Upgrade the impedance model in RCS of CSNS

The Rapid Cycling Synchrotron (RCS) in the China Spallation Neutron Source (CSNS) is a high-intensity proton accelerator that is susceptible to collective instabilities and performance limitations due to its impedance. With impedance study of key components and new component installations, the impedance model should be updated. This article presents a comprehensive estimate of the coupling impedance and outlines the impedance model of the RCS.


Introduction
The China Spallation Neutron Source (CSNS) [1][2] is a high intensity proton accelerator-based facility.The accelerator complex includes a negative hydrogen (H-) linac and a rapid cycling synchrotron (RCS).The RCS accelerates the beam up to 1.6 GeV with a repetition rate of 25 Hz, aiming to provide a 100 kW output beam power with an intensity of 1.56 × 10 13 at an injection energy of 80 MeV.At present, the beam power has reached 140 kW with an intensity of 2.18 × 10 13 , 40 % more than the nominal power.An impedance model has been provided in the Ref [3], and new vacuum components have been installed in the RCS to enhance beam power, such as the installation of a magnetic alloy cavity as the second harmonic cavity in summary 2022.During beam commissioning, a fast oscillation of beam position in the transverse plane was observed [4], prompting a re-investigation of the impedance to identify the source of the instability.This paper presents the key upgrade of the coupling impedance based on the existing model [3].A new impedance model in the RCS is obtained.

Impedance calculation
The impedance was calculated using analytical formulas along with numerical simulations and measurement.The vacuum components taken into account in this calculation include stainless-steel chamber, transverse collimator, and steps.The ABCI [5], CST [6] and IW2D [7] were used in the simulation.For the measurement, the bench method of the coaxial line and twin-wire was mainly adopted and the measurement was carried out since the extraction kicker measurement in 2012.The vacuum chamber of the RCS is 227.92 m and it comprises of ceramic, stainless-steel and Cu chamber.The Cu chamber is implemented at RF cavity and collimator.

Ceramic chamber
The ceramic vacuum chambers have been implemented in the dipole, the quadrupole and the injection painting magnet in the RCS to reduce the eddy current effects and the large ohmic loss.The physical appearance of ceramic chamber is exhibited in the reference [8].The shape, length and thickness are summarized in Table 1, with an elliptic chamber adopted in the dipole magnet and circular chambers used in others.The total length of ceramic chamber is 129.9 m.The inner surface of the ceramic vacuum chambers is coated with 100 nm TiN film to decrease the secondary electron yield, with a resistivity of the TiN at about 1.7 × 10 -7 Ωm and a relative dielectric constant of about 10.An RF shield comprised of 0.4 mm thick Cu plates waterjet-cut into a width of 5 mm and spacing of 5 mm is utilized to reduce the impedance of the image current.The impedance of ceramic chamber was calculated by the IW2D and the total impedance is summarized in Fig. 1 and Fig. 2.
Table 1.Parameters of the RCS Ceramic Chamber

Stainless-steel chamber
Two types of the stainless-steel chambers (304 and 316L) are utilized in the RCS and their impedance is calculated in Ref [3].It should be noted that a permeability of μr=55 was used for magnetic shielding against the leak field in a special region, which is actually closed to 1 in the RCS.Additionally, apart from the ceramic and Cu chamber, the length of the stainless-steel chamber is about 70 m, which is much shorter than the previous calculation.The stainless-steel chamber is circular with an average radius of 10 cm and the impedance has been re-estimated.The longitudinal and transverse impedances are 0.2 Ω and 2 kΩ/m at 1 MHz, respectively.

Magnetic alloy cavity
The magnetic alloy cavity is a self-developed device at CSNS and one such magnetic alloy cavity was installed in 2022.The impedance measurement has been conducted since the development of the onecell protype with connecting a power source.After the real cavity with three cells was fabricated, the impedance was checked.The longitudinal and transverse impedances were measured by using the wire method.There is not impedance peak in transverse plane within the RCS beam concerned range.The longitudinal impedances of the one-cell protype and the real cavity are displayed in Fig. 3.The fundamental mode at about 2 MHz is also given.It is worth noting that the parasitic impedance for the protype appears at about 50 MHz, but it shifts to 21.65 MHz for the real cavity.With the increased number of cells in the real cavity, the inductance and capacity are bigger than those of the prototype, which may result in the frequency shift of the parasitic impedance.The resonant frequency of the parasitic impedance is very closed to the cut-off frequency of RCS beam, and the study on the beam effects is currently underway.

Ferrite loaded cavity
The fundamental acceleration system in the RCS is comprised of eight ferrite-loaded cavities [9].With a classical and conservative NiZn load-ferrite, the coaxial resonant cavity has two accelerating gaps with single end [10].The ferrite loaded material used is Ferroxcube 4M2, with 56 ferrite cores installed in a cavity.The maximum voltage offered by eight cavities is 165 kV.In 2021, the impedance was simulated, and the transverse impedance was found to be serious.Subsequently, the impedance was checked by using the two-wire method.The measured impedance is different from the simulation result [11] because the simulated mode cannot accurately represent the real cavity.The total measured impedance is shown in Fig. 4. In longitudinal plane, the fundmental mode at about 0.8 MHz is removed and the high order mode at 9.9 MHz is clear but the peak value is small.In transverse plane, there are four resonances between 17 MHz and 22 MHz, and the peak values are also small.

Quadrupolar BPM
In 2022, a quadrupolar BPM was installed to measure the space charge tune shift [12].The simulation mode is shown in Fig. 5, where the left side of the figure displays the external cavity shape and connection ports of the BPM, while the right side shows the four electrodes inside the cavity.The vacuum aperture in the BPM is 220 × 165 mm.The transverse impedance can be ignored and the longitudinal impedance of the BPM is very small as shown in Fig. 6.

Extraction kicker
The fast extraction kicker system is one of the critical accelerator components and the primary source of impedance in the RCS of the CSNS.Conventional and improved wire methods were employed to measure the impedance as shown in Fig. 7, before its installation into the tunnel [13].The small oscillation sideband of the longitudinal and vertical impedance is due to the mismatching between the kicker magnet and the pulse form network.The measured impedance of the kicker magnet is explained by comparison with simulation, and the results are in good agreement.The average longitudinal, vertical, and horizontal impedance of the kicker system is about 7+j6.5 Ω, 1.2+j2.7 kΩ/m and 0.4+j4.5 kΩ/m, respectively.The measurement shows the kicker does not cause RCS beam effects.

Tune kicker
There are two strip-line kickers to excite the beam for tune measurement [14].The kicker is made of a lossy material, stainless-steel, resulting in a higher or lower deposit of power in the electrode, although most of the power is absorbed by the terminal matching load.The impedance is simulated by CST as shown in Fig. 8.A wide impedance appears at 84 MHz, but its peak value is very small.For the gaussian beam with sigma length of 10 m in the RCS, the loss factor is about 1 × 10 -7 V/pC.For 100kW beam power in the RCS, the power loss is small as about 2 W.

Summary
Based on previous model with the calculation above, the total impedance in the RCS is displayed in Fig. 9 and the impedance model is given in Eq. ( 1), Eq. ( 2) and Eq.(3).
a it means horizontal × vertical half-length of elliptic chamber.b it means horizontal × vertical thickness of elliptic chamber.c it is circular at quadrupole magnet except circular #5 at injection magnets.

Figure 1 .
Figure 1.Longitudinal impedance of ceramic chamber in RCS.

Figure 2 .
Figure 2. Horizontal and vertical impedance of ceramic chamber in RCS.

Figure 3 .
Figure 3.The longitudinal impedance of the protype and real magnetic alloy cavity.

Figure 4 .
Figure 4. Measured impedance of the ferrite cavity.

Figure 5 .
Figure 5. Simulation mode of the quadrupolar BPM.

Figure 7 .
Figure 7.The longitudinal (left) and vertical (right) impedance of the extraction kicker.

Figure 8 .
Figure 8. Simulated impedance of the tune kicker.

Figure 9 .
Figure 9.Total impedance of the RCS.