## 2 Materials and Methods
### Analytic Calculation of \(y_{\text{D}}\)  
In the present work, the calculation of \(y_{\text{D}}\) relies upon the application of an analytical microdosimetry model proposed by [PERSON] and colleagues [28; 29], which has been exploited in space applications [31; 38; 39]. This model aims to estimate the distribution of deposited energy inside microscopic volumes, taking into account both direct and indirect events. The term 'direct event' refers to an ion that crosses the target and deposits an amount of energy inside of it, while the term 'indirect event' refers to the energy deposition in the target from \(\delta\) rays that are produced from an ion that misses the target. The imparted energy (\(\epsilon\)) from a particle that crosses the target, traversing a path length (\(l\)) inside of it,is described by a probability density function \(f(\varepsilon)\). This function can be expressed as a convolution of an energy loss straggling distribution, \(\mathrm{p}(\varepsilon,l)\), with a path length distribution, \(c(l)\)[29]:  
\[f(\varepsilon)=\int\mathrm{p}(\varepsilon,l)\;c(l)\;dl. \tag{12}\]  
If the particle's path is considered a straight line, \(c(l)\) expresses the chord length distribution, which is a geometrical parameter known for some simple geometries [40; 41; 42]. For spherical volumes of interest, with the diameter \(d\), the chord length distribution is given by:  
\[c(l){=2l/d^{2}}, \tag{13}\]  
while the distribution of energy loss straggling, in this model, is approximated by a log-normal distribution [29]:  
\[\mathrm{p}(\varepsilon,l)= \frac{1}{\sqrt{2}\pi\sigma_{\varepsilon}\varepsilon}exp\left[- \left(\frac{ln(\varepsilon)-\mu_{\varepsilon}}{\sqrt{2}\sigma_{\varepsilon} }\right)^{2}\right], \tag{14}\]  
where \(\mu_{\varepsilon}\) is the mean value of the distribution and \(\sigma_{\varepsilon}\) is the standard deviation which are related to the mean deposited energy to the target (\(\overline{\varepsilon}\)) and its relative variance (\(V_{\varepsilon}\)) according to the expressions [29]:  
\[\mu_{\varepsilon}=ln(\overline{\varepsilon})-0.5\sigma_{\varepsilon}^{2}, \tag{15}\]  
\[\sigma_{\varepsilon}=\sqrt{ln(1+V_{\varepsilon})}. \tag{16}\]  
In the case of direct events, the mean deposited energy and its relative variance can be expressed through Equations (6) and (7), respectively, as:  
\[\overline{\varepsilon}_{ion}{=f_{ion}}\overline{\mathrm{LET}}_{ion}\overline{ \mathrm{T}}, \tag{17}\]  
\[\mathrm{V}_{\varepsilon,ion}{=V_{l}}{+V_{\delta,ion}}, \tag{18}\]  
considering that, for monoenergetic proton beams, the LET fluctuations are negligible (\(V_{\mathrm{LET}}=0\)). The relative variance of an ion's path length is calculated by Equation (8) as:  
\[V_{l}=\frac{l_{D}}{\overline{l}}-1, \tag{19}\]  
where \(l_{D}\) is the weighted average chord length (for spherical targets, \(3d/4\)), \(\overline{l}\) is the mean chord length (for spherical targets, \(2d/3\)) [33], and the relative variance of energy loss straggling is given by Equation (9), where \(\delta_{2,ion}\) is calculated from [29]:  
\[\delta_{2,ion}{=A\;\Delta^{B}}. \tag{20}\]  
The parameter \(\Delta\) is the cut-off energy which represents the energy of secondary electrons with a range equal to the mean chord length (\(\overline{l}\)) of the target, and the parameters \(A\) and \(B\) are constants that depend on the type of the particle and the material [29]. The cut-off energy of secondary electrons in liquid water spheres with a diameter of 1 \(\upmu\)m was found by [PERSON] et al. [43] to be \(\Delta=5.562\) keV, while the parameters \(A=0.195\) and \(B=0.610\) were found empirically from [PERSON] et al. [27] by fitting Equation (20) to data derived from the Geant4-DNA code.