Cavity Design

Based on a comparison of the beam dynamics, a 2.6-cell structure was adopted for further optimization. By adjusting the structural parameters of the gun, changes in important RF parameters, including the surface electric field and quality factor, were observed, which provide a valuable reference for the optimal design of electron guns.

In Fig. 5 and Fig. 6, A represents the axis of the ellipse at irises along the z-axis, B represents the axis of the ellipse at irises along the radial direction, C and R are the beam hole radius and chamfer radius at the cell walls, E_cath represents the E field at the cathode, and E_max represents the maximum E field at the iris. Figure 2(b) shows that the electric field intensity at the iris is quite high in the electron gun operating in π mode. In the preliminary consideration of the RF breakdown rate and dark current emission phenomenon, it is necessary to optimize the shapes of the cell walls and iris to minimize both the surface electric and magnetic fields. In the design of the electron gun, the iris aperture and rounding were adopted to guarantee a higher shunt impedance and lower surface electric field. Meanwhile, the influence of the iris length between adjacent cavities on the coupling factor was also considered to maintain the mode separation within an acceptable range.

Fig. 5Full-screen View

　Q₀, mode separation, E_cath/E_max change with B/A

Fig. 6Full-screen View

(Color online) Q₀ and mode separation change with C (a) and R (b)

Figure 5 shows that each RF parameter has a better performance when the ratio of the horizontal and vertical axis of the ellipse contour is controlled near 1.3~1.4. This is because this ratio realizes a seamless connection between the rounding of the cell walls and iris aperture, which effectively increases the surface area, thus reducing the surface electric field and power loss. Because the field emission is exponentially related to the surface electric field, this helps reduce the contribution of the dark current at the iris. Based on the change in radius of the circular chamfer of the model shown in Fig. 6, it can be found that this design is useful to achieve a higher quality factor, which helps to reduce the RF power dissipation on the cell walls and the RF power needed.

To reduce the error during installation and decrease the dark current at the cathode, a new electron gun with a TM₀₂-mode cathode cell was designed. The optimized results can provide a reference for the full cells and later coupler designs. The new gun structure is shown in Fig. 7. In the TM₀₂ mode, the radial electric field at the cathode plane is shown in Fig. 8. The minimum electric field was observed over a circular region of radius (r = 21.616 mm). Because field emission often occurs near the groove at the edge of the cathode plug (Cu plug), the edge of the cathode plug can be set at this position, which means that the electric field at the cathode edge can be effectively reduced, thereby reducing dark current generation. Because the cavity design still needs to consider the subsequent power-coupling factors, it is a continuous iterative process. Here, we propose a new electron gun structure, and the final optimized structural parameters are presented in the following section.

Fig. 7Full-screen View

(Color online) New electron gun structure with a TM₀₂ mode cathode cell

Fig. 8Full-screen View

(Color online) Electron field at the cathode plane(a) and electric field along the radial direction(b)

Suppression Of High Field On Surface

The gradient on the cathode of the electron gun was designed to reach 200 MV/m under the working conditions. Although the material has a higher hardness at low temperatures, it is necessary to consider the RF breakdown phenomenon, which may be caused by the high electric field on both the cathode and cell walls. To predict and analyze the performance of the structure at a high gradient, the modulated Poynting vector [42] was calculated as follows: ${\overrightarrow{S}}_{\text{c}}=Re\left\{\overrightarrow{S}\right\}+g_{\text{c}}\cdot Im\left\{\overrightarrow{S}\right\},$ (1) where ${\overrightarrow{S}}_{\text{c}}$ is the modulated Poynting vector and g_c is the weighting factor. In a previous study, g_c was experimentally measured to be 1/6 as an independent parameter [42]. In addition, RF pulse heating is calculated using the following equation [26]: $\text{Δ}T={\left|H\right|}^2\sqrt{t_{\text{p}}}\frac{R_{\text{s}}}{\sqrt{\pi \rho c_{\epsilon }k}},$ (2) where $\text{Δ}T$ is the maximum temperature rise caused by the electric field on the cell walls, $t_{\text{p}}=2$ μs is the RF pulse length, R_s is the RF surface resistance, ρ= 8.94×10³ is the density, $c_{ϵ}=59$ is the specific heat capacity, and k = 1841 is the heat conductivity. We assumed that these parameters did not change during the pulse and that the structural parameters were optimized based on these two aspects.

After optimizing the parameters (A, B, and C), the surface electric field was reduced, as shown in Fig. 9. S_c in the RF gun was estimated while working at room temperature, and the maximum S_c= 4.84 W/ m², which is less than the 5 W/ m² recommended in experimental studies [42]. The maximum S_c is located near the iris because of the reactive power flow oscillating along the electromagnetic field in each cycle of the standing wave structure. The maximum RF pulse heating was 19.98 K, which was less than the limit of 50 K at room temperature.

Fig. 9Full-screen View

(Color online) High field suppression on cavity surface

Coupler Design on the Cathode Cell

Considering factors such as beam dynamics and multipole field, it is innovative to feed RF power into the cathode cell by adjusting its shape and size rather than using the traditional coaxial coupling or inserting the coupler into the full cell. Feeding RF power into the cathode cell allows the solenoids to be placed closer to the cathode to achieve a better beam quality. However, owing to the limited size of the cathode cell, significant efforts have been made to explore the proper structure of the cathode cell working in the TM₀₂ mode, with the other two full cells still maintaining the TM₀₁ as their working mode. It is worth mentioning that the TM₀₂ mode can achieve fewer errors during installation owing to its stability and insensitivity compared to the traditional TM₀₁ mode. Therefore, we propose two designs, as shown in Fig. 10: one uses a J-shaped waveguide to directly insert the coupling holes into the plane where the cathode is located, and the other widens the outer contour of the cathode cell to place the coupling ports and feed RF power with a T-shaped waveguide. In contrast, the T-shaped waveguide coupling facilitates a cathode plug. Here, we selected the traditional dual-feed scheme for further optimization. The cathode feed scheme will be studied later as an alternative.

Fig. 10Full-screen View

(Color online) TM₀₂ cathode cell with a J-shaped waveguide (a), modified TM₀₂ cathode cell with a T-shaped waveguide (b) and model modification

Because of the TM₀₂ working mode in the cathode cell, these structures have more possibilities. In the selected scheme, the shape of the cathode plane can be optimized to accommodate alternative cathodes(Cu plugs). Relevant experiments have shown that the field emission occurring in the groove around the cathode plug is often the main contributor to the dark current [43]. Because the TM₀₂-mode electric field has a minimum electric field in the radial direction of the cathode insertion position and exists in a circular profile, the edge of the cathode can be set at this profile to effectively reduce the field emission. After testing many coupler port shapes, a dual feed on the cathode cell with Z-coupling slots was adopted [18]. The length of the coupling slots was consistent with that of the wall of the cathode cell, and the required radius was easily fabricated. By adjusting the radius of the chamfers (r1,r2) at the coupling slots, the locally generated pulse heating can be controlled. To facilitate the power feed, an external waveguide was designed by connecting the coupler to a T-shaped waveguide structure through a transition section. By setting the two ports at the T-waveguide external power feed port and at the transition interface, the S₂₁ parameter was adjusted to approximately -3 dB in working mode. Because the beam load was low, the coupling coefficient was set to 1 at the operating frequency of 5.712 GHz. Under this condition, the S₁₁ parameter in the working mode is close to -35 dB, and power feeding is realized as shown in Fig. 11.

Fig. 11Full-screen View

(Color online) S parameters in the two schemes

Optimization Of Multipole Mode

The axisymmetric structure of the cavity was destroyed by the addition of a coupler. Therefore, the electric and magnetic fields exhibit an uneven azimuthal distribution, which destroys the symmetry of the electron beam at the exit of the electron gun. To reduce the influence of the multipole components on the quality of the electron beam, the electric field along the beam axis in each cell in the electron gun was derived from a small circle to observe the multipole field introduced by the coupler ports.

The dual-feed scheme effectively reduced the dipole mode in the cathode cell. The shape of the racetrack gun shown in Fig. 12, which can suppress the quadrupole field, was adopted based on the TM₀₂ mode cavity to reduce the influence on the transverse momentum and emittance of the beam.

Fig. 12Full-screen View

(Color online) Race-track cavity design and optimized gun structure with parameters

Figure 13 shows that the multipole components are simulated on a circle with radius 5 mm and as can be observed, the multipole filed in the full cells is much weaker than that in the cathode cell. By optimizing the racetrack arc separation, d, the intensity of the quadrupole field was controlled to approximately 0.01 % in the gun. In addition, it is necessary to analytically evaluate the effect on beam quality. Considering that the strongest multipole field mainly exists in the cathode cell, the transverse momentum imparted to an electron beam by the RF field in the cathode cell was calculated using the Panofsky–Wenzel theorem [44, 45] as follows: $p_{\perp }=\frac{e}{\omega }Re\left({\displaystyle {\int }_{-\frac{L}{2}}^{\frac{L}{2}}}i{\nabla }_{\perp }{E}_z\text{d}z\right),$ (3) where L is the cell length (here, L = 0.6 × λ/2) and Ez is the accelerating electric field along the longitudinal axis. If we consider only the necessary TM_n02 modes in the half cell, the electric field can be expressed as follows: $E_z=e^{i\left(\omega t+{\varphi }_{\text{0}}\right)}{\displaystyle \sum _{n=0}^2}{E}_n\cos \left(kz\right)J_n\left(k_{\text{c}}r\right)\cos \left[n\left(\theta -{\theta }_{\text{0}}\right)\right],$ (4) where r and z are the radial and longitudinal coordinates, θ is the azimuthal angle, ${\theta }_{\text{0}}$ is the polarization angle, ${\varphi }_{\text{0}}$ is the RF phase, k is the RF wave number, k_c is the radial wave number, and $J_{n\left(x\right)}$ is the Bessel function. When the quadrupole field is the dominant multipole field (an RF gun with two holes) and the dipole field can be neglected, the transverse momentum and transverse emittance growth [22] can be obtained as $p_{n\mathrm{,}y}=-\left[a_{\text{1}}+\left(2a_{\text{2}}-\frac{k_{\text{c}}^{\text{2}}}{2}\right)y\right]\alpha L\sin \left({\varphi }_{\text{0}}\right)\widehat{y},$ (5) ${\epsilon }_{n\mathrm{,}\text{y}}^{\text{020}+\text{220}}=\left(\frac{k_{\text{c}}^{\text{2}}}{2}-2a_{\text{2}}\right)\alpha L\cos \left({\overline{\phi }}_{\text{0}}\right){\sigma }_{\text{y}}^{\text{2}}{\sigma }_{\phi },$ (6) where a₁,a₂ are parameters characterizing the relative strength of the quadrupole field to the monopole field, $\alpha =\frac{\left(e{E}_{\text{0}}\right)}{2m{c}^{\text{2}}k}$ is the normalized RF field strength, L is the length of the half-cell, ${\overline{\phi }}_{\text{0}}$ is the ensemble average of ${\phi }_{\text{0}}$ along the electron distribution in the bunch, and σ_y and σφ are the rms beam size and rms beam length (in the RF phase). The effect of the RF field on the transverse emittance can be obtained by calculating the transverse momentum with an axial electric field, and the overall effect of the monopole and quadrupole fields is 0.5mmmrad. At this time, the emittance influence is greater because the selected center point is close to the cathode and the emittance of the entire bunch is also at 1mmmrad. It is worth mentioning that the radial wave number is independent of z in the pillbox cavity, whereas in the curved cavity, k_r is dependent on z. However, because of the unique structural design of the half-cell, it is difficult to find out the dependency between k_r and z. Here, considering that most of the electrons only move in the inner RF field approximate to the TM₀₁ mode, the model is simply approximated to the full cell working at TM₀₂ to roughly calculate the influence of the RF field on the transverse emittance.

Fig. 13Full-screen View

(Color online) Multipole field suppression results

Finally, the T-shaped waveguide was changed to an asymmetrical waveguide to study its stability and insensitivity. Because the cathode cell of the TM₀₁ mode is generally not used for power coupling, the influence on the microwave field inside the TM₀₂ cathode cell from the length of the external waveguide is analyzed. In the optimized structure, hc = 10 mm. The electric field derived from the circle in the cathode cell was observed by varying hd as shown in Fig. 14. Figure 15 shows that the quadrupole field gradually transforms into the dipole field with the change of hd, which means that the millimeter-magnitude error caused by installation will affect the suppression effect of the multipole field. The deviations represent the error (hd-hc) that may have been introduced during the installation. The multipole field intensity introduced by the hd change is shown in Fig. 15. After the calculation, the suppression effect of the quadrupole field strength can still reach approximately 0.03 % within a range of variation in hd of 2 mm.

Fig. 14Full-screen View

(Color online) Length of the waveguide connecting to the transition sections

Fig. 15Full-screen View

(Color online) Normalized electric field with different deviations and multipole field jitter introduced from waveguide asymmetry

RF Properties under Low Temperatures

The RF performance of the electron gun structure needs to be tested after machining, and some of its key parameters must be predicted [34, 46-52]. To achieve gun operation below 40 K, the effect of cryogenic temperature on the RF properties of the cavities must be analyzed. Several important parameters must be considered as the temperature decreases. The first parameter is the residual resistivity ratio (RRR), which is often defined as the ratio of the resistivity of materials at room temperature and that at absolute zero. For copper with different RRRs (or purities), the resistivity is different at low temperatures. In the following calculations, RRR = 300 was selected to obtain the surface resistance of copper in a C-band gun operating at 5712 MHz. $RRR=\frac{R_{\text{298K}}}{R_{\text{4K}}}$ (7) The second parameter is the coefficient of thermal expansion, which refers to the change in material length per unit temperature in iso pressure situations. As the temperature decreases, the coefficient of thermal expansion of Cu changes nonlinearly, leading to a change in the working frequency. The third parameter is thermal conductivity, which measures the heat transfer capacity of the cryogenic cavity. Higher thermal conductivity implies a lower temperature increase caused by pulse heating, resulting in a lower breakdown rate (BDR). To achieve cryogenic operation, it is necessary to predict the frequency of the electron gun at room temperature. Here, using the coefficient of thermal expansion, the working frequencies of the three individual cavities and the overall electron gun can be predicted. Figure 16(a) presents the frequency shifts of the cathode cell, middle cell, and right cell. The working frequency shifts of the three cavities are 18.1 MHz, 17.1 MHz, and 15.7 MHz from room temperature to 40 K. It can be concluded that the three cavities have the similar working frequency curves, which means they can reach the expected working point (5712 MHz) after cooling. As shown in Fig. 16(b), the changes of thermal conductivity and thermal expansion coefficient with temperature were evaluated for a simple calculation. In addition, a TM₀₁ mode cathode cell model was simulated to obtain a frequency shift of approximately 17.71 MHz for comparison with the TM₀₂ mode cathode cell. It appears that the introduction of the TM₀₂ mode in the cathode cavity has little effect on the frequency shift of the electron gun after the temperature change. Based on the thermal coefficient, the structural parameters at T to those at 293 K are defined as the coefficients of structural expansion in Fig. 17(a). It can be concluded that when 40 K was selected as the working temperature, the volume of the structure decreased to 96.77 % of the original structure after cooling. In the simulation, the maximum displacement of the structural parameters of the electron gun and coupler models was 0.55 mm.

Fig. 16Full-screen View

(Color online) Frequency shifts of single cavities (a), thermal conductivity and mean thermal expansion coefficient (b)

Fig. 17Full-screen View

(Color online) Coefficient of structural expansion (a) and change of electric conductivity with temperature (b)

Considering that the conductivity of Cu also changes significantly with temperature, its relationship with the anomalous skin effect(ASE) [53] is calculated in Fig. 17. As the temperature decreases from 293 K to 40 K, the coefficient of thermal expansion decreases, whereas the frequency increases from 5693.51 MHz to 5712 MHz. The quality factor, shunt impedance, and peak RF power can be obtained with the electrical conductivity below 40 K. The parameters of the 2.6-cell electron gun are presented in Fig. 17 and Table 3.

Cryostat Considerations

To operate the gun in cryogenic state, a stable cryostat is required to maintain the gun structure at 40 K. Power was transmitted to the 2.6-cell electron gun through a T-shaped waveguide connected to the RF power source. The vacuum requirements of the cryogenic platform are 1×10^-6 Pa at room temperature and 1×10^-7 Pa at cryogenic temperatures, which can be provided by molecular and ion pumps. The engineering diagram and cryostat design of the 2.6-cell electron gun are illustrated in Fig. 18. In the near future, a 2.6-cell gun will be manufactured and tested on a low-temperature platform.

Fig. 18Full-screen View

(Color online) Engineering diagram of electron gun structure (a) and cryostat design (b)