energy Transfer Rate in e-He Scattering
## 2 Methods
### Optical Potential  
At relatively low energies, the study of electron scattering on atoms becomes increasingly complex due to the necessity of accounting for multiparticle interactions. To address this challenge, the optical potential method simplifies the analysis by reducing electron scattering to a consideration of the pair interaction potential \(V_{\text{opt}}(r)\). This potential can be expressed as a sum of three key components: static, correlation-polarization, and exchange interactions:  
\[V_{\text{opt}}(r)=V_{S}(r)+V_{P}(r)+V_{\text{ex}}(r). \tag{6}\]The static potential \(V_{S}(r)\) represents the Coulomb interaction between an electron and the nucleus, as well as the other electrons of the target atom, characterized by its electron density \(\rho(r)\) and charge number \(Z\). When the electron density at position \(r^{\prime}\), denoted as \(\rho(r^{\prime})\), is derived by solving the Schrodinger equation, the potential is referred to as the Hartree-Fock potential:  
\[V_{HF}(r)=-\frac{Ze^{2}}{4\pi e_{0}r}+\frac{e^{2}}{4\pi e_{0}}\int\frac{\rho(r^ {\prime})}{|r-r^{\prime}|}d^{3}r^{\prime}. \tag{7}\]  
In scenarios where relativistic effects are significant, the Dirac equation can be employed, resulting in the Dirac-Hartree-Fock-Slater potential \(V_{DHFS}(r)\). For our analysis, we utilized the analytical form of the static potential \(V_{DHFS}(r)\) as presented by [PERSON] et al. [31]:  
\[V_{DHFS}(r)=-\frac{Z}{r}\sum_{i=1}^{N}A_{i}e^{-a_{i}r}, \tag{8}\]  
where \(A_{i}\) and \(\alpha_{i}\) are fitting parameters specific to the target atom, with their values for helium provided in Table 1.  
The incoming electrons induce a polarization of the charge cloud of the target atom, which leads to an induced dipole moment that, in turn, acts back on the electrons. The semi-empirical potential used as the polarization potential is presented in the work of [PERSON] et al. [32]:  
\[V_{P}(r,r_{c})=-\frac{e^{2}}{2r^{4}}\left(1-\exp\left(-\left(\frac{r}{r_{c}} \right)^{6}\right)\right), \tag{9}\]  
where \(r_{c}\) is the cutoff parameter, determined by fitting it to reproduce previously known experimental or theoretical data. For the e-He interaction, the cutoff parameter is \(r_{c}=0.526a_{0}\), and the polarizability of the atom is \(\alpha_{d}=1.38a_{0}^{3}\), with \(a_{0}\) being the Bohr radius. The cutoff function in Equation (9) reduces the potential at short distances (\(r<r_{c}\)), while restoring the asymptotic form \(-\frac{\alpha_{d}}{2r^{4}}\) for \(r>r_{c}\). This asymptotic form is derived within the framework of the adiabatic approximation, and the given equation accounts for non-adiabatic effects [33; 34]. When the incoming particle is an electron, we must also consider rearrangement collisions, where the incoming electron swaps place with an atomic electron. To account for exchange effects, a local exchange potential was used as described by [PERSON] et al. [35]. This potential is approximated in the context of a free electron gas [36]:  
\[V_{\text{ex}}^{r}(r)=V_{\text{ex}}^{M}[r,K(r)]=-\frac{2}{\pi}K_{F}(r)F(\eta), \tag{10}\]  
where the function \(F(\eta)\) is given by the following:  
\[F(\eta)=\frac{1}{2}+\frac{1-\eta^{2}}{4\eta}\ln\left|\frac{\eta+1}{\eta-1} \right|, \tag{11}\]  
and \(\eta\) is defined as the ratio of the momentum of the incoming electron to the maximum momentum corresponding to the surface of the Fermi sphere in momentum space: \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Z** & \(A_{1}\) & \(A_{2}\) & \(\alpha_{1}\) & \(\alpha_{2}\) \\ \hline
2 & \(-0.2259\) & \(1.2259\) & \(5.5272\) & \(2.3992\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters for Helium [31]. \[\eta=\frac{K(r)}{K_{F}(r)}, \tag{12}\]  
where the Fermi wave vector \(K_{F}(r)\) is defined as follows:  
\[K_{F}(r)=[3\pi^{2}\rho(r)]^{1/3}. \tag{13}\]  
The value of \(K(r)\) is chosen as given by [PERSON] et al.