# Extension of the Discrete Electron Transport Capabilities of the Geant4-DNA Toolkit to MeV Energies
## 2 Materials and Methods
### Relativistic Corrections  
In the context of the Relativistic Plane Wave Born Approximation (RPWBA), the DCS is commonly written as the sum of a longitudinal and transverse term:  
\[\frac{d\sigma_{\text{RPWBA}}(E;T)}{dE}=\frac{d\sigma_{\text{ Longitudinal}}(E;T)}{dE}+\frac{d\sigma_{\text{Transverse}}(E;T)}{dE}. \tag{8}\]  
An in-depth discussion of RPWBA calculations for condensed matter systems is provided by [PERSON] et al. [57]. Similar to the non-relativistic limit, the longitudinal term in Equation (8) is determined by an integration of the ELF over \(q\). However, due to relativistic kinematics, the final expression differs from Equation (2) and reads as follows:  
\[ \begin{split}&\frac{d\sigma_{\text{Longitudinal}}(E;T)}{dE}=\sum \limits_{j,k}^{ioniz,excit}\frac{d\sigma_{\text{Longitudinal}}^{(j,k)}(E;T)}{dE} \\ &=\frac{1}{\pi a_{0}Nmc^{2}\beta^{2}}\Bigg{\{}\sum\limits_{j,k}^{ ioniz,excit}\int_{q_{min,rel}}^{q_{max,rel}}\text{ELF}^{(j,k)}(E,q)\frac{dq}{q} \Bigg{\}}\\ &=\frac{1}{\pi a_{0}Nmc^{2}\beta^{2}}\int_{q_{min,rel}}^{q_{max, rel}}\left(\frac{c^{2}q}{\sqrt{c^{2}q+(mc^{2})^{2}}}\right)\frac{1+\frac{Q(q)}{mc^{2}}}{1+ \frac{Q(q)}{2 mc^{2}}}ELF(E,q)\frac{dq}{Q(q)},\end{split}  \tag{9}\]  
where \(Q=\sqrt{c^{2}q^{2}-(mc^{2})^{2}}-mc^{2}\) is the relativistic free-recoil energy (with \(c\) the speed of light) and \(\beta^{2}=\frac{v^{2}}{c^{2}}=1-\frac{1}{\sqrt{\left(\frac{T}{mc^{2}}+1\right)}}\). The limits of the \(q\)-integration at relativistic energies are as follows:  
\[q_{max,rel/min,rel}=\frac{1}{c}\bigg{[}\sqrt{T(T+2 mc^{2})}\pm\sqrt{(T-E)(T-E+2 mc^{2})}\bigg{]}. \tag{10}\]  
Following the criterion discussed above (see Figure 1), the longitudinal term, Equation (9), replaces Equation (2) in the energy Regimes II, III, and IV. Note that in Regime II the Mott-Coulomb corrections are applied to the Longitudinal term of the RPWBA, Equation (9), which should replace the \"PWBA\" term in Equations (5)-(7). For the transverse term in Equation (8), we follow common practice and use [PERSON]'s small angle scattering approximation [PERSON] (1961). This approximation makes the transverse term depend only on the optical limit of ELF as follows:  
\[ \begin{split}&\frac{d\sigma_{\text{Transverse}}(E;T)}{dE}=\sum \limits_{j,k}^{ioniz,excit}\frac{d\sigma_{\text{Transverse}}^{(j,k)}(E;T)}{dE} \\ &=\frac{1}{\pi a_{0}Nmc^{2}\beta^{2}}\Bigg{\{}\sum\limits_{j,k}^{ ioniz,excit}\text{ELF}^{(j,k)}(E,q=0)\Bigg{\}}\Big{[}\text{ln}\Big{(}\frac{1}{1- \beta^{2}}\Big{)}-\beta^{2}\Big{]}.\end{split}  \tag{11}\]  
Note that Equation (11) vanishes at the non-relativistic limit. Following the criterion discussed above (see Figure 1), the transverse term, Equation (11), is considered only in the energy Regimes III and IV. An additional relativistic correction that sets in at condensed media and energies above the rest-mass energy of the projectile is the Fermi density effect correction [PERSON] (1961). It is related to the weakening of the Coulomb force experienced by distant atoms due to the enhancement of screening that arises from length contraction. This correction affects only distant (\(q=0\)) transverse interactions. Note that the density effect correction is not included in the recent relativistic version of DNA-Opt4 (called DNA-Opt4 Rel) [PERSON] (1961). We have implemented the Fermi density effect correction to the DCS following the formalism of [PERSON] et al. [PERSON] et al. (2016), which yields the following expression:  
\[\frac{d\sigma_{\text{Transverse},\delta}(E;T)}{dE}=\delta_{F}\frac{d\sigma_{ \text{Transverse}}(E;T)}{dE}, \tag{12}\]  
where \(\delta_{F}\) is the density effect parameter. According to [PERSON] (1961); [PERSON] (1961), \(\delta_{F}\) is linked to parameters related to the optical oscillator strength or, equivalent, to the ELF(\(E,q=0\)). Inthe present work, we have used the analytical approach by [PERSON] et al. [61] with parameters related to liquid water medium [58]. \[\delta(X)=\left\{ \begin{matrix}4.6052X+\alpha\left(X_{1}-X\right)^{m}+C,\ X_{0}<X<X _{1}\\ 4.6052X+C,\ X>X_{1}\end{matrix}