# Extension of the Discrete Electron Transport Capabilities of the Geant4-DNA Toolkit to MeV Energies
## 2 Materials and Methods
### Implementation of the Plane Wave Born Approximation (PWBA)  
The analytic representation of the dielectric function \(\varepsilon(E,q)=\varepsilon_{1}(E,q)+i\varepsilon_{2}(E,q)\), with \(E\) being the energy transfer and \(q\) the momentum transfer, is based on the same methodology used in DNA-Opt2 and DNA-Opt4. Specifically, a sum of Drude-type functions is used to parameterize the experimental optical data (\(q=0\)) for \(\varepsilon_{2}(E,q=0)\) and \(\varepsilon_{1}(E,q=0)\) which are then extended to non-zero \(q\) (for arbitrary \(E\)) by analytic dispersion relations implemented into the Drude coefficients, which ensure the proper asymptotic limits at both  
Figure 1: Schematic of the different correction terms implemented into the present inelastic model (DNA-Opt4X). To reduce the computational burden, corrections are turned “ON/OFF” only within the energy range (regime) in which they have an effect larger/smaller than 1% to the electronic stopping power (SP). small- and large-\(q\). The contribution of individual ionizations and excitation transitions to \(\epsilon_{2}(E,q)\) is obtained by an in-house deconvolution algorithm [43; 44] for \(\epsilon_{2}(E,q)\), which introduces threshold energies (e.g., shell binding energies), conserves the sum-rules, and allows \(\epsilon_{1}(E,q)\) to be obtained analytically by the Kramers-Kronig relation. Then, the complete ELF along with its partitioning to ionizations and excitations reads as follows:  
\[\text{ELF}=\text{Im}\bigg{[}-\frac{1}{\epsilon(E,q)}\bigg{]}=\sum\limits_{j}^{ ioniz}\frac{\epsilon_{2}^{(j)}(E,q)}{\frac{\epsilon_{1}^{2}(E,q)+\epsilon_{2}^{2}(E,q)}{ \text{ELF}_{simi}}}+\sum\limits_{k}^{excit}\underbrace{\frac{\epsilon_{2}^{(k )}(E,q)}{\epsilon_{1}^{2}(E,q)+\epsilon_{2}^{2}(E,q)}}_{\text{ELF}_{excit}}, \tag{1}\]  
where \(j\), \(k\) correspond to the ionization and excitation transitions, respectively, which include four ionization shells (\(j=1,2,\ldots 4\)) and five discrete excitation levels (\(k=1,2,\ldots,5\)) as described elsewhere [43; 44; 50]. These outer (or valence) inelastic channels are expected to exhibit condensed-phase properties (e.g., long-range screening) and, therefore, most appropriately treated within the present dielectric-based ELF model of Equation (1). Within the PWBA all relevant physical quantities that are needed as input to MCTS simulations can be calculated from the ELF of Equation (1) by suitable integrations over the \(E\) and \(q\) variables. The partitioning of ELF described in Equation (1) is particularly important to MCTS simulations because it allows the calculation of inelastic cross sections (both differential and integral) for the individual ionization shells and excitation levels. For example, the inelastic energy-transfer differential cross section (DCS), which is the most fundamental quantity in MCTS codes since it determines the single-collision energy-loss spectrum of the incident charged particle, is obtained by an integration of the ELF of Equation (1) over \(q\):  
\[ \begin{array}{rl}\frac{d\sigma_{PWBA}(E,T)}{dE}=&\sum\limits_{j}^{ ioniz}\frac{\omega_{PWBA}^{(j)}(E,T)}{dE}+\sum\limits_{k}^{excit}\frac{ \omega_{PWBA}^{(k)}(E,T)}{dE}=\\ &=&\frac{1}{\pi\alpha_{0}\text{NT}}\Bigg{\{}\sum\limits_{j}^{ ioniz}\int_{q_{min}}^{q_{max}}\text{ELF}_{ioniz}^{(j)}(E,q)\frac{dq}{q} \\ &+&\sum\limits_{k}^{excit}\int_{q_{min}}^{q_{max}}\text{ELF}_{excit}^{(k)}(E,q )\frac{dq}{q}\bigg{\}},\end{array}  \tag{2}\]  
where \(\alpha_{0}=5.29\times 10^{-11}\) m is the Bohr radius, \(N=33.4\times 10^{27}\) m\({}^{-3}\) is the density of water molecules in liquid water, and the limits of the momentum transfer (\(q\)) in the non-relativistic case are  
\[q_{min/max}=\sqrt{2m}(\sqrt{T}\pm\sqrt{T-E}). \tag{3}\]  
Subsequent integration of the DCS of Equation (2) over \(E\) yields the total (integral) cross section (TCS) for inelastic collisions:  
\[