Fano cavity test and investigation of the response of the Roos chamber irradiated by proton beams in perpendicular magnetic fields up to 1 T

Objective. The aim of this work is to investigate the response of the Roos chamber (type 34001) irradiated by clinical proton beams in magnetic fields. Approach. At first, a Fano test was implemented in Monte Carlo software package GATE version 9.2 (based on Geant4 version 11.0.2) using a cylindrical slab geometry in a magnetic field up to 1 T. In accordance to an experimental setup (Fuchs et al ), the magnetic field correction factors kQB⃗ of the Roos chamber were determined at different energies up to 252 MeV and magnetic field strengths up to 1 T, by separately simulating the ratios of chamber signals MQ/MQB⃗, without and with magnetic field, and the dose-conversion factors Dw,QB⃗/Dw,Q in a small cylinder of water, with and without magnetic field. Additionally, detailed simulations were carried out to understand the observed magnetic field dependence. Main results. The Fano test was passed with deviations smaller than 0.25% between 0 and 1 T. The ratios of the chamber signals show both energy and magnetic field dependence. The maximum deviation of the dose-conversion factors from unity of 0.22% was observed at the lowest investigated proton energy of 97.4 MeV and B⃗ = 1 T. The resulting kQB⃗ factors increase initially with the applied magnetic field and decrease again after reaching a maximum at around 0.5 T; except for the lowest 97.4 MeV beam that show no observable magnetic field dependence. The deviation from unity of the factors is also larger for higher proton energies, where the maximum lies at 1.0035(5), 1.0054(7) and 1.0069(7) for initial energies of E0 = 152, 223.4 and 252 MeV, respectively. Significance. Detailed Monte Carlo studies showed that the observed effect can be mainly attributed to the differences in the transport of electrons produced both outside and inside of the air cavity in the presence of a magnetic field.


B 
of the Roos chamber were determined at different energies up to 252 MeV and magnetic field strengths up to 1 T, by separately simulating the ratios of chamber signals without and with magnetic field, and the dose-conversion factors  in a small cylinder of water, with and without magnetic field.Additionally, detailed simulations were carried out to understand the observed magnetic field dependence.Main results.The Fano test was passed with deviations smaller than 0.25% between 0 and 1 T. The ratios of the chamber signals show both energy and magnetic field dependence.The maximum deviation of the doseconversion factors from unity of 0.22% was observed at the lowest investigated proton energy of 97.4 MeV and B  = 1 T. The resulting k Q

Introduction
Proton therapy represents a promising approach in cancer treatment.Compared to x-ray beam radiotherapy, it is considered as more advantageous regarding the low entrance dose, the distinctive steep increase of dose deposition towards the end of the particle range (the so-called Bragg-Peak); and the diminishing dose at the distal side of the Bragg-Peak.These favorable properties of protons can be exploited to achieve high-conformal dose distribution, while sparing the organs-at-risk situated at the entrance or exit side of the beam (Suit et al 2003).On the one hand, these geometrical superiorities associated with proton beams appear to be very favorable.On the other hand, anatomical changes of the patient, such as inter-and intra-fractional motion, change in tumor size or patient weight and geometrical uncertainties related to patient positioning, might potentially lead to range uncertainties and deviations in the desired dose distribution.As a result, the tumor volume may be missed or the intended dose to tumor is deposited in adjacent critical organs (Lomax 2008, Paganetti 2012, Engelsman et al 2013).
Following the development of MR-guided x-ray beam radiotherapy (MRgXRT) and acknowledging the clinical potential provided by real-time MR imaging to account for inter-and intra-fractional anatomical variations, the development of MR-guided proton radiotherapy (MRgPRT) has been reported recently (Mackay 2018, Hoffmann et al 2020).Several working groups have investigated the influence of the magnetic fringe field on the pencil beam transport before entering the patient and the direct impact of the strong magnetic field within the patient geometry (Moteabbed et al 2014, Oborn et al 2015).Like MRgXRT, the impact of strong magnetic fields on the detector's dose response in proton beams must also be investigated thoroughly to guarantee the safe clinical implementation of MRgPRT in the future.
Up to date, studies related to dose measurements in proton beams in the presence of magnetic fields are very scarce.In these early investigations, films were utilized to study the dosimetry effect caused by magnetic fields (Padilla-Cabal et al 2019, Gebauer et al 2023b).Only recently, an experimental study on the impact of magnetic fields on the response of active point-detector's has been published (Fuchs et al 2021).Nevertheless, the authors emphasized in their study that the underlying mechanisms are yet to be fully understood and these aspects require further studies.In a recent study, Monte Carlo and experimental investigations on the dose response of a plane-parallel chamber (Advanced Markus chamber, PTW Freiburg, Germany) have been reported, where the proton energy and magnetic field dependent dose response of the chamber has been reported (Gebauer et al 2023a).
The main aim of this work is to evaluate the response of the Roos chamber (PTW Freiburg, Germany) irradiated by proton beams in a magnetic field with the help of Monte Carlo simulations.Firstly, a Fano cavity test was performed with the Geant4 Monte Carlo code to identity the optimum set of transport parameters and physics list.Thereafter, the response of the chamber was simulated for a wide range of proton energies and magnetic fields in the same configuration as used in the empirical study performed by Fuchs et al (2021).Special focus has been placed in this study to understand the underlying mechanisms leading towards the empirical and Monte Carlo observations using detailed Monte Carlo simulations by differentiating the contribution from different particles species and their origins.These results provide insights into the magnetic field dependence as well as of the role of chamber's geometry towards its response in a magnetic field.

Investigated detector
The Roos chamber (type 34001, PTW Freiburg, Germany) is a plane-parallel air-filled ionization chamber with a sensitive volume of 0.35 cm 3 .The sensitive volume has a radius of 7.8 mm and a thickness of 2 mm.It is surrounded by an air guard-ring with a width of 4 mm.The collecting electrode of the Roos chamber has the same radius as the sensitive volume, the chamber wall mainly consists of PMMA.A schematic drawing of the Roos chamber is given in figure 1.

Magnetic field correction factor
According to Van Asselen et al (2018) the magnetic field correction factor k Q B  is defined as


is the detector-independent dose-conversion factor expressed as the ratio of the absorbed dose-to-water in the absence of the detector, with and without magnetic field; and is the detector-dependent ratio of the simulated deposited energy in the detector sensitive volume without and with magnetic field.Both the quantities have been obtained as described in section 2.3, from which the magnetic field correction factors are calculated according to equation (1).

Monte carlo setup
Monte Carlo simulations were performed in GATE (version 9.2) (Jan et al 2004, Jan et al 2011) based on the Geant4 toolkit (version 11.0.2) (Agostinelli et al 2003).The setup of the simulations corresponds to that of the experimental setup used by Fuchs et al (2021).A schematic drawing of the simulation setup is given in figure 2.
The monoenergetic proton beam used has a field size of 10 cm × 10 cm with initial energies of E = 97.4,152, 223.4 and 252 MeV.The Roos chamber was modelled according to the detailed constructional drawing from the manufacturer and orientated with its entrance window perpendicular to the beam's axis (−x).The reference point of the detector was placed at z = 2 cm depth in water.The magnetic field was orientated perpendicular to the beam's axis with nominal magnetic field strengths B  = 0.25, 0.32, 0.5, 0.75 and 1 T. At the position of the chamber, which is located 9.5 cm further beam upstream from the center of the magnetic field poles, the measured field strengths correspond to 0.24, 0.31, 0.49, 0.73 and 0.97 T.
The quantities M Q and M Q B  (per primary) were scored as the energy deposited in the sensitive volume of the chamber, without and with magnetic field, respectively.The quantities D w Q , and D w Q B ,


were scored as the energy deposited in a 0.25 mm thick water cylinder of 1 cm radius, as already employed in previous studies (Gomà et al 2016, Wulff et al 2018, Gomà and Sterpin 2019, Baumann et al 2020, Kretschmer et al 2020, Baumann et al 2021a, 2021b).During the simulations, an extra geometry encompassing a 0.5 cm margin from the chamber or water cylinder outer dimensions in each direction was created (cutbox), in which the production cut for secondary particles was set to 1 μm.This small production cut was chosen following previous publications (Baumann et al 2020, Kretschmer et al 2020, Kretschmer et al 2022) to guarantee accurate transport within the chamber geometry.Elsewhere, a production cut of 1 mm was used.A summary of the chosen settings in GATE is given in table 1.In this study, the default I-values (updated according to key data of ICRU 90) of all materials were used.Padilla-Cabal et al (2020) have reported negligible influence of the magnetic field transport parameters (δ and ε).Therefore, their default values were used in this study.Furthermore, the uncertainty associated with the choice of these parameters can be assessed from the results of the Fano tests performed in a magnetic field (see section 3.1).

Fano cavity test
The Fano cavity test is employed to test the correctness of charged particle transport and the choice of associated parameters based on condensed history technique implemented in the Monte Carlo simulations by means of validating the Fano theorem (Fano 1954).In this work, the Fano cavity test was performed for the GATE/Geant4 implementation to identify the appropriate simulation settings in a magnetic field up to 1 T. The slab geometry as described in Wulff et al (2018) was used with density correction turned off.The slab geometry consists of three layers: two layers of wall consisting of normal water with a thickness z wall = 22 cm and a density ρ water = 1 g cm −3 enclosing a layer of low-density water with a thickness of z gas = 0.2 cm and a density of ρ gas = 1.2048 mg cm −3 .The thickness z wall is larger than the maximum range of the initial protons.An isotropic line source of monoenergetic protons with an energy of E 0 = 150 MeV was located at the center of the geometry.The radius r = 207.53m of the geometry corresponds to the maximum range of the charged particles in the low-density layer.The intensity of the source was adjusted according to the local mass-densities of the three layers.A schematic drawing of the simulation setup is shown in figure 3.
The quantity Q defined as (Wulff et al 2018): was calculated, where ∆E is the deposited energy in the low-density layer with N histories.The deviation of the calculated Q from unity was evaluated in the Fano test.
As the starting point, simulation settings reported in Kretschmer et al (2020) used for the simulations of proton beams without magnetic field were adopted.In the first step, the influence of the parameter dRoverRange that limits the maximum energy loss in one single step was studied.Its value was stepwise reduced from the default values of 0.2 for electrons and 0.1 for protons to 0.05 and 0.02 for both particles.In the second step, the ).An isotropic line source of monoenergetic protons with an energy of E 0 = 150 MeV was located at the center of the geometry.parameter maximum step size was reduced from no restriction to 1 mm, 0.1 mm and 0.01 mm.The Fano test was performed without magnetic field and with magnetic field B  = 0.5 and 1 T.

Investigations of the underlying mechanisms
To understand the observed magnetic field dependence of the Roos chamber, the following detailed simulations were performed in GATE.

The role of chamber's components
The simulations of the quantities as described in section 2.2 have been repeated by stepwise modification of the chamber geometry (ref.figure 1).In the first step, the chamber wall was replaced by water.In a second step, the chamber wall was still replaced by water and additionally, the guard-ring was also considered as sensitive (refer to figure 1), that is, the complete air volume with a radius of 11.8 mm was regarded as the chamber's sensitive volume.

Contribution of particle species
The total deposited energy, M , total consists of contributions from protons M , p electrons M , e -and to a small extend, various fragments M : the sensitive volume.To quantify each of these contributions, the chamber simulations described in section 2.2 were repeated with particle filters for electrons and protons.To further differentiate the electron contributions into electrons produced inside and those produced outside the sensitive volume, separate simulations were performed by increasing the production cut of electrons outside of the sensitive volume to 1000 km, so that only electrons that were produced within the sensitive volume contributed to the deposited energy, that is, M .
e in -The difference between the two simulations with normal and increased production cut represents the contribution originated from outside the sensitive volume, that is, M .e out -It is noteworthy here that the proton contribution refers to all energy transfers to electrons with energies corresponding to ranges below the chosen production cut of 1 μm.These electrons were not produced, and their entire energy being absorbed, at the point of interaction.
For electrons with energies that correspond to values above the production cut of 1 μm, their trajectories might be affected by the presence of a magnetic field.It is also expected that the influence depends on their energy, initial trajectories, and points of origin.To gain insights on these aspects, the spectral distribution of the electrons at the measurement depth for each initial proton energy was simulated according to the same setup as described in section 2.2 utilizing the energy spectrum actor and an electron particle filter.Furthermore, their angular distributions were scored within a water sphere of 1.5 mm diameter placed at the measurement depth of 2 cm for three energy ranges (see section 3.5).Based on this information, electron sources comprising the simulated energy and angle distributions were used to study the influence of the magnetic field on their trajectories in the sensitive volume.Three separated sources representing their respective origins were created:

Fano cavity test
The results of the Fano cavity test are presented in figure 4 showing the Q-values computed according to equation (2). Figure 4(a) shows that, for a magnetic field strength of 1 T and a maximum step size of 1 mm, Q does not depend on dRoverRange and is rather constant with a value of 1.002.Figure 4(b) shows that the smallest Q-value of 1.0009 ± 0.0003 is achieved when the maximum step size was not restricted for B  = 0 T. Subsequently, the Fano test with magnetic field was performed with the default settings of dRoverRange and two variants of maximum step size: without limitation and with 1 mm limitation (figure 4(c)).Considering the results over the entire range of magnetic field, from B  = 0 up to 1 T, the best results were obtained with a maximum step size of 1 mm, where the agreement of Q-value lies within 0.25%, that is between 1.0024(3) without magnetic field and 0.9983(2) at B  = 1 T.

Dose-conversion factor
The detector independent dose-conversion factors are presented in table 2. For each energy, the dose-conversion factors increase with increasing magnetic field, which indicates that the dose at the measurement depth increases with magnetic field.This effect becomes less prominent for higher energies.The maximum value is 1.0022(1) for the lowest energy of 97.4 MeV and highest magnetic field of B  = 1 T studied in this work. of the Roos chamber increase with increasing proton energy.A magnetic field dependence can also be observed for E = 152, 223.4 and 252 MeV: firstly, the correction factors increase up to a maximum at around B  = 0.5 T, after which the correction factors decrease again.The highest correction factor is found to be k Q B  = 1.0069( 7) for E = 252 MeV and B=  0.5 T.

The role of chamber components
The role of the chamber wall on the detector-ratios was investigated by replacing the chamber wall by water.Additionally, the influence of the guard-ring was studied by considering the guard-ring also as the sensitive volume.Both results are presented in figure 6, exemplarily for one proton energy of E = 152 MeV.For comparison, the results from figure 5(a) with the complete chamber model are also presented (red closed circles).All three curves show similar magnetic field dependence within the simulation uncertainty, as reported in section 3.3, where the ratios start to increase from B  = 0 to a maximum, after which the factors descend again.

Contribution of particle species
The results of the investigation of the individual contributions from protons, secondary electrons, and fragments to the total deposited energy in the chamber's sensitive air volume are presented in figure 7, exemplarily for the proton energies of (a) 152 and (b) 252 MeV.The ratios for the total deposited energy (black closed circles) correspond to the values presented in figure 5(a).While protons contribute to almost 60% to the total deposited energy, the ratios of proton contributions, ( )  for E = 97.4, 152, 223.4 and 252 MeV and magnetic fields between 0.25 and 1 T. The number in the brackets indicates the change of the last digit for 1 S.D. fluctuate within the simulation uncertainty around unity for both energies with a maximum deviation of 0.3% from unity.The dose component attributed to the fragments amounts to less than 3% of the total dose for all cases.Therefore, their contributions towards the magnetic field dependence of is expected to be negligible.
Both the electrons originated from outside and inside the sensitive volume demonstrated strong magnetic field dependence.While the ratio of the electrons originated from the outside (grey close circles) increases with increasing magnetic field, the ratio of the electrons originated from the inside (green close circles) increases slightly up to about B=  0.3 T and then decreases with further increasing magnetic field.The contributions of all electrons originated both from inside and outside the sensitive volume (blue close circles) that together contribute to around 40% of the deposited energy resulted in an increasing portion before the maximum and a decreasing portion after the maximum.
Figure 8(a), shows exemplarily the spectral distribution of the electrons for the 152 MeV proton beam at the measurement depth of 2 cm.The angular distributions of the electrons, differentiated in three energy ranges: (b) below 100 keV, (c) between 100 and 200 keV and (d) above 200 keV, show that the electrons with higher energy of the Roos chamber evaluated for the complete chamber (red closed circles), the chamber, where the wall was replaced by water (dark grey closed circles), and for the case where the sensitive volume was expanded by the guard-ring region (light grey closed circles), exemplarily for E = 152 MeV. are more forward peaked, that is, they travel preferentially in the same direction as the primary protons.Nevertheless, approximately 65% of all electrons fall into the first energy range below 100 keV with broader angular distribution having a significant portion with angles exceeding 90°(backscattered).
Table 3 shows the trajectories of the electrons from three sources derived from the energy and angular distributions as presented in figure 8 placed separately at the entrance surface (upper row), (ii) inside (middle row), and (iii) at the exit surface (last row), of an air cavity representing the sensitive volume of the Roos chamber with a radius of 7.8 mm and a thickness of 2 mm, without (first column), with B  = 0.5 T (middle column) and with B  = 1 T (last column).In the absence of a magnetic field, the electrons' trajectories starting from the for the total deposited energy, and the individual contributions from protons, all electrons (total), electrons from inside and electrons from outside the sensitive air volume for initial proton energies of (a) E = 152 MeV and (b) E = 252 MeV.entrance surfaces are mostly straight throughout the air cavity.In a magnetic field, the trajectories are curved with some being deflected back towards the entrance surface by the resulting Lorentz force.This effect becomes more prominent with increasing magnetic field up to the studied 1 T, where most electrons are deflected back with almost no electrons reaching the exit surface.
Similar behavior can be observed for the electrons emitted at the exit surface (last row), where those backscattered electrons (>90°) could enter the sensitive volume.On the one hand, as these electrons possess lower kinetic energy, they experience more scattering within the air cavity without magnetic field.On the other hand, they are also more susceptible to the influence of magnetic field, so that they also get deflected back to the exit surface, like the electrons originated at the entrance surface.
In the middle row, one can observe that the trajectories of the electrons originated within the sensitive volume are also strongly modified in the presence of a magnetic field.Without magnetic field, most of these trajectories are straight so that they eventually leave the air cavity.In a magnetic field, the electrons will engage in a spiral motion.With increasing field strength (comparison between 0.5 and 1 T), most of these electrons would be trapped within the air cavity hence depositing their entire energy within it.

Fano cavity test
The Fano cavity test passed with an agreement better than 0.25% up to 1 T using the simulation settings in Kretschmer et al (2020) with default dRoverRange and 1 mm maximum step size.Our results showed that stricter dRoverRange and maximum step size did not always yield better results, especially in a magnetic field.Henceforth, the settings in Kretschmer et al (2020) were considered adequate and can be applied in further chamber simulations in proton dosimetry in magnetic fields.Marot et al (2023) reported Fano test results within 0.82% ± 0.42% for 150 MeV proton beam in 1.5 T magnetic field using the TOPAS/Geant4 package within a slab geometry.Although they implemented a stricter step size limitation (dRoverRange and finalRange), their production cut corresponds to energy of 1 keV in the materials.

Magnetic field correction factor
The results presented in figure 5  is larger than unity for the Roos chamber, where its magnitude increases with increasing proton energy.Similar behavior for the Roos chamber has been reported in the empirical study performed by Fuchs et al (2021), where the experimentally determined ratio for E = 252.7 MeV are 1.0053(10), 1.0067(10), 1.0026(10) and 0.9989(20), as compared to our values of 1.0052(7), 1.0066(7), 1.0056(7) and 1.0036( 7), for 0.25, 0.5, 0.75 and 1 T, respectively.
Table 3. Trajectories of electrons from the entrance surface (upper row), inside the air cavity (middle row) and the exit surface (lower row) for one exemplary proton energy E = 152 MeV and B  = 0 (left column), 0.5 (middle column) and 1 T (right column).These electrons represent the secondary electrons generated by a proton beam directed perpendicular to the cavity entrance surface.

Influence of the chamber's components
The results presented in figure 6 demonstrated no significant influence of the components of the Roos chamber on its behavior in a magnetic field.By replacing the chamber wall of the chamber, as well as by considering the guard-ring also as the sensitive volume resulted in similar behavior of Therefore, it can be deduced that the magnetic field dependence of the Roos chamber is mainly caused by the low-density air volume.It is at first sight not surprising as the chamber wall of the Roos chamber is mainly made of PMMA with a density of ρ PMMA = 1.19 g cm −3 and is therefore close to water-equivalent.
The comparison between the simulations without and with the guard-ring included in the sensitive volume, where the radius is increased from 7.8 to 11.8 mm, reveals no significant difference hinting that the radius of the chamber is not a decisive factor for the magnetic field dependent chamber's response.

Contributions from different particle species
As discussed above, the particle transport within the low-density air volume plays the central role that leads to the magnetic field dependence of the Roos chamber.Nevertheless, it is not immediately evident of why the ratios and consequently the factors k , show a maximum at around B  = 0.5 T. The results of our detailed simulations, presented in figure 7 and table 3, have revealed the rather complex underlying mechanisms that are discussed in the following: A. The contribution of the protons towards the behavior of k Q B  is negligible since the deflection of their trajectories by the magnetic field within the dimensions of the chamber (2 mm thickness) is small.B. The contribution of the fragments is negligible due to their small overall contribution to the total energy deposition within the chamber.
C. The observed behavior of the k Q B  factors can be almost entirely attributed to the contribution of the electrons.
D. Figure 7 demonstrated, on the one hand, that the ratios owed to the electrons originated from outside the sensitive volume increase with magnetic field.The explanation is illustrated in table 3 (first and last rows), where on average the trajectories of the electrons are shortened in a magnetic field as a large portion of these is being deflected back to the entrance and exit surfaces by the Lorentz force.In other words, the fluence of these electrons is decreased and correspondingly, the deposited dose in the cavity ), which is the product of the fluence and the stopping power, also decreases.This phenomenon shares similarities to the so-called electron-return effect often described in the literature within the context of dosimetry in a magnetic field encountered at boundaries with low-density tissues (Raaijmakers et al 2008, Tekin et al 2020, Cervantes et al 2022).
E. Figure 7 also demonstrated, on the other hand, the ratios owed to the electrons originated from inside the sensitive volume show a slight increase up to around B  = 0.3 T, but then decrease with further increasing magnetic field.As shown in table 3 (middle row), the electrons released within the air cavity are also subject to deflections by the resulting Lorentz force.Up to around 0.3 T, M Q B  decreases slightly owing to the same explanation as in D, where the electrons trajectories, especially those near the surfaces, are shortened as they are being deflected back to the surfaces leaving the sensitive volume.However, further increment of the magnetic field will result in a spiral down trajectory for most of these electrons as can be best observed in the case of 1 T of the middle row in table 3.As a result, the electrons are confined by the magnetic field within the air cavity, increasing their dose contribution M , increases with the proton energy in this case.This observation can be mainly attributed to the fact that higher magnetic field is required to confine the trajectories of the electrons released within the sensitive volume as their gyroradius increases with the proton energy.Therefore, the increase of M Q B  as described in E for higher magnetic fields becomes less prominent for higher proton energy.

Conclusion
This study presents extensive Monte Carlo investigations on the response of the Roos chamber in magnetic fields up to 1 T irradiated with proton beams of energies between 97.4 and 252 MeV.Firstly, a Fano test for proton radiation in a magnetic field was performed with the software package Geant4/GATE, with which the appropriate simulation parameters were investigated.In the second part of this study, the magnetic field is mainly attributed to the low-density air cavity of the chamber.Furthermore, the electrons originated from the outside and inside of the sensitive volume are shown to cause the observed chamber behavior in a magnetic field, while the protons and the fragments do not play a significant role for these observations.These insights are invaluable for the further studies to assist the comprehension of the influence of magnetic field on the response of detectors with different geometries, such as thimble-type chambers or diodes, as well as other modalities, such as heavy ions.

Figure 1 .
Figure 1.Schematic cross-section of the Roos chamber, showing the sensitive volume, guard-ring and the chamber wall.

Figure 2 .
Figure 2. Schematic drawing of the simulation setup, showing a side view (a) and a view in beam direction (b).The magnetic field in −y-direction leads to a Lorentz force in +z-direction acting on the protons.

Figure 3 .
Figure 3. Schematic drawing of the slab geometry for the Fano cavity test consisting of two layers of water (ρ water = 1 g cm −3 ) enclosing one layer of low-density water (ρ gas = 1.2048 mg cm −3).An isotropic line source of monoenergetic protons with an energy of E 0 = 150 MeV was located at the center of the geometry.
The electrons can be further differentiated into electrons produced outside (out), M , e out -and inside (in), M , e in - (a) plane source placed at the entrance surface of the sensitive air volume, (b) homogeneous source within the sensitive air volume, and (c) plane source at the exit surface of the sensitive air volume.Thereby, sources (a) and (c) provide insights to the contributions of electrons originated outside of the sensitive volume M , e out -while source (b) provides insights to the contributions of the electrons originated inside the sensitive volume M .

3. 3 .
Magnetic field correction factor According to equation (1), the magnetic field correction factors k figures 5(a) and (b), respectively.Regarding the energy dependence, the magnetic field correction factors k Q B 

Figure 4 .
Figure 4. Results of the Fano cavity test given by Q-values. (a) changing of dRoverRange, (b) changing of the maximum step size (c) with magnetic field of B  = 0.5 T and B  = 1 T.

Figure
Figure 5. Simulated ratios / M M Q Q B  (a) and the calculated magnetic field correction factor k Q B  (b) for the Roos chamber for E = 97.4,152, 223.4 and 252 MeV in magnetic fields up to B  = 1 T.

Figure 8 .
Figure 8. Upper panel (a): simulated electron spectrum for the initial proton energy of 152 MeV; lower panels: the angular distributions of the electrons in three energy ranges: (b) below 100 keV, (c) between 100 and 200 keV and (d) above 200 keV, with 0°c orresponds to the initial direction of the proton beam.
demonstrated that the magnetic field correction factors k Q B  of the Roos chamber depend on the magnetic field as well as the initial proton energy.Although a small magnetic field dependence of the dose conversion factor / , especially at the lowest investigated energy that can be largely attributed to the deflection of the primary proton beam by the magnetic field, the behavior of k can be considered as constant within the uncertainty of the results up to 1 T.For higher energies, the k Q B  factors increase up to a maximum at around B  = 0.5 T and descend again when the magnetic field is increased further.Generally, k Q B  combination of D and E resulted in the overall observed behavior of k Q B  with an increasing portion towards a maximum at around B  = 0.5 T and a decreasing portion with further increasing magnetic field.G.As the gyroradius of the electrons increases with their energy, the effects described in both D and E are energy dependent.Consequently, k Q B  is also energy dependent, where k Q B  (1).The magnetic field dependence of the factors k Q B  is shown to depend on the energy of the proton beam with a maximum at around B  neglible, except for the lowest investigated proton energy of 97.4 MeV, where a ratio of 1.0022(1) was obtained at B  = 1 T. Detailed simulations have revealed that the behavior of k Q B 

Table 1 .
Simulation settings in GATE/Geant4 used in this study.