Charge collection efficiency of commercially available parallel-plate ionisation chambers in ultra-high dose-per-pulse electron beams

Objective. This investigation aims to experimentally determine the charge collection efficiency (CCE) of six commercially available parallel-plate ionisation chamber (PPIC) models in ultra-high dose-per-pulse (UHDPP) electron beams. Approach. The CCE of 22 PPICs has been measured in UHDPP electron beams at the National Metrology Institution of Germany (PTB). The CCE was determined for a dose per pulse (DPP) range between 0.1 and 6.4 Gy (pulse duration of 2.5 μs). The results obtained with the different PPICs were compared to evaluate the reproducibility, intra- and inter-model variation, and the performance of a CCE empirical model. Main results. The intra-model variation was, on average, 4.0%, which is more than three times the total combined relative standard uncertainty and was found to be greater at higher DPP (up to 20%). The inter-model variation for the PPIC with 2 mm electrode spacing, which was found to be, on average, 10%, was also significant compared to the relative uncertainty and the intra-model variation. The observed CCE variation could not be explained only by the expected deviation of the electrode spacing from the nominal value within the manufacturing tolerance. It should also be noted that a substantial polarity effect, between 0.914(5) and 1.201(3), was observed, and significant intra- and inter-model variation was observed on this effect. Significance. For research and pre-clinical study, the commercially available PPIC with a well-known CCE (directly measured for the specific chamber) and with a small electrode spacing could be used for relative and absolute dosimetry with a lower-limit uncertainty of 1.6% (k = 1) in the best case. However, to use a PPIC as a secondary standard in UHDPP electron beams for clinical purposes would require new model development to reduce the ion recombination, the polarity effect, and the total standard uncertainty on the dose measurement.


Introduction and purpose
FLASH radiotherapy dosimetry is known to be challenging (Schüller et al 2020, Romano et al 2022).Active dosimeters routinely used for conventional radiotherapy dosimetry, such as ionisation chambers and solid-state detectors, exhibit a nonlinear dose-response dependency.In the case of ionisation chambers, the deviation from dose-response linearity is due to a significant decrease in the charge collection efficiency (CCE).This decrease in CCE is caused by an increase in ion recombination, which is known not to behave as expected from the wellaccepted Boag model (Petersson et al 2017).Investigations in Ultra-high dose-per-pulse (UHDPP) electron beams, defined by a dose delivery greater than 0.6 Gy within a single pulse, can be viewed as an excellent opportunity to study the CCE of parallel-plate ionisation chambers (PPICs) and improve our understanding of ion recombination mechanisms.At the moment, different studies have shown that the CCE is primarily dependent on the dose per pulse (DPP) delivered, the polarising voltage (Petersson et al 2017) and the electrode spacing (gap) of a PPIC (Kranzer et al 2021).Other factors, such as the delivered dose pulse structure, i.e. the fluctuation of the instantaneous dose rate within the pulse, can have an impact (Paz-Martín et al 2022), but it is a secondary factor compared to the three others mentioned above.
So far, investigations in UHDPP electron beams using PPICs are performed with chambers that are not commercially available (Kranzer et al 2021, Gómez et al 2022) or are usually limited to the Advanced Markus (Petersson et al 2017, Bourgouin et al 2020, Liu et al 2023) and the Roos (McManus et al 2020, Paz-Martín et al 2022) models.Since these chambers have different electrode spacing, evaluating the inter-model variation for PPICs is therefore impossible.The publications are also usually limited to a low number of chambers tested.Most of the time, only one chamber per model is investigated, three at best (Petersson et al 2017), limiting the intra-model variation assessment.A recent publication by Liu et al (2023) investigated the response of four commercially available ion chambers and one prototype.They aimed to evaluate the response of ion chambers in electron FLASH radiotherapy beams with substantially different designs.Therefore, none of the models were similar, and only one chamber per model was investigated.
This experimental investigation aims to determine the CCE of a large number of commercially available PPICs in the DPP range between 0.1 Gy and about 6.4 Gy in 20 MeV UHDPP electron beams to evaluate intraand inter-model variation.The CCE in UHDPP electron beams cannot be derived from the two-voltage-analysis method, the theoretical Boag models including free electrons (Boag et al 1996), or the Burns and McEwen (1998) semiempirical equation (Petersson et al 2017, Paz-Martín et al 2022).Therefore, the CCE is determined using the ion chamber's absorbed dose to water equation, while the absorbed dose to water is known from alanine dosimetry.To determine the intra-and inter-model variation, 22 PPICs from six commercially available models have been characterized in PTB's UHDPP electron beams for an extensive range of polarizing voltage.The second aim of this investigation is to evaluate the use of a generic empirical model to determine the CCE and its impact on dosimetry performed with PPIC.This investigation also aims to provide the scientific community with a large dataset of CCE measurements (data available on the Zenodo server; http://10.5281/zenodo.8164568), which will hopefully improve our understanding of the ion recombination mechanisms.

Charge collection efficiency determination
The CCE is the reciprocal of the ion recombination correction factor, k , ion and is determined using the following equation: where N D,w,Q 0 is the specific chamber calibration coefficient determined in the reference beam of quality Q , 0 and D w,Q is the absorbed dose-to-water delivered by a beam of quality Q at the reference point of measurement.
The measurand, M , raw is the uncorrected charge collected from the PPIC under investigation and M leak. is the charge leakage measured while the PPIC is connected to the reading system without being irradiated.The following k are correction factors and will be described below.
The beam quality correction factor, k , Q,Q 0 accounts for the different responses of the PPIC between the reference beam of quality Q , 0 usually 60 Co, and the beam of quality Q used during measurement.The correction factor depends primarily on the water-to-air stopping power ratio and, therefore, on the electron energy spectrum at the measurement reference point.This correction factor can be evaluated using the beam quality specifier R 50 and fitting equations found in literature, e.g.Muir et al (2012), or in the International Code of Practices, e.g. International Atomic Energy Agency (2000).Although the correction factor is dose rate independent, the primary data used to obtain the k Q,Q 0 in literature are based on results obtained in reference conditions, i.e. in a radiation field with well-defined flatness, geometry and source-to-surface distance of 100 cm.The UHDPP electron beams used in this investigation do not fulfil the requirement to be considered a reference beam using the conventional definition.Also, the PTB's reference UHDPP electron beams are known to have similar beam quality specifiers, within 0.7 mm, while having different electron energy spectra, an average difference of 1 MeV (Bourgouin et al 2022b).For these reasons, it was decided to evaluate the required k Q,Q 0 using Monte Carlo simulations within this investigation (details in section 2.6).
The temperature-pressure correction factor, k , TP is applied to correct for the change of the air mass contained in the sensitive volume of the PPIC due to the variation of these influence magnitudes from their reference condition value, namely 20 °C and 101.325 kPa.The correction factor is based on the ideal gas law and is obtained following the code of practice TRS-398 (International Atomic Energy Agency 2000).All temperature and pressure recorded during the presented measurements lay within the intervals (18.6 ± 1.0) °C and (100.5 ± 1.3) kPa, respectively, whose centres correspond to the average values.
The electrometer calibration factor, k , elec accounts for any difference between the absolute electrical charge collected and the readout obtained from the electrometer.This calibration factor was evaluated using a reference standard constant current source.Independently of the nonlinear response of the PPICs with the DPP, the electrometer readout may deviate from a linear response with respect to the actual electrical charge due to the high instantaneous electrical current input from the detector, which potentially exceeds the maximum electrometer dynamical range.An electrometer correction factor, k , elec ¢ was used to account for this effect.More details on this correction factor are provided in section 2.2.
The polarity correction factor, k , pol accounts for the change in raw signal magnitude when the PPIC polarising voltage is reversed.Since each CCE measurement during this investigation was performed for both polarities, this effect is taken into account in the analysis by using the average values obtained from both polarities.The correction factor presented in the results section is calculated using the following equation: where Q + are the positive charges collected and Q -is the negative charges collected when the polarity between the electrodes is reversed.The radial beam profile correction factor, k , rp is a correction factor to account for the volume averaging effect from the radial beam profile variation (McEwen et al 2014).This correction factor was evaluated from Monte Carlo simulations and relative measurements.The field correction factor, k , field has been added to the equation to account for the difference between the detector and the reference point of measurement position, i.e. at a depth z ref in the irradiation field centre.On average, this correction factor equals 1.000 as the PPICs were positioned at z ref and profile measurement was performed with all PPICs before the CCE measurement.As the beam centre is known to move slightly between the linac setting to enable different DPP (Bourgouin et al 2022b), an uncertainty of 1.5 mm on the beam centre was considered and propagated to the standard uncertainty (Type-B) of k .
rp The uncertainty of this correction factor was also based on a depth positioning uncertainty of 0.5 mm.The standard uncertainties reported here follow the recommendations of the Guide to the Expression of Uncertainty in Measurements (GUM-2008(GUM- 2008)).The number in brackets following a reported value throughout the text represents the standard uncertainty (k = 1) on the corresponding digits.

Ionisation chamber and reading system
During this investigation, six different models of commercially available PPIC were used, as listed in table 1.The table also provides some PPIC parameters, such as the nominal gap, i.e. the electrode spacing, which defines the collecting air volume thickness of the chamber.At least two chambers were investigated for the different models, and up to six were used.The specific PPIC calibration factor, N , D,w,Q 0 was determined for each PPIC listed in table 1 at the reference 60 Co beam of the National Metrology Institution of Germany, the Physikalisch-Technische Bundesanstalt (PTB), with a relative standard uncertainty of 0.25% (k = 1).
The PPICs were connected to a Keithley Model 616 electrometer.As mentioned in the above section, the electrometer readout may deviate from a linear response with respect to the actual electrical charge (or current).This deviation could lead to an erroneous readout providing the wrong raw measurement and, therefore, an incorrect determination of CCE.This issue is highly dependent on the electrometer manufacturer and model.To keep the input voltage below the maximum linear response range of the electrometer, a capacitor can be connected in parallel to the electrometer input.The capacitor size required is dependent on the signal measured and the range used.Unfortunately, at the initial moment of the measurement campaigns, this problem remained unnoticed as the nonlinear response from the PPICs was already expected.As the measurement campaigns involved a vast number of measurements, including the CCE measurements, beam characterisation and absolute dose-to-water calibration, it was decided to use a correction factor, k , elec ¢ in the analysis instead of repeating the measurements.However, using a capacitor during measurements is recommended rather than a correction factor.
The electrometer's nonlinear response depends on the PPIC current pulse shape and height.The required capacitor size to reduce the input voltage to the electrometer can be calculated if the input resistance for the electrometer measurement ranges and the maximum voltage to ensure linear response are known.For a Keithley Model 616 electrometer, this information is available in the manufacturer user guide.The required capacitance value was evaluated to be 33 nF with a maximum range used of 10E+07.To validate that the capacitor size was sufficient, measurements using larger capacitors were performed.As the signal did not vary significantly, the measurement was performed with a 33 nF capacitor in parallel to the electrometer input.The correction factor was determined based on measurements with four PPIC models (PPC05, Advanced Markus, PPC40 and Roos) at two different polarising voltages (±100 V and ±450 V) for both polarities and a range of DPP (same as used during the investigation).The correction factor is defined as the ratio between the measurement with and without the capacitor.The results were fitted with second-degree polynomial curves.The dependent variable is the correction factor k elec ¢ and the independent variable is the raw charge measurement without a capacitor.Three polynomial curves were fitted, one representing the average and the other two representing the distribution's upper and lower limits.The correction factor and its associated uncertainty are then obtained, during the analysis of the CCE, using the three polynomial curves.

Radiation field and beam setup
The PPIC's CCE measurements were performed at the PTB Metrological Electron Accelerator Facility (Schüller et al 2019).The presented results of this investigation were obtained from two separate measurement campaigns.
One was performed before establishing reference UHDPP electron beams and one after, using both reference UHDPP electron beams described by Bourgouin et al (2022b).For both measurement campaigns, the pulse duration was 2.5(1) μs, and the pulse rate frequency was 5 Hz.The beam setup used during the first investigation was similar to the reference UHDPP electron beam using a source-to-surface distance (SSD) of 70 cm, with the exception that a scattering foil, 0.52(7) mm thick, was positioned at the exit window of the linac.The scattering foil consists of an aluminium alloy plate with a thin layer of scintillating material used for monitoring purposes through video surveillance from the control room.The first measurement campaign was performed on a lower number of PPICs.The different beam setup parameters and beam characteristics in water are provided in table 2. The reference depth, as defined in TRS-398 (International Atomic Energy Agency 2000) and the beam size in water at the z ref are defined by the full width at half maximum (FWHM) of the Gaussian radial beam profile listed in table 2. The method to determine the absorbed dose-to-water per pulse will be described in section 2.5.The SSD is defined as the distance between the linac 0.1 mm copper exit window and the entrance window of the water tank.

Measurement procedure
For both experimental campaigns, the depth-dose curve (PDD) and radial beam profile were determined previously to the CCE measurement.The polarising voltage was applied by an external high-voltage source which was changed automatically by the software controlling the measurement system.The list of polarising voltages was applied randomly, with magnitudes ranging between 75 and 450 V in the first measurement campaign and 25-450 V during the second.The evaluation of the CCE for a specific chamber at a specific DPP and polarising voltage was based on 100 measurements performed in a row, with one pulse being delivered per measurement.The measurements were performed in a PMMA water tank (30 × 30 × 30 cm 3 ).
For the second measurement campaign, the following protocol was used.A flashDiamond detector prototype (Kranzer et al 2022a) was used prior to CCE measurements with PPIC to perform relative measurement (Bourgouin et al 2022b) to verify that the linac output was stable through the measurement campaign.Once relative measurement was performed with the flashDiamond, a first PPIC was installed in the water phantom, and relative measurements were repeated for positioning purposes.Once the PPIC was at the reference point of measurement, the chamber readings were recorded for a range of polarising voltage and DPP.A second PPIC was installed in the water tank, and the procedure was repeated.Once measurements with the beam setup SSD70-00 were finished, the whole procedure was repeated for the beam setup SSD90-02.The beam setup using a scattering aluminium foil was always used at last in the day for radioprotection purposes (potential risk related to activation of the scattering plate).Three PPICs were measured at the beginning and the end of the second measurement campaign to verify reproducibility throughout the second measurement campaign.Also, the six PPICs used in the first measurement campaign were re-measured during the second.Finally, as one of the PPICs, the Advanced Markus SN2176, was provided by the Federal Institute of Metrology in Switzerland (METAS), it was decided to compare the CCE obtained during this investigation with the CCE they obtained in their reference UHDPP electron beams, which is calibrated using their primary standard, Fricke dosimeter (Vörös and Stucki 2007).Their reference UHDPP electron beam is a 15 MeV electron beam with a pulse rate repetition frequency of 1.5 Hz, and a pulse duration of 3.0 μs generated by a microtron accelerator from Scanditronix (Vislanda, Sweden).The DPP range achieved is between 0.1 and 0.87 Gy for a beam size of about 14 cm at the reference depth of 3.5 cm.

Monitoring system and absolute dose-to-water calibration
The PTB's UHDPP electron beams, generated by a dedicated research linac, are monitored by an in-flange integrating current transformer (ICT) (Bergoz, Saint-Genis-Pouilly, France) (Schüller et al 2017).The ICT signal was between 30 and about 230 nC.The statistical standard relative uncertainty is 0.1%, and the absolute standard uncertainty is 0.015 nC.To determine the dose-to-water at the point of reference, D , w,Q the ICT signal was calibrated against a series of alanine secondary standard measurements, which has been shown to be consistent with the PTB water calorimeter primary standard (Bourgouin et al 2023).During the second measurement campaign, two alanine calibrations for a range of DPP were obtained, one at the beginning and one at the end of the measurement campaign.In addition, three alanine dosimeter measurements were performed at a single DPP throughout the measurement campaign.The results of the ICT's absolute dose to water calibration of the second investigation, using the beam setup SSD70-00 and SSD90-02, are presented in Bourgouin et al (2022a).The combined relative standard uncertainty assigned to the dose determined using the alanine calibration was 0.85%.The same method was used to calibrate the beam setup SSD70-01Sc at the beginning of the first measurement campaign.

Monte Carlo simulations
Monte Carlo simulations were used during this investigation to determine the beam quality, k , Q,Q 0 and the radial beam profile, k , rp correction factors for the two reference UHDPP electron beam setups.All Monte Carlo simulations were performed using the application egs_chamber from EGSnrc software (Kawrakow et al 2000) (released version 2020).A summary of the Monte Carlo parameter details can be found in table 3.
The PPIC models were provided by Dr B Muir (Muir and Rogers 2014) for the chambers from IBA and PTW manufacturers.The PPICs model of the SNC350p was provided by Alissa et al (2023).To separate the effects of the non-uniform radial beam profile from the beam quality correction factor, a second cylinder of air with a radius of 0.1 cm was added in the centre of the sensitive volume.The radial beam profile correction factor, k , rp was defined as the ratio between the dose evaluated in the volume with a radius of 0.1 cm to the dose evaluated in the sensitive volume of the PPIC.For the beam quality correction factor, the dose deposited in the chamber is therefore defined by the dose deposited in the volume with a radius of 0.1 cm, centred in the sensitive volume of the PPIC, and extending to the same height.A systematic standard uncertainty (Type-B) of 0.7% (Muir and Rogers 2014) was assigned to the beam quality correction factor.The beam model of the research accelerator used in this investigation was presented and validated by Bourgouin et al (2022b).For the beam setup SSD70-01Sc, the same model was used as for the setup SSD70-00 with an additional scattering plate as described in section 2.3.The water tank was simulated as a 30 × 30 × 30 cm 3 water cube at the same SSD as during measurements.The absorbed dose to water at the measurement reference point was defined as the dose deposited in a cylindrical scoring volume of 0.1 cm thick by 0.1 cm radius centred at the reference depth.The reference 60 Co beam was simulated as a collimated 10 × 10 cm 2 square field at 100 cm from a point source.The energy fluence spectrum from Mora et al (1999), available with the EGSnrc distribution, was used.To mimic the water tank used during the measurement, a water cube of 30 × 30 × 30 cm 3 was simulated at 95 cm from the point source.For the 60 Co, the absorbed dose to water was calculated in a cylindrical scoring volume of 0.1 cm thick by 0.25 cm radius centred at the reference depth.

Charge collection efficiency model
As stated in the introduction, one of the aims of this investigation is to evaluate the use of a generic model to determine the CCE and its impact on absolute and relative dosimetry performed with commercially available PPIC.The model should be simple and reproduce the observed CCE over a range of DPP of 0.12-6.4Gy for PPIC with electrode spacing ranging between 0.6 and 2 mm.To determine the CCE, different models are available for dose rates greater than conventional, i.e.DPP > 0.1 cGy per pulse.Di Martino et al (2005) provided an analytical model based on the Boag theory, considering the free electron fractions (Boag et al 1996).This analytical model was, however, validated for a DPP range below the targeted range of this investigation.Also, the Boag theory has been shown to be invalid for the DPP range of interest in this investigation (Paz-Martín et al 2022).A numerical model has been developed by Paz-Martín et al (2022), which was validated against a Roos chamber for a range of polarising voltage and DPP similar to the one used in this investigation.However, this model is not currently publicly available.Petersson et al (2017) presented an empirical model based on the CCE measurement of three Advanced Markus chambers for three polarising voltages.The model has been shown to reproduce the CCE measured in a DPP ranging between 0.005 and 10 Gy, which is the range of interest for this investigation.Therefore, the empirical model, although not based on physics principles, seems well designed for the aim of the model for this investigation.However, this model is only tested against one PPIC model and does not consider the electrode spacing in the PPIC, which is known to affect the CCE (Kranzer et al 2022b).Therefore, it was decided to modify the empirical equation to include it.Also, the CCE should be proportional to the inverse ratio between the polarising voltage applied to the chamber with the square of the PPIC's gap (Kranzer et al 2022c).Therefore, the generic empirical model that was tested is the following: where DPP is the absorbed dose-to-water per pulse [mGy], U is the polarising voltage across the chamber [V], G is the PPIC's gap between the electrodes [mm].The fitting parameters , µ , b g and d will be presented in the results section.The variable M¢ is the product of the raw measurement with the PPIC calibration coefficient and the electrometer correction factors and is defined as:

Correction factors and uncertainty budget
The beam quality correction factors, k , Q,Q 0 used during this investigation are presented in table 4, along with the radial beam profile correction factor, k .
rp The radial beam profile correction factors were compared to the value derived from beam profile measurement and the stated PPIC's radius.The simulated values were all in agreement, within k = 2, with values derived from measurements.The average difference between the beam quality correction factors obtained for both reference UHDPP electron beams (SSD90-02 and SSD70-00) is 0.2%.This value agrees with the change in the water-to-air Spencer-Attix stopping power ratio (table 7.V of International Atomic Energy Agency 2000) considering the average difference of 1 MeV between the two beams (table 3

of Bourgouin et al 2022b).
The polarity correction factor evaluated for the difference PPIC investigation is presented in figure 1 for a nominal polarising voltage of ±300 V.The values presented are the results for one PPIC for each model in the left panel and all the Advanced Markus on the right panel.The polarity correction factor is substantial for DPP greater than 1 Gy per pulse, as much as 1.201(3).It can also be observed that the polarity effect can broadly vary between PPICs, even among the same model.The strongest correlation found with the measured polarity effect was with the DPP delivered (or the CCE) and the polarising voltage, which will be presented in section 3.5.
The electrometer calibration factors, k , elec obtained were found to be stable, with the calibration performed a year and a half earlier.The difference between the correction factors was used to estimate a contribution to uncertainty as Type-B.The correction factor, k , elec ¢ was determined for each measurement using the polynomial fit described in section 2.2.For the Advanced Markus, k elec ¢ was always below 1.0025 at ±300 V.For the PPC05, the correction factor was always below 1.015 at ±300 V.The correction factor could reach a value as large as 1.053 at ±300 V for the 2 mm gap PPIC models.This correction rapidly decreases with the DPP and is below 1.010 for all chambers for a DPP of less than 1 Gy.The uncertainty budget for determining the CCE is presented in table 5.The value presented is the average obtained for all PPIC at all DPP delivered at ±300 V.All components of equation (1) are presented except for the charge measurement, i.e.M M .raw leak.

( ) -
The uncertainty associated with the fluctuation of the PPIC's charge signal measured with the electrometer is implicitly included in the polarity effect uncertainty and, therefore, not included in table 5.

Reproducibility
The comparison between CCE results obtained at the PTB facility and the METAS facility for the Advanced Markus SN2176 are listed in table 6 and figure 2. As presented in table 6 and shown in the inset of figure 2, the results of CCE obtained at the PTB facility (blue) and the 'raw' results obtained at the METAS facility (orange) are not consistent with each other within the stated uncertainty.The results obtained at the METAS facility are systematically larger than those obtained at the PTB facility, and the difference is not constant through the range of DPP measured.Therefore, the source of the difference is not a systematic error.After a thorough review of the  analysis and the measurement conditions, it was found that the most significant difference between the two experiments was the pulse length and the average ambient temperature and pressure.At the METAS facility, the average temperature and pressure were 20.5 °C and 96.4 kPa, whereas, at the PTB facility, the ambient conditions were 18.6 °C and 100.5 kPa.The difference in temperature and pressure would most likely impact the ion mobility and, therefore, the CCE.It was decided to use the numerical model developed at the Universidad de Santiago (Paz-Martín et al et al 2022) to determine a correction factor, k , ion,TP for the CCE measured at the METAS facility as the temperature-pressure correction factor, k , TP does not correct for such effect.The correction factor was defined as the ratio between the CCE determined by numerical analysis for PTB's experimental conditions (pulse duration, temperature, and pressure) and the one from the METAS's experimental conditions.The k ion,TP correction factor was numerically determined for each DPP listed in table 6.The resulting CCE measured at the METAS facility and corrected is shown in figure 2 in yellow and correction factors applied to the METAS dataset are provided in table 7.As shown, the two sets of results are now consistent within k = 1 of the stated uncertainty for each set of measurements.
The impacts of the pulse duration and the ambient air conditions on the CCE were evaluated separately using the numerical model.The results of the simulations were that pressure was the primary factor in the difference in the CCE measured in both facilities.However, the pulse duration had an impact as well.About onefourth of the correction for the highest DPP was from the difference in the pulse duration.This finding is not inconsistent with the finding of Petersson et al (2017), who concluded that the total dose deposited in the beam was the most determinant parameter rather than the instantaneous dose rate within the pulse for Advanced Markus chambers.The change in the CCE from the pulse duration is below the uncorrelated measurement uncertainty and, therefore, cannot be significantly observed through experimental measurements.
Following the findings presented above, as the temperature and pressure fluctuated during the measurement campaigns and this investigation aimed to evaluate the intra-and inter-model variation, it was decided to correct all measurements to a single reference temperature and pressure.All the following CCE results presented have Table 5.The average relative standard uncertainty for the component of equation (1) and the combined uncertainty estimated to determine the CCE.The average is determined from every PPIC and DPP measured at ±300 V.The range is provided in the last row when applicable.

Relative standard uncertainty (%)
Type-B evaluation been corrected using a numerically determined k , ion,TP to the average temperature of 18.6 °C and 100.5 kPa.The typical reference ambient conditions of 20.0 °C and 101.325 kPa were not used as it is far from the experimental conditions.As the numerical model is not fully validated against a range of PPIC models and ambient conditions, keeping the correction k ion,TP small was preferred.The correction was, on average, of the order of 0.35%, and the largest correction was of the order of 0.88%.The CCE measurement will be presented along with the trace of the empirical fit model obtained from fitting equation (3) on the results presented.

Component of equation (1)
The CCE measurement reproducibility obtained at the PTB facility between the two measurement campaigns and within the second measurement campaign is presented in figure 3. The results obtained at ±300 V for PPICs measured in both campaigns are presented in the left-hand panel.For every PPIC, the average difference in the CCE obtained for the range of DPP is within k = 2 of the uncorrelated uncertainty, i.e. all elements listed in table 5 except for the specific calibration factor, N .D,w,Q 0 The same observation is found for the reproducibility within the second measurement campaign.The average difference for the range of DPP measured between the initial and duplicate measurements are all within k = 2 of the uncorrelated uncertainty, i.e. all elements listed in table 5 except for N , D,w,Q 0 k Q,Q 0 and D w,Q as they are obtained from the same set of alanine absolute calibrations.

Intra-model variation
The obtained CCE for the six Advanced Markus and the five PPC05 chambers are presented in figure 4. The average and largest (maximum deviation from the average CCE obtained at a specific DPP) intra-model variation observed for the different PPIC models is listed in table 8 along with the uncorrelated relative standard  As shown in figure 4 and table 8, the intra-model variation observed is significant compared to the uncorrelated uncertainty for all PPIC models and the level of reproducibility found during this investigation.The numerical model developed by Paz-Martín et al (2022) was used to verify if the fluctuation of the PPIC's gap within the manufacturing tolerance could explain the variation in the CCE measured.For all PPIC models used during this investigation, the fluctuation of the relative CCE with the thickness of the chamber was evaluated at 6 Gy per pulse for a pulse duration of 2.5 μs at a room temperature of 18.6 °C and an atmospheric pressure of 100.5 kPa.The results of these calculations are presented in table 8, along with the average and maximum intra-  Table 8.Sensitivity of the CCE, determined numerically, for the different PPIC models with the gap thickness at 6 Gy per pulse for a pulse duration of 2.5 μs at a room temperature of 18.6 °C, an atmospheric pressure of 100.5 kPa and a polarising voltage of 300 V along with the CCE relative intra-model variation observed in these conditions with the corresponding gap variation according to the sensitivity of the CCE.

PPIC model
Average model variation measured in the range between 5.5 Gy per pulse and 6.5 Gy per pulse and its associated gap predicted from the CCE sensitivity.
As shown in table 8, the sensitivity analysis from the numerical model and the measured intra-model variation at 6 Gy per pulse result in a considerable gap variation, exceeding the manufacturing tolerance by one or two orders of magnitude.Therefore, it can be concluded that the fluctuation in the CCE measured is not exclusively explained by the expected variation in the gap between PPIC of the same model, and a more significant unknown effect is the primary source of the observed intra-model variation.

Inter-model variation
The obtained CCE for all PPIC models at ±300 V is presented in figure 5.As shown in the figure and as expected from other publications (Kranzer et al 2021), the CCE increases with the decreases in the electrode spacing.On the right-hand panel of figure 5, the CCE for all PPIC models is presented when the ratio between the applied polarising voltage and the square of the electrode distance is about 125 V•mm −2 .As shown in figure 5(B), a similar CCE is found when the ratio of the applied polarising voltage and the square of the PPIC gap are similar, consistent with the finding of Kranzer et al (2022c).
Figure 5(A) shows a significant variation in the obtained CCE for the different PPIC models with an electrode spacing of 2 mm.The average inter-model variation obtained is 10%.Although a few chambers were used for some PPIC models in this investigation, the inter-model variation observed is significantly larger than the intra-model variation observed.As shown in table 8, the numerical model predicts that the CCE would vary by about 0.04% per μm at 6 Gy per pulse for the PPIC with a 2 mm gap.Therefore, considering that inter-model variation for these PPIC models at 6 Gy per pulse was evaluated to be, on average, ∼10% and the maximum deviation was 32%, the same conclusion as in the intra-model section can be made.The gap variation can partially explain the variation in the CCE measured between PPIC, but it would rather be a secondary effect.To study further the impact of the PPIC gap thickness on the CCE, measurement with PPICs with a range of electrode spacing around 2 mm (known with high precision and accuracy) should be performed.Also, it should be noted that the SNC350p and the Roos have the same air volume geometry (nominal gap and radius); therefore, the CCE should be consistent.The CCE measured for both models is consistent between 0.1 Gy and 1 Gy per pulse.However, for the DPP larger than 1 Gy, the CCE obtained with the SNC350p significantly differs from the CCE of the Roos chambers.

Polarising voltage
Figure 6 presents the effect of the polarising voltage on both the CCE and the polarity effect for the PPC05 SN1178.The results of only one PPIC chamber are presented as the same behaviour was observed for all models and chambers.As expected, the CCE decreases with the polarising voltage, which is explained by the increase in the free electrons' collecting time, increasing the probability of recombining.For the polarity effect, the strong correlation between the polarity effect with the magnitude of the polarising voltage suggests that the electrical field has an important effect on the polarity effect.

Empirical fit results and relative measurement
As seen in figures 2-6, the lines shown, derived from equation (3) of the empirical fit models, reproduce the CCE curvature observed in most PPIC models.However, for PPC40 (figure 3) and SNC350p, the fit appears to have a greater curvature than the measured one.Table 9 lists the average and maximum deviations between the measured CCE and the empirical model for a ±300 V polarity.
The relative measurements performed with commercially available PPICs, if not corrected for ion recombination, can result in significant errors in determining the depth of R 50 and the beam size (FWHM).In both beam setups, SSD70-00 and SSD90-02, the depth of R 50 is consistently overestimated by 4-20 mm, and the FWHM by 4-80 mm.Results are significantly improved by applying ion recombination correction factors.The depth of R 50 is evaluated to be within −1.3 to 6 mm of the expected value, and the FWHM is within −8 to 12 mm.These results are obtained when correction factors are derived from the empirical model fitted using measurements from all PPICs of the same model.Further improvements are observed when ion recombination is assessed using a third-order polynomial fit for each PPIC, as opposed to a single equation generated for a PPIC model.For both beam setups, SSD70-00 and SSD90-02, the depth of R 50 is within 0.2-4 mm of the expected value.The FWHM is found to be overestimated by 3-9 mm maximum.This finding suggests that CCE can be evaluated based on a fitted equation; however, this equation must be directly fitted to measurements taken with the specific PPIC, even when the objective is to correct relative measurements.

Conclusions
For this investigation, 22 PPICs were used to characterize the charge collection efficiency across a range of DPP values, from 0.1 to 6.4 Gy, using 20 MeV UHDPP electron beams.The intra-and inter-model variation of the CCE obtained has shown to be significantly larger than the total combined standard uncertainty, 1.3%, and reproducibility achieved during this investigation.The CCE variation within and among PPIC models could not be explained by the expected variation of electrode spacing within the manufacturing tolerance.
The CCE of one Advanced Markus chamber used in this investigation was also evaluated at the Swiss Federal Institute of Metrology.The comparison in the results revealed evidence of a dependence of CCE on ambient temperature and atmospheric pressure during measurements.At a dose rate of approximately 0.8 Gy per pulse, the CCE exhibited a variation of approximately 5% due to a difference of 1.9 °C in temperature and 4.1 kPa in atmospheric pressure.These findings emphasise the necessity of systematically reporting ambient conditions in future investigations using PPIC in UHDPP electron beams.To mitigate the impact of this dependency, a novel  Significant polarity effects, reaching a correction value of 1.201(3), were observed and reported during this investigation.It was also observed that the polarity effect broadly varies between PPICs, even among the same model, although good reproducibility was found on average.The current theory does not explain this important polarity effect observed.It would be, therefore, recommended to report and correct for this effect systematically for investigation in UHDPP electron beams.These findings underscore the need to study further the polarity effect, which seems to depend on the DPP and the polarising voltage.
A generic empirical fit model based on measured data was obtained and used on relative measurement.The results indicated that, on average, measured depth of R 50 and the beam profile are within a few millimetres of the expected values.However, it is important to note that utilising a generic CCE empirical model would only serve as an approximate evaluation of beam parameters and would not yield the level of accuracy required for direct comparison with other detectors or Monte Carlo simulations.Given its inherent limitations in precision and reliability, absolute and relative dosimetry using an empirical model is strongly discouraged for clinical and research purposes.
In conclusion, the investigation indicates that commercially available parallel plate ionisation chambers are not suitable as secondary standards for determining the absolute absorbed dose to water in clinical settings for UHDPP electron beams.This conclusion is based on the total relative standard uncertainty achieved in the ion recombination correction factor, 1.3%.Even with potential improvements, such as using a primary standard for absolute dose-to-water calibration, achieving a total relative standard uncertainty below 1% would be challenging.For research and pre-clinical investigations, the available and validated secondary standards are not real-time detectors, can involve time-consuming procedures, and their relative standard uncertainties exceed 1% in UHDPP electron beams (Romano et al 2020).Therefore, parallel plate ionisation chambers with smaller electrode spacing could serve as suitable secondary standards in that context.However, it is crucial to evaluate the CCE of each chamber by measuring it against a primary or secondary standard dosimeter that has shown no dependence on dose per pulse.For relative measurements, other commercially available detectors offer better alternatives and require fewer corrections than parallel plate ionisation chambers in UHDPP electron beams.

Figure 2 .
Figure 2. The CCE of the Advanced Markus SN2176 measured at the PTB and the METAS facilities.Lines represent the results from the empirical fit models (equation (3)).

Figure 3 .
Figure 3.The CCE obtained at ±300 V for the PPC05-SN1551, Advanced Markus-SN1279, and the PPC40-SN1888 during both measurement campaigns (A), and in (B), the CCE obtained in the second campaign for the PPICs measured twice, namely the Advanced Markus-SN1279, PPC40-SN0162, and NACP02-SN20652.Lines represent the results from the empirical fit models (equation (3)).

Figure 4 .
Figure 4. CCE obtained at ±300 V for (A) the six Advanced Markus and (B) the five PPC05 PPIC used during this investigation.Lines represent the results from the empirical fit models (equation (3)).

Figure 6 .
Figure 6.Polarising voltage effect on the CCE (A) and the polarity correction factor (B).The results presented are for the PPC05 SN1178.The label of each set of data is the measured polarising voltage applied to the PPIC in [V].Lines represent the results from the empirical fit models (equation (3)).

Table 1 .
List of PPIC models used during this investigation with some physical properties, such as the gap between the two electrodes, i.e. the inner surface of the front window and the collector's upper surface.
A Bourgouin et al

Table 2 .
Beam parameters of the three UHDPP electron beams used during this investigation.
a Pure aluminium (99.99%).b Total thickness of the aluminium alloy plate with the scintillating layer.

Table 3 .
Alissa et al (2023)14)8) methods used in the Monte Carlo simulations(Sechopoulos et al 2018).Cobalt spectrum:Mora et al (1999)Ion chambers model:Muir and Rogers (2014)andAlissa et al (2023)Source descriptionNo phase-space files were used.Each input file included beam geometry and particle sources.Descriptions for the 60 Co and electron beams are provided in the text above is the energy deposited per unit fluence in the defined scoring volume as defined in the text above Number of historiesThe number of particles varied across simulations, ranging from 3.08E+06 to 3.53E+07 for Cobalt beams and from 5.0E+06 to 4.8E+07 for electron beams Statistical uncertaintyThe Type-A statistical uncertainty limit for all Monte Carlo simulations was set with the aim of achieving a final combined relative uncertainty 0.1% Statistical methodsDefault settings PostprocessingThe deposited energy is converted to dose by dividing it by mass.The air density used is 1.20478997E-03, and volume is calculated from input file geometry.Postprocessing for radial beam profile and beam quality correction factors is explained in the text above

Table 6 .
Results of the CCE obtained at the METAS and PTB facility of the Advanced Markus SN2176.

Table 7 .
The k ion,TP correction factors were applied to the METAS measurements to generate the yellow dataset in figure 2.

Table 9 .
The average and maximum relative deviation between the measured CCE and the expected value from the empirical model for an applied polarity of ±300 V. Phys.Med.Biol.68 (2023) 235002 A Bourgouin et al correction factor, k , ion,TP has been introduced.Further comprehensive studies are recommended to explore this relationship in controlled environments.