The Design of a High-Intensity Deuteron Radio Frequency Quadrupole Accelerator

Abstract

This paper presents the design of a high-intensity 10 mA deuteron RFQ accelerator that generates a 2.1 MeV beam in a continuous wave (CW) mode. The operation frequency is 162.5 MHz. The results of beam dynamics simulations demonstrate excellent output beam quality, achieving a transmission efficiency of 98.63%. The beam tracking results indicate that the RFQ is capable of managing errors within reasonable tolerances. In addition, the RF electromagnetic design and optimization are based on an RFQ model. Multiphysics simulations are then performed for the CW mode. Vacuum calculations suggest that the RFQ requires four 1200 L/s vacuum pumps and one 440 L/s ion pump to attain a vacuum pressure of 10⁻⁶ Pa.

1. Introduction

Since the 1990s, deuteron radio frequency quadrupole (RFQ) accelerators have attracted international interest due to their wide range of applications, encompassing accelerator-driven nuclear energy systems (ADSs) [1] and spallation neutron sources (SNSs) [2]. In particular, RFQs operating in high duty factor or even CW mode have become a research hotspot. The Spiral-2 CW RFQ was designed to accelerate deuterons from 0.04 MeV to 1.5 MeV; in 2016, the commissioning of a facility with the beam showed that it could accelerate 1.34 mA CW He²⁺ [3]. The Phase II SARAF RFQ was designed to accelerate 5 mA deuterons from 0.04 MeV to 2.54 MeV; in 2017, the commissioning of a facility with the beam showed that it could accelerate 1.15 mA CW D⁺ [4]. The “973” RFQ was intended to accelerate deuterons from 0.05 MeV to 1 MeV; in 2018, experiments substantiated its capability to accelerate 1.78 mA CW H₂⁺ [5,6]. The IFMIF CW RFQ was designed to accelerate 125 mA deuterons from 0.1 MeV to 5 MeV, and the facility successfully accelerated 125 mA deuterons at a duty factor of 0.1% in 2019 [7,8]. Several other CW RFQs, such as the FRIB RFQ [9] and CMIF RFQ [10], have also demonstrated notable achievements. All of these RFQs have played a significant role in the advancement of accelerator science and technology.

In this project, the RFQ accelerator was designed to accelerate 10 mA CW D⁺ ions from 0.04 MeV up to 2.10 MeV. Following this, the propelled beam was introduced into a medium-energy beam transport (MEBT) system and subsequently directed into a drift tube linac (DTL). Three-dimensional simulations were performed employing CST [11] and COMSOL [12], encompassing electromagnetic (EM) and vacuum analyses. The graphical representation of the RFQ modeling, realized through the utilization of SolidWorks [13], is shown in Figure 1. Spanning a length of approximately 3.8 m, the RFQ exhibits a well-defined quadrilateral structure. The provision of power and maintenance of the necessary vacuum level (10⁻⁶ Pa) for the cavity are achieved by means of two power couplers and a set of five pumps.

2. Main Parameters of the RFQ

The main parameters of the RFQ are summarized in

Table 1. Main parameters of the cavity.

Ion	D+
Frequency (MHz)	162.5
Duty factor	100%
Vane length (m)	3.7
Average aperture (mm)	4.23
Input/output energy (MeV/u)	0.020/1.05
Beam current (mA)	10
Intervane voltage (kV)	60
$E_k$ (MV/m)	18.98 (1.40 Ek)
Input trans. ${\epsilon }_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m},\mathrm{r}\mathrm{m}\mathrm{s}}$ (mm · mrad)	0.200
Output ${\epsilon }_{\mathrm{x},\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m},\mathrm{r}\mathrm{m}\mathrm{s}}$ (mm · mrad)	0.201
Output ${\epsilon }_{\mathrm{y},\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m},\mathrm{r}\mathrm{m}\mathrm{s}}$ (mm · mrad)	0.202
Output ${\epsilon }_{\mathrm{z},\mathrm{r}\mathrm{m}\mathrm{s}}$ (mm · mrad)	0.180
Transmission efficiency (Elimit = 0.08 MeV)	98.63%

. LANL code phase and radial motion in transverse electric quadrupoles (PARMTEQM) v.3.05 [14] and CEA code TOUTATIS [15] were used for dynamics design, while TraceWin [16] was used for verification. The maximum peak surface electric field ($E_{max}$) is 1.40 times the Kilpatrick limit electric field [17] of 13.56 MV/m. $E_k$ is calculated through the following equation:

$$f=1.643{E_k}^2{e}^{-\frac{8.2}{E_k}}$$

(1)

where $f$ is the operating frequency (MHz). Lowering the ratio $k_p={E_{max}/E}_k$ decreases the risk of discharging. The low-energy demonstration accelerator (LEDA) RFQ [18,19] is a milestone high-power CW accelerator, whose beam power and power density are enormous, and its $k_p$ is 1.8. Based on practical experience, we argue that the $k_p$ of this RFQ is suitable for CW mode.

3. Design Consideration and Strategy

For the beam dynamics design, we adopted the conventional dynamics design strategy known as “four-stage theory” [20]. A constant electrode voltage and a steady average aperture were chosen to facilitate the machining of the RFQ structures. This approach has the potential to enable the utilization of a basic 2D cutter for machining vane modulations. Furthermore, during the EM analysis phase, the adjustment of voltage to align with the designed value is a more straightforward process compared with that of variable voltage and aperture design [21]. To increase mode separation, we ensured that the electrode length was twice the wavelength [22]. Consequently, the RFQ operates without the necessity of any $\pi$-mode stabilizers (PSL) or dipole-mode stabilizer rods (DSRs) to sustain cavity stability, while concurrently maintaining a mode separation of 3.3 MHz. Moreover, the following aspects were taken into consideration:

(a)

Transmission efficiency: Deuterons lost in the acceleration process may collide with the electrode surface, causing instability and reduced efficiency over long-term operation. The loss of D⁺ ions may also generate unintended neutrons through charged-particle activations with the apparatus. Especially for D⁺ losses in higher energy sections, neutron radiation is more of a concern.

(b)

Intervane voltage: A higher intervane voltage offers the advantage of reducing the RFQ length; however, it also increases the power consumption per unit length and peak field strength, thereby raising the risk of RF sparking. A lower intervane voltage of 60 kV requires a smaller average aperture to generate enough transverse focusing factor (B), and a smaller average aperture needs higher requirements on the machining accuracy of the electrode. The “973” RFQ’s successful facility commissioning substantiates the viability of employing an intervane voltage of 60 kV for CW operation [6,23]. It also operates at a frequency of 162.5 MHz.

(c)

Input energy: The input ion energy was optimized at 40 keV. It affects the cavity length, space–charge force, and initial cell length. Lowering the input energy enlarges the space charge effects, negatively influencing beam transmission in the low-energy beam transport (LEBT) section. Additionally, the length of the initial segment shortens and impedes electrode processing. Conversely, an excessive input leads to an elongated bunching section, thereby increasing the cavity length and the associated construction and operational costs.

(d)

Large transverse acceptance: Due to the space–charge force, the size of the beam increases during transport, especially for low-energy beams. Consequently, adopting a large input aperture can enhance transverse acceptance.

The main PARMTEQM parameters for the RFQ are shown in Figure 2. The synchronous phase ${\mathsf{\phi }}_{\mathrm{s}}$ shifts gradually from −90° to −29°, effectively preventing the loss of particles in the longitudinal direction. The modulation factor m increases smoothly from 1 to 1.9, while the average aperture R0 shrinks gradually to 4.2 mm. A large entry aperture method is adopted to enhance the transverse acceptance by approximately 4%. The method achieves improvements by reducing the input B, as shown in Figure 3 and

Table 2. Twiss parameters of different designs.

Twiss Parameter	Typical Design	Large-Input Aperture Design
α	0.99	0.75
β (cm/rad)	3.53	4.25

.

Figure 4 illustrates the normalized beam density derived from a TraceWin simulation, involving 100,000 macroparticles with a current of 10 mA. It indicates that the ion losses are primarily concentrated within the bunching section. To facilitate the matching of the beam into the following focusing channel, a transition region with a modulation factor (m) of one is used to end the RFQ vane tips [24]. The transition region provides a convenient way to control the orientation of the output transverse phase–space ellipses, as depicted in Figure 5. The length of this transition section is 30.2 mm.

The 3D EM field generated via CST was employed to validate the beam dynamics design. The transmission efficiency was found to be 98.17% for ions accelerated from 0.04 MeV to 2.11 MeV. Notably, the emittance growth at the RFQ exit remained relatively minor, aligning with the PARMTEQM design. The results are presented in

Table 3. Result comparison of PARMTEQM and CST + TRACEWIN.

	PARMTEQM	CST + TRACEWIN
Output energy (MeV)	2.11	2.11
Beam current (mA)	10	10
Transmission (%)	98.65	98.17
Transverse normalized RMS emittance of input beam $(\pi ·mm·mrad$) (x/y)	0.20/0.20	0.20/0.20
Transverse normalized RMS emittance of output beam $(\pi ·mm·mrad$) (x/y)	0.201/0.202	0.203/0.209
Longitudinal normalized RMS emittance of output (degree · MeV)	0.0625	0.0646

and Figure 6.

Figure 7 displays the Hofmann resonance diagram. The red line with dots indicates the evolution of the working points. To avoid the horizontal and vertical coupling caused by the resonance, the beam must quickly traverse or circumvent the formant region [25]. This proved that our design meets this requirement, as there were no obvious transverse or longitudinal beam resonances.

4. Error Study

Errors arising from a nonideal beam, machining, and alignment are inherent and unavoidable. These errors impact and contribute to degradation of beam quality. A robust accelerator should be able to manage many different types of error conditions [26]. TraceWin code was used to study the tolerances and sensitivity of the RFQ. As error sources are complex, a single-error simulation was conducted initially, followed by the consideration of comprehensive errors.

The error study involved an input Twiss parameter, beam transverse offset, and tilt, as shown in Figure 8. When the TWISS parameter of the input beam was widely varied, the transmission efficiency always remained higher than 98%. This was also true when the transverse offset was less than 0.5 mm or the beam tilt was less than 20 mrad.

Further error analysis was carried out involving the input beam emittance, energy, and currents, with the results illustrated in Figure 9, Figure 10 and Figure 11. It was found that as long as the transverse emittance was less than 0.35 mm mrad, the input energy was between 0.0395 MeV and 0.0410 MeV, and the current was within 15 mA, the transmission efficiency could be maintained above 98%.

Moreover, the effects of the alignment, intervane voltage, and dipole components were investigated to analyze the sensitivity of the RFQ. As the RFQ is divided into four segments about 1 m long during processing, misalignments inevitably occurred during assembly, as depicted in the left side in Figure 12. The right side in Figure 12 presents the simulation of these displacement errors. It was observed that even for a displacement of 50 μm (within the typical error range), there was no major alteration in transmission efficiency.

During actual RFQ operation, the occurrence of intervane voltage deviations is a conceivable phenomenon. By introducing voltage errors between electrodes, we analyzed the effect of a nonuniform electric field on beam transmission. Firstly, we examined the impact of voltage tilt on beam transmission. Secondly, we assessed the influence of higher-order voltage harmonics on beam transmission, which are defined as follows:

$$V\left(z\right)=V_0\left(z\right)+∆Vcos\left(\frac{n\pi z}{2l}\right)$$

(2)

where n is a harmonic number (n = 1, 2, 3, ……), $V_0$ is the nominal intervane voltage, $l$ is the length of vane, $∆V$ is the amplitude of residual voltage error, and $∆V$ is set to 3%$V_0$. The error modes of the two voltages are shown in Figure 13a,b. The influence of the intervane voltage tilt on the RFQ transmission efficiency is shown in Figure 13c. A decrease in the voltage between the entrance poles leads to inadequate transverse focusing strength and results in beam loss. For increased voltage, the initial beam loss is less pronounced. For multiple harmonics shown in Figure 13d, raising the harmonic number has no significant impact on beam transmission.

Furthermore, we also studied the influence of the dipole field on beam transmission. Figure 14a illustrates the dipole filed distribution, while Figure 14b reveals the influence of a dipole field on beam transmission. Dipole field errors below 5% maintain a high transmission efficiency. As a consequence of processing and assembly errors, the four quadrants of the RFQ may be asymmetric, giving rise to the presence of a dipole field. The perturbation component of the dipole field imparts a lateral deflection of the beam, leading to oscillation of the beam center, triggering instability, lateral beam loss, and reduced transmission efficiency. To more clearly show its influence, we added a dipole field component of 14% onto the intervane voltage for comparison with the no-dipole-field condition. In this case, the amplitude of center oscillations reaches 0.6 mm, as shown in Figure 15. For real RFQ measurement tuning, the dipole component incurred by machining or assembly imperfections is mitigated by tuner adjustments.

The comprehensive error analysis considered 100 RFQs with different error sizes.

Table 4. Parameters of comprehensive error analysis.

Error Type	Max Value
Input beam alignment position (mm)	0.15
Input beam angle displacement (mrad)	5
Input beam emittance growth	40%
Input beam Twiss mismatch (α)	15%
Input beam Twiss mismatch (β)	15%
RFQ vane voltage jitter	2%
RFQ longitudinal vane profile (μm)	15
RFQ transverse vane curvature (μm)	15
Horizontal and vertical segment displacement (μm)	10
Parallel and perpendicular vane tilt (μm)	10

displays all error types along with their corresponding maximum value. For each RFQ, the beam transmission error size is uniformly distributed within the range of the specified maximum. The input beam contains 100,000 macroparticles in a standard “water-bag” distribution. The results show that the transmission efficiency can be maintained above 90.4% under the various errors.

5. Electromagnetic (EM) Analysis of RFQ

Our RFQ adopts a four-vane structure. In comparison with a four-rod structure, a four-vane RFQ is more compact, mechanically robust, and convenient for incorporating water-cooling pipes. A four-vane structure also has a large Q value, high specific shunt impedance, and low surface current density, which is conducive to long-term, stable operation in CW mode [27].

The goal of the RF electromagnetic field design is to satisfy the beam dynamics requirements for a correct resonant frequency and intervane voltage distribution. Consequently, optimizing the RF structural components is crucial to obtaining a high specific shunt impedance, reduced surface power density, and low peak field strength, mitigating the risk of ignition and ensuring stable operation of the RFQ in CW mode over the long term.

Figure 16 illustrates the structure of our RFQ and its 1/4 cross-sectional parameters. The quadrilateral-shaped cavity was chosen for its ease of machining and assembly, which involves the installation of couplers and vacuum pumps. While the average aperture of the RFQ gradually decreases at the entrance, the electrode tip radius remains a constant 3.19 mm (0.75 times the average aperture) across the entire RFQ. For subsequent simulations, the electrodes were modulated, and frequency tuners were introduced. By adjusting the transverse dimension of the cavity, denoted as L_max, the resonant frequency requirement of the cavity was found.

Table 5. Parameters of the cavity cross-section.

Parameter	Value	Unit
$r_0$	4.23	mm
$r_T$	3.19	mm
$\rho$	0.75
$L_1$	33.43	mm
$L_2$	11.15	mm
${\theta }_1$	10	degree
${\theta }_2$	10	degree
$R_v$	30	mm
$R_w$	30	mm
$L_{max}$	172.5	mm

shows the optimized parameters of our RFQ.

5.1. Undercutting and Tuning

The longitudinal distribution of the electric field within the RFQ is highly sensitive to the undercutting at both ends of the electrode. Hence, the field is best adjusted by optimizing the undercut parameters. Upon introducing undercutting, the magnetic field between adjacent quadrants may form a closed loop, increasing the electric field at both ends. Figure 17 gives a schematic of the RFQ undercutting, and

Table 6. Undercutting parameters.

Parameter	Value	Unit
$\mathrm{Input} \alpha$	55	degrees
Input D	86.8	mm
$\mathrm{Output} \alpha$	55	degrees
Output D	78.3	mm

lists the input and output undercutting parameters. By optimizing the undercutting angle α (degrees) and undercutting depth D (mm), a smoother electric field was obtained between electrodes. For an RFQ operating in CW mode, a large α value can lead to hot spots at the root of the undercutting and cause deformation at extremities due to poor water cooling.

The optimized off-axis field is shown in Figure 18, where the voltage deviation between electrodes is less than 0.5%.

Errors occurring during machining and assembly of the cavity can lead to mode-mixing, disrupting the field distribution of the operational quadrupole mode. The tuners can compensate for these errors by returning the field distribution and frequency to their designated values. Indeed, both the inductance-tuning and capacitance-tuning components are capable of altering the local frequency of the cavity. In our four-vane RFQ cavity, the electric field is concentrated in the small area of the electrode head, while the magnetic field is widely distributed in the cavity, so the tuner is inserted mainly to disturb the magnetic field, that is, inductance tuning. The local inductance decreases as the tuner is inserted deeper into the cavity, leading to a higher local frequency and smaller electric field. Our RFQ has a total of 64 tuners, with 16 in each quadrant. As shown in Figure 19, the tuners are evenly distributed about the horizontal axis, with a diameter of 50 mm each. Figure 20 illustrates the linear relationship between the cavity frequency and tuner insertion depth. The tuner insertion depth ranges from 20 mm initially to 50 mm at maximum, spanning −0.82 MHz to +1.65 MHz, with increments of 53 kHz/mm.

5.2. Mode Separation

The four-vane RFQ possesses many harmonics, encompassing adjacent resonant frequencies apart from the operating mode. This includes dipole modes (${\mathrm{T}\mathrm{E}}_{11\mathrm{n}+}$, ${\mathrm{T}\mathrm{E}}_{11\mathrm{n}-}$) and quadrupole modes (${\mathrm{T}\mathrm{E}}_{21\mathrm{n}}$). The quadrupole modes in Figure 21 explain the H field of ${\mathrm{T}\mathrm{E}}_{21\mathrm{n}}$ and ${\mathrm{T}\mathrm{E}}_{21\mathrm{n}}$ modes. The color of the different quadrants in the plot indicates that the left one is ${\mathrm{T}\mathrm{E}}_{21\mathrm{n}}$ mode and the right one is ${\mathrm{T}\mathrm{E}}_{11\mathrm{n}}$ mode. Given the lowest mode of fixed frequency, longer vanes reduce the spacing between the modes. The properties can be described by the following equations [22,28], where $l_v$ is the vane length:

$$\begin{gathered} f_{T{E}_{21n}}=\frac{c}{2\pi }\sqrt{{\left(\frac{2\pi }{c}\cdot f_{T{E}_{210}}\right)}^2+{\left(\frac{n\pi }{l_v}\right)}^2} \\ n=\mathrm{0,1},2,\dots k_n=\frac{n\pi }{l_v} \end{gathered}$$

(3)

$$\begin{gathered} f_{T{E}_{11n+}}\cong f_{T{E}_{11\left(n+1\right)-}}\cong \frac{c}{2\pi }\sqrt{{\left(\frac{2\pi }{c}\cdot f_{T{E}_{110-}}\right)}^2+{\left(\frac{\left(n+1\right)\pi }{l_v}\right)}^2} \\ n=\mathrm{0,1},2,\dots k_n=\frac{n\pi }{l_v} \end{gathered}$$

(4)

Table 7. Mode frequencies.

Mode	Frequency (MHz)
TE210	162.38
TE110	158.34/158.40
TE111	165.67
TE211	167.28
TE212	181.09
TE213	202.06

shows the cavity frequency of different modes. The mode spacing is best when the electrode length is an even multiple of the RF wavelength. For this study, the length of the RFQ electrode was chosen to be approximately 3.7 m, which is about twice the RF wavelength. The corresponding mode separation was found to be 3.3 MHz, which is sufficient for stable operation of the cavity.

5.3. Power Loss

With the RF structure in mind, the water-cooling design and thermal response of the cavity were analyzed to ensure stable operation in CW mode at high power levels. Initially, we investigated the power loss distribution within the RFQ cavity to determine the best water-cooling scheme. As shown in the heat density map in Figure 22, a strong magnetic field was present in the undercut sections, leading to large power losses. The waterway should thus be constructed as close to the undercut as possible.

The power-loss values are summarized in

Table 8. Power distribution by RFQ part.

Component Name	Section Number	Power Loss (kW)	Percentage (%)
Cavity wall with electrode	1	13.2	21.69
	2	12.7	20.87
	3	12.7	20.87
	4	13.5	22.18
Electrode head	1	1.12	1.84
	2	1.13	1.85
	3	1.14	1.87
	4	1.16	1.91
Tuners	All	3.39	5.57
Coupling ring	1	0.068	0.11
	2	0.067	0.11
End plate	1	0.29	0.48
	2	0.4	0.66
Sum		60.86	100

. We calculated a cavity power consumption of 60.86 kW, beam power consumption of 20.6 kW, total power consumption of 81.46 kW, and Q value of 1.54 × 10⁴. As the surface heat loss mostly concentrates around the vane and wall areas of the cavity, the water-cooling system should be designed to accommodate these areas. At the exit of the undercut, the maximum surface power density of the cavity was 14.6 W/cm².

5.4. Multiphysics Analysis

In our multiphysics analysis, we separately considered numerical simulations involving electromagnetic, thermal, and deformation effects. The steps of our process are outlined in Figure 23. Firstly, the RF electromagnetic field was simulated, yielding the electromagnetic field distribution and heat loss within the cavity. Subsequently, the surface heat loss within the cavity was determined via a simulation of radio frequency effects and used in a thermal analysis module to compute the temperatures within the cavity. From these values, we were able to determine the amount of deformation due to heat loss.

For the thermal analysis, the temperature distribution was based on [30]:

$$\mathrm{T}=T_0+\frac{P}{A·h}$$

(5)

where T is the surface temperature, T₀ is the initial water temperature, A is the contact area between the water and cavity, and P is the RF power loss. The heat transfer coefficient h describes the convection process on the surface of a conductor and is obtained from the following formulas:

$$\mathrm{R}\mathrm{e}=\frac{\rho vd}{\mu }$$

(6)

$$\mathrm{P}\mathrm{r}=\frac{C_p\mu }{k}$$

(7)

$$\mathrm{N}\mathrm{u}=0.023·{Re}^{0.8}·{Pr}^{0.4}$$

(8)

$$\mathrm{h}=\frac{Nu·k}{d}$$

(9)

where k is the heat conduction coefficient of water, d is the diameter of water pipe, $v$ is the velocity of water, $\rho$ is the density of water, $\mu$ is the coefficient of kinetic viscosity, $C_p$ is specific heat capacity, Re is the Reynolds number, Pr is the Prandtl number, and Nu is the Nusselt number. In our design, each wing features three water cooling channels 16 mm in diameter, as illustrated in Figure 24. The water velocity within these channels is 2.5 m/s, which corresponds to 26.5 L/min, a Reynolds number of 39,920, and a heat transfer coefficient of 9010 W/m²/K. An additional waterway positioned at each corner of the cavity wall increases the flow rate to 30.2 L/min. In this case, the ambient temperature and water temperature are both 20 °C.

Table 9. Parameters of water cooling.

Channel Type	Vane	Body
Velocity (m/s)	2.5	2.5
Channel number	48	16
Pipe diameter (mm)	15	16
Flow rate (L/min) (each channel)	26.51	30.16
Flow rate (L/min) (all channel)	1272.48	482.55
Fluid density (kg/m3)	998	998
Fluid viscosity	0.001	0.001
Fluid thermal conductivity (W/m · K)	0.6	0.6
Prandtl number	7	7
Reynolds number	37,425	39,920
Nusselt number	228	240
Heat transfer coefficient (W/(m2 · K))	9128	9011

lists further parameters of our water-cooling design.

Table 10. Result of the multiphysics analysis.

Parameter	Value	Unit
Ambient temperature	20	°C
Water temperature	20	°C
Flow velocity	2.5	m/s
Maximum temperature rise	18.1	°C
Maximum deformation	86.1	μm
Maximum stress	53.5	MPa
Frequency drift	9.9	kHz

gives the results of the multiphysics field simulated using CST. The cavity temperature and deformation distributions are shown in Figure 25 and Figure 26. The maximum temperature differential was 18.2 °C, and the maximum deformation was 86.1 μm, observed at the end of the electrode head. The corresponding frequency drift was 9.9 kHz, indicating the reasonable accuracy of the multiphysics analysis. The simulation also revealed that temperature-induced frequency drift leads to a smaller mode separation of the cavity.

6. Beam-Loading Effect and Power Coupler Design

6.1. Beam-Loading Effect [31,32]

The existence of beam-loading effects represents a significant challenge for achieving stable operation in high-current RFQs. This factor demands diligent consideration, particularly in relation to power-coupling strategies. Otherwise, the beam-loading effects lead to an amplitude decrease and a phase offset of the electric field in the cavity. For most RFQs, the current is low enough that the beam power consumption is less than that of the cavity, allowing loading effects to be ignored. In our case, however, the beam power consumption (Pb) is approximately 21 kW, whereas the simulated cavity power consumption (Pc) is 61 kW for a beam load ratio (Pb/Pc) of about 0.34. This relatively high beam load ratio is compensated for by adjusting the coupling coefficient. If detuning occurs, its effects can be counteracted by adjusting the driving frequency. The detuning frequency is calculated below.

The power consumption of the cavity and the external power consumption are denoted by P_c and P_ex, respectively. The coupling coefficients of the coupler the cavity are then defined as a ratio of their power consumption,

$$\mathsf{\beta }=\frac{P_{ex}}{P_c}$$

(10)

By the definition of the Q value, the relationship between on-load Q_L, no-load Q₀, and external Q_ex is

$$\frac{1}{Q_L}=\frac{1}{Q_0}+\frac{1}{Q_{ex}}$$

(11)

$$Q_0=\left(1+\mathsf{\beta }\right)Q_L, \mathsf{\beta }=\frac{Q_0}{Q_{ex}}$$

(12)

Due to microwave effects, the voltage reflection coefficient $\mathsf{\Gamma }$ of the cavity load is expressed as

$$\mathsf{\Gamma }=\frac{\beta -1}{\beta +1}$$

(13)

In order to feed all of the power from the source into the resonant cavity, the reflection coefficient should be zero. According to the formula, the coupling coefficient is then one. When the cavity resonates, the equivalent circuit becomes a resistance circuit. For an input power Pg, the cavity power is

$${\mathrm{P}}_{\mathrm{C}}={\mathrm{P}}_{\mathrm{g}}\left(1-{\mathsf{\Gamma }}^2\right)={\mathrm{P}}_{\mathrm{g}}\frac{4\mathsf{\beta }}{{\left(1+\mathsf{\beta }\right)}^2}$$

(14)

For a loaded beam, the optimal matching parameters for a single coupler are listed in

Table 11. Best matching parameters for a single coupler.

Parameter	Symbol and Formula	Value
Frequency (MHz)	f	162.5
Beam current (A)	I	0.01
Cavity power (kW)	Pc	60.97
Beam power (kW)	Pb	20.78
Generator power (kW)	Pg = Pc + Pb	81.75
Optimum coupling factor	${\beta }_m=1+Pb/Pc$	1.34
Unloaded quality factor	Q0	14,124
Loaded quality factor	QL = Q0/(1+βm)	6036
Accelerating voltage (MV)	$V_0={\int }_0^{L}Ez dz={\sum }_1^n{A}_{i}V$	3.122
Transit-time factor	T	$\mathsf{\pi }/4$
Cavity voltage (MV)	$\mathrm{Vc}=V_0$T	2.452
Beam injection phase	$\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\phi }=\mathrm{P}\mathrm{b}/\mathrm{I} \mathrm{V}\mathrm{c}$	32.06°
Detuning angle	$\mathrm{t}\mathrm{a}\mathrm{n}\mathsf{\psi }=-\frac{{\beta }_m-1}{{\beta }_m+1}tan\phi$	−5.20°
Effective shunt impedance ($\mathrm{M}\mathsf{\Omega }$)	$\mathrm{Rs}={Vc}^2/Pc$	98.61
Beam-induced voltage (MV)	$\mathrm{V}\mathrm{b}=\frac{I·Rs·cos\psi }{1+\beta }$	0.420
Generator-induced voltage (MV)	$\mathrm{V}\mathrm{g}=\frac{2\sqrt{\beta ·Pg·Rs}}{1+\beta }\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\psi }$	3.418
Detuning factor	$\mathsf{\gamma }=-\mathrm{t}\mathrm{a}\mathrm{n}\mathsf{\psi }$	0.091
Detuning frequency (kHz)	$∆\mathrm{f}\approx \mathrm{f}·\mathsf{\gamma }/2\mathrm{Q}\mathrm{L}$	1.225

. The beam-loading effect for a beam load ratio of 0.34 is compensated for by adjusting the optimal coupling coefficient to 1.34 and raising the driving frequency by 1.225 kHz.

6.2. Coupling Ring Area

A coaxial power coupler was designed for the transmission of high-frequency power into the RFQ. Magnetic coupling through the coupling ring is most effective since there is a large magnetic field in this cavity. Our power coupler design involves three components: a coupling ring, transition section, and radio frequency matching section. The coaxial impedance of each port is fixed at 50$\mathsf{\Omega }$. Only the coupling ring and transition section are described in this paper.

An eigenmode solver was used to simulate the external Q value ($Q_{ex}$) of the coupling ring. The coupling coefficient for different coupling ring areas is obtained from:

$$\beta =\frac{Q_0}{Q_{ex}}$$

(15)

The left side in Figure 27 illustrates the shape and position of the coupling ring, while the right side in Figure 27 shows the coupling coefficient varying with coupling ring area. When $\beta$ = 1, S_ring is 926 mm². During construction, the two coupling rings are symmetrically inserted into the cavity to maintain uniformity of the quadrupole field and ensure that the two working phases remain consistent.

6.3. Transition Section

The transition section is a crucial part of the coupler assembly, connecting the power input waveguide with the coupling ring while also separating the cavity vacuum environment from the atmospheric conditions of the feed tube. For this project, we opted for a tubular ceramic window due to its shape-conducive mechanical strength, ease of processing, and gradual impedance transition.

Figure 28 shows the transition section and material specifications in the CST model. The feed tube that connects to the power source adheres to the standard size of EIA6-1/8. The inner conductor of the feed tube has an outer diameter of 32.5 mm, while the outer conductor has an inner diameter of 74.65 mm, yielding an impedance of 49.89 $\mathsf{\Omega }$. The simulated value matched well with calculated value, as shown in Figure 29. It is critical to achieve impedance matching at each longitudinal position, because the places in which the impedance changes sporadically may generate heat and are more prone to ignition. The S11 and S21 parameters are plotted in Figure 30 and Figure 31. For a frequency of 162.5 MHz, the S11 and S21 parameters are −39 dB and −0.0022 dB, respectively.

7. Vacuum Design

For a high-power RFQ operating in CW mode, improving the degree of vacuum can effectively reduce the risk of unintended sparking [33,34]. To ensure stable operation, the vacuum level was set at 10⁻⁶ Pa. Since there is limited gas conductance between quadrants, attaining vacuum uniformity is challenging. Previous RFQ experiments with similar vacuum system [35] demonstrated that two adjacent quadrants can maintain similar pressures by using only one vacuum window, as shown in Figure 32 for our vacuum system.

7.1. Vacuum Gas Load Calculation

The gas load Q of the RFQ is divided into three variables [36]:

(1)

The surface outgassing rate of the vacuum material (Q1) is defined as the product of the outgassing rate of the material and the surface area of the system. For an RFQ made of oxygen-free copper vacuum material, the outgassing rate is 7.98 × 10⁻⁹ Pa · L · s⁻¹ cm⁻². For a total surface area of about 94,199 cm², the Q1 value is then 7.52 × 10⁻⁴ Pa · L · s⁻¹.

(2)

The system leakage and permeability (Q2) is typically 5–10% of the total gas load Q1 in an ultra-high vacuum system. For a conservative estimate, Q2/Q1 = 10%.

(3)

The outgassing capacity of other components added to the system (Q3) mainly consists of vacuum gauges outgassing. We considered Q3/Q1 = 10%.

Summing over all source variables, the total gas output of the RFQ system is approximately 9.02 ×10−4 Pa · L · s⁻¹.

During the passage of the beam, the continuous release of hydrogen gas from the ion source into the LEBT section contributes to a larger gas concentration into the RFQ cavity. Here, the beam pipes in both the low-energy and medium-energy segments are cylindrical, as shown in Figure 33. The capacity for the gas to pass through the vacuum pipe is referred to as its conductance and is defined as the amount of gas that flows through the pipeline under a unit pressure difference. For conductance C and pressures at either end of the vacuum pipe p1 and p2, the flow rate is

$$\mathrm{Q}=\mathrm{C}∆\mathrm{p}$$

(16)

The mean free path of the gas is larger than the inner diameter of the tube. As a result, gas particle collisions with the inner wall of the pipe are common and dominated by free molecular flow. When the length of the pipe is more than 20 times its own diameter, the pipe conductance can be calculated with the long flow conductance equation [36]:

$$\mathrm{C}=\frac{1}{6}\sqrt{\frac{2\pi RT}{M}}·\frac{d^3}{L}$$

(17)

where d is the pipe diameter (m), L is the pipe length (m), R is the molar gas constant (8.3143 J/(K · mol)), and T is the gas temperature (K). In 1932, Clausing gave a simplified flow conductance formula C (L/s) for short, cylindrical tubes (L/d < 20) at 20 °C,

$$C_{20℃}=11.6 A_0\alpha$$

(18)

where $A_0$ is the area cross-section of the pipe (cm²), and $\alpha$ is the Clausing coefficient. The smaller the L/d ratio, the closer $\mathsf{\alpha }$ is to unity. In order to guarantee the vacuum pressure, $\mathsf{\alpha }$ was chosen to be one. The vacuum pressures of the LEBT and MEBT are about 3 $\times$ 10⁻⁵ Pa and 1 $\times$ 10⁻⁵ Pa, respectively. The diameter of the beam input pipe was designed to be 1 cm, and the exit diameter was set to 2.2 cm. Details of the conductance parameters and air intake for both the low-energy and high-energy sections are provided in

Table 12. Conductance parameters.

Parameter	LEBT Value	MEBT Value
D (cm)	1	2.2
S (cm2)	0.79	3.80
$C_{air}$ (L/s)	9.11	44.09
${\mathrm{C}}_{H2}$ (L/s)	35.26	170.65
$P_{RFQ}$ (Pa)	10−6	10−6
$P_{system}$ (Pa)	3 × 10−5	10−5
$∆\mathrm{P}$ (Pa)	2.9 × 10−5	0.9 × 10−5
Q (Pa · L/s)	1.022 × 10−3	1.536 × 10−3

. The ion source employs hydrogen gas, for which the conductance with air is:

$$C_{H_2}=3.87C$$

(19)

The gas loads amount to 1.022 × 10⁻³ Pa · L/s for the LEBT and 1.536 × 10⁻³ Pa · L/s for the MEBT. The gas load between sections is determined using the formula for the H₂ flow conductance.

7.2. COMSOL Simulation

Our vacuum design was conducted using a COMSOL simulation. As mentioned, in the presence of ultra-high vacuum conditions, the mean free path of gas molecules largely exceeds the characteristic length of the vacuum cylinder. Consequently, the free-molecular flow solver is chosen.

Initially, the vacuum of an individual cavity is computed without considering the influence of preceding or subsequent components. In COMSOL, using a tetrahedral grid gives too small of a step size for cavity electrode band modulation, leading to excessive grid division and a slow calculation. To improve computational efficiency, we carried out the electrode simulation without modulation. The thermal desorption rates for the connecting pipe, coupler, tuner and cavity structure of the vacuum pump were all set to 7.98 × 10⁻⁹ Pa · L/(cm² · s), in accordance with oxygen-free copper parameters. As shown in Figure 34, pairs of vacuum pumps, set to a speed of 600 L/s each, were placed within the first and last cavities. The simulation included the gas loads over all surfaces and the gas pressure on the beam line.

Accounting for the effects of the LEBT and the MEBT sections on the vacuum, the gas load was set to 1.022 × 10⁻³ Pa · L/s and 1.536 × 10⁻³ Pa · L/s separately. The speed of the pumps was set to 1200 L/s, and an ion pump with 440 L/s was added to the second cavity. The 3D vacuum level of the simulated gas is depicted in the left side of Figure 35. Here, the primary concentration of gas molecules at the RFQ entrance indicate that the low-energy transport section wields the largest influence. The right side of Figure 35 shows the pressure along the central axis of the cavity. The pressure at the RFQ end plate is as high as 1 × 10⁻⁴ Pa, whereas the pressure along the central axis is approximately 1.2 × 10⁻⁶ Pa. To increase the vacuum level, it is necessary to reduce the pressure at the entrance of the low energy section.

In our first simulations, which focused on a single cavity without any gas load from LEBT or MEBT sections, we observed that a total of four sets of vacuum pumps were necessary, each with a pumping speed of 600 L/s. In this configuration, after reaching a steady state, the vacuum level reaches 10⁻⁶ Pa. When factoring in the influence of both the LEBT and MEBT sections, however, a more robust vacuum system is required. By employing four sets of vacuum pumps with a pumping speed of 1200 L/s and an additional pump with a capacity of 440 L/s, a vacuum level of 1.2 × 10⁻⁶ Pa can be obtained. To achieve this standard, the molecular pump is set to FF-200/1200, while the ion pump is set to CF150/SIP500.

8. Conclusions

RFQ dynamics design, electromagnetic design, and multiphysics were completed. The RFQ, operating in CW mode, can accelerate a deuteron beam at 10 mA from 40 keV to 2.1 MeV over 3.7 m, with a transmission of 98.63%. Error analyses showed that the accelerator is suitable for handling nonideal beams and various types of machine errors. The cavity, which consists of a four-vane structure, offers mechanical stability, convenient cooling, and a high specific impedance. Due to the cavity length being an even multiple of the wavelength, no additional tuning systems are required, and the mode separation can reach 3.3 MHz. Through optimized tuning and undercut parameters, the cavity satisfies all resonant frequency, longitudinal electric field, and tuning specifications.

For transmission, the electric field computed from the RF structure yields outcomes consistent with the beam dynamics design. The total cavity power is 82 kW, which is fed into the cavity through 6-1/8 couplers. Gas loads calculations for the RFQ system, encompassing both low- and medium-energy segments, were undertaken. Through a COMSOL simulation, it was found that an optimal configuration requires four 1200 L/s molecular pumps and one 440 L/s ion pump.

Based on the RFQ design outlined in this paper, further processing of the acceleration cavity will commence. The device, which is set to be built in Chongqing, will represent an advanced high-current, CW deuteron RFQ.