# The Design of a High-Intensity Deuteron Radio Frequency Quadrupole Accelerator
## 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 Figures 9-11. 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, intervene 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 \(\upmu\)m (within the typical error range), there was no major alteration in transmission efficiency. Figure 8: Transmission efficiency of the RFQ versus Twiss parameter, beam offset, and tilt of the input beam. Figure 9: Transmission efficiency with different transverse emittances. 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 effect of a nonuniform electric field on beam transmission. Secondly, we examined the effect of a nonuniform electric field on beam transmission. Figure 11: Transmission efficiency with different beam currents. Figure 12: (**Left**) Schematic diagram of misalignment errors; (**Right**) consequent error analysis. Figure 10: Transmission efficiency with different input beam energies. 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:  
\[V(z)=V_{0}(z)+\Delta Vcos\Big{(}\frac{n\tau z}{2l}\Big{)} \tag{2}\]  
where \(n\) is a harmonic number (\(n\) = 1, 2, 3,....), \(V_{0}\) is the nominal intervane voltage, \(l\) is the length of vane, \(\Delta V\) is the amplitude of residual voltage error, and \(\Delta V\) 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.