Dielectric assist accelerating structures for compact linear accelerators of low energy particles in hadrontherapy treatments

Dielectric Assist Accelerating (DAA) structures based on ultralow-loss ceramic are being studied as an alternative to conventional disk-loaded copper cavities. This accelerating structure consists of dielectric disks with irises arranged periodically in metallic structures working under the TM₀₂-π mode. In this paper, the numerical design of an S-band DAA structure for low beta particles, such as protons or carbon ions used for Hadrontherapy treatments, is shown. Four dielectric materials with different permittivity and loss tangent are studied as well as different particle velocities. Through optimization, a design that concentrates most of the RF power in the vacuum space near the beam axis is obtained, leading to a significant reduction of power loss on the metallic walls. This allows to fabricate cavities with an extremely high quality factor, over 100,000, and shunt impedance over 300 MΩ/m at room temperature. During the numerical study, the design optimization has been improved by adjusting some of the cell parameters in order to both increase the shunt impedance and reduce the peak electric field in certain locations of the cavity, which can lead to instabilities in its normal functioning.

1 Introduction

Over many years, extensive research has been dedicated to room-temperature disk-loaded copper radio frequency (RF) structures, which have found diverse applications spanning fundamental science [1], cancer therapy [2], and various industrial activities [3]. Nevertheless, one of the foremost challenges in the realm of RF cavities for accelerators lies in achieving high accelerating gradients while minimizing energy consumption. Notably, the Compact Linear Collider (CLIC) project [4] accomplished a remarkable milestone by attaining a gradient of 100 MV/m with X-band normal conducting copper structures. This High Gradient (HG) technology is currently undergoing a seamless transition from linear colliders to various domains, including compact linear accelerators tailored for Hadrontherapy treatments [5–7]. Offering a promising avenue to enhance the energy efficiency of traditional disk-loaded copper structures is the adoption of dielectric-loaded accelerating (DLA) structures [8–10].

A DLA structure consists of a dielectric tube surrounded by a conducting cylinder. The dielectric decreases the phase velocity as well as the ratio of the peak electric field to the average accelerating gradient, which is about unity [11, 12]. In dielectric breakdown studies, a surface field threshold of 13.8 GV/m was observed for 30–330 fs pulse length at THz frequencies [13]. Concerning cavity testing, no instances of breakdown were observed at X-band with a 200 ns pulse length at an accelerating gradient of 8 MV/m [14] and 15 MV/m [15] for DLAs. In contrast, a significantly higher accelerating gradient of 102 MV/m was achieved without breakdown for a 10 ns pulse length in the case of Dielectric Disk Accelerating (DDA) cavity [16], within the X-band frequency range. However, multipactor discharges were observed in DLA and DDA [17, 18]. Thus, the main issues limiting the performance of these structures are surface multipactor and RF breakdowns due to strong local field enhancement in micro-scale vacuum gap in the dielectric joint [19, 20].

The concept of DLA structure was proposed in the 1940’s [21–24], and first experimental measurements were carried out in the 1950s [25–27]. However, disk-loaded metallic structures were more successful in that time due to their high quality factor and field holding capabilities. Recently, thanks to a remarkable progress in new ceramic materials with high dielectric permittivity (ɛ_r > 20), low loss (tan δ < 10^–4) [28–30], and ultra-low loss (tan δ < 10^–5) [31–33], DLA structures are again being studied for multiple applications such as dual-layered DLA structure [34], a hybrid dielectric and iris-loaded accelerating structure [35], and a DDA structure [36]. Some examples of these dielectrics are fused silica, chemical vapor deposition (CVD) diamond or alumina, among other ceramics, some of which have been experimentally tested with high-power wakefield at Argonne National Laboratory [37, 38].

Based on these technologies, a Dielectric Assist Accelerating (DAA) structure proposed by Satoh et al. [39–41] at C-band frequency is of particular interest since it achieved extremely high quality factor and shunt impedance. Later, this design was studied at X-band as a proposal for future linear accelerators [42] due to its high field holding capability. Building on these developments, a DAA structure for low beta particles operating at low frequency (S-band) is studied for the first time in this work, as a solution for compact linear accelerators of low energy and low beam current, such as medical accelerators for Hadrontherapy treatments. Unlike the case of study of [42], hadrontherapy treatments make use of very low current (0.1–1 nA) [43]. Consequently, wakefields are not excited in this kind of accelerators. In addition, the choice of S-band, at the cost of slightly decreasing the electromagnetic performance of the cavity compared with higher frequencies, is much more accessible for industrial production at lower cost. As well, the larger size of the ceramic iris at lower frequencies increases the strength and rigidity of the disks.

Section 2 describes a numerical study of an efficient S-band DAA structure operating under the TM₀₂-π accelerating mode for four different dielectrics (CVD Diamond, MgO, MgTiO₃, BaTiO_x) and different particle velocities (β = v/c = {0.4, 0.5, …, 1}), where c is the speed of light in vacuum. This mode allows to reduce power loss on the conducting wall, achieving very high quality factor Q₀, and shunt impedance Z_eff, at room temperature if the right dielectric material is chosen. An improvement in the optimization approach, by taking into account the iris thickness allows to enhance the Q₀ by approximately 15% and energy efficiency by around 50%. A comparison with high-gradient copper-disk loaded structure for compact linear accelerators for medical use is shown.

2 Methodology

On the contrary to conventional disk-loaded copper structures, that operate in a TM₀₁ mode and achieve high Z_eff/Q₀, DAA structures operate under a TM₀₂ mode in order to reduce the surface field and increase the quality factor. The dielectric allows to decrease the size of the structure and concentrate the electromagnetic energy near the beam axis, which consequently reduces copper losses and increases the shunt impedance. Thus, it is crucial that the dielectrics cost low power loss, show good thermal conductivity and withstand high electromagnetic fields.

The DAA structures [39, 42] consist of axially symmetric dielectric cylinders with irises periodically arranged in a metallic enclosure operating in standing wave π-mode, as illustrated in Figure 1.

FIGURE 1

FIGURE 1. (A) Conceptual schematic of an infinite DAA structure. (B) Longitudinal cross section geometry of a regular cell of a DAA acelerator structure.

DAA design starts by optimizing parameters for the regular cell in order to maximize the Q₀ and the Z_eff for the resonant frequency of interest. The longitudinal cross section of the regular cell can be seen in Figure 1, where r₀ is the aperture radius, r_c is the corner fillet radius, a₁ is the inner radius, b₁ is the outer radius, c₁ is the copper waveguide radius, d₁ is the dielectric disk thickness, also known as iris, and L₁ is the constant periodic length.

Once L₁, r₀ and r_c are selected based on criteria explained later, the combination of a₁, b₁, c₁ and d₁ determines the figures of merit in the cavity, such as resonant frequency, quality factor and shunt impedance of the accelerating mode TM₀₂-π. Thus, in this paper a new step is added to the optimization analysis, looking for the value of d₁ which, in combination with the three radius selection, maximizes the shunt impedance of the cavity, in spite of fixing this value to $d_1={\lambda }_0/(4\sqrt{{\epsilon }_r})$ where λ₀ = c/f₀ is the free space wavelength, as was done in previous studies [39, 42]. Thanks to the axial symmetry, optimum parameters can be calculated using the SUPERFISH tool [44], in addition results have been cross checked using HFSS [45]. Periodic boundary conditions were applied to regular cell in order to simulate an infinite long structure.

The resonant frequency goal was fixed at f₀ = (3,000 ± 2) MHz, L₁ = βλ₀/2, r₀ = 2 mm for comparison with high gradient copper structures and r_c = d₁/2. Then, in order to find the best values for d₁ and the radii a₁, b₁, c₁, a two step scan needs to be done. First, d₁ is fixed at its initial value $d_1=d_0={\lambda }_0/(4\sqrt{{\epsilon }_r})$, so the resonant frequency f₀ is determined by the combination of a₁, b₁ and c₁. Once a₁ and c₁ are fixed, the value of b₁ can be recalculated for the given frequency f₀ using the numerical solver. Following this methodology, optimum parameters can be found by sweeping through a₁ and c₁, finding the corresponding value of b₁, Q₀ and Z_eff/Q₀. This is shown in Figure 2 for MgTiO₃ with ɛ_r = 16.66 and tan δ = 0. It must be noted here that for low electric permittivity and low particle velocity, it might be impossible to find a geometry of a regular cell with the desired resonant frequency.

FIGURE 2

FIGURE 2. Numerical unloaded quality factor Q₀ (A), shunt impedance over quality factor Z_eff/Q₀ (B) and geometric parameter b₁ (C) as a function of geometrical parameters a₁ and c₁ for a regular cell using ideal dielectric MgTiO₃ and β = 0.6. White region is due to the absence of a valid solution for the geometry.

This process is repeated for each value of a second swept in d₁ = ξd₀, where ξ is the normalized iris thickness. This allows to find a better solution in terms of Z_eff, as illustrated in Figure 3 for MgTiO₃ with ɛ_r = 16.66 and tan δ = 3.43 × 10⁻⁵.

FIGURE 3

FIGURE 3. Numerical unloaded quality factor Q₀ and shunt impedance over quality factor Z_eff/Q₀ as a function of normalized iris thickness for a regular cell using real dielectric MgTiO₃ and β = 0.6.

The ratio of the peak electric field E_p and the peak magnetic field H_p to the average accelerating electric field E_a usually limits the achievable accelerating gradient for conventional iris-loaded metallic structures, where

$E_a=\frac{1}{L_1}{\int }_{-L_1/2}^{L_1/2}{E}_z\left(\mathrm{0,0},z\right)\cos \left(\omega \frac{z}{\beta c}\right)dz,(1)$

where E_z is the longitudinal component of the electric field, ω = 2πf is the angular frequency and z is the longitudinal spatial coordinate. Field profiles for this structure are illustrated in Figure 4.

FIGURE 4

FIGURE 4. Electric field distribution E/E_a (A) and magnetic field distribution H/E_a (B) for the accelerating mode TM₀₂-π in a half regular cell using MgTiO₃, β = 0.6 and ξ = 2.

An optimization was done using four different ceramics that have been already used for RF dielectric cavities [11, 16, 40], whose electromagnetic properties can be found in Table 1, and particle velocity β = {0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1}. During these studies, it was observed that the geometry optimization depends mainly on the particle velocity β and the relative electric permittivity ɛ_r of the ceramic, while loss tangent tan δ determines the final value of Q₀ as well as Z_eff.

TABLE 1

TABLE 1. List of dielectrics studied in the optimization [12, 16, 39].

Energy ranges for Hadrontherapy treatments vary between 70 and 230 MeV for protons and 100–430 MeV/u for carbon ions, which correspond to particle velocities between 0.37–0.60 and 0.43–0.73, respectively [46]. Final results for the figures of merit for these designs, taking into account dielectric losses, are compared with an extension for all particle velocities of a high-gradient standing wave copper cavity designed for protons with β = 0.38 [47], as it can be seen in Figure 5.

FIGURE 5

FIGURE 5. Final results for Q₀ (A), Z_eff/Q₀ (B) and Z_eff (C) for the different values of particle velocity and different material in the ideal case (dashed lines), and taking into account dielectric losses (solid lines). Vertical dashed lines correspond to the maximum energy of protons (blue) and carbon ions (orange). The results are compared with a high-gradient cell coupled linac CCL-HG copper cavity (purple line) [47].

A summary of the geometrical and electromagnetic parameters is included in Table 2.

TABLE 2

TABLE 2. List of optimum results for a selection of geometries.

Note the difference on the performance between the ideal and relativistic case. For the ideal case, the Q₀ is over two orders of magnitude higher compared to HG copper structures. As it can be seen in Figure 5, the Q₀ increases with ɛ_r and it is very sensitive to losses in the dielectric, though results are better than normal copper cavities. One caveat of this design is that a high amount of electrical energy is stored inside the dielectric, which is not going to be used to accelerate the beam. As a consequence, energy efficiency worsens, resulting in lower values of Z_eff/Q₀. Energy efficiency improves for larger ɛ_r and, as expected, does not depend on the dielectric tan δ. The performance of the structure will be given by the shunt impedance, which is a compromise between Q₀ and Z_eff/Q₀. For the ideal case, the final result will be better for higher ɛ_r. However, due to the high sensitivity of Q₀ with dielectric losses, the characteristic value of tan δ of the material is crucial on the real performance of the final structure. In addition, the performance increases also for higher particle velocities.

3 Results

Once the optimization of the geometry has been performed, the electromagnetic performance of the regular cell has to be considered as a component of a real accelerator system. In order to do so, electromagnetic losses are studied in detail as well as the dispersion relation of the regular cell. In addition, the field instabilities and singularities which can lead to multipactor or RF breakdown discharges are taking into account to minimize risk at high power test. Besides, the consequences of using coating to suppress multipactor in the RF performance is also deliberated.

3.1 Dielectric loss tangent

The advantage of working under the TM₀₂-π mode, is that metallic losses are highly suppressed and, consequently, the DAA regular cell performance will be determined mainly by the quality of the dielectric in terms of its tan δ.

Dielectric and metallic losses are given by [39],

$P_d=\frac{1}{2}\omega {\epsilon }_0{\epsilon }_r\tan \delta {\int }_V|\mathbf{E}{|}^{2}d\tau ,(2)$

$P_c=\frac{1}{2}{R}_s\int |\widehat{\mathbf{n}}\times \mathbf{H}{|}^{2}dS,(3)$

respectively, where $R_s=\sqrt{\omega {\mu }_0/(2\sigma )}$ is the surface resistance, ɛ₀ɛ_r is the electric permittivity of dielectric, E is the electric field, H is the magnetic field, $\widehat{\mathbf{n}}$ is the unitary normal vector to the surface, μ₀ is the magnetic permeability of vacuum and σ = 5.8 × 10⁷ S/m is the copper electric conductivity.

A graphical representation of both surface and volumetric loss densities are illustrated in Figure 6. As the loss tangent depends strongly on the manufacturing process for the ceramic fabrication, values of Table 1 are just references from previous experimental measurements. Therefore, it is of great importance to study the dependence of regular cell performance as a function of the tan δ of the material, as illustrated in Figure 6, where an exponential decrease of the cavity performance can be seen for values of tan δ > 10^–5.

FIGURE 6

FIGURE 6. Surface metallic losses (A) and volumetric dielectric losses (B) density for accelerating mode TM₀₂-π in a regular cell using MgTiO₃, β = 0.6, stored electromagnetic energy W = 1 J and ξ = 2. Shunt impedance as a function of tan δ (C) for different materials and β = 0.6. Experimental tan δ values for each material are shown.

3.2 Dispersion relation

The overlapping between adjacent modes is a typical problem from the tunability and operational point of view for periodic RF accelerating structures, which is the case for the standing-wave accelerating structure studied in this work. In addition, good electromagnetic coupling between consecutive cells is also a key factor in order to determine the maximum number of cells per cavity.

Electric coupling between consecutive cells improves for lower electric permittivity and particle velocity, as it is illustrated in Figure 7. It can be observed that the TM₀₂ mode is strongly electrically coupled, so there is no need for coupling cells between regular cells.

FIGURE 7

FIGURE 7. Electrical coupling bandwidth for different materials and particle velocities with normalized iris thickness ξ = 1.5. The bandwidth is defined as BW = (f_π − f₀)/f_π ⋅ 100%, where f_π and f₀ are the resonant frequencies of π and 0 mode respectively.

Dispersion curves for the second order mode and the next higher frequency axisymmetric mode are depicted in Figure 8 for each material for two different normalized iris thickness. It can be seen that for low electric permittivity material, the higher order mode crosses the 3 GHz point and, therefore, overlapping cannot be avoided. In addition, as the iris becomes thicker, it can be seen that higher order modes with a phase advance of π get closer to the resonant frequency and they can even cross this point for thicker irises. Consequently, electric permittivity and normalized iris thickness are bounded by the overlapping process.

FIGURE 8

FIGURE 8. Dispersion curve of the accelerating mode TM₀₂ (solid line) and next higher frequency axisymmetric mode (dashed line) for different dielectric materials and β = 0.6 and normalized iris thickness ξ = 1 (A) and ξ = 1.5 (B).

A field distribution of the higher frequency modes are shown in Figure 9 for a phase advanced of 0°, which corresponds to the mode TM₀₃ at f = 3.354 GHz and for a phase advance of 180°, corresponding with a dielectric mode at f = 3.184 GHz.

FIGURE 9

FIGURE 9. Next higher frequency axisymmetric mode for Δϕ = 0° at f = 3.354 GHz (A) and Δϕ = 180° at f = 3.184 GHz (B) for D16, β = 0.6 and ξ = 1.5.

3.3 Surface field studies

As it is shown in Figure 10, E_p/E_a for the optimal geometry decreases with the material permittivity and particle velocity and it is always below 4 which is the value obtained for CCL-HG cavity [47]. This ratio is one of the main limiting factors for HG cavities, since it is related with breakdowns production. Besides, it was observed that this ratio also decreases for thinner irises. Therefore, the performance of the DAA regular cell improves with particle velocity and higher electric permittivity being able to improve current values for room-temperature copper cavities.

FIGURE 10

FIGURE 10. Ratio of peak electric field and average accelerating field for different materials and particle velocities.

The existence of sharp angles and triple junction points in the original design can lead to singularities in the surface electric field. As a consequence, field instabilities, RF breakdowns or multipactor discharges can emerge.

Regarding the triple junction point (point B in Figure 11), assuming zero conductivity in dielectric and flat metal wall, the electric field increases as |E|∝ rⁿ⁻¹, where r is the radial distance to the triple junction point and n follows [19]:

$\cot n\alpha +{\epsilon }_r\cot n\left(\pi -\alpha \right)=0(4)$

where α is the angle of vacuum between dielectric and metal and ɛ_r is the relative electric permittivity of the dielectric.

FIGURE 11

FIGURE 11. (A) Geometry modification. Electric field magnitude along path CD (B) and AB (C) for different corner radius for D16. Electric field is normalized for 1 MV/m of accelerating voltage.

From Eq. 4 it can be concluded that if α < 90° then n < 1, leading to infinitely large electric field in the junction, whereas if α > 90° then n > 1, leading to null electric field. Only the case with α = 90° leads to n = 1, implying a non-zero and non-singular value. However, we are just interested in avoiding singularities, which can be achieved by adjusting α ≥ 90°, which was already satisfied in the original design. In addition, sharp metallic corners are another source of field singularities which must be avoided.

Regarding the dielectric corners in the junction between the dielectric ring and the iris, it was observed that sharp geometries also lead to field divergences. Thus, the geometry was changed as shown in Figure 11 and the surface electric field for different round corners was studied for a fine mesh, as illustrated in Figure 11. In addition, this changes produced a slight increase in the Z_eff.

3.4 Coating effects

Amorphous Carbon (a-C) and Diamond Like Carbon (DLC) coatings were studied at Conseil Européen pour la Recherche Nucléaire (CERN) for Secondary Electron Yield (SEY) reduction in order to avoid multipactor discharges [48]. However, surface losses on the coating will have an impact on the electromagnetic performance of the cavity. These losses are given by

$P_s=\sigma {\int }_V|\mathbf{E}{|}^{2}d\tau =\frac{1}{2R}\int |\widehat{\mathbf{n}}\times \mathbf{E}{|}^{2}dS,(5)$

where R is the sheet surface resistance of the coating.

Figure 12 illustrates Q₀ as a function of the sheet resistance (in ohms per □) for different cases: no coating, dielectric fully covered with coating, internal coating (which corresponds with coating in region CD) and external coating (which corresponds with coating in region AB). The surface resistance of DLC coating was above 1 MΩ per □ and could not be measured, while a-C samples measurements are marked with black dashed lines.

FIGURE 12

FIGURE 12. (A) Graphical representation of coating losses in logarithmic scale for a sheet resistance Rs = 37,000 Ω per □ for D16, β = 0.6, ξ = 2 and W = 1 J. (B) Comparison of unloaded quality factor of a regular cell partially coated in the external or internal regions of the cavity, full covered and without coating as a function of the sheet resistance.

As it can be observed in Figure 12, Q₀ rapidly decreases for low resistance coatings and therefore thin films or materials with high resistivity are useful coatings to improve the electromagnetic performance.

3.5 Thermal simulations

In order to estimate the required cooling system and the mechanical stress and deformation induced by RF losses, thermal simulations were carried out using the ANSYS software [49]. Volumetric and surface losses were used as input for steady thermal simulations with 3 cm of copper wall fixing the external temperature at 22°C as boundary condition. Simulations were done for different geometries and ceramics, as illustrated in Table 3. Ultra high pure alumina was used for simulations instead of MgO and BaTiO_x because of its higher thermal conductivity in order to evaluate three different meaningful values.

TABLE 3

TABLE 3. List of dielectrics used for thermal simulations.

For numerical simulations, an accelerating gradient of 50 MV/m was used, with a duty cycle D = 0.075 × 10⁻³. The duty cycle is defined as

$D=\frac{\tau }{T},(6)$

where τ is the pulse width and T is the total period of the signal. A graphical solution for Al₂O₃ for β = 1 is shown in Figure 13.

FIGURE 13

FIGURE 13. Steady temperature distribution for Al₂O₃ and β = 1.

As shown in Figure 13, the maximum temperature is reached close to the aperture of the ceramic, with a decreasing temperature gradient towards the copper metallic enclosure, which barely changes thanks to its high thermal conductivity (κ = 400 W⋅ m⁻¹⋅ K⁻¹). The maximum temperature reached for different geometries and materials is illustrated in Figure 14.

FIGURE 14

FIGURE 14. Maximum temperature reached for different ceramics and particle velocities.

The temperature is higher for lower particle velocity while it seems to saturate at around β = 0.7 following the behaviour of the Z_eff. In addition, very low thermal conductivity, as in MgTiO₃, leads to temperatures beyond acceptable limits regarding stress and deformation tolerances, even though we are still far from the fusion point.

4 Conclusion

DAA structures for low β particles have been studied for the first time, proving the potential to improve the performance of current room-temperature copper cavities. This study shows improvements in the optimization process and optimal results for an S-band DAA cavity as a solution for compact linear accelerators for Hadrontherapy treatments.

Working under the TM₀₂-π mode, copper ohmic losses can be highly reduced by accumulating electrical energy inside the dielectric. From these studies we could conclude that cavity efficiency increases for higher particle velocity and electric permittivity. However, due to the high energy density inside the dielectric, the cavity performance will be limited by dielectric losses. Therefore, reaching low dielectric loss tangent is a fundamental key in the fabrication of ceramics in particular for DAA cavities.

Iris thickness plays also a fundamental role in the cell optimization by increasing the accelerating voltage and also by reducing the electric energy density inside the ceramic by decreasing dielectric losses. As a consequence, materials with higher loss tangent have thicker optimum irises than ideal geometries.

In addition, the low E_p in metallic walls in combination with high breakdown threshold of dielectrics, potentially allow DAA cavities to reach higher gradients without producing RF breakdowns, after multipactor suppression. This ratio decreases for high particle velocity, high electric permittivity and thin irises.

High electric coupling between consecutive cells has been observed for all kind of geometries. In addition, it was shown that coupling improves for lower particle velocity and lower electric permittivity. However low electric permittivity materials, such as CVD diamond, suffer from mode overlapping. In addition, thicker irises produce the excitation of more modes whose resonant frequencies are close to our operational frequency. As a result, the final design must find a compromise between an optimum electromagnetic design, which is achieved for thicker irises and low mode overlapping and low peak electric field, which improve for thinner irises.

Dielectric corners have been rounded in order to smooth the surface electric field. Moreover, stability studies of triple junction point were performed concluding that in order to avoid electric field singularities, the vacuum angle between dielectric and copper must be α ≥ 90° and metallic sharp angles must be avoided.

Multipactor is one of the main limitations of DLA cavities due to high SEY of ceramics. Because of that, thin coating with low SEY is used for multipactor suppression. However, surface resistance of coating will have an effect on RF performance that must be studied in advance. Numerical simulations showed that low resistance coatings are unacceptable from an electromagnetic point of view, which implies that only high resistance materials or very thin coatings can be used in order to reduce multipactor.

Finally, thermal conductivity of the ceramic is found to be a crucial parameter also on the design to avoid overheating of the cavity leading to high deformation and stress. Thus, a lower bound value is set around 20–30 W⋅ m⁻¹⋅ K⁻¹ depending on particle velocity and duty cycle.

Data availability statement

The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.

Author contributions

PM-R: Conceptualization, Formal analysis, Software, Writing–original draft. DE: Conceptualization, Supervision, Funding acquistion, Writing–review and editing. AG: Conceptualization, Methodology, Supervision, Funding acquistion, Writing–review and editing. BG: Conceptualization, Supervision, Funding acquistion, Writing–review and editing. CB: Writing–review and editing. DG-I: Writing–review and editing. NF-M: Writing–review and editing. PM-L: Writing–review and editing. EM: Writing–review and editing. AM: Writing–review and editing. JF: Funding acquistion, Writing–review and editing.

Funding

The author(s) declare financial support was received for the research, authorship, and/or publication of this article. Work supported by Ministerio de Universidades (Gobierno de España) under grant number FPU19/00585 and EST22/00739. The authors declare that this study received funding from CERN.

Conflict of interest

AG was employed by CERN.

The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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