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 Q ₀, 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 Q ₀ 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.

On the contrary to conventional disk-loaded copper structures, that operate in a TM₀₁ mode and achieve high Z _eff /Q ₀, 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.

DAA design starts by optimizing parameters for the regular cell in order to maximize the Q ₀ 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 r ₀ is the aperture radius, r _c is the corner fillet radius, a ₁ is the inner radius, b ₁ is the outer radius, c ₁ is the copper waveguide radius, d ₁ is the dielectric disk thickness, also known as iris, and L ₁ is the constant periodic length.

Once L ₁, r ₀ and r _c are selected based on criteria explained later, the combination of a ₁, b ₁, c ₁ and d ₁ 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 d ₁ which, in combination with the three radius selection, maximizes the shunt impedance of the cavity, in spite of fixing this value to d 1 = λ 0 / ( 4 ε r ) where λ ₀ = c/f ₀ 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 f ₀ = (3,000 ± 2) MHz, L ₁ = βλ ₀/2, r ₀ = 2 mm for comparison with high gradient copper structures and r _c = d ₁/2. Then, in order to find the best values for d ₁ and the radii a ₁, b ₁, c ₁, a two step scan needs to be done. First, d ₁ is fixed at its initial value d 1 = d 0 = λ 0 / ( 4 ε r ) , so the resonant frequency f ₀ is determined by the combination of a ₁, b ₁ and c ₁. Once a ₁ and c ₁ are fixed, the value of b ₁ can be recalculated for the given frequency f ₀ using the numerical solver. Following this methodology, optimum parameters can be found by sweeping through a ₁ and c ₁, finding the corresponding value of b ₁, Q ₀ and Z _eff /Q ₀. 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.

This process is repeated for each value of a second swept in d ₁ = ξd ₀, 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.

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 = 1 L 1 ∫ − L 1 / 2 L 1 / 2 E z 0,0 , z cos ω z β c d z , (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.

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 Q ₀ as well as Z _eff .

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.

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

Note the difference on the performance between the ideal and relativistic case. For the ideal case, the Q ₀ is over two orders of magnitude higher compared to HG copper structures. As it can be seen in Figure 5, the Q ₀ 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 /Q ₀. 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 Q ₀ and Z _eff /Q ₀. For the ideal case, the final result will be better for higher ɛ _r . However, due to the high sensitivity of Q ₀ 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.

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.

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.