1 Introduction

The RFQ is an essential component of the linac and has a significant advantage with respect to the acceleration of beams with low velocity in the typical range of approximately 0.01–0.06 times the speed of light [1]. In recent years, many compact accelerators have utilized RFQ as injectors in many fields such as cancer therapy research and other radiation-based applications. For instance, hardon radiotherapy has gradually been developed in clinics, and many facilities have been built around the world such as in Germany [2], Japan [3, 4] and China [5, 6]. The injection and extraction energy parameters of some of these RFQ injectors are listed in Table 1 [7].

Recently, a new linac injector for the carbon ion therapy facility was proposed and designed by IMP. It consists of an ECR ion source, a 162.5 MHz RFQ, a compact Inter-digital H-mode Drift-Tube-Linac (IH-DTL), and beam transport lines. The layout of the linac is shown in Fig. 1.The RFQ operates at 0.1% duty factor, and accelerates a ¹²C⁴⁺ beam from 8 keV/u to 600 keV/u with a peak current of 0.05 pmA(0.2 emA). In the design process, the peak current is set to 0.1 pmA to maintain current tolerance.

Figure 1:Full-screen View

(Color online) Layout of the cancer therapy facility.

The dynamics design of the RFQ, which is mainly directed against space-charge effects, has been extensively investigated worldwide over many years [8-12]. However, the requirement of a compact structure is considered to be more important in most radiotherapy facilities than the suppression of the space-charge effect. In such facilities, the structure size impacts cost and can pose construction challenges. In this report, the optimization of the RFQ design will be investigated based on the Gentle-bunching (GB) section of the Four-section design. A new fast-bunching section is introduced to replace the GB section. This results in a compact design of the RFQ while keeping particle loss and emittance growth well-controlled.

2 Dynamics design

The dynamics design of the RFQ can be performed using the code PARMTEQM [13] which is widely used [14, 15] and important parameters such as the input energy and the inter-vane voltage should be defined in advance.

In the design process, a high input energy results in a long RFQ cavity due to the cell length and the effect of the bunching process, which generally leads to a decrease of the accelerating gradient of the RFQ. The ion source extraction voltage is usually chosen to be 20 to 30 kV for the C⁴⁺ ions as shown in Table. 2 [7]. The extraction voltage of this source should be chosen to match two conditions: one is an adequate beam intensity with a relatively good quality and the other is a reduction of the sparking risk. A suitable inter-vane voltage is required in an RFQ to effectively maintain the radial focusing strength and accelerating gradient. However, this value is restricted by the maximum surface field of the electrode. Finally, the intervane voltage is chosen in the range of 70 to 80 kV.

A basic dynamics design of the proposed RFQ was performed by PARMTEQM after reiterating the important parameters previously mentioned. However, a few problems were encountered. For example, the beam size could not be well controlled when it entered the GB section as shown as in Fig. 6, which impacts the transmission efficiency and the beam quality due to abrupt changes in certain parameters and phase advance resonances. On the other hand, the length of the electrode is 2.7 m. The fabrication of such an electrode in one segment will be difficult in standard machining environments. Thus, the best approach is to reduce the length or separate the electrodes into segments for machining.

Figure 6:Full-screen View

(Color online) Transverse phase space projection at the entrance and exit of the RFQ. The upper three figures refer to the entrance while the lower three refer to the exit. For the input beam, zero energy spread and 360° phase width are assumed. The points in the top right box are located on the line dW=0.00.

The optimization procedure was repeated to achieve an optimal design with a maximum transmission efficiency, and a minimum emittance growth and structure length. Some specifications and constraints are listed as follows:

1. For the proposed RFQ operating with 0.1% duty factor, the maximum surface field is constrained to lower than 27.2 MV/m. This corresponds to a Kilpatrick factor [16] of less than 2.0 at 162.5 MHz.

2. The structure length should be less than 2.5 m to minimize construction costs and power requirement.

3. Beam transmission efficiency should be as high as possible.

4. Phase advance resonances should be avoided.

Parametric resonance and coupling between the transverse and the longitudinal phase space are avoided by changing the focusing factor B along the cells. The B changes with the transverse RF defocusing force to maintain the transverse beam movement in dynamic balance [10, 20]. In addition, abrupt changes of parameters at the junctions are smoothed to improve the transmission efficiency [8].

The traditional RFQ dynamics design is based on Kapchinsky’s adiabatic bunching condition [22]. Under this condition, the geometric length of the separatrix is almost constant in spite of the increase in the velocity of particles, which can reduce the impact of the space charge effect. In a traditional design, the RFQ is divided into four sections: radial matching, shaping, gentle bunching(GB), and accelerating sections. The GB-section where the Kapchinsky’s condition is applied could be further divided into pre-bunching (small m and slow m ramping) and bunching (fast m ramping) sections [23].

The space-charge effect is not strong in the proposed RFQ for a compact linac design, and the GB-section can be shortened to preserve length for the accelerating section. A short fast-bunching section is introduced to substitute for the long adiabatic pre-bunching section. The parameter S which is almost proportional to the separatrix area is used for optimization of the fast-bunching section. S is defined as [23]:

where,

A is the accelerating efficiency, V is the inter-vane voltage, ϕ_s is the synchronous phase angle, ψ is the phase width, β is the synchronous velocity, γ is the Lorentz factor, q is the charge, and m is the rest mass of the ion.

An increase in S is important in forming a good beam bunch under the influence of a strong space-charge force. However, for low-current dynamics, this parameter could be kept at a constant, which facilitates faster beam bunching with sufficiently high transmission [23].

The parameter S can be kept almost constant in the fast-bunching section through optimization of the modulation factor m and the synchronous phase angle ϕ_s. The modulation factor grows faster compared to the case of a traditional design. Fig. 2 shows the optimized parameter curves of the fast-bunching section. When the optimized m factor of the fast-bunching is compared with a traditional m factor, the platform at the pre-bunching section almost vanishes. Moreover, the entire length of the RFQ is shortened based on the premise of ensuring particle transport, and the RFQ acceleration efficiency is increased.

Figure 2:Full-screen View

(Color online) The parameter curves of the optimized fast-bunching section. The m curve of the pre-bunching section is shown using dots for comparison. The start and end points of the fast bunching and pre-bunching sections are indicated by the solid orange circles.

Additionally, the shaping section is optimized to allow the beam to transmit effectively during the fast-bunching process. In this section, the m factor is optimized to a relatively low value, which decreases the separatrix height in the RFQ entrance and maintains a sufficient acceptance for the input beam.

Fast-bunching is used to create a more compact structure by ignoring the space-charge effect. As the beam current is increased, the fast-bunching process terminates, therefore, the current limits the application range of the fast-bunching design. Fig. 3 shows the impact of an increase of the current in the fast-bunching process. The beam experiences dispersion in the longitudinal phase space, since the longitudinal phase advance, decreases as the space charge effect increases. This impacts the beam transmission and quality in the bunching process. The current should be lower than 1.0 pmA in the design of the 162.5 MHz RFQ. This can be realized by analyzing the influence of the current increase on the fast-bunching process.

Figure 3:Full-screen View

(Color online) The longitudinal phase space at the 140th cell (near the end of the bunching section) for different beam currents.

Fig. 4 and Table. 3 show the parameters of the 162.5 MHz RFQ dynamics design after optimization. The final Kilpatrick factor of the design is 1.83. In the case of the fast-bunching design, phase advance resonances are avoided, the transmission efficiency is increased, and the structure length is shortened. A transmission efficiency of 99.3% is obtained with an RFQ length of 230 cm.

Figure 4:Full-screen View

(Color online) The final result of the fast-bunching design parameters as a function of position z.

The simulation results of the PARMTEQM are shown in the Fig. 5 and Fig. 6. The number of particles in the simulation is 10000. The particles are judged to be lost when their energy deviates from that of the synchrotron particle by more than 5%. The sum of the lost particles and transverse emittance is clearly reduced in the optimized design as shown in Fig. 7. It is observed that the loss peak in the optimization design occurs at approximately cell-180, because the fast-bunching process causes the separatrix to shrink quickly. The particles at the edge of the separatrix will be lost longitudinally when the beam enters the accelerating section.

Figure 5:Full-screen View

(Color online) Beam envelope evolution along the RFQ under final optimized design simulated by PARMTEQM. Plots from top to bottom are the beam profile in x and y-planes, phase and energy spectra respectively.

Figure 7:Full-screen View

(Color online) Comparison of beam loss (shaded bar graphs) and transverse emittance growth (dashed and solid curves) between the traditional and optimized design.

3 Error analysis

The tolerance of the RFQ for a non-ideal input beam directly influenced the beam stability and quality. Both the operating stability of the ion source and the ripple of the LEBT power supplies impact the beam parameters at the RFQ entrance [17, 18-19]. It is difficult to adjust the parameters of the beam to satisfy dynamics design during actual operation. Consequently, it is necessary to perform the error analysis at the entrance of the RFQ. The results are shown in Fig. 8.

Figure 8:Full-screen View

(Color online) Transmission efficiency of the RFQ versus input beam parameters. The red-square line represents the emittance. The green-triangle line represents the beam energy spread, and the blue-circle line represents the beam current.

The design values of the beam parameters at the entrance of the RFQ are εt =0.2 πmm⋅mrad (Norm. RMS), the beam current of 0.1 pmA, and the energy of 8 keV/u with zero energy spread and 360° phase width. The controlling variables method was used during the simulation. One of the parameters such as the emittance, current or energy spread of the input beams is adjusted while the other parameters are kept constant in the simulations. The Fig. 8 shows the relationship between some input beam parameters and the transmission efficiency.

A large emittance which is indicative of a large beam size and divergence, will decrease the transmission efficiency. The transmission efficiency is approximately 95% with 6% energy spread when the emittance is 0.35 πmm⋅mrad. This energy spread is much greater than that produced by the LEBT. When the beam current is 0.8 pmA, the transmission efficiency is maintained at 96%. This current tolerance satisfies the requirement of the accelerator and provides a large space for subsequent upgrading.

Except for the impact of a non-ideal input beam on the RFQ, there are some errors which result from fabrication, assembling, and operation. Some of these errors may adversely affect the beam transmission in the RFQ. As such, it is necessary to perform an error analysis of the RFQ. In this regard, we performed 1000 simulations using the TraceWin code. The error setting includes five items as shown in Table. 4. The errors are uniformly distributed in the ranges listed in Table 4.

The dR value refers to the error of the electrode pole radius, and the d refers to the error of the depth of the modulation curve. Both of these values are set to 0.1 mm based on prior experience with SSC-Linac fabrication. The ϕ refers to the error of the RF phase, which is determined by the precision of the RF phase control system. The Δ T and Δ L are the cavity position offsets produced during installing, which are set by the precision of the collimation. Based on the analysis of 1000 sets of errors, the probability distribution of the beam loss is determined using TraceWin simulations as shown in the Fig. 9. The calculated transmission efficiency is slightly higher than that of PARMTEQM, because the electric field and the beam loss criteria are different [24]. As shown in Fig. 9(a), it is 70% more likely to maintain the transmission efficiency above 97% when the fabrication and operation errors are considered. In Fig. 9(b), an emittance growth of less than 15% is highly feasible. This value will exceed not 30% when the RFQ error is taken into account. The emittance growth satisfies the operational requirement, and therefore, this dynamics design could be used for fabrication.

Figure 9:Full-screen View

(Color online) The relative frequency of the beam loss and transverse emittance increase for the error setting

4 Summary

A new RFQ design method which uses a fast-bunching section instead of the traditional GB section is developed for a compact, low-intensity linac. The design is constrained by the condition that the S parameter could be kept almost constant when the space-charge effect is not strong. In the case of IMP cancer therapy RFQ, very low beam loss is achieved using a fast longitudinal bunching process. Finally, bunching design is adopted to meet the strict requirements of a short length. This proposed method could benefit the next generation of RFQs for medical and nuclear physics applications where the current is relatively low, and the space-charge is not important.