Investigation of TL and OSL detectors in ultra-high dose rate electron beams

Objective. This work aims at investigating the response of various thermally stimulated luminescence detectors (TLDs) and optically stimulated luminescence detectors (OSLDs) for dosimetry of ultra-high dose rate electron beams. The study was driven by the challenges of dosimetry at ultra-high dose rates and the importance of dosimetry for FLASH radiotherapy and radiobiology experiments. Approach. Three types of TLDs (LiF:Mg,Ti; LiF:Mg,Cu,P; CaF2:Tm) and one type of OSLD (Al2O3:C) were irradiated in a 15 MeV electron beam with instantaneous dose rates in the (1–324) kGy s−1 range. Reference dosimetry was carried out with an integrating current transformer, which was calibrated in absorbed dose to water against a reference ionization chamber. Additionally, dose rate independent BeO OSLDs were employed as a reference. Beam non-uniformity was addressed using a matrix of TLDs/OSLDs. Main results. The investigated TLDs were shown to be dose rate independent within the experimental uncertainties, which take into account the uncertainty of the dosimetry protocol and the irradiation uncertainty. The relative deviation between the TLDs and the reference dose was lower than 4 % for all dose rates. A decreasing response with the dose rate was observed for Al2O3:C OSLDs, but still within 10 % from the reference dose. Significance. The precision of the investigated luminescence detectors make them suitable for dosimetry of ultra-high dose rate electron beams. Specifically, the dose rate independence of the TLDs can support the investigation of the beam uniformity as a function of the dose rate, which is one of the challenges of the employed beam. Al2O3:C OSLDs provided high precision measurements, but the decreasing response with the dose rate needs to be confirmed by additional experiments.


Introduction
The interest in radiotherapy using ultra-high dose rates (UHDR) has increased in recent years, as evident by the various studies involving electrons, photons, protons, and heavier ions (Montay-Gruel et al 2018, Beyreuther et al 2019, Tinganelli et al 2022. Compared to conventional radiotherapy (average dose rate ∼0.1 Gys −1 ), radiotherapy with average dose rates > 40 Gy s −1 has been shown to potentially reduce the side effects on the healthy tissue with the same tumor control (so-called FLASH effect) (Favaudon et al 2014. Experiments with different particle beams aim to investigate the biological and chemical processes inducing the FLASH effect. Nevertheless, challenges related to beam monitoring and dosimetry still need to be addressed, particularly for the translation to the clinics (Schüller et al 2020, Romano et al 2022. Ionization chambers, the reference detectors for conventional radiotherapy (IAEA 2000), show decreasing ion collection efficiencies with increasing ionization densities (Di Martino et al 2005. Several studies investigated possible strategies to overcome their limitations, due to their key role for the clinical implementation of FLASH radiotherapy (Christensen et al 2016, McManus et al 2020, Kranzer et al 2021, Gómez et al 2022. Radiochromic films have been shown to be dose rate independent in electron  and proton beams (Darafsheh et al 2020, Villoing et al 2022. Alanine detectors, which are used for radiation processing dosimetry (ISO 2013), have been demonstrated to be dose rate independent up to an instantaneous dose rate of 1 × 10 10 Gy s −1 (Kudoh et al 1997). Other detectors have also been investigated for UHDR dosimetry; an overview of their performances and limitations can be found in Romano et al (2022).
Among these detectors, thermoluminescence (TL) and optically stimulated luminescence (OSL) detectors (TLDs/OSLDs) emerge as candidates for UHDR dosimetry due to their widespread use in personal and medical dosimetry, as well as advantages such as wide dose linearity range, small sizes, low cost, high precision and accuracy when specific protocols are used (Izewska et al 2002, Kry et al 2020, Yukihara et al 2022. Theoretical studies indicate the possibility of dose rate effects in OSL and TL materials (Chen and Leung 2001), but experimental studies have shown the opposite. LiF:Mg,Ti TLDs and BeO OSLDs were demonstrated to be dose rate independent at extremely high instantaneous dose rates: up to 4 × 10 9 Gy s −1 for electron beams (Karsch et al 2012) and up to 6 × 10 5 Gy s −1 for photon beams (Zorloni et al 2020). LiF:Mg,Ti has been also used, together with alanine and radiochromic films, to support biological experiments for FLASH radiotherapy (Jorge et al 2019). Al 2 O 3 :C OSLDs and LiF:Mg,Ti TLDs have been found to be dose rate independent in proton beams with average dose rates up to 9000 Gy s −1 (Christensen et al 2021) and 4500 Gy s −1 (Motta et al 2023), respectively.
Only a few TL and OSL materials were investigated in UHDR beams relevant for FLASH radiotherapy. Nevertheless, other TL and OSL materials with properties similar to LiF:Mg,Ti and BeO are available and can be potentially suitable for UHDR dosimetry. Additionally, Al 2 O 3 :C OSLDs have not been investigated in UHDR electron beams.
Therefore, this work aims at extending the investigation of possible dose rate effects in UHDR electron beams to various luminescence materials, specifically to those that are mostly employed for personal and medical dosimetry. Moreover, we studied the precision of the developed dosimetry protocol applied to the investigated materials, as well as the uncertainties associated to the irradiation in the UHDR electron beam used. In this way, the precision of our study can be compared to data available in the literature. Three types of TLDs (LiF:Mg,Ti, LiF:Mg,Cu,P, CaF 2 :Tm) and one type of OSLD (Al 2 O 3 :C) were studied. Additionally, dose rate independent BeO OSLDs (Karsch et al 2012) were used to monitor the beam delivery. A matrix of TLDs/OSLDs was used to cope with the beam non-uniformity, which could not be avoided. Reference dosimetry was performed with an integrating current transformer calibrated in absorbed dose to water against reference ionization chambers. The response of the reference chamber at UHDR was corrected for ion recombination effects based on measurements with alanine detectors.

TL and OSL dosimeters
The investigated TL and OSL materials and the reference BeO OSLDs are listed in table 1 along with their sizes, manufacturer and commercial names. Al 2 O 3 :C OSLDs were prepared from the same detectors used in Luxel TM dosimeters (Landauer Inc., USA), consisting of a polyester film containing Al 2 O 3 :C powder (Bøtter-Jensen et al 2003). Before irradiation, the TLDs and OSLDs were reset by thermal treatment (annealing) or optical treatment (bleaching), according to the standard protocols summarized in table 1 (McKeever et al 1995, Yukihara and McKeever 2011). Muffle furnaces were used for the annealing (models F48020-33-80 and FB1310M-33, Thermo Scientific Thermolyne) and a green light-emitting diode (LED, 520 nm, 24 W m −2 , Advanced Illumination, USA) to bleach the Al 2 O 3 :C OSLDs.
Each investigated material was arranged in 4 × 6 matrices (∼17.5 mm × 26.5 mm) and packaged in a 3Dprinted detector holder made of polylactic acid (PLA, polyester with a density of 1.24 g cm −3 ), specifically designed to fit the experimental setup and minimize the detectors motion (figure 1). A single 3D-printed package contained only one type of TL or OSL material, plus four BeO OSLDs placed at the corners of the TL/ OSL matrix. A 0.5 mm thick PLA layer was used as a cover. The packages were sealed with glue to protect the detectors from water. Figure 1 shows that the 3D-printed holder was designed such that the TL/OSL matrix is not symmetric with respect to the beam axis, indicated by the red dot. With this configuration, we wanted to measure the beam non-uniformity in the vertical direction. Nevertheless, the matrix could not be extended further on the bottom part due to space limitation in the PMMA holder hosting the 3D-printed package.
After irradiation, the 3D-printed packages were handled under red light to prevent a light-induced loss of the OSLD signal. The TLDs and OSLDs readouts started one week after the irradiations with the electron beam. Due to practical constraints, the readouts of the detectors irradiated at 195 kGy s −1 started one day after the irradiation. (1 mm).
The readout was carried out with two available lexsyg smart automated TL/OSL readers (Freiberg Instruments GmbH, Freiberg, Germany), both equipped with a UV-VIS photomultiplier tube (model 9235QB, Electron Tubes Inc.), a 90 Sr/ 90 Y beta source, and various optical filters to match the emissions of all the investigated materials. The readers are identical, except for the available optical filters. The TL and OSL readout parameters, specific to each material, are listed in table 1, along with the irradiation time (t R ) using the built-in source utilized in the readout protocol, as explained below.
For each detector, the readout protocol consisted of the sequence: 1. Readout after irradiation to obtain the signal S.
2. Irradiation of the detector using the built-in beta source in the reader, for a duration t R .
3. Readout after the irradiation in the reader to obtain the signal S R .
The ratio S/S R was calculated for each TLD and OSLD. The described protocol was applied to the detectors irradiated in the UHDR electron beam, as well as for the calibration of the detectors using the reader built-in source. For TLDs, the obtained TL curves (steps 1 and 3) were fitted with single TL components (peaks) given by the first-order model (Randall andWilkins 1945, Horowitz andYossian 1995), as shown in figure 2. The area under specific components was chosen as the dosimetric signal: peaks 4 and 5 for LiF:Mg,Ti, peak 4 for LiF:Mg,Cu,P, and peaks 3 and 5 for CaF 2 :Tm.
For OSLDs, the Continuous-Wave OSL (CW-OSL) technique was employed for Al 2 O 3 :C and the Linear Modulation OSL (LM-OSL) for BeO. In the CW-OSL technique, the light stimulation intensity is constant, resulting in a signal that decays continuously ( figure 3(a)). In the LM-OSL technique (Bulur 1996), the light stimulation intensity increases linearly with time, resulting in a peak-like curve ( figure 3(b)). For the same dose, a reduced maximum intensity is obtained by LM-OSL, which prevents the photomultiplier tube saturation in case of intense signal at high doses. Therefore, given the available detection filters in the readers and the photomultiplier tube saturation threshold, we extended the BeO OSLDs dose range measurable by using LM-OSL.
In both CW-an LM-OSL, the signal was calculated as the sum of the OSL counts in the stimulation time interval after background subtraction. For CW-OSL, a constant background corresponding to the average of the last 10 points was considered. For LM-OSL, an increasing background with maximum value equal to the counts at 240s was considered. Examples of CW-and LM-OSL curves are shown in figure 3, as well as the dosimetric signal (gray area) and the subtracted background (black area).

Irradiations with electron beam at METAS
The irradiations were carried out with a Scanditronix microtron accelerator (Uppsala, Sweden) at the Swiss Federal Institute of Metrology (METAS). A 15 MeV electron beam with 3 μs pulse duration was delivered in all irradiation sessions. The pulse repetition frequency was varied between 1 Hz and 25 Hz, as indicated in table 2.
When dealing with UHDR beams, one must be careful to distinguish between average dose rate and instantaneous dose rate, as most of the studies are carried out with pulsed beams. The dose rates reported in this work are instantaneous dose rates, calculated as the ratio of the dose per pulse and the pulse duration.
In each irradiation session, one single dose rate in the (1-324) kGy s −1 range was delivered by changing the accelerator settings. The relevant dosimetric properties and the beam parameters are listed in table 2 for each irradiation session. The session at 1 kGy s −1 aimed at determining the calibration coefficients for each detector type, as described in section 2.2.3.
At 1 kGy s −1 , the beam uniformity is obtained through flattening filters, which are made of stacked disks of stainless steel and are located in the treatment head. To achieve higher instantaneous dose rates, namely higher doses per pulse, the scattering target, made of different material (gold and brass) and thickness (from 0.05 to 0.10 mm), is changed and the flattening filter is removed. In addition, the beam current is increased by removing a scattering foil inside the microtron (electron flag, used to reduce the beam current for conventional radiotherapy applications). The highest dose rate (∼350 kGy s −1 ) is achieved with a scattering target made of brass and gold (total thickness of 0.075 mm). The removal of the scattering and filtering elements allows higher dose rates to be achieved, but at the expense of the beam uniformity and slightly changed reference depth.
The irradiations were carried out in a 62.0 cm × 65.0 cm × 58.0 cm water phantom (WP 700, Wellhöfer, Würzburg, Germany) located in front of the treatment head. The source-to-surface distance was 100.0 cm with a field size of 15.0 cm × 15.0 cm at the phantom surface. This setup is used for the calibration of ionization chambers for radiotherapy in electron beams, according to the TRS-398 Code of Practice (IAEA 2000). The 3Dprinted package containing the TL/OSL matrix and the monitor BeO OSLDs was inserted into a PMMA holder, with 0.8 mm front and back plate. The PMMA holder was immersed in the water phantom and aligned (vertically and horizontally) such that the center of the 3D-printed package, indicated by the red dot in figure 1, coincided with the beam axis. The lateral position of the beam axis was assumed as the reference position for all dose rates, both for the TL/OSL package positioning and for the reference ionization chambers alignment.
Due to the different scattering targets used to modify the dose rate, the electron beam depth dose curve in water varies with the dose rate. From the definition of the reference depth in water z ref (Almond et al 1999, IAEA 2000, a variation in the electron beam depth dose curve results in a variation of z ref . In our experiment, the depth in water was defined as the distance between the water phantom surface and the upstream surface of the detectors. Table 2 lists the chosen values obtained from simulations (PTB 2021).

Reference dosimetry
The treatment head is equipped with an Integrating Current Transformer (ICT, BERGOZ Instrumentation, France). For each irradiation session, the ICT was calibrated in absorbed dose to water against a reference ionization chamber, centered with the beam axis. The obtained calibration curves were used to determine the dose delivered in the irradiations with the TL/OSL matrices, based on the charge measured by the ICT. Moreover, the ICT was used to calculate the delivered instantaneous dose rate as the ratio of the measured dose per pulse and the pulse duration.
At 1 kGy s −1 , an ionization chamber (NACP02, IBA Dosimetry GmbH, Schwarzenbruck, Germany) was employed as reference chamber. At higher dose rates, an Advanced Markus chamber (TM34045, PTW-Freiburg  Table 2. Beam parameters for the four irradiation sessions. D pulse and D total indicate the dose per pulse and the total delivered dose, respectively, as derived from the integrating current transformer, calibrated in absorbed dose to water against a reference ionization chamber. The instantaneous dose rate D  inst is the ratio of the dose per pulse and the pulse duration (3 μs). u(D total ) is the relative uncertainty of the delivered dose and z ref is the reference depth in water chosen for the irradiations.

# pulses
Frequency GmbH, Freiburg, Germany) was used. To ensure the traceability, both chambers were calibrated in absorbed dose to water using a 60 Co source at METAS and the beam quality correction factor k Q was applied for the irradiations with electrons, following the TRS-398 Code of Practice (IAEA 2000). Given the decreasing ion collection efficiency of the Advanced Markus chamber with the dose per pulse, the ion recombination correction factor k s was previously characterized by comparison measurements with respect to alanine detectors. For this purpose, three alanine dosimeter packages were irradiated for each of the four different accelerator settings leading to a different dose per pulse. For each setting, k s was calculated as the ratio of the dose per pulse measured by alanine and the dose per pulse measured by the ionization chamber uncorrected for ion recombination. The results were subsequently fitted with the empirical model proposed by Petersson et al (2017) and plotted in figure 4 as a function of the uncorrected dose per pulse. The fitted ion recombination correction factors were then used in this study to determine the absorbed dose to water measured by the Advanced Markus chamber for the ICT calibration.

TLD and OSLD calibration
The TLD and OSLD calibrations were carried out by irradiating groups of five annealed/bleached detectors with the 90 Sr/ 90 Y source present in the lexsyg smart reader. Each group was irradiated for a chosen irradiation time in the (2-200) s range, depending on the material. Following the dosimetry protocol described in section 2.1, the S signal was recorded and the detectors were then re-irradiated during a time t R (defined in table 1). The S/S R signals were hence determined.
The calibration curves were obtained by fitting the S/S R data as a function of the irradiation time. A linear function was used for LiF:Mg,Ti, LiF:Mg,Cu,P, and BeO, a quadratic for CaF 2 :Tm, and a saturating exponential (S S a bt 1 exp R ( ( ) ) = --) for Al 2 O 3 :C. This relationship allows the interpolation from S/S R to the indicated value M t , which corresponds to the irradiation time with the reader built-in source necessary to generate the same S/S R value as the dosimeters irradiated in the field.
To correlate the source irradiation time to absorbed dose to water, a calibration coefficient was obtained for each material by irradiating the TL and OSL packages in the reference electron beam at METAS at 1 kGy s −1 (indicated hereafter by the subscript '0'). The reference absorbed dose to water D 0 was determined from the ICT, calibrated against the NACP02 ionization chamber (see section 2.2.2). The TLD/OSLD indicated value M t,0 was determined from the measured S/S R signal based on the calibration curves obtained with the reader source. The calibration coefficient N was then calculated as

Dose formalism and uncertainties estimation
For each irradiation session, the TLD/OSLD indicated value (M t,0 or M t ) was determined from the S/S R signal measured after the irradiation at METAS, based on the calibration curves obtained with the reader 90 Sr/ 90 Y beta source. The indicated value was then converted into absorbed dose to water with the calibration coefficient (see equation 1), according to the following equation Additional correction coefficients were assumed to be negligible, as discussed below. Fading correction factors were neglected, because the chosen dosimetric signal, as defined in section 2.1, is already stable after one day and is equal (within the uncertainties) to the signal after one week (see fading study in the Supplementary Material, section II).
The targets employed to vary the dose rate affected the beam energy spectrum. However, in the energy range relevant for this work, variations of less than 0.5 MeV in the maximum electron energy do not influence the TLD and OSLD response (Robar et al 1996, Yukihara et al 2008).
In the TL/OSL matrix, the four central detectors (defined by the dotted rectangle in figure 1) were considered for a direct comparison with the ICT reference dose. For each material and for each dose rate, the average TL/OSL dose D M was calculated based on the n = 4 × m detectors irradiated in the same irradiation conditions, where m is the number of packages.
Given the dose formalism in equation 2, the uncertainty in the dose measured by the central TLDs/OSLDs was calculated as The uncertainty in the TLD/OSLD indicated value (u(M t,0 ) and u(M t )) is calculated through the error propagation formula applied to the fitting functions (linear, quadratic, and saturating exponential). It consists of two components: the uncertainty related to the fitting parameters and their covariance term (u(fit)), and the uncertainty related to the measurement (u(exp)), depending on the standard deviation of the mean signal S/S R of the central TLDs/OSLDs. More details about the error propagation for the different fit models can be found in the Supplementary Material, section V.I.
The uncertainty of the delivered dose, u(D 0 ) at 1 kGy s −1 and u(D) at UHDR, includes the uncertainty in the ICT measurements, the uncertainty in the ICT calibration (dose measurement with the reference ionization chamber and ICT measurement during the calibration), and the uncertainty in the TLD/OSLD matrix positioning. Its value is reported in table 2 under the column u(D total ) for the four irradiation sessions.
All the uncertainties reported in this work are given for a coverage factor k = 1.

Beam uniformity evaluation
The beam uniformity decreases in the UHDR configurations, due to the removal of the flattening filter and the use of thinner scattering targets. To distinguish the effect of the beam non-uniformity from possible dose rate effects, a matrix of TLDs and OSLDs was employed. Simultaneously, the beam non-uniformity was monitored by reference BeO OSLDs at the corners of the matrix, whose results are presented in the Supplementary Material (section III). The beam uniformity was also independently monitored with Gafchromic ® EBT3 films (Ashland ISP Advanced Materials, Kearny, NJ, USA). They were cut into strips of ∼34 mm × 200 mm, installed on the PMMA holder and irradiated at the beginning or at the end of each irradiation session. The analysis of the EBT3 films is inlcuded in the Supplementary Material (section III).

Calibration
The LiF:Mg,Ti calibration curve is illustrated in figure 5. Each point is the mean S/S R value of five detectors irradiated with the same irradiation time. The error bars show the standard deviation of the data and visually have a size comparable to the marker size. The precision (standard deviation) of S/S R is better than 3 % , whereas the precision of the signal S can be as high as 13 %. These results show that the S/S R protocol eliminates the dependency of the signal on the detector sensitivity.
The residuals of thelinear fit are also illustrated in figure 5. In the dose range relevant for the experiment ( 1.2 Gy), the data are well-approximated by the linear fit as the residuals are within ± 1 %, with a maximum deviation of − 2.2 % at 40 s.
The calibration coefficient N for LiF:Mg,Ti, to convert the TLD indicated value M t to absorbed dose to water, was determined to be (42.3 ± 1.1) mGy s −1 , where the associated uncertainty is the standard deviation of the data. The calibration curves and calibration coefficients were determined also for the other TL and OSL materials (see Supplementary Material, section I).

Beam uniformity
The TLDs and OSLDs matrices reflected the beam non-uniformity, as shown in figure 6. The figure shows the dose maps obtained with LiF:Mg,Cu,P matrices (1 package per dose rate), where the color code is the dose normalized to the maximum dose of each package. The dose map at 324 kGy s −1 shows that the highest dose is measured in the top-left corner, in agreement with the EBT3 film (see Supplementary Material, section III).
The results justify the choice of using the four central TLDs/OSLDs for the comparison with the reference ICT dose, to distinguish the effects of the beam non-uniformity from possible dose rate effects. The results also prove the usability of a matrix of detectors to map the beam distribution.
3.3. Dose rate dependence 3.3.1. TL and OSL curve shapes Figure 7 shows the TL curves at the investigated dose rates, where one detector per dose rate was considered, normalized by the maximum intensity for comparison.  Discrepancies in the TL curves are observed, but they do not show any trend with the dose rate. Such discrepancies are smaller or comparable to the discrepancies observable in the detectors irradiated at the same dose rate (see Supplementary Material, figure S7). The variability of the TL curves results from sample-to-sample variability, as no pre-selection of the detectors was carried out. Given the dosimetry protocol used, which includes the irradiation with the reader beta source (see section 2.1), sample-to-sample variability can be observed in the TL curves, but it does not affect the dosimetric signal S/S R .
Also for Al 2 O 3 :C, no variation of the OSL curve with the dose rate is shown in figure 8. Unlike the TL materials, Al 2 O 3 :C OSLDs were prepared from the same batch, which explains the reduced observed variability compared to TLDs.

Comparison with the reference BeO OSLDs
To understand if the TLDs/OSLDs are dose rate dependent, first they were compared to the reference BeO OSLDs. To account for the beam non-uniformity, the four TLDs/OSLDs at each corner of the matrix were compared to the BeO OSLDs in the same corner (see figure 1). Figure 9(a) shows the ratio between the response of the LiF:Mg,Cu,P TLDs at the corners and of the corresponding BeO OSLDs (D corner /D BeO ), where each subplot refers to one corner. The dose measured by LiF:Mg,Cu,P is in agreement with the BeO OSLDs at the investigated dose rates, indicating the dose rate independence of LiF:Mg,Cu,P within ± 10 %. Similar results were obtained for the other investigated TL materials (see Supplementary Material, figure S8). Figure 9(b) compares the Al 2 O 3 :C OSLDs and the reference BeO. The Al 2 O 3 :C OSLDs showed an average under-response for all corners, but less than 10 % as for the other materials.
Both in the case of TLDs and of OSLDs, no monotonic trend was observed in the response relative to BeO OSLDs as a function of the dose rate.  Figure 10 shows the comparison between the central TLDs/OSLDs response with the reference ICT dose, where the ratio D center /D ICT is presented as a function of the dose rate. The uncertainty in the delivered dose is not included here, as it will be discussed in section 3.4. The best agreement between the TLDs/OSLDs and the ICT is achieved at 1 kGy s −1 , as this irradiation session was used to determine the calibration coefficient N (see section 2.2).

Comparison with the reference ICT
Except for Al 2 O 3 :C OSLDs, the response of the investigated TL materials at 42 kGy s −1 is higher (∼3 %) than the reference dose for all packages, as observed for the BeO OSLDs (see Supplementary Material, figure S5). The underestimation of the reference dose is due to the underestimation of the recombination coefficient extracted from the fit (see figure 4), compared to the experimental one.
No trend in the response of the investigated TL materials as a function of the dose rate is observed within the one standard deviation band. Indeed, for each TL material, the responses of the packages irradiated at the different dose rates are in agreement within their uncertainties, which only take into account the reproducibility (standard deviation) of the TL measurement. Al 2 O 3 :C OSLDs show an under-response at 195 kGy s −1 and 324 kGy s −1 of at most 8 %. This could indicate a possible dose rate dependence, but given the small deviation and the uncertainties involved, the results are not conclusive, particularly given that a dose rate dependence is not observed when the BeO OSLDs are used as a reference ( figure 9(b)).

Estimated uncertainties
The TLDs and OSLDs uncertainties were evaluated separately for the three irradiation sessions with UHDR. The results are reported in tables 3(a)-(d), where all the relative uncertainties are expressed for a coverage factor k = 1. The tables show the number of detectors (n) per session , the uncertainty of the TLD/OSLD indicated value (u r (M t,0 ) and u r (M t )), including the fit and the experimental components, the uncertainty in the delivered dose (u r (D 0 ) and u r (D)), and the combined uncertainty in the dose measured by the TLDs/OSLDs (u r (D M )), calculated from equation 3.
The reduced number of LiF:Mg,Ti TLDs used at 195 kGy s −1 is the cause of the higher experimental uncertainty compared to the other sessions: for the 195 kGy s −1 irradiation, two LiF:Mg,Ti packages were irradiated with more than two pulses (one additional pulse was passing while the beam shutter was closing). Therefore, they werediscarded from the calculation of D M and u r (D M ).
The major contribution to the uncertainty in the TLD/OSLD indicated value M t comes from the experimental uncertainty, which takes into account the reproducibility of the TL/OSL technique and the reproducibility of the beam, calculated based on the detectors irradiated in different packages with the same beam parameters. To reduce this uncertainty, it would be necessary to increase the number of irradiated detectors n to reduce the standard deviation of the mean S/S R signal. Indeed, in the uncertainty of the calibration indicated value M t,0 , the experimental uncertainty is lower for most of the materials, as a larger number of detectors was employed in the calculations (entire matrix instead of the central TLDs/OSLDs).
The uncertainty derived from the fitting parameters is, instead, less than 0.7 % for all the materials, indicating that the calibration data are accurately described by the chosen fitting models. The uncertainty in the delivered dose increases with the dose rate, as a consequence of the increased uncertainty in the recombination correction factor, which is the major contribution to the irradiation uncertainty. A correction factor of 1.4 with a 2.7 % uncertainty was estimated at 324 kGy s −1 . At dose rates lower than 324 kGy s −1 , the uncertainty in the delivered dose (u r (D)) is comparable to the uncertainty in the dose measured by the TLDs/OSLDs (u r (D M )), except for LiF:Mg,Ti at 42 kGy s −1 , due to the high contribution of the experimental uncertainty in u r (M t ). At the highest dose rate, instead, the uncertainty in the delivered dose is the main limitation to the precision of our study. The TLDs/OSLDs are able to measure with an uncertainty u r (D M ) better than 2 % at all dose rates, but a 3 % uncertainty on the delivered dose is present at the highest investigated dose rate.
The results of the uncertainty evaluation were then included in the comparison between the central TLDs/ OSLDs and the reference ICT dose. Figure 11 shows the deviation of the central TLDs/OSLDs relative to the reference dose as a function of the dose rate, for the investigated materials. D M is the average dose response Figure 9. Comparison between (a) LiF:Mg,Cu,P TLDs and (b) Al 2 O 3 :C OSLDs with the BeO OSLDs placed at each corner, as a function of corner location (figure 1) and dose rate. Each bar represents the average response of the four detectors at the corresponding corner, divided by the response of the associated BeO OSLD, for one single package. The error bars represent the uncertainty of the ratio, which includes both the uncertainty of the TLDs/OSLDs (calibration and standard deviation of the data) and the uncertainty of the BeO OSLDs (calibration). The dotted lines indicate the average ratios, the colored bands show one standard deviation around the mean, whereas the dashed lines define unity. calculated according to equation 2, while the error bars show the relative uncertainties of the ratio D M /D ICT , calculated according to the uncertainty evaluation in table 3.
An over-response of the investigated TLDs is observed at 42 kGy s −1 . This is due to an underestimation of the ion recombination coefficient from the logistic model (see section 2.2.2), which results in an underestimation of the delivered dose. Overall, the relative deviation from the reference dose is lower than 4 % for the investigated TL materials at all dose rates. The response of Al 2 O 3 :C OSLDs might indicate a dose rate dependence, but the uncertainties are large and the maximum relative difference with the reference dose is lower than 7.5 %. Figure 10. Comparison of the response of the investigated TL and OSL materials against the reference dose estimated from the ICT for each dose rate. Each bar illustrates the average response of the four central TLDs or OSLDs divided by the reference dose, and the error bars represent the standard deviation of the data. The dotted lines indicate the average ratios, the colored bands show one standard deviation around the mean, whereas the dashed lines define unity. Figure 11. Deviation of the investigated TL and OSL materials relative to the dose estimated from the reference ICT. The TLD/OSLD measured dose D M is calculated considering the central detectorsand according to the formalism described in section 2.3. The error bars represent the uncertainties (k = 1) of the ratios, which include the uncertainties of the dose measured by the TLDs/OSLDs and the uncertainty of the delivered dose (see table 3). A horizontal offset was introduced to visually distinguish the data and the error bars.

Conclusions
This work aimed at investigating possible dose rate effects in various TLDs and OSLDs, irradiated with UHDR electron beams with instantaneous dose rates in the (1-324) kGy s −1 range. The study showed that no dose rate effects were observed in the investigated TL materials, with average relative deviations from the reference dose smaller than 4 %. Al 2 O 3 :C OSLDs showed a decreasing response with the dose rate, but still within 10 % from the reference dose. This result remains unexplained, but could be caused by experimental factors (e.g. light-induced fading, detector calibration). Additional experiments are necessary to confirm this result.
The discrepancy between the measured luminescence detector doses and the reference dose at 42 kGy s −1 , due to the underestimation of the ionization chamber ion recombination correction factor, demonstrates the importance of a correct estimation of this factor. Also, we proved that the uncertainty associated to the correction factor is the main contribution to the delivered dose uncertainty.
Among the investigated materials, Al 2 O 3 :C OSLD was shown to have the lowest uncertainty in the dosimeter reading (u r (M t ) < 1 %), which adds to its advantages, such as the possibility of cutting the film in various sizes, the completely optical readout, and the simple analysis of the OSL signal. The investigated TL materials show comparable uncertainties in the dosimeter reading (u r (M t ) < 2 %), but their readout is more demanding than OSL, particularly for the calculation of the dosimetric signal. However, the TLDs used here have the advantage of being insensitive to visible light. Table 3. Estimated uncertainties for the investigated TL and OSL materials. All uncertainties are expressed as relative uncertainties in percentage with a coverage factor k = 1. M t,0 and M t denote the TLD/OSLD indicated value as derived from the calibration curves, for 1 kGy s −1 and UHDR irradiations, respectively. The respective uncertainties (u r (M t,0 ) and u r (M t )) include the uncertainty in the fit of the calibration curves (u r (fit)) and the experimental uncertainty (u r (exp)). n is the number of irradiated detectors. u r (D 0 ) and u r (D) denote the uncertainties of the delivered dose, at 1 kGy s −1 and UHDR, respectively. u r (D M ) is the uncertainty of the dose measured by the investigated TLDs/OSLDs, which is the combination of u r (M t ), u r (M t,0 ) and u r (D 0 ). 1.5 1.2 2.9 u r (D) 1.5 1.2 2.9