Study for Space Charge Effect in tune Space at CSNS-II/RCS

CSNS-II is an upgrade project of China Spallation Neutron Source (CSNS), requiring an increase in beam power from 100 kW to 500 kW. In order to find a suitable working points area in advance and evaluate the influence of the space charge effect on the Rapid Cycling Synchrotron (RCS) of CSNS-II, the measurements of beam loss with various tunes on CSNS/RCS was carried out and the beam loss simulation in the transverse tune space at CSNS-II/RCS has been performed. Finally, four candidate working points are compared.


Introduction
CSNS is a high-power pulsed spallation neutron source consisting of an accelerator, a target station and 3 spectrometers [1].The accelerator complex primarily includes a negative hydrogen (H-) Linac and a RCS ring.The H-beams at 80 MeV is injected into the RCS via a multi-turn charge-exchange process and accelerated to 1.6 GeV with approximately 20,000 turns in the ring and finally is extracted to the target.The design beam power is 100 kW in CSNS and 500 kW in CSNS-II, corresponding to the number of particles stored in RCS from 1.56 × 10 13 to 7.8 × 10 13 .The RCS beam commissioning started in May 2017 and it reached the designed beam power in February, 2020 [2].In order to achieve the goal of beam power with 500 kW, it is planned to use a radio frequency ion source to raise the peak current from 15 mA to 30 mA, a number of superconducting cavities were proposed to increase the linac energy from 80 MeV to 300 MeV.In the RCS ring, the injection area will be modified and 3 magnetic alloy cavities will be added to reduce the influence of space charge effect on the beam.
The working point is an important parameter of the ring accelerator, which is determined by the lattice structure of the accelerator.The improper working point will cause a strong transverse oscillation, and even beam loss during crossing resonance line.The uncontrollable beam loss for high intensity proton machine will lead to severe radiation problems and increase the difficulty of maintenance.Furthermore, the space charge tune shift and spread will cause the particles to easily approach or cross the resonance line and reduce the available tune area.In addition, the resistive-wall head-tail instability found in CSNS is closely related to the working point, which also makes it more difficult to find suitable tunes [3].Requirements on the working point at CSNS-II/RCS are higher because of the greater beam power and stricter limits on beam loss.It is therefore necessary to find a suitable working point region in advance through a large range of tune scanning and reevaluate the impact of the space charge effect on CSNS-II/RCS.In this paper, the beam experiments are performed with different working points at CSNS/RCS, then the relation between various working points and the beam loss at CSNS-II/RCS is given by numerical simulations, and the effect of space charge on high intensity beam is also analyzed.

Tune experiment at CSNS/RCS
CSNS/RCS is a 4-fold symmetrical structure and adopts a triplet cell as the fundamental unit of the lattice model.The layout of the magnets in one super-period is shown in Fig. 1, which consists of 6 dipoles and 4 sets of triplet focusing structures.The 48 quadrupole magnets in the whole ring are powered by 5 sets of power supplies and the specific parameters are shown in Table 1.The magnetic focusing structure of CSNS-II/RCS remains unchanged.The design working points was (Q x , Q y )=(4.86,4.80) for CSNS/RCS in order to avoid the major low-order structure resonance and the influence of space charge tune shift.However, it was found that a very strong horizontal coherent oscillation occurs when the beam power exceeds 40 kW.Generally, the beam instability can be suppressed by sextupole magnets with increasing the chromaticity.But the beam oscillation is hard to suppress when the beam power exceeds 80 kW.Finally, the tunes (Q x , Q y )=(4.80,4.86) was adopted in the actual machine operation.The horizontal coherent oscillation was relatively weak at this operating point and can be suppressed by the sextupole magnets with little beam loss.The beam power of CSNS-II is increased by a factor of 5, and the coherent oscillation will be stronger.In order to suppress the instability, we will replace the DC power supplies of the sextupole magnets with more powerful bipolar AC power supplies.However, it is difficult to predict precisely whether the instability can be suppressed in the actual beam commissioning.Besides, the working points (4.80, 4.86), (4.86, 4.80) and nearby are small-split cases, easily affected by the Montague coupling resonance, the resonance condition for which is 2Q x -2Q y = 0 [4], resulting in emittance exchange.
It is important to understand the low intensity beam behavior before studying the high intensity effects.In order to provide the candidate working points for CSNS-II/RCS, we carried out a series of experiments with different tunes and focused on the beam loss and instability.The partial results are shown in Fig. 2, which is the beam position oscillation measured by turn-byturn BPMs.The left and right sides correspond to the horizontal and vertical plane, respectively.Figure 2 shows that the coherent oscillation mainly occurs in the horizontal direction when Q x > 4.8, and the oscillation time is delayed with the increase of the horizontal working point.A detailed discussion on the coherent oscillation can be found in the literature [3].Table 2 displays the beam survival fraction and instability with some working points.The PyORBIT tracking program based on the PIC algorithm was adopted to calculate the theoretical beam loss.As can be seen in Table 2, the simulated and measured transmissions are very close, which also demonstrates the accuracy of the PyORBIT program.Furthermore, in Table 2, there are no coherent oscillations at some operating points below half-integer, which is a good reference for operating point selection for CSNS-II/RCS.According to the upper limit of the actual power supply, the scanning range of the working points is determined as 4 < Q x < 5.5, 4 < Q y < 5.5 (the actual range is 4.08 ˜5.43) with a scan interval of 0.05.
The corresponding numerical simulation still adopts the PyORBIT tracking program.For the simulation, the number of macro particles is set to 2 × 10 5 using a transverse grid of 128 × 128 × 128 to minimize grid noise.In order to accurately predict the beam loss and find better operating points, the simulation starts from the injection time and tracks 3000 turns, while the whole cycle is about 22000 turns.The aperture is set according to the actual machine, and the maximum transverse acceptance is 540 π mm mrad.The space charge effect is particularly critical for CSNS/RCS according to a large number of simulations and beam experiments.Therefore, this paper only considers the beam loss caused by the space charge effect at different operating points without taking into account other factors such as magnetic field error.
Figure 3 shows the simulated beam loss map in transverse tune space 4 < Q x < 5.5, 4 < Q y < 5.5 at 500 kW.Different colors indicate different beam survival fraction.When the vertical working point is close to an integer or half-integer, the beam survival fraction decreases significantly or even loses all.In the regions of 4 < Q x < 4.5, 4 < Q y < 4.5, there is a very large amount of beam loss, which is primarily due to the particles close to the structure resonance of Q x = 4 or Q y = 4 .In addition, the beam survival fraction of several large areas approaches 100%, such as 4.33 < Q x < 5.23, 4.73 < Q y < 4.88 and 4.88 < Q x < 5.28, 4.33 < Q y < 4.43.In general, the beam power of 500 kW is easy to achieve without taking the magnetic field error and other factors into account, and the lattice has the ability to adjust the operating points in a large range.

Discussion
The space charge effect is the most important factor limiting the power increase of the high intensity proton synchrotron.Generally, the space charge tune shift is used to characterize the strength of the space charge effect.The expression of the space charge tune shift is where r p = 1.53 × 10 −18 is the classical proton radius, N is the number of particles in the accelerator, ε is the transverse emittance, β, γ are the usual relativistic quantities, and B f is the bunching factor.For CSNS-II/RCS, the injection energy is 300 MeV, the total number of particles is 7.8 × 10 13 , the bunching factor is approximately 0.42 and the emittance is around 250 π mm mrad.According to Eq. 1, the incoherent tune shift is about 0.2.For high intensity synchrotron, half integer resonance is among the strongest effects limiting the achievable maximum beam power.A large number of theories and experiments have confirmed that the half-integer resonances caused by space charges are directly related to coherent resonances rather than incoherent resonances [5,6].In general, the coherent resonance condition has the form where ∆v is the incoherent space-charge tune shift from its zero-current value (v 0 ), ∆Ω m is the coherent space-charge tune shift of the mth-order mode, and m = 2 stands for the half-integer resonance.The corresponding coefficient C mk can be easily extracted from the literature [5].According to Eq. 2, we roughly estimate that the coherent tune shift for CSNS-II/RCS is about 0.1.So the nominal operating point is at least outside the range of ± 0.1 to avoid the halfinteger resonance.As shown in Fig. 3, when the vertical working points are close to the integer or half-integer range ± 0.1, the beam survival fraction is significantly reduced, which is also consistent with the coherent resonance theory.On the basis of the above simulation scan results and the beam experiments at CSNS/RCS, four candidates of operating points for CSNS-II/RCS are found and compared, as listed in Table 3.In both cases, the beta functions of the modes (4.80, 4.86) and (4.80, 4.86) are relatively small, and the emittance is small as well.However, the two modes are affected by the Montague resonance, causing strong emittance exchange.Furthermore, it is not completely sure whether the instability can be suppressed by the modified AC sextuple magnets with the enhancement of coherent oscillation.The working points of (5.30, 4.38) and (4.38, 5.30) have large splitting with weak coupling, and no coherent oscillation associated to them has been observed in CSNS/RCS experiments.In addition, the simulated beam emittance is within the acceptance range in both cases.Overall, the last two working points in Table 3 may have a large advantage in the future.

Summary and Conclusion
To find an appropriatee working point area for CSNS-II/RCS, the measurements of beam loss with different tunes at CSNS/RCS are carried out and the beam loss simulations in transverse tune space at CSNS-II/RCS have been performed by PyORBIT code.At last, four candidates of operating points for CSNS-II/RCS are found and compared.At present, the simulation work only considers the influence of the space charge effect and the magnet errors will be taken in to count in the next step.

Figure 1 .
Figure 1.The layout of magnets in one super-period.

Figure 2 .
Figure 2. Turn by Turn beam position with different tunes.

Table 1 .
Main parameters of quadrupole magnets.

Table 2 .
Beam survival fraction with different tunes.
3. Tune scan simulation at CSNS-II/RCSThe program MAD-X was used to match different working points with the magnet fields of five groups of main quadrupole magnets as variables.The matching is based on the magnet strength of adjacent modes to ensure the continuity of the magnetic field and Twiss parameters.

Table 3 .
Comparison of working points.