Simulation study on the slow extraction for the improvement of the beam shill structure at J-PARC Main Ring

J-PARC Main Ring (MR) delivers a slow extracted 30 GeV proton beam to the Hadron Experimental Facility using third-order resonance. One of the critical properties required for the proton beam is the flatness of the time structure of the extracted beam (spill structure). We performed a simple beam simulation of the MR slow beam extraction to investigate the effect of the current ripples of the main magnet power supplies on the beam spill structure. In addition, we investigated in the simulation the effects of the feedback control system on the betatron tune using fast Q magnets and the transverse RF system to improve the spill structure. The simulation qualitatively reproduced the measured spill structures. Now the attempts to optimize their parameters for better spill structure are ongoing.


Introduction
The Main Ring (MR) at J-PARC (Japan Proton Accelerator Research Complex) [1] is a proton synchrotron that accelerates protons from 3 GeV to 30 GeV.One of the two MR's extraction modes is slow extraction using third-order resonance [2] toward the Hadron Experimental Facility, where various particle and nuclear physics experiments are conducted.
One of the essential properties required for the slow extracted beam is the flatness of the beam spill structure.Various efforts to improve the spill structure are being made in accelerators around the world where slow beam extraction is performed [3][4][5][6][7].In J-PARC MR, the main magnet power supplies have current ripples of about 10 −5 with respect to the flat-top current values, mainly in the frequency range below several hundred Hz [8], which causes a large fluctuation of the beam spill during the extraction.
Thus, we performed a simple beam simulation of the slow extraction at J-PARC MR to investigate the effect of the current ripples in the main magnet power supplies on the beam spill structure.In addition, the effects of feedback control on the extraction using fast Q magnets and a transverse RF kick using stripline kickers were investigated in the simulation for the search for better operation parameters.

Slow extraction at J-PARC MR and beam spill structure
In the slow extraction at J-PARC MR, the third-order resonance is excited by eight sextupole magnets in the ring, and the horizontal betatron tune is made close to 22. 333  resonance.The betatron oscillation amplitudes of the particles which enter the unstable region increase, and the particles are finally kicked by the electrostatic septa [9].In J-PARC MR, a spill length of about 2 s in a 5.2 s repetition cycle is realized by gradually increasing the horizontal betatron tune toward the third-order resonance.If the betatron tune has a large ripple, the beam spill has a corresponding time structure, and the flatness of the beam spill is impaired.Figure 1 shows the measured beam spill structure without countermeasures against the betatron tune ripple.

Figure 1.
The measured time structure of the extracted beam when only the macroscopic spill structure shaping was performed with a fixed polynomial function current pattern on the tune control quadrupole magnets.The upper panel is the number of particles in the ring measured by the DCCT.The lower panel is the extracted beam rate measured by the spill monitor made of a plastic scintillator and PMT.The binning is 10 kHz.
To evaluate the flatness of the spill structure, we adopted spill duty factor I 2 / I 2 , where I stands for beam rate and bracket means time average.The spill duty factor is 100% for an ideally flat beam.The spill duty factor for Fig. 1 was about 4%.
To improve the flatness of the beam spill structure, we utilize feedback control on horizontal betatron tune using fast Q magnets and transverse RF kicks by stripline kickers [10][11][12].Figure 2 shows the outline of the spill regulation system in J-PARC MR.
The measured beam spill structure with the spill feedback system and transverse RF kicks is shown in Fig. 3.The spill duty factor was significantly improved to about 46% but still far from the ideal, and the further improvement of the spill structure is one of the major issues in the J-PARC MR.

Simulation for slow extraction at J-PARC MR
We performed a simple slow extraction simulation to reproduce the beam spill structure following the below recipe.
(i) Generate normalized horizontal coordinates (X, X ) of each particle according to a normal distribution.The normalized coordinates (X, X ) are related to the horizontal position x and angle x by the following formula with twiss parameters.The dispersion in the straight section of J-PARC MR is almost zero.The chromaticity is also set to be almost zero.Thus the momentum distribution was not taken into account.The dynamic bump scheme [2] was not included in the simulation, either.The particle motion in the We input the betatron tune ripple calculated from the measured current ripples of the main magnet power supplies in the simulation to reproduce the beam spill structure, In the caluculation of tune ripple, the high-frequency components above 400 Hz were reduced according to the measured relation between the current ripple and magnetic field ripple [13].
The obtained beam spill structure from the simulation is shown in Fig. 5.The spill duty factor in this result is 5%.The distribution of the number of remaining particles in the ring well reproduced the stair-step-like changes in the measured DCCT data in Fig. 1.
As the next step, we implemented the spill feedback system using fast Q magnets and transverse RF kicks in the simulation.The feedback signals for the fast Q magnets during the beam operation were generated by a DSP (Digital Signal Processor).The function of the DSP was described in C language, which can be easily implemented in the computer simulation.We have two types of fast Q magnets, which we call EQ (Extractoin Q-magnet) and RQ (Ripple Q-magnet) [10].The delay times from the feedback signals to the magnetic field responses were set to be 400 µs for EQ and 50 µs for RQ based on the measurements.
In the beam operation, we used two stripline kickers.The bands of the input RF frequency tunes were 248.3268 − 248.3273 for one and 1.250 − 1.414 for the other.In this simulation, the frequency tune band of 248.3268 − 248.3273, which has a larger effect on the spill structure, was implemented as a single frequency of 248.32705.The results for the spill structure are shown in Fig. 6.The obtained spill duty factor was 31%.
We compared the spill duty factors between the simulation and the measurements in the following three cases; without any spill regulation, with only the spill feedback system using fast Q magnets, and with both the spill feedback system and transverse RF kick.Table 1 shows the results.The simulation qualitatively reproduced the measured results.
As an example of parameter exploration using simulation, the results of the search for the kick angle of the transverse RF are shown in Fig. 7.In the previous beam operation, the kick angle of the transverse RF was 1µrad.However, in the simulation, a better spill structure can be obtained with a kick angle of 0.6 − 0.8µrad.

Summary and outlook
We have constructed a simple simulation for the slow extraction of J-PARC MR, which qualitatively reproduces the spill structure caused by the current ripple of the main magnet    power supply.The next tasks are to reproduce the quantitative spill duty factor by improving the accuracy of the simulation and to find a better feedback algorithm and parameters of the transverse RF system using the simulation.
The main magnet power supplies in the MR were upgraded from 2021 to 2022, and the tunings of the power supply controls are underway.Improvement of the current ripples of the power supplies is expected [8].We will also proceed with simulations using the current ripples of the new power supplies as input.
ii) Kick the particles by sextupole magnet strength S and advance the betatron oscillation phase by betatron tune Q n .The eight sextupole magnets are replaced by one equivalent

Figure 2 .
Figure 2. Schematic picture of the spill regulation systems for the slow extraction at J-PARC MR.

Figure 3 .
Figure 3.The measured time structure of the extracted beam with the spill feedback system and transverse RF kicks.

Figure 4 .
Figure 4.The behavior of the particles' (X, X ) in simulation with the horizontal betatron tune Q n of 22.322.The ribbons of the electrostatic septum are located at X = −0.004.

Figure 5 .
Figure 5.The simulated time structure of the extracted beam without the spill feedback system or the transverse RF system.The binning is 10 kHz.

Figure 6 .
Figure 6.The simulated time structure of the extracted beam with the spill feedback system and the transverse RF kick.

Figure 7 .
Figure 7.The result of the search of the kick angle of the transverse RF for the better spill duty factor.

Table 1 .
Comparison of spill duty factor between measurements and simulation eesults.