1 Introduction

To achieve excellent performance in the research of nuclear and atomic physics, the Heavy Ion Research Facility in Lanzhou (HIRFL) was upgraded with a multi-functional Cooler Storage Ring (CSR) at the end of 2007 [1]. As the only injector of the CSR, the Sector Focusing Cyclotron (SFC) is still required to provide ion beams for downstream experiment terminals and the Separated Sector Cyclotron (SSC). The CSR has to be shut down when the SFC provides the beams to experiment terminals or the SSC, which results in the low utilization of the CSR. Furthermore, the SFC has served for nearly 60 years, and it cannot fulfill new experimental requirements, such as the beam current. To solve these problems, a room temperature heavy ion linac called CSR-LINAC is proposed as a new injector of the CSR to replace the SFC [2], as shown in Fig. 1. The CSR-LINAC consists of a superconducting high charge state Electron Cyclotron Resonance (ECR) Ion Source, a normal-conducting IH-RFQ, and six Interdigital H-type Drift Tube linac (IH-DTL) cavities [3].

Fig. 1.Full-screen View

(Color online) Layout of the HIRFL-CSR with CSR-LINAC.

Since its invention by Kapchinsky and Teplyakov [4] in the last century, the RFQ accelerator has become an essential part of the hadron accelerators in low energy section[5, 6] due to its high efficiency. In the recent decades, several RFQs have been designed and constructed for heavy high charge state ion beams. For example, the HLI-RFQ at GSI, which accelerates high charge state U²⁸⁺ ions with a duty factor of 50% [7, 8], the SSC-LINAC RFQ of IMP, which is a continuous-wave (CW) mode accelerator for accelerating ions [9, 10], and the MAFF IH-RFQ, which is a pulsed accelerator with a duty factor of 10% that can accelerate ions with mass-to-charge ratios up to 6.3 [11, 12]. The proposed CSR-LINAC RFQ can accelerate 3 emA ²³⁸U³⁴⁺ beams to 300 keV/u. The maximum duty factor is 0.4% so that this RFQ can be cooled easily. In this paper, the NFSP strategy is adopted to improve the transmission and shorten the cavity length. An error analysis is completed for verifying the error tolerance of the RFQ. The IH-RFQ cavity is employed to achieve excellent RF performance.

2 Beam Dynamics

Being upstream of the IH-DTL, the RFQ accelerator supplies the function of changing the initial DC beam to suitable bunches at the desired energy for the requirement of the DTL. The beam dynamics design is based on ²³⁸U³⁴⁺ beams with current of 3.0 emA and transverse emittance of 0.15 mm⋅mrad (normalized RMS), which is carried out by the code RFQGen [13]. Some important parameters, such as the operating frequency and inter-vane voltage, V, must be determined at the first step.

For an RFQ, a higher operating frequency is useful for designing a compact cavity. On the other hand, a lower frequency signifies a larger beam acceptance of the RFQ channel, a better transmission performance, and freer selection of structure parameters [2]. 108.48 MHz is set as the operating frequency of CSR-LINAC RFQ.

The RFQ inputs kinetic energy, at 4 keV/u, and is chosen since the corresponding ECRIS extraction voltage is in an acceptable range of 12 kV ~ 28 kV for the stability and reliability of ECRIS. To ensure the machinability and operation stability of the DTL, the DTL beginning cell length should be longer than 20 mm. Then, the output energy of the RFQ can be designed to be 300 keV/u, which will lead length of the shortest DTL cell of 35 mm. The inter-vane voltage of 68.8 kV is used for the compact RFQ structure and to reduce the risk of RF sparking. The corresponding Kilpatrick coefficient is 2.

The Four-Section Procedure (FSP) [14] developed by LANL (Los Alamos National Laboratory)is a conventional design strategy, which divides a RFQ into four sections: a radial-matcher (RM) section to adapt the time-independent input beam to the time structure of the focusing system, a shaper (SH) section for pre-bunching, a gentle-buncher (GB) section to help change the input beam to proper, small bunches, and an accelerator (ACC) section for a fast energy gain. As the essence of this design method, the GB section keeps the zero current longitudinal phase advance, σ_0L and the geometric length of the separatrix, Z_ψ, constant, which can provide excellent control of the emittance growth induced by the space-charge. The σ_0L and Z_ψ are determined by the following formulas:

where A₁₀ is the acceleration efficiency, V is the inter-vane voltage, ϕs is the synchronous phase, M/q is the mass-to-charge ratio, m₀ is the rest mass per nucleon, β is the velocity divided by the speed of light, and ψ is the angular width, which is related to the synchronous phase by tanϕ_s = (sinψ-ψ)/(1-cosψ).

However, there are remarkable disadvantages: the synchronous phase, ϕ_s, and the acceleration coefficient, A₁₀, rise slowly in the GB section, which results in a long cavity, and the unsmooth evolution of the phase advance leads to beam mismatch and even particle losses. To improve the beam transmission and obtain one compact cavity for the CSR-LINAC RFQ, an unconventional design approach, the New Four-Section Procedure (NFSP) [2, 15, 16] has been employed. The NFSP strategy carefully considers the match between the beam size and the acceleration channel. The transverse focusing factor, B, is varied along with the RFQ to balance the transverse defocusing effect and avoid the parametric resonance [17], especially the case of σ_0T<σ_0L in the traditional FSP (see Fig. 2(c)). σ_0T and B are given by:

Fig. 2.Full-screen View

(Color online) The comparisons of beam dynamics of the FSP and NFSP designs.

where λ is the RF wavelength, a is the minimum aperture radius, X is focusing efficiency. The non-resonant stable region requires $B>2\sqrt{3}\pi {\sigma }_{\text{0L}}$. In the new SH section, the parameter B increases gradually as the transverse defocusing factor due to the growing acceleration coefficient, A₁₀. The ascending B in the SH section and the variable Z_ψ allow ϕ_s and A₁₀ to rise much more rapidly than in the original GB section. In the ACC, section B should decrease since the transverse defocusing factor is reduced because of the rapidly increasing beam energy.

The comparisons of beam dynamics between the FSP and NFSP designs of CSR-LINAC RFQ are shown in Fig. 2. As shown in Fig. 2(a), ϕ_s and A₁₀ in the NFSP GB section increase much more rapidly than in the FSP GB section, and the cavity length of the NFSP design is 58 cm shorter than the FSP design. Fig. 2(b) shows that the comparison of the transverse focusing factor, B, and peak electric field, E_s, in the FSP and NFSP design. The maximum value of B in the NFSP design is larger than in the FSP design, while the maximum value of E_s in the NFSP design is smaller. In the FSP GB section, the σ_0L is larger than σ_0T (the dash curves in Fig. 2(c)), which causes parametric resonance and significant particle losses in the transverse plane (the red dots in Fig. 2(d)). The solid curves shown in Fig. 2(c) represent σ_0T and σ_0L in the NFSP design, which shows that the curves of σ_0T and σ_0L have been separated. Beam losses are also diminished, as shown in Fig. 2(d) (the black dots).

Figure 3 shows the beam transmission along the RFQ. Particle losses occur mainly at the end of the GB section (around the 177th cell) and the ACC section (around the 225th cell). For ²³⁸U³⁴⁺ beams with a current of 3 emA, the transmission efficiency of 98.0% is reached. Fig. 4 shows the beam phase space projection at the entrance and exit of RFQ. The FWHM energy spread is about 2.4% at the exit of the RFQ.The main parameters of the CSR-LINAC RFQ are listed in Table 1.

Fig. 3.Full-screen View

(Color online) Beam transmission along the RFQ. Plots from top to bottom are the beam profiles in the horizontal and vertical planes, the phase, and the energy spectra, respectively.

Fig. 4.Full-screen View

(Color online) The beam phase space projection at the entrance and exit of the RFQ. (a), (b), and (c) are the horizontal, vertical, and longitudinal phase space at the entrance respectively; (d), (e), and (f) are the horizontal, vertical, and longitudinal phase space at the exit, respectively.

3 Error Analysis

In actual operation, the beam parameters at the entrance of the RFQ and the inter-vane voltage have some differences with the design values, which may significantly influence the beam transmission in the RFQ channel. Hence, it is valuable to analyze the impact on the RFQ sensitivity and stability caused by non-ideal input beams and vane voltage deviation. The design values of the beam parameters at the entrance of the RFQ are the beam transverse emittance ϵ=0.15 mm⋅mrad (normalized RMS), α=0.6, β = 2.6 cm/rad, the beam current of 3 emA, and the energy of 4 keV/u with zero energy spread and 360° phase width. The design inter-vane voltage is 68.8 kV. One parameter was changed and others were kept constant during the error analysis. In Fig. 5, the current changes from 0 to 20 emA, and the efficiency remains relatively high for values less than 7.0 emA. As such, this design leaves a large margin for future updates and extensions. Fig. 6 presents that the input beam energy deviation has an effect on the transmission efficiency. The transmission slowly declines when the absolute value of the input energy deviation is less than 3%. As Fig. 7 shows, for the design Twiss parameters, a large emittance, which means a large beam size, induces a transmission decay, and a lower emittance causes transmission efficiency decreases as well because it enhances the space charge effect due to the smaller beam size. Fig. 8 shows the plots of the transmission efficiency and output energy versus inter-vane voltage. To meet the design requirements of transmission and output energy, the inter-vane voltage must be more than 67 kV. Fig. 9 is the contour map of the transmission efficiency and transverse emittance growth for different input Twiss parameters. The white point in the figure represents the design value. Fig. 10 and 11 are contour maps of the transmission efficiency and transverse emittance growth with input beam center and beam center angle offset, respectively. To get high transmission efficiency and the emittance growth to be less than 10%, the input beam center and beam center angle offset must be in the range of ±0.4 mm and ±20 mrad, respectively.

Fig. 5.Full-screen View

Transmission efficiency versus input beam current

Fig. 6.Full-screen View

Transmission efficiency versus input energy deviation

Fig. 7.Full-screen View

Transmission efficiency versus input emittance

Fig. 8.Full-screen View

Transmission efficiency and output energy versus inter-vane voltage

Fig. 9.Full-screen View

(Color online) Transmission efficiency (a), horizontal emittance growth (b), and vertical emittance growth (c) versus input beam Twiss parameters.

Fig. 10.Full-screen View

(Color online) Transmission efficiency (a), horizontal emittance growth (b), and vertical emittance growth (c) versus beam center offset at the entrance.

Fig. 11.Full-screen View

(Color online) Transmission efficiency (a), horizontal emittance growth (b), and vertical emittance growth (c) versus beam angle offset at the entrance.

4 RF Design

The CST Microwave Studio (MWS) [18] code was used for the RF design of the CSR-LINAC RFQ with respect to resonance frequency, field distribution, and power losses. The impact of the non-ideal field distribution on the beam dynamics is investigated by the TraceWin [19] code.

The IH type RF structure was chosen since this structure was demonstrated to have a higher shunt impedance and mechanical stability [20, 21]. Fig. 12 shows the CST model of the IH-RFQ, which consists of a cylinder, 4 mini-vane rods, 31 stems, and 2 supporter plates. There is strong RF coupling along the cavity by the longitudinal magnetic field, which leads to a homogeneous distribution of wall losses. The overall power loss in the cavity walls is comparatively small, which results in a high shunt impedance. The electrical conductivity of the copper was set to 5.0 × 10⁷ S in the electromagnetic simulation based on the consideration of RF contact resistance in the installation plane. The RF parameters, simulated by the MWS code, are presented in Table 2. The RF power of 123 kW is needed to accelerate ²³⁸U³⁴⁺ beams to the design energy, and then a 200 kW RF transmitter with some margin is required for the RF power supply.

Fig. 12.Full-screen View

The CST model of the IH-RFQ

The beam dynamics design requires a flat inter-vane voltage distribution. However, in practice the field is different from the dynamics requirement, which can be described as the quadrupole field unflatness and the dipole field. The quadrupole field unflatness is mainly caused by nonuniform capacitive loading along the whole cavity and stray capacitances between the wall and the rods. It can be calculated with the following formula:

where $\overline{V}$ is the average value of V; V=(V₁+V₂+V₃+V₄)/4; V₁, V₂, V₃, and V₄ are inter-vane voltages of the 1st, 2nd, 3rd and 4th quadrants, respectively. Each quadrant in the RFQ is shown in Fig. 13. The dipole field is caused by the asymmetry of the RF structure, which can be represented by:

Fig. 13.Full-screen View

The number of each quadrant. This figure shows the view from the entrance of the cavity.

As Fig. 14(a) shows, the quadrupole field unflatness is between -21% to 12% in the preliminary RF design, which is not satisfied for the beam transmission. Furthermore, the beam dynamics simulation with this unflatness indicates that it can cause an additional energy deviation at the RFQ exit. This result does not meet the requirement of the DTL structure. At the same time, the quadrupole field unflatness changes the Kilpatrick coefficient from 2 to 2.24 when leads to the increased risk of RF sparking. To the optimization of the quadrupole field unflatness, the tuning method of the magnetic flux induction by variation of the girder that undercuts lengths at both tank ends, is applied, as shown in Fig. 12. Fig. 14(b) shows the quadrupole field unflatnesses after optimizing. When the height of undercuts is 40 mm, the lengths of the undercut at the entrance and exit are 120 mm and 170 mm respectively, and the quadrupole field unflatness is at a minimum (±2.5%), which then has no effect on the output energy.

Fig. 14.Full-screen View

The quadrupole field unflatness optimization. Plots from left to right are the quadrupole field unflatness distribution before optimizing and the effect of undercuts on quadrupole field unflatness respectively.

The dipole field will cause the oscillation of the beam center. Fig. 15(a) shows that the distribution of the dipole field along the RFQ ranges from -3.2% to -2.1%. The beam dynamics simulation results show that the dipole field of -3.2%∼-2.1% causes a slight oscillation of the beam center, as shown in Fig. 15(b). The transmission efficiency is still 98.0%. This means that the dipole field does not need to be decreased any more.

Fig. 15.Full-screen View

The dipole field and its effect. The left figure is the distribution of the dipole field, and the right is the distribution of beam envelope and center with dipole field. The blue line is the beam envelope, and the red line is the beam center.

5 Conclusion

The beam dynamics study and RF design of an IH-RFQ for CSR-LINAC are finished at the Institute of Modern Physics. This RFQ accelerates heavy ion ²³⁸U³⁴⁺ beams with a current of 3.0 emA from 4 keV/u to 300 keV/u. The NFSP design method was applied to achieve the transmission efficiency of 98.0% with a 3.07 m long cavity. The transmission of the ²³⁸U³⁴⁺ beams for different input beam parameters and inter-vane voltages was investigated. The simulation results show that this beam dynamics design has a wide compatible margin for non-ideal beams. The RF performance of the IH-RFQ was simulated by the CST MWS code. The RF power for the acceleration of ²³⁸U³⁴⁺ beams is 123 kW. The quadrupole field unflatness of ±2.5% and the dipole field of -3.2%∼-2.1% were obtained, which have little effect on the beam transmission.