## 2 Materials and Methods
### Charge Recombination  
Ion recombination is a complex phenomenon that depends on the detector characteristics and on several properties of the beam: Linear energy transfer (LET), particle type, mode of radiation (continuous, pulsed or pulsed-swept), and fluence rate [23; 24; 25]. Different theoretical models were used to describe the collection efficiency in parallel-plate ICs for the two main types of recombination: the initial and the volume recombination. The initial one occurs between ions created in the same ionization track; it is independent of the ionization current but depends on the ionization density within the track. The second one is the volume recombination which takes place between ions originating from different ionization tracks and depends on the ionization current; thus, it depends on the beam fluence rate. Several theories were developed to describe the ion collection efficiency as a function of the applied voltage for a constant ionization rate [26; 27; 28; 29; 30]. For the volume recombination, in the limit of beam pulses of duration much larger than the charge collection time (continuous beams), [PERSON] and [PERSON] [29] derived the following relation for the detector collection efficiency (\(f\)), which is defined as the ratio of the charge collected at the electrodes (\(Q^{\prime}\)) and the ionization charge generated in the gas (\(Q\)) in the same time interval:  
\[f\ =\ \frac{Q^{\prime}}{Q}\ =\ \frac{2}{1+\sqrt{1+\frac{2}{3}\xi^{2}}} \tag{1}\]  
with  
\[\xi\ =\ \sqrt{\frac{\alpha}{ek_{1}k_{2}}}\frac{\sqrt{q}d^{2}}{V}\ =\ m\frac{\sqrt{q}d^{2}}{V}\ =\ \sqrt{5}\frac{d^{2}}{V}\,\quad m\ =\ \sqrt{\frac{\alpha}{ek_{1}k_{2}}} \tag{2}\]where the constant \(m\) is characteristic of the gas because \(k_{1}\) and \(k_{2}\) are the mobilities of positive and negative ions, \(\alpha\) is a gas-specific recombination coefficient, and e is the electric elementary charge. Additionally, \(\xi\) depends on the electrode gap width \(d\), the detector bias voltage \(V\) and the rate of volumetric ionization charge produced in the gas \(q\) (C\(\cdot\)cm\({}^{-3}\cdot\)s\({}^{-1}\)). The quantity \(S=m^{2}q\) thus depends linearly on the rate of volumetric ionization. For beam pulse duration much smaller than the charge collection time and for an interval between pulses larger than the same quantity, [PERSON] [30] derived the following relation for the detector efficiency (\(f\)):  
\[f\ =\ \frac{1}{u}ln(1+u)\ \ \text{with}\ u\ =\ \frac{\alpha/e}{(\kappa_{1}+ \kappa_{2})}\bigg{(}\frac{rd^{2}}{V}\bigg{)}\ =\ \mu r\bigg{(}\frac{d^{2}}{V}\bigg{)} \tag{3}\]  
where \(\mu\) is a gas-specific constant for pulsed dose rate, involving the recombination coefficient and the mobilities, and r is the charge ionization density (C\(\cdot\)cm\({}^{-3}\)) produced by each pulse. Both these models include a dependence of the efficiency on the ratio \(V/d^{2}\), which was multiplied by a parameter depending either on the rate of ionization (\(S^{1/2}\)) for continuous beams or on the total ionization per pulse (\(\mu r\)) for pulsed beams. It should be observed that these theories refer to a uniform irradiation field covering the sensitive area of the detector, leading to  
In order to determine the calibration curve, measurements with the MGD need to be performed by varying beam intensities and detector bias voltages to change the rate of recombination. The corresponding collection efficiencies can either be determined using a reference detector (e.g., a Faraday cup) or, as in the present study, using the analytical models described in the previous section to fit the data, as explained in the following. For a given beam current, the counts of the three ICs corresponding to the same number of ions were acquired varying the bias voltage and were plotted as a function of V/d\({}^{2}\). To improve the precision, the average over repeated measurements at the same voltage was used, and the uncertainty was evaluated as \(\sigma/\sqrt{\text{N}}\).