# Extension of the Discrete Electron Transport Capabilities of the Geant4-DNA Toolkit to MeV Energies
## 2 Materials and Methods
### Stopping Power (SP) and Range (R)  
Knowledge of the inelastic cross section for all ionization shells and excitation levels allows the calculation of the (electronic) SP, i.e., the mean energy loss per pathlength, from first principles, using the expression:  
\[\mathrm{SP}(T)=-\frac{dT}{dx}=N\sum_{j}^{ioniz.}\int_{E_{min,j}}^{E_{max,j}} \frac{d\sigma_{ioniz.}^{(j)}(E;T)}{dE}dE+N\sum_{k}^{excit.}E_{k}\sigma_{excit. }^{(k)}(T). \tag{14}\]  
In the present work, depending on the electron incident energy (see Figure 1), different corrections enter into Equation (14) via the expressions for \(d\sigma_{ioniz.}^{(j)}/dE\) and \(\sigma_{excit.}^{(k)}\) MCTS simulations of SP are practically equivalent to Equation (14) as long as the distance traveled (\(dx\)) by the particle is sufficiently small so that \(dT\ll T\). At sufficiently high particle energies (e.g., electrons above \(\sim\)1 keV), a convenient approximation to Equation (14) is offered by the Bethe SP formula which is also used by ICRU [59]. Importantly, the Bethe formula allows the calculation of the (electronic) SP without explicit knowledge of the inelastic cross sections, i.e., the \(d\sigma_{ioniz.}^{(j)}/dE\) and \(\sigma_{excit.}^{(k)}\) (see Equation (14)). Instead, the only non-trivial quantity in the Bethe SP formula is the so-called \(I\)-value of the medium which depends upon the ELF at the limit of \(q\to 0\). The latter quantity can be both measured and calculated. On the basis of such data, a reasonably accurate \(I\)-value for liquid water has been recommended by ICRU equal to 78 eV [58] and used to calculate SP values via Bethe'sformula with an uncertainty of 0.5-1% for \(T>100\) keV, 1-1.5% for \(100\geq T\geq 10\) keV, and 1.5-5% for \(10>T\geq 1\) keV. Due to the approximations used in [PERSON]'s formula, uncertainties increase rapidly at lower energies and, therefore, ICRU does not provide SP values below 1 keV.  
Another important quantity associated solely with the inelastic cross sections is the average pathlength traveled by the particle, also called the continuous slowing down approximation (CSDA) range (hereafter simply range or \(R_{CSDA}\)), which is defined by the following:  
\[R_{CSDA}(T)=\int_{T_{final}}^{T_{initial}}\frac{1}{\mathrm{SP}(T)}dT. \tag{15}\]  
MCTS simulations of the average pathlength of the particle, as it slows down from \(T_{initial}\) to \(T_{final}\), are practically equivalent to the CSDA range of Equation (15); although energy-loss straggling, which is automatically accounted for in the MCTS simulations, may cause Equation (15) to be somewhat larger than the MCTS values at low particle energies. Due to the limitations of the Bethe SP formula below 1 keV (discussed above), ICRU calculates the CSDA range by using Equation (15) only down to \(T_{final}\) = 1 keV and the remaining range is estimated very roughly using a linear interpolation of SP from 1 keV down to 0 eV, i.e., \(R_{\mathrm{ICRU}}=R_{\mathrm{c}\mathrm{a}\mathrm{d}\mathrm{a}}(T_{initial} \to 1\ \mathrm{keV})+R_{\mathrm{linear\ SP}}(T_{1\ \mathrm{keV}}\to 0)\) where \(R_{\mathrm{linear\ SP}}(T_{1\ \mathrm{keV}}\to 0)=\frac{1}{2}\frac{1\ \mathrm{keV}}{SP(T-1\ \mathrm{keV})}\).