Study and simulation of cryogenic bi-periodic accelerating structure with TM02 mode

To further enhance the accelerating gradient of accelerators, we designed a cryogenic C-band standing wave bi-periodic accelerating structure for the Shanghai Soft X-ray Free Electron Laser Facility (SXFEL). According to the low-temperature environment, material characteristics and technological conditions, the design is completed and it is decided to design the accelerating structure into a bi-periodic magnetic coupling structure. It is a 17-cell structure consisting of 9 accelerating cavities and 8 coupling cavities. To guarantee the symmetry of the field, the structure is doubly-fed. Operating with the π/2 mode standing wave, it is much less sensitive than the standing-wave structure of π-mode. Additionally, the microwave mode is TM02 in coupling cavities that are larger and even less sensitive than the traditional bi-periodic structure. The shape of the coupling cavity can be redesigned to make it tunable.


Introduction
A high gradient accelerating structure is an important part of accelerators.In recent years, many studies have shown that the cryogenic accelerating structure can effectively reduce the radio frequency (RF) breakdown rate and thus improve the accelerating gradient [1,2,3].With the analysis of the material and RF properties of the cavities at cryogenic temperatures, the design of the cryogenic accelerating structure is completed for SXFEL energy upgrading.
Considering the stability and accelerating efficiency at low temperature, we finally decided to use the bi-periodic accelerating structure, which operates at 20-40K, and the accelerating gradient can be raised to 80MV/m.To improve the accelerating efficiency, the structure with nose cones is used to improve the shunt impedance, requiring the beam hole to be as small as possible.However, this results in very weak on-axis coupling between cavities.Therefore, magnetic coupling is used.Many scientists have researched bi-periodic structures [4,5].But in this paper, the mode in the coupling cavity is set to TM02 innovatively, which further reduces the cavity's sensitivity and helps dissipate heat.Table 1 lists some parameters of the accelerating structure.We designed the RF cavities in the bi-periodic mode by RF simulation.

Design for the cryogenic structure
The design of a cryogenic accelerator needs to consider many factors, in which the most important is the selection of temperature and materials.Through data analysis, the design  [6], the surface resistance of the copper cavity decreases with the temperature (Fig 1).However, low temperature will improve the cost of the cryostat.Assuming that the Carnot cycle coefficient η is 30% and the ambient temperature T a is 293.15K,equation 1 can be used to determine the relationship between the cooling power P c needed for each watt of heat removed and the cavity's operating temperature T .
As can be seen from Fig 1, the required cooling power rises sharply as the temperature drops.This puts enormous pressure on cryogenic plants and consumes huge amounts of energy.Therefore the operating temperature should be optimized with respect to the required cooling power.
In addition, the high Q value caused by the decrease of surface resistance also increases the field building time in the cavity.For the travelling-wave accelerating structure, the group velocity of microwaves is slower in the cavity with a higher Q value, which makes the pulse width longer, thus increasing the breakdown rate.When the coupling degree is 1, the higher the value of Q, the longer the field construction time.
Considering all factors comprehensively, 40K is selected as the working temperature.Considering the accelerating gradient, frequency drift, surface resistance changes, and the current technical conditions, the C-band accelerating structure was decided.

Residual resistance ratio(RRR)
The choice of cavity material also needs to be considered.Today, copper is still the main material used in high-gradient accelerating structures.The purity of copper can be expressed by residual resistance ratio (RRR), which relates the resistivity at room temperature to the one at 4K.The temperature rise caused by pulse heating can be represented by [7] where H is the magnetic field strength, R S is the surface resistance of the corresponding frequency of the material, t p is pulsing time, ρ d is the material density, c ϵ is specific heat capacity, and k is the thermal conductivity.Fig 2 shows the relationship between normalized temperature rise and RRR.It can be seen that when RRR is high, the change of temperature rise tends to be gentle.Considering that copper with a high RRR is expensive, common copper with RRR=500 in the market is selected.

Cavity design
The dispersion relation of the bi-periodic accelerating structure is [8] ( where ω is the resonant frequency of the φ mode of the cavity, ω a is the resonant frequency of the accelerating cavity, ω c is the resonant frequency of the coupling cavity, k is the coupling coefficient between the adjacent accelerating cavity and the coupling cavity, k a is the coupling coefficient between two adjacent accelerating cavities, and k c is between coupling cavities.This is obtained when If ω a ̸ = ω c , a band gap appears on the bi-periodic dispersion curve.If ω a = ω c , the band gap disappears and the double passband coupling resonates.To improve the accelerating efficiency, the accelerating cavity is designed in the shape of a nose cone and the beam hole is reduced [9].The cavities are magnetically coupled and the coupling holes are designed into curved pill shapes.Compared with conventional bi-periodic structures, the microwave is transmitted in TM02 mode in the coupling cavities.The radius of the coupling cavities should be larger than than the accelerating cavities.So we designed it to be thin in the middle and thick on the outside, so that it becomes a tunable cavity (Fig 3).Through simulation, the field distribution and dispersion curve of the periodic structure were calculated (Fig 4).
The cavities are magnetically coupled to each other using two coupling holes.Each pair of coupling holes is rotated by 90°from their neighbouring coupling holes so that the symmetry of the field distribution is ensured as far as possible.The shape of the coupling holes is designed as a fan shape with rounded corners.
A model of a repeating unit of this two-cycle structure was developed.This model starts in the middle of the accelerating cavity and ends in the middle of the next one.The field distribution and dispersion curves in this model were then calculated and the size of the cavity was optimised to form a passband (see Fig 4).In theory, the model starts at one end of a complete accelerating cavity; however, our model starts in the middle of the accelerating cavity.Therefore, the point in the dispersion curve where the frequencies are equal is π rather than π/2.
We designed a cryogenic standing wave accelerating structure with a total length of 1m and four accelerating sections.Each accelerating section consists of 9 accelerating cavities and 8 coupling cavities, which are fed from the middle cavity in a double-fed way.Through simulation, the field distribution of a single accelerating segment is calculated, and the microwave frequency and coupling degree are optimized.To ensure the symmetry of the structure, the two coupling holes connected with the waveguide differ by 45°from the coupling holes between the left and right cavities (

Conclusion
The accelerating gradient of the cryogenic structure is significantly higher than that of normal temperature accelerating structures.Considering the temperature, Q value, material characteristics, cooling power, and other factors, the design of the cryogenic structure is completed.A bi-periodic structure is established, and TM02 is innovatively adopted as the mode in the coupling cavity.Such a structure has the advantages of low sensitivity, good heat dissipation, and tunability.The 17-cell cavity was designed and the coupling degree and the field distribution on the axis were optimized.

Figure 1 .
Figure 1.The surface resistance curve of RRR=3000 copper in different wavebands varies with temperature (Black); the relationship between the cooling power required for every 1 watt of heat removed and the working temperature of the cavity (Red).

4 Figure 2 .
Figure 2. The normalized temperature rise of the copper with different RRR at 40K.
Fig 5).By optimizing the radius of each cavity, the field distribution in the accelerating tube is uniformly adjusted, as shown in Fig 6.By adjusting the coupling hole, the degree of coupling is set to 1 (Fig 7).

Figure 3 .Figure 4 .
Figure 3.A repeating unit of bi-periodic structure((a): The overall model of the repeating unit; (b): Cross-section of the unit; (c): Crosssection of the coupling holes.)

Figure 5 .
Figure 5. (a): The 17-cell cavity; (b): a side view of the structure, from which it can be seen that the waveguide is tilted concerning the coupling holes between the cavities.

Figure 6 .
Figure 6.(a): The electric field distribution inside the accelerating structure; (b): The normalized electric field distribution along the z-axis.

Figure 7 .
Figure 7. S-parameter and Smith chart of the 17-cell cavity.

Table 1 .
The Design Parameters of the Accelerator.