Research on longitudinal dynamics and design of RF parameters of synchrotron for XiPAF-upgrading project

Xi ’an 200 MeV Proton Application Facility (XiPAF) will be upgraded to proton and heavy ion synchrotron recently. In order to ensure the enough life of heavy ion beam, the synchrotron requires ultra-high vacuum, and the designed static vacuum is better than 5×10-10 Pa. In order to place enough vacuum pumps in the synchrotron, the circumference of the synchrotron was increased from 30.9 m to 39.96 m. The revolution frequency range of heavy ions is 0.49∼1.03 MHz. In order to reduce the engineering difficulty and improve the lower limit of frequency bandwidth requirement of the RF system, the harmonic of h=2 is used to capture and accelerate the heavy ions, and the frequency bandwidth range of the RF system is adjusted from 1∼6 MHz to 0.8∼5.0 MHz. In this paper, the longitudinal dynamics parameters of the upgraded synchrotron are designed, and the simulation calculation is carried out. Finally, the parameter requirements of the RF system are proposed.


Introduction
Xi 'an 200 MeV Proton Application Facility (XiPAF) is composed of a linear injector of negative hydrogen ion, a medium energy beam transport line (MEBT), a proton synchrotron, a high energy beam transport line (HEBT) and a target station.In 2020, the synchrotron achieved the first successful beam generation, the number of extracted particles reached 4.3×10 10 , the extraction time was adjustable from 1~10s, the extraction energy was adjustable from 10~200 MeV [1], and the beam spot size at the target station was adjustable from 1 cm×1 cm to 10 cm×10 cm.Now it is planned to upgrade the XiPAF to add heavy ion beam line to provide heavy ion beam current [2], and upgrade the synchrotron to make its injection and extraction system, RF system and vacuum degree meet the requirements of various ion injection, capture, acceleration and extraction processes.
The new circumference of the proton and heavy ion synchrotron is 39.96 m, and the maximum magnetic rigidity of the dipole magnets is 2.32 Tm.It can complete the injection, capture, acceleration and extraction process of the required Bi 32+ , Kr 18+ , He 1+ , Ar 11+ , Si 8+ , C 4+ and H + in the synchrotron.Finally, the energy of each particle reaches the demand of the application.The main parameters of particles are shown in table 1.

Research on longitudinal dynamics
In the operation of synchrotron, the frequency of RF system is h times the revolution frequency (h is a positive integer, called harmonic number).Considering the low frequency corresponding to the injection energy of heavy ions, which is not within the working bandwidth range of the RF system, the harmonic wave of h=2 can be used to capture the heavy ion beam for acceleration, and the harmonic wave of h=1 to capture the proton for acceleration.According to the height formula and the area formula of the bucket (the area where particles can exist stably in longitudinal motion), the harmonic wave of h=2 has a lower height and smaller area than the bucket of the wave of h=1 under the same voltage amplitude.
Since the original RF system on XiPAF device can only stably provide RF voltages within the bandwidth of 1~6 MHz and the maximum amplitude of 800 V, when the capture and acceleration of heavy ions with h=2 (proton still takes h=1) is adopted, the demand for the RF system is: the bandwidth range is 0.9~4.3MHz and the maximum amplitude is 1200 V.For the h=2 and h=1, suitable RF parameters can be designed to make the acceleration efficiency basically the same.In practical engineering, it is easier to increase the power of RF system to meet the requirement of h=2 than to reduce the lower limit of frequency bandwidth to meet the requirement of h=1.Therefore, in the physical design of RF parameters, it is considered to use the harmonic of h=2 for the capture and acceleration of heavy ions (h=1 is still adopted for proton).
In order to ensure a certain capture efficiency and acceleration efficiency, the RF parameters must meet certain conditions.Under these conditions, the bucket area of the longitudinal phase space must be larger than the longitudinal emittance of the beam.From the perspective of transverse and longitudinal coupling, the bucket of longitudinal phase space should not be too high, otherwise the momentum dispersion of some particles will be too large, and then the coupling will cause the transverse envelope to be too large and then cause beam loss.The area of the bucket whose synchronous phase is not 0 is mainly obtained by multiplying the static bucket area (synchronous phase is 0) and the area factor [3], which changes with the synchronous phase.According to the initial emittance of the beam in longitudinal phase space, appropriate RF parameters are designed so that the bucket area after capture can cover the initial emittance.With the acceleration process, the decrease of the beam longitudinal emittance caused by the increase of beam energy and the decrease of the static bucket area caused by the increase of beam energy cancel out.It is only necessary to consider that the area of the bucket decreases due to the movement of the synchronous phase in the acceleration process, so that the area of the bucket can also cover the longitudinal emittance of the beam under the maximum synchronous phase.This ensures that when the bucket area shrinks with the increase of the synchronous phase during the acceleration process, the motion of the particles is also within the phase stability region.

Simulation results
According to the design principles mentioned in the previous section, combined with the dipole magnets strength rise curve, RF voltage amplitude, phase and frequency curves can be set up.
Combined with the above table, the RF system should provide a voltage amplitude of no less than 1200 V and a frequency bandwidth covering 0.9~4.3MHz.The voltage amplitude, phase and RF frequency curves are given by taking Bi32+ and H+ as figure2 shows.
According to the position of the element and the size of the vacuum pipe, the position and aperture limits are set in simulation calculation.Harmonic of h=2 is adopted to calculate the capture and acceleration process, and the efficiency of each ion is obtained in table 3.
Figure 3 and figure4 respectively show the particle loss at the scraper and longitudinal phase space distribution of Bi 32+ .It can be seen from the simulation results that the particle loss is mainly concentrated in the first 20 ms, and the beam loss location is mainly in the inner direction of the synchrotron at the scraper, that is, the loss is due to the aperture limitation of 28 mm.We also carried out simulation calculation for protons.Different from heavy ions, proton must consider the influence of space charge force during simulation calculation.We adopted the 2.5D space charge force model of PyOrbit [4].The injection working point is (1.730/2.110)and the extraction working point is (1.678/2.110).The number of particles before capture is 6.78e10, and the simulation results are shown in table 4.  The capture efficiency of proton acceleration to 60 MeV in XiPAF is 73% and the acceleration efficiency is 82% [5].The acceleration efficiency in XiPAF is higher due to the following reasons: 1.The acceleration time and final energy are different.Due to the power supply and radiation protection safety of the temporary laboratory where XiPAF is located, the experiment was mainly carried out for the acceleration to 60MeV, but the acceleration time to 60MeV is short (300ms) and the impact of space charge effect is limited.
2. Synchrotrons have different acceptances.XiPAF-uprading project, in order to meet the vacuum degree requirements, the vacuum tube wall needs to be baked, resulting in a small acceptance in y direction.
Compared with heavy ions, the capture and acceleration efficiency of H + is lower due to the existence of space charge effect.On the one hand, the number of protons is 1~2 orders of magnitude larger than that of heavy ions.On the other hand, the charge-mass ratio of protons is larger than that of heavy ions (for example, the charge-mass ratio of protons is 3 times that of C 4+ ).The tune shift caused by space charge effect is positively related to the charge-mass ratio.However, due to different particle terminal application requirements, the current strength requirements provided by different types of ion sources are also different, and there is no definite relationship between the number of particles and the charge-mass ratio.
On the one hand, the beam loss is caused by the low acceptance at scraper.On the other hand, the space charge effect of proton at low energy is strong, and the acceptance in x-direction and y-direction is nearly 10 times different after beam injection.At this time, the x-y coupling caused by space charge effect causes the emittance in y-direction to increase rapidly, resulting in beam loss at the entrance of the dipole magnets.The simulation results shown in table 5 are the results of preliminary optimization.We have adjusted the working point of the injection beam, and changed the number of particles of the injection painting to make the particle number as large as possible after the acceleration.In the next step, we will optimize the working points of injection and extraction and the changing path of the working points during acceleration so as to avoid the resonant lines that cause more beam loss.

RF system parameters
According to the longitudinal dynamics design results of the synchrotron and leaving a certain margin, the parameters of the RF system are given in the table 5.

Conclusion
In the plan of XiPAF upgrading project, the initial revolution frequency of heavy ions on the synchrotron is only 0.49 MHz, and the working bandwidth of the original RF system is 1~6 MHz.The difficulty of upgrading the RF system can be reduced by adopting h=2 harmonic to capture and accelerate the heavy ions.Through the analysis of longitudinal dynamics, the RF parameters of the design are given, and the simulation calculation is carried out.Finally, the RF parameter requirements are given.The new RF parameters increase the power requirement by 5 times, so it is necessary to make a new set of power amplifier.The magnetic alloy cavity is still considered to reuse the previous one.Later, it is necessary to carry out a test on the cavity to confirm that the surface temperature rise and impedance of the magnetic ring can meet the requirements under high voltage.

Figure 3 .
Figure 3. Particle loss at the scraper of Bi 32+

Figure 4 .
Figure 4. Longitudinal phase space distribution of Bi 32+ (From top to bottom, from left to right, the distribution is before capture, after capture, after acceleration and after voltage drop)

Table 1 .
Main Parameters Of Particles

Table 3 .
Simulation Results

Table 4 .
Simulation results of H +

Table 5 .
Parameters of RF system