Design of a 162.5 MHz Superconducting Radio-Frequency Quadrupole for High-Intensity Proton Acceleration

Abstract

Superconducting (SC) radio-frequency quadrupoles (RFQs) have exhibited outstanding performance in transmitting and accelerating high-current continuous-wave (CW) ion beams. They can complete beam acceleration at a much higher gradient and with much lower power consumption compared with normal conducting (NC) RFQs. In this study, we introduce a novel SC RFQ scheme operating at 162.5 MHz to accelerate 10 mA proton beams from 30 keV to 2.5 MeV. It will be used as a crucial component for a neutron source dedicated to Boron Neutron Capture Therapy (BNCT) and neutron imaging projects. For efficient transmission of proton beams, we selected a relatively high inter-vane voltage of 240 kV, and the beam dynamics design yielded satisfactory results. Subsequently, RF design and multi-physics analysis were carried out to validate the reliability of the design. A 30-centimeter-long cavity was specifically designed for the vertical test and allowed for a thorough evaluation of the performance of the SC RFQ after post-treatments. Additionally, the tuning design of the 30 cm cavity was also carried out.

1. Introduction

1.1. RFQ Concept

RFQ accelerators can simultaneously focus, bunch, and accelerate low-energy and high-intensity beams; high-intensity RFQ accelerators have a number of applications such as a common front-end injector for high-energy accelerators [1,2], accelerator-based diagnostics in magnetic fusion devices [3,4], and a compact neutron source for radiography and cancer therapy [5,6].

The basic types of RFQ include the four-vane type and the four-rod type, with the former being more suitable for high-frequency ranges and acceleration of light ions [7]. Consequently, we chose the four-vane structure for this 162.5 MHz proton RFQ. The electrode configuration of the four-vane RFQ is illustrated in Figure 1, operating in a TE210 mode. As we can see from Figure 1a, the four electrodes symmetrically encircle the axis and are excited with $\pm V/2$ RF voltages, generating an electric quadrupole field for transverse beam focusing. As the electric force is independent of velocity, the RFQ is especially effective in the focusing and bunching of low-velocity ions compared with the conventionally used magnetic force [8]. To establish a longitudinal accelerating field, the electrodes need to be modulated, i.e., periodic transverse displacements relative to the axis. Figure 1b illustrates the varying distances between the electrode and the axis, denoted as a at the closest point and ma at the farthest, where m represents the modulation parameter. The length of a unit cell is $\beta \lambda /2$, which is half of the varying period. When m = 1, the electric field has no longitudinal component and cannot accelerate the beam, while the focusing effect is the strongest. As m increases, the acceleration effect of the accelerator becomes better, but the focusing effect becomes weaker. The modulation phase difference between the horizontal and vertical electrodes is $\pi$ [9].

Figure 1c provides a visualization of the electromagnetic (EM) field distribution within the cross-section of the RFQ, showing the transverse distribution of the electric field and the longitudinal distribution of the magnetic field. Since the electric field of a four-vane RFQ is mainly distributed in the region close to the axis, and the magnetic field is mainly distributed in the region close to the cavity wall, we can effectively characterize these fields using a simplified lumped circuit model. Each quadrant can be analyzed as an individual resonant cavity, which is represented by capacitance C′ and inductance L′ as illustrated in Figure 1d. The resonant frequency of the four-vane RFQ can be expressed mathematically as

$$f=\frac{1}{2\pi \sqrt{L^{\prime }{C}^{\prime }}}.$$

(1)

Within the cavity, C′ (F/m) is associated with the electric field, while L′ ($\mathrm{H}·\mathrm{m}$) corresponds to the magnetic field. Before convenient simulation software was available, semi-empirical formulas were used to estimate the values of C′ and L′ of cavities, so as to obtain the resonant frequency of them [10].

1.2. SC RFQ

The RFQ accelerator is well suited for the acceleration and focusing of low-energy beams, with NC RFQs demonstrating robust operating stability under pulse mode [11]. However, as large-scale scientific devices increasingly shift towards CW operation and higher current requirements, some RFQs’ power consumption even reaches the level of megawatts [12,13,14], which restricts the stability and economy of RFQs’ long-term operation. Fortunately, superconductivity offers a solution by significantly reducing power losses and enabling CW operation; the combination of these two technologies presents a pathway for the more efficient acceleration of high-current beams [15].

Compared with NC RFQs, an SC RFQ has the following advantages:

• Reduced Power Loss: The high quality factor (Q) of the SC cavity results in significantly lower power loss, thereby diminishing the operating costs of linear accelerators [16]. The Q value of the SC cavity is typically five orders of magnitude higher than that of the NC cavity. Even taking the power consumption to maintain the low-temperature environment required for the superconducting state into account, the total power requirement of an SC cavity is about two orders of magnitude lower than that of an NC cavity.
• Higher Accelerating Gradient: The accelerating field of the RFQ increases as the inter-vane voltage applied between the electrodes increases. As discussed in Section 2, the SC RFQ can withstand higher voltages between electrodes without being constrained by RF power loss. This capability results in a significantly higher accelerating gradient compared to the NC RFQ, which shortens the cavity’s length and makes the device more compact [17].
• Enhanced Beam Aperture: The beam-focusing force is proportional to the inter-vane voltage and inversely proportional to the aperture [18]. The SC RFQ, without being constrained by RF power loss, is capable of accommodating higher voltages between electrodes. This flexibility allows for a significantly larger aperture compared to NC RFQs, which has two benefits: firstly, it increases the transverse acceptance of the accelerator, promoting the efficient transmission of high-intensity beams; secondly, it mitigates beam instability arising from the beam–cavity interaction.

As discussed above, SC RFQs are going to be an indispensable part of the low-energy end of high-intensity accelerators. However, due to numerous challenges in the design and construction of SC RFQs, only the National Institute for Nuclear Physics (INFN) in Italy successfully built two 80 MHz SC RFQs around 2000, which have been in steady operation for a long time [19,20,21,22,23]. It is urgent to conduct further research on SC RFQs.

This paper describes the design of a 162.5 MHz 10 mA CW proton SC RFQ and is organized as follows: Firstly, a comprehensive introduction of the beam dynamics design is given, focusing on optimizing acceleration and transmission efficiency; secondly, the RF design of the cavity is presented, emphasizing structural enhancements to minimize peak surface electric and magnetic fields; thirdly, the paper encompasses a multi-physics analysis and a tuning study of a 30 cm model cavity, providing conclusive evidence regarding the accelerator’s reliability.

2. Beam Dynamics Design

2.1. Design Considerations

The design goal of the SC RFQ is to attain optimal beam acceleration and transmission efficiency. Acceleration efficiency depends on the accelerating gradient $E_{acc}$, which is defined as the ratio of the maximum accelerating voltage $V_{acc}$ experienced by the charged particle to the effective acceleration length $d$ of the cavity:

$$E_{acc}=\frac{V_{acc}}{d}=\frac{1}{d}\left|{{\displaystyle \int }}_0^d{E}_z\left(z\right)e^{i{\omega }_{0}z/\beta c}dz\right|,$$

(2)

its unit is V/m. Here, ${\omega }_0$ is the angular frequency of the cavity mode, $\beta$ denotes the relativistic velocity of the ions, and c is the speed of light. In the RFQ accelerator, the axial electric field distribution is given by

$$E_z=\frac{kAV}{2}{I}_0\left(kr\right)coskz,$$

(3)

where A and V are the accelerating parameter and inter-vane voltage of the RFQ, respectively, and $k=\frac{2\pi }{\beta \lambda }$ is the wave number. As described above, $E_{acc}$ is proportional to the inter-vane voltage $V$. However, the peak surface electric field $E_{sp}$ and magnetic field $B_{sp}$ of the cavity also increase with the inter-vane voltage. Excessive peak surface fields may lead to cavity quenching or multipacting effects. Consequently, the maximum accelerating gradient of an SC cavity is constrained by these fields. Therefore, in the design phase, it is imperative to minimize the peak surface electric and magnetic fields while preserving the highest achievable accelerating gradient. For SC cavities, the risk of quenching is pronounced when $B_{sp}$ exceeds 80 mT under 4 K conditions. However, advancements in post-treatment technology enhance the cavity’s ability to withstand larger surface electric fields. Consequently, the magnetic field, rather than the electric field, stands as the ultimate limiting factor determining the performance of SC cavities [24].

Transmission efficiency is affected by various dynamic design parameters, which are crucial for controlling the beams. For SC accelerators, achieving a transmission efficiency as high as 100% is imperative to prevent quenching resulting from beam interactions with the cavity wall.

Maintaining the superconducting state requires thorough consideration of cavity design to prevent quenching. Consequently, our design approach prioritizes simplicity by adopting a constant focusing strength B, a dimensionless parameter defined as [25]:

$$B=\frac{q{\lambda }^{2}V}{m_0{c}^2{r}_0{}^2},$$

(4)

where q and $m_0$ are the charge and the rest mass of the accelerated ions, $\lambda$ is the RF wavelength, and $r_0$ is the average aperture of RFQ. When B remains unchanged, $r_0$ also remains unchanged, which can simplify subsequent processing and facilitate cavity tuning.

We employed the traditional four-step method [26] developed at Los Alamos National Laboratory (LANL) for the beam dynamics design. This method divides the RFQ into four distinct sections: the radial matching section, the shaper section, the gentle buncher section, and the accelerator section. The input beam undergoes a process of gradual bunching in the first three sections, then will be rapidly accelerated to the designed energy in the final section.

The dynamics simulation was conducted using the PARMTEQM code [27]. PARMTEQM requires various input parameters, including operating frequency, ion species, beam current, input energy, inter-vane voltage, and synchronous phase, etc. In the simulation, the RFQ is divided into unit cells of length $\beta \lambda /2$, and the input parameters are gradually changed in each cell. The optimal accelerator structure generated by PARMTEQM is then selected to achieve the desired particle distribution. The design goal is to attain adequate radial focusing, transverse emittance growth, overall length, and peak surface field of the accelerator.

2.2. Choice of Beam Dynamics Parameters

2.2.1. Input Beam Emittance

At first, we should give the input beam emittance. Beam emittance characterizes the distribution of particles in a phase space, serving as a measure of beam quality. Emittance in a specific plane is linked to the area A occupied by all particles within that phase space, typically measured in $\mathrm{mm}·\mathrm{mrad}$ or $\mathrm{cm}·\mathrm{mrad}$. In our design, the input emittances of the proton beam in the x, y, and z planes are determined by the output emittances of the ion source, which are ${\epsilon }_x={\epsilon }_y=0.020 \mathrm{cm}·\mathrm{mrad}$, ${\epsilon }_z=0$.

2.2.2. Synchronous Phase

The electric field generated by the RFQ is time-varying, with opposite polarities in adjacent cells. When a particle traverses precisely half an RF period within one cell, consistently entering each cell at the same phase, it can experience continuous acceleration at a constant phase during transmission. Such a particle is referred to as the synchronous particle, and its phase is called the synchronous phase ${\phi }_s$ [28].

Choosing ${\phi }_s=-90°$ provides the RFQ with the maximum longitudinal acceptance. Therefore, we set the initial synchronous phase to −90° to effectively capture and bunch the beam. Then, we slowly increase the synchronous phase to bring it closer to the crest of the electric field waveform to enhance acceleration efficiency. In order to maintain optimal bunching and accelerating effects, the final synchronous phase is set at $-30°$.

2.2.3. Inter-Vane Voltage

In the subsequent stage of the design procedure, determining the optimal inter-vane voltage V becomes important to achieve a high accelerating gradient $E_{acc}$ while ensuring that the peak surface electric field $E_{sp}$ remains within acceptable limits. Both $E_{acc}$ and $E_{sp}$ are proportional to V. In RFQ cavities, the power consumption per unit length P/L is expressed as [29]:

$$\frac{P}{L}=\frac{E_{acc}{}^2}{\frac{R}{Q_0}·Q_0},$$

(5)

where $\frac{R}{Q_0}$ is the geometric shunt impedance in $\mathsf{\Omega }/\mathrm{m}$; it only depends on the geometry of the structure. Given that the quality factor $Q_0$ of SC cavities is typically 5 orders of magnitude higher than that of NC cavities, approximately in the range of ${10}^9–{10}^{10}$, relatively large inter-vane voltages can be applied to SC cavities and achieve remarkably low power consumption.

We have investigated the $E_{sp}$ achieved in the vertical tests of international SC half-wave resonators (HWRs). Due to advancements in processing and post-treatment technology, experimental $E_{sp}$ values have increased a lot. For instance, in the SC HWR study at the Institute of High Energy Physics (IHEP) in 2013 and the Rare Isotope Science Project (RISP) study at the Korea Institute of Basic Science in 2016, the achieved $E_{sp}$ values when field emission occurs were approximately 66.2 MV/m and 72 MV/m, respectively [30,31]. In the SC HWR experiment at Argonne National Laboratory in 2017, $E_{sp}$ reached 134 MV/m [32]. Based on the experimental data, in order to ensure the stable operation of our SC RFQ, we opted for a scheme with an inter-vane voltage of 240 kV, yielding an $E_{sp}$ of approximately 40 MV/m.

2.2.4. Input Energy

Subsequently, we experimented with different input energies of proton beams to optimize both transmission efficiency and cavity length, aiming for a high accelerating gradient. The choice of input energy significantly impacts the design parameters. The lower the input energy of the particles, the stronger the space charge effect between them, which will cause emittance growth. Higher input energies require longer shaper and gentle buncher sections to effectively bunch the beam, consequently leading to an extended cavity length [33]. Transmission efficiency, influenced by numerous factors, exhibits variations across different input energy levels.

Finally, the optimal balance is achieved when the input energy is set at 30 keV, resulting in a transmission efficiency of 100% and a relatively short cavity length of 2.03 m.

2.2.5. Final Design Parameters

Figure 2 provides an overview of the main parameters in the SC RFQ design. The cavity is divided into 112 cells. To get an optimal performance, we set the focusing strength B to be 13, ensuring effective beam focusing for high-intensity beams and an improved transmission efficiency. Additionally, we carefully control the maximum value of the modulation parameter m to approximately 2.07. This control enhances our ability to manage the beam envelope and facilitates the efficient transmission of high-intensity beams. The design strategically achieves a smooth variation of various parameters along the SC RFQ, mitigating abrupt changes in the accelerator that could lead to beam instability.

The final dynamics design parameters are summarized in

Table 1. Dynamics design parameters of the SC RFQ.

Parameters	Value (SC)	Value (NC)
Frequency (MHz)	162.5
Particles	proton
Beam current (mA)	10
Input energy (MeV/u)	0.030
Output energy (MeV/u)	2.55	2.54
Inter-vane voltage (kV)	240	70
Focusing parameter B	13.0	10.0
Average aperture $r_0$ (mm)	8.18	4.86
Minimum aperture a (mm)	5.10	2.12
Synchronous phase ${\phi }_s$ ($°$)	−90~−30
Peak surface electric field (MV/m)	41.8	20.8
Transmission efficiency (%)	100.0	99.1
Cell number	112	190
Cavity length (m)	2.03	4.20
Q	$2.52\times {10}^9$	9949
Power consumption (W)	2.48	$1.49\times {10}^5$

, where we compared the SC and NC situations. Both the average aperture $r_0$ and the minimum aperture a of the SC cavity surpass those of the NC cavity, significantly reducing beam loss. Despite the nearly threefold increase in inter-vane voltage under the SC condition, resulting in a larger $E_{sp}$, it remains within established safety limits. Moreover, the transmission efficiency of the SC cavity reaches 100.0%, higher than the value under the NC condition, while the length of the SC cavity is only half of the NC length, which can significantly enhance acceleration efficiency and save construction costs. As for the power loss of the cavity, the SC case demonstrates a power consumption about five orders of magnitude lower than that in the NC case, as detailed in Section 1.

In addition, we conducted a comparative analysis of key parameters in this SC RFQ design with two counterparts, SRFQ1 and SRFQ2, proposed by INFN, as presented in

Table 2. Parameters comparison between this design and INFN SC RFQs [22,34].

Parameters	This Design	INFN-SRFQ1	INFN-SRFQ2
Frequency (MHz)	162.5	80	80
Particles	proton	${}_{}{}^{238}U{}^{28+}$	${}_{}{}^{238}U{}^{28+}$
Inter-vane voltage V (kV)	240	148	280
Average aperture $r_0$ (mm)	8.18	8.00	15.30
$E_{sp}$ (MV/m)	41.8	24.0	25.5
$B_{sp}$ (mT)	27.5	25.0	30.0

. The INFN-SRFQs are built to accelerate heavy ions with a charge-to-mass ratio (Z/A) of 8.5, and its cavity operating frequency is lower than our design, at 80 MHz. In RFQ cavities, a lower frequency corresponds to a smaller tolerable $E_{sp}$, and thus, INFN SC RFQs are configured with $E_{sp}$ values around 25.0 MV/m. According to Equation (3), $r_0$ increases with V if the wavelength (i.e., the frequency of the cavity) is held constant, and it increases with the wavelength (i.e., decreasing cavity frequency) if V remains constant. The $r_0$ values for these three cavities listed in the table adhere to this principle. The $B_{sp}$ values for all three SC RFQs are consistent and all far below the upper limit of the magnetic field.

2.3. Simulation Results of Beam Transport

The simulation results generated by PARMTEQM are plotted in the figures below, utilizing 10,000 macro particles with a 4D water-bag distribution. Figure 3 displays the transverse and longitudinal beam envelopes along the RFQ, revealing focused beams within a confined region of the cavity and achieving a remarkable 100.0% transmission efficiency. In Figure 4, the phase space projection at the entrance and exit of the RFQ demonstrates well-controlled emittance growth in both transverse and longitudinal planes. The full width at half maximum (FWHM) phase spread and energy spread at the RFQ exit are approximately 15° and 0.05 MeV, respectively.

The normalized rms transverse emittances of the output beam are ${\epsilon }_x={\epsilon }_y=0.024 \mathrm{cm}·\mathrm{mrad}$, both of which represent a 20% increase from the input values. Additionally, the output normalized rms longitudinal emittance is ${\epsilon }_z=0.058 \mathrm{cm}·\mathrm{mrad}$, equivalent to $0.106 \mathrm{MeV}·\mathrm{deg}$. Illustrated in Figure 5, the small emittance growth along the RFQ suggests effective beam focusing and bunching. Notably, the initial rapid emittance growth of transverse planes in the first few cells is attributed to the rise of B from a minimal to maximum value in these cells. The insufficient focusing force at the beginning of the RFQ fails to effectively constrain the high-intensity proton beam, resulting in a rapid increase in beam emittance. However, in the subsequent cells, B is kept at a maximum value of 13, which is sufficient to limit the growth of the beam emittance and thereby achieve a small transverse emittance at the end of the RFQ. In the longitudinal plane, due to the slow change of the synchronous phase from −90° to −30°, its emittance only increases significantly in the accelerator section when the synchronous phase reaches −30°, but it can still achieve a good bunching effect and obtain low emittance growth.

2.4. Tolerance Analysis

In practice, the distribution of input particles may deviate from the ideal state, prompting an investigation into the accelerator’s ability to maintain optimal performance under such conditions. The impact of input beam mismatch on transmission has been systematically examined to assess the tolerance of the RFQ structure [35], as shown in Figure 6.

Figure 6a reveals that, for a substantial range of input beam transverse Twiss parameters $\left(\mathsf{\alpha },\mathsf{\beta }\right)$ near the design point, the transmission efficiency consistently exceeds 99.0%. In Figure 6b, the transmission efficiency is plotted against alignment errors in the input beam. To ensure stable operation, displacements in both x and y directions are preferably maintained within $\pm 2$ mm. Figure 6c illustrates that, even with input transverse emittance as large as 1.60 $\mathrm{mm}·\mathrm{mrad}$, the transmission efficiency remains above 99.0%. Figure 6d outlines the sensitivity to input beam current, indicating that transmission efficiency surpasses 99.0% when the current is below 18 mA. Figure 6e demonstrates the transmission efficiency versus input energy of proton beams. Specifically, when the input energy ranges from 28.0 to 32.5 keV, the transmission efficiency remains consistently above 99.0%.

The tolerance analysis yields positive outcomes, revealing that a reasonable deviation from the ideal input beam distribution has a negligible impact on transmission efficiency. Consequently, the beam can be stably accelerated. These results affirm the robustness of the SC RFQ design, paving the way for subsequent structural design.

3. RF Structure Design

The simulation software employed for this study is CST MWS 2023 [36], a comprehensive and highly integrated professional tool designed for 3D electromagnetic (EM), circuit, temperature, and structural stress simulations. CST MWS, a module within the CST software suite, stands out as the fastest and most accurate simulation tool in the field of high-frequency applications.

For an SC cavity, on the one hand, simplicity in structure is essential to prevent the occurrence of large surface electric and magnetic fields, which could lead to quenching. On the other hand, the high Q value of the SC cavity necessitates mechanical stability to mitigate sensitivity to frequency fluctuations [37]. Considering these factors, we opted for the four-vane RFQ structure due to its superior mechanical and thermal stability, complemented by a circular shell design. In order to save the superconducting material niobium, the RFQ is constructed as a 3-millimeter-thick thin-wall structure, as shown in Figure 7.

The four electrodes are installed in the four quadrants of the cavity, and their structural diagrams are shown in the figures below. Achieving the design resonant frequency of the cavity requires an iterative optimization of the electrode parameters, as outlined in Figure 8a. Simultaneously, to ensure a uniformly flat electric field distribution along the central axis of the cavity so as to achieve stable beam acceleration, it is necessary to optimize the undercut structure as illustrated in Figure 8b. It is essential to note that the values of cutz differ at the front and end of the vanes.

It is also crucial to minimize the peak surface electric field and magnetic field of the SC cavity to avoid sparking and quenching. To achieve this, we implemented chamfers on both the electrodes and vanes, as demonstrated in Figure 8 and Figure 9.

Ultimately, the structural parameters of the electrodes were fine-tuned to the values specified in

Table 3. Structural parameters of the electrodes.

Parameters	Values
v1x (mm)	17.00
v1y (mm)	45.00
v2x (mm)	31.50
vaneh (mm)	208.00
$\theta$ (deg)	4.30
cut1 (mm)	62.00
cut2 (mm)	82.00
cutz (mm)	84.00 (front)/76.00 (end)
$r_1$ (mm)	20.00
$r_2$ (mm)	10.00
$r_3$ (mm)	30.00
$r_4$ (mm)	1.20

. At this stage, the cavity could be excited by an EM field of a specific frequency, generating the expected quadrupole electric field.

After modeling the cavity, we need to select the appropriate solver. For frequency and mode calculation of resonant cavities with high Q values, the Eigenmode Solver is typically employed. Then, the hexahedral mesh is adopted to obtain higher precision for complex model calculations. Given the finite integration (FIT) method used in the simulation, the mesh settings play an important role in both speed and accuracy. The global mesh significantly impacts frequency calculation results, while the local mesh in regions with large electric fields influences peak surface electric field outcomes, so a mesh convergence analysis was conducted. The simulation results indicate that with a mesh number around 3,000,000 and a local mesh step width of less than (3,3,3) in the central part of the cavity, both frequency and peak surface electric field converge.

The EM field simulation results reveal that the operating frequency of the cavity is 162.53 MHz (refer to Figure 10), and it operates in TE210 mode. Figure 11 illustrates the EM field mode of the cavity. As shown in Figure 10, $E_{sp}$ is located at the end of the electrodes and has a value of 56.82 MV/m (the electric field values shown in Figure 10 are not normalized, and the actual $E_{sp}$ value of the cavity given by the normalized calculation in post-processing is 56.82 MV/m), which is slightly higher than the dynamics design result. This deviation may be attributed to structural differences between the actual electrode configuration and the dynamics design for ease of fabrication. Nevertheless, the electric field remains within the safe range. The $B_{sp}$, which is located at the vane’s undercut, is 27.46 mT, significantly below the upper limit of 80 mT.

The 3D EM field simulation results of this SC RFQ are consistent with the dynamics design results, which verifies the safety and feasibility of the dynamics design. The actual construction of the SC RFQ can be carried out in accordance with this validated structure afterward.

4. 30 cm Model Cavity Simulation

4.1. Electromagnetic Field Simulation

We plan to construct a 30-centimeter-long cavity for a vertical test, aiming to measure the quality factor Q of the cavity under SC conditions at different stored energies. This test is essential to validate the performance of the SC RFQ cavity after surface treatments, cooling, and evacuation. The model cavity is depicted in Figure 12. On the one hand, the operating frequency and EM field mode of the RFQ are affected by the cavity’s dimensions. As long as we choose the right size, the 30 cm cavity can produce the same resonant frequency and mode as the full-length cavity [38]; on the other hand, we have selected a section of the electrodes with m near its average value for the electrodes of this 30 cm cavity. As a result, the performance of this short cavity is basically consistent with that of the full-length cavity.

The simulated cavity frequency given by CST MWS is 162.52 MHz (refer to Figure 13), and the calculated normalized $E_{sp}$ is 53.13 MV/m, which is close to the simulated values of the full-length cavity. However, the $B_{sp}$ is 47.29 mT, which is higher than the simulated value of the full-length cavity. This discrepancy can be attributed to the emergence of the peak surface magnetic field at the undercut location, as indicated by the red arrow in Figure 14. The size of the undercut in the short cavity differs from that of the long cavity, resulting in inconsistent magnetic field values.

While the electromagnetic (EM) field values of this 30 cm cavity exhibit slight variations from those of the full-length cavity, their excitation mode and distributions remain consistent. Importantly, all peak values fall within the established safe limits. This proves that the results obtained from the vertical test of this cavity can also demonstrate the performance of the full-length cavity.

4.2. Post-Treatment Processes

Next, we conducted simulations to assess the impact of post-treatment processes on cavity frequency. Firstly, it provides theoretical guidance for subsequent frequency control of the SC cavity. Secondly, it helps in structural reinforcement of the SC cavity by designing stiffeners in easily deformed positions to improve the mechanical stability of the cavity. In the simulation procedure, we utilized both CST and COMSOL Multiphysics software v6.2 [39], the latter being a comprehensive platform offering fully coupled multi-physics and single-physics modeling capabilities, particularly useful for temperature, solid, and EM field coupling calculations.

During the stamping and machining phases, the surface of the 3 mm high-purity niobium plate used for cavity fabrication may be damaged, and the typical damaged layer is 100 to 200 $\mathsf{\mu }\mathrm{m}$. This damaged surface needs to be removed after the cavity processing. The standard procedure involves ultrasonic cleaning, buffering chemical polishing (BCP) of 150 $\mathsf{\mu }\mathrm{m}$, hydrogen removal baking, additional BCP of 30 $\mathsf{\mu }\mathrm{m}$, high-pressure water rinsing (HPR), ultra-clean-room assembly, and low-temperature baking [40,41]. Among them, BCP can affect the frequency of the cavity. After surface treatments, the cavity undergoes cooling to 4.2 K and evacuation for vertical tests, both of which can affect the working frequency of the cavity. The simulation procedure flowchart is illustrated in Figure 15.

4.2.1. BCP Processes

BCP is one of the most important steps in the post-processing of SC cavities, used to remove impurities, contaminants, and mechanical damage on niobium surfaces. This process can significantly enhance the RF performance of SC cavities. Employing CST MWS and the COMSOL Electromagnetic Waves (EMW) module, we conducted precise calculations for the BCP processes within the cavity, assuming a uniform polishing of the inner surface. The simulations focused on the changes in cavity frequency resulting from polishing the inner surface by 150 $\mathsf{\mu }\mathrm{m}$ and an additional 30 $\mathsf{\mu }\mathrm{m}$, as depicted in the calculation models presented in Figure 16. In order to minimize the simulation errors between the two programs, we set the CST mesh to a tetrahedral mesh which is consistent with COMSOL for simulation calculations, and the results are shown in Figure 17 and Figure 18. Polishing the inner surface by 150 $\mathsf{\mu }\mathrm{m}$ yielded a frequency increase of 1356 kHz by CST and 1450 kHz by COMSOL, while an additional 30 $\mathsf{\mu }\mathrm{m}$ polishing resulted in a frequency increase of 268 kHz by CST and 180 kHz by COMSOL. Due to differences in calculation methods and mesh density used by the two programs, there are also differences in the frequency variations of the resonator obtained from them. However, this total difference is only 0.003% of the resonant frequency, which indicates that the simulation results of these two programs are basically consistent. The frequency sensitivity to the polishing thickness is about 9.04 kHz/μm.

4.2.2. Cooling Process

To enable superconducting operation of the accelerator, it undergoes a cooling process to achieve the necessary temperature for superconductivity. Using COMSOL Multiphysics, we simulated the deformation of the niobium cavity as it transitions from room temperature 293.15 K to the superconducting temperature of 4.2 K. This simulation allows us to calculate the frequency of the cavity after deformation. The simulation model includes the superconducting cavity shell, 4-vane structure, and the vacuum inside the cavity (refer to Figure 19).

The coupling calculation between the cavity’s deformation after cooling down to 4.2 K and the corresponding eigenfrequency change involves three modules in COMSOL: Solid, EMW, and Deformation Geometry (DG). Within the Solid module, we solve for the shell’s deformation u resulting from thermal expansion, using the linear momentum balance equation and the stress–strain relation:

$$\nabla ·S+{\mathit{F}}_V=0,$$

(6)

$$S=C:{\mathit{ϵ}}_{el},$$

(7)

where S is the second Piola–Kirchhoff stress tensor, and ${\mathit{F}}_V$ is the force per unit volume. The elasticity tensor $C$ is typically defined by Young’s modulus and the Poisson coefficient. In the absence of any plastic effects, the elastic strain tensor ${\mathit{ϵ}}_{el}$ carries the temperature dependence via the thermal strain tensor ${\mathit{ϵ}}_{th}$, as expressed by

$${\mathit{ϵ}}_{el}={\mathit{ϵ}}_{tot}-{\mathit{ϵ}}_{th},$$

(8)

$${\mathit{ϵ}}_{tot}=\frac{1}{2}\left[{\left(\nabla u\right)}^T+\nabla u\right],$$

(9)

$${\mathit{ϵ}}_{th}=\mathit{\alpha }\left(T-T_{ref}\right),$$

(10)

where ${\mathit{ϵ}}_{tot}$ represents the total strain tensor, $\mathit{\alpha }$ is the coefficient of thermal expansion that characterizes the ability of the material to contract and expand due to temperature variations, $T=4.2 \mathrm{K}$ is the environment temperature, while $T_{ref}=293.15 K$ is the reference temperature.

The boundary of the vacuum undergoes deformation through the specified mesh displacement within the DG module. During the deformation calculation, fixed constraints are applied to both ends of the SC cavity (refer to Figure 15) to prevent rigid displacement. The material properties employed in the calculation are listed in

Table 4. The material properties of niobium [42].

Property	Value
Density (g/cm3)	8.57
Young Modulus (GPa)	105
Poisson Ratio	0.4
Thermal Expansion (K−1)	7.3 × 10−6
Thermal Conductivity (W/(m·K))	54

. After deformation, the Frequency Domain EMW module is utilized to further analyze the vacuum and determine the eigenfrequency of the deformed SC cavity.

The simulation results reveal that the niobium cavity experiences a maximum deformation of 0.56 mm on its outer surface upon cooling to 4.2 K. The frequency of the deformed cavity is 164.78 MHz, which is 1450 kHz higher than the initial frequency (refer to Figure 20).

4.2.3. Evacuation Process

To ensure SC operation, the accelerator requires a vacuum environment. Simulating the evacuation process involves applying atmospheric pressure to the cavity’s surface, calculating resultant deformations, and determining the cavity’s frequency after deformation.

Utilizing the same simulation model, calculation modules, and material properties employed during the cooling process, the Solid module now addresses stress and deformation induced by atmospheric pressure on the cavity surface. The equations governing this process are as follows:

$$\nabla ·\mathrm{S}+{\mathit{F}}_V=0,$$

(11)

$$\mathrm{S}·n=-\mathrm{p}n,$$

(12)

where $n$ represents the normal direction of the surface, and $p=1.01\times {10}^5 \mathrm{Pa}$ denotes atmospheric pressure.

The DG and EMW modules function consistently with their roles in the cooling process simulation. The results indicate that the maximum deformation of the niobium cavity during evacuation to a vacuum is 1.28 mm, predominantly occurring on the vanes. The frequency of the deformed cavity is 164.35 MHz, reflecting a decrease of 430 kHz compared to the cavity frequency after cooling (refer to Figure 21).

The total frequency changes given by the simulation processes are listed in

Table 5. The simulation results of frequency change after post-treatment processes.

Post-Processing	Frequency (CST)/MHz	Frequency (COMSOL)/MHz	Frequency Change (Average)/kHz
Initial state	162.580	162.580	−
BCP of 150 $\mathsf{\mu }\mathrm{m}$	163.936	164.030	+1403
BCP of 30 $\mathsf{\mu }\mathrm{m}$	164.204	164.210	+224
Cool down to 4 K	-	164.780	+570
Vacuum out	-	164.350	−430

. Observing the outcomes, it is evident that following a sequence of post-treatment procedures, the cavity’s frequency has experienced a net increase of 1.77 MHz. This indicates that in order to achieve the final frequency of 162.5 MHz, the initial 30 cm cavity frequency should be set to 160.73 MHz. Additionally, it will be helpful to design stiffeners in positions with large deformation to enhance the overall stability of the cavity.

4.3. Cavity Tuning

Due to inherent discrepancies such as displacement, rotation errors during the assembly process of the cavity, and external environmental disturbances during the operation of the cavity, the frequency may deviate from the designed frequency.

For resonant cavities, a widely employed tuning method is to squeeze the cavity to cause its deformation, hence changing the operating frequency. This method is also known as the perturbation method. The mathematical expression for this method is defined as

$$\frac{\omega -{\omega }_0}{{\omega }_0}\approx \frac{\Delta W_m-\Delta W_e}{W},$$

(13)

where ${\omega }_0$ signifies the initial frequency, $\omega$ represents the frequency after perturbation, and W is the energy stored within the cavity. To maximize the tuning effect, it is crucial to select locations for deformation where the magnetic energy $W_m$ and electric energy $W_e$ exhibit significant discrepancies.

Since the traditional tuning method of adding tuning blocks has the risk of contaminating the cavity, which is not conducive to the cleaning of the SC cavity, we use the approach of squeezing two end plates to change the distance between the interior of the end plates and the end of the vane tips to tune the cavity [24]. However, it is crucial to note that this distance is determined by the beam dynamics design. Modifying it requires adjustments to the dynamics design. To preserve the change in the beam dynamics design while achieving the purpose of tuning, a small disk with a radius of 60 mm is reserved centered on the beam axis. Only the distance between the outer end plates and the end of vane tips is modified, creating a groove in the center of the end plate [43], as shown in Figure 22. This innovative tuning method allows for effective cavity tuning without compromising the previously determined beam dynamics design.

Subsequently, we employed CST MWS to simulate the impact of the end plates’ moving distance on the cavity frequency. In the outer end-plate region, where magnetic energy surpasses electric energy, increasing the aforementioned distance increases inductance. This, in turn, leads to a decrease in the cavity frequency as indicated by Equation (1), and there is a linear relationship between moving distance and cavity frequency [44]. The result is displayed in Figure 23, illustrating a linear correlation between cavity frequency and the movement at each end, with a frequency change of approximately 331 kHz/mm. This degree of tuning is sufficient for correcting frequency deviations caused by machining errors. Additionally, Figure 24 shows the influence of various squeezing amounts on the off-axis electric field distribution within the cavity. Notably, the electric field distribution remains consistently flat under different squeezing amounts, with deviations not exceeding 2%, which further proves the robustness and reliability of our innovative tuning method.

5. Conclusions

The beam dynamics design and structural design of a 162.5 MHz 10 mA CW proton SC RFQ have been successfully completed. In the dynamics design phase, we carefully selected various input parameters to achieve efficient transmission and acceleration. The results demonstrated that this SC RFQ can effectively accelerate proton beams from 30 keV to 2.5 MeV over a length of 2.03 m, achieving a remarkable 100.0% transmission efficiency. In the EM field simulation, the cavity’s operating frequency was calculated to be 162.53 MHz, consistent with the design specifications. The peak surface electric field and magnetic field were optimized to 56.82 MV/m and 27.46 mT, respectively, both under the safety limit to mitigate the risk of quenching.

Prior to constructing the full-length cavity, we planned to construct a 30-centimeter-long cavity for vertical tests under SC conditions to verify the performance of the SC cavity after surface treatments, cooling, and vacuum extraction. Simulation results for the 30 cm model cavity indicated a total frequency change of approximately 1.77 MHz due to post-treatment processes. Consequently, setting the initial frequency of the 30 cm cavity to 160.73 MHz is advised, with additional stiffeners in areas experiencing substantial deformation to enhance cavity stability. Finally, the tuning study of the short cavity revealed that squeezing the end plates to change the distance between them and the vane tips is an effective tuning method. The frequency sensitivity was about 331 kHz/mm, providing enough capability for correcting frequency deviations caused by machining errors.

The next phase involves initiating the construction of the 30 cm testing cavity for vertical tests to validate the SC cavity’s ability to achieve the desired performance.