2.1 Center geometry characteristics

To get better RF performance, two geometries of center inner conductor are often used in the low beta coaxial resonators, i.e. RT (the race track) and RS (the ring shaped)^[9,10], as shown in Figs. 2(a) and 2(c).

Fig.2Full-screen View

Inner conductors: ring shaped (a), elliptical shaped (b) and race track (c).

For the ring shaped center conductor, the cavity has higher G*R/Q (G is the geometry factor and R/Q is shunt impedance) and lower normalized peak magnetic field of B_peak/E_acc, but it is hard to get lower normalized peak electric field of E_peak/E_acc on the ring edge. However, the situation is reversed for the race track. The loft transition, which is asymmetric stem from the race track cross section to the round base of inner conductor top radius, produce a uniform magnetic field distribution. Therefore, combining features of the race track and the ring shaped, an elliptical shaped (ES) center inner conductor is proposed (Fig. 2b).

The ring shaped center conductor is a special case of the elliptical shaped center conductor. When the semi major axis (MR_ES) equals to the semi minor axis of the elliptical shaped, the elliptical shaped model converts to ring shaped model. For comparing the race track model, the semi minor axis of the elliptical shaped is named as ICHW_ES (Fig. 2b). Thus, we only illustrate the design processes of the RT and ES in the following sections. For RF design, geometry parameters of the magnetic and electric groups are used to optimize B_peak/E_acc and E_peak/E_acc.

2.2 Magnetic field domain

The magnetic area was optimized by fixing the inner conductor top radius (ICTR) at 110 mm and changing the cavity top radius (CVTR) from 225 mm to 255 mm. As shown in the Figs. 3(a) and (b), An increase of CVTR affects positively the RF properties. However, a larger cavity top radius would increase the cost of fabrication and cooling equipment.

Fig. 3Full-screen View

The normalized magnetic fields (B_peak/E_acc, a&c) and R/Q*G (b&d), as function of ICTR (inner conductor top radius) and CVTR (cavity top radius).

By changing ICTR from 100 mm to 135 mm under fixing CVTR at 240 mm, the B_peak /E_acc and G*R/Q dropped about 19% and 17%, separately, (Figs.3c and 3d), but a larger ICTR would increase the inner surface area, hence the decrease of surface current density; while decreasing the RF volumes would decrease G*R/Q.

Fig.3 shows that the elliptical shaped has a smaller B_peak/E_acc and larger G*R/Q than those of the race track model.

2.3 Inner center conductor

The RF parameters are not sensitive to variations of the inner conductor sizes along with the stem direction. Therefore, the optimization mainly focused on the center inner conductor half width (ICHW) and inner conductor thickness (ICT). As shown in Figs. 4(a) and 4(c), the E_peak/E_acc decreases first and then rises with increasing ICT for both the geometries. An increase of ICT is favorable to obtaining higher G*R/Q for the race track, while it is opposite for the elliptical shaped when ICT >75 mm. From Figs. 4b and 4d, increasing ICHW will lower the G*R/Q, but the effect is around 6% for both of the center inner conductor. E_peak/E_acc will take its minimum at a certain value of ICHW for the race track model.

Fig. 4Full-screen View

E_peak/E_acc and G*R/Q of the race track (RT) and the ring shaped (ES), as function of inner conductor thickness (ICT) and inner conductor half width (ICHW).

2.4 Results and comparison

Through the RF simulation, three HWR cavities of different inner conductors were optimized. The results are given in Table 1, where parameters of the medium beta (β=0.53) HWR with inner conductor of the conventional race track at MSU were given, too. Considering advantages of the RT and RS, the ES can lower the normalized surface fields and increase R/Q. With the same geometry of magnetic areas, E_peak/E_acc of the RS model is about 18% higher than the others. In addition, the RT model has the lowest R/Q and E_peak/E_acc. The B_peak/E_acc of HWR with elliptical center conductor is 19.5% lower than that of the conventional HWR for the FRIB. While the G*R/Q of HWR with elliptical center conductor is 27% higher than that of the conventional HWR (hence higher acceleration efficiency), smaller RF power dissipation and less thermal load to the cryogenic system. Thus, the elliptical center conductor is chosen as the RF structure for further study. For the time transit factor of ≥ 0.7, this HWR can accelerate beams from the beta of 0.37 to the elliptical cavity region.

3.1 Monopole mode

For simplification, the bunch was treated as a point like charge. When a point charge q travels along the beam axis, the excited monopole mode voltage is ^[12]:

where ω_n is the angular frequency of the mode n and q=61.5 pC. Considering the velocity dependent transit time factor, the shunt impedances (R/Q)_n (β) of the monopole modes can be defined the same as the fundamental accelerating mode:

where U_n is the stored energy of mode n, and E_n,z is the electric field along the beam axis. For estimating the maximum value of excited voltages at the range of β(0.37, 0.63), the maximum value of R/Q are given in Table 2.

When a bunch passes the cavity, the excited voltage starts to oscillate by exp(iω_nt) and decay by exp(-t/T_d,n). The decay time constant is T_d,n=2Q_L,n/ω_n, with Q_L,n = 1/(Q₀^-1+Q_ex^-1) being the load quality factor, and the value of Q₀ and Q_ex are given in the Table2. For the CW operation, the equilibrium voltage can be derived by evaluating the cumulative voltages directly:

where T_b = 6.15 ns. Form Eq.(3), one can figure out that when f_N=hf_b (h is an integer, hf_b is the principal machine line), the HOM will be excited on resonance. The HOM frequency is far away from its closest machine lines, it can hardly excite the resonance with the CW beam, as shown in Table 2.

The excited HOMs increases the heat load of the SC cavity, and the dissipated power can be calculated by Eq.(4):

The sum of maximum dissipated power by the HOMs is less than 1 mW for the CW operation at 10 mA beam current. Therefore, the effect of the excited monopole HOMs to the SC cavity can be neglected.

3.2 Dipole mode

The dipole modes are excited by the off-axis bunches. They deflect the particles. When the polarization axis is along the x axis, the “transverse voltage” can be expressed from the Panofski-Wenzel theorem ^[12,13]:

where q is the charge of the bunch with x off the center axis. The transverse shunt impendence (R/Q)_n(β) is independent of the off axis value. Similar to the monopole modes, the transverse shunt impendence for dipole HOMs can be given as ^[12]:

where k_n=ω_n/c, and the E_n,z (ρ=a) is the electric field parallel with z axis with off z axis value a. The maximum values of (R/Q)_n are given in Table 3, calculated with β =0.37–0.63.

The sum of transverse voltages and dissipated powers of dipole modes were calculated using Eqs. (3) and (4) with off center value of x = 1 mm. If the bunches off the Z axis at 1 mm, the maximum dissipated powers of dipole modes were evaluated for the CW operation of CADS. As shown in Table 3, the HOMs resonant frequencies are far away from their closest machine lines and one can figure out that the total of maximum dissipated powers is less than 1 nW at 10 mA CW proton beams. Thus, the HOMs couplers for the HWR cavity are not necessary.

4.1 General consideration

The common approach to meet the requirement of structure safety and the frequency stability, is to make the cavity more rigid by adding stiffer ribs on the outside wall. Adding the ribs, however, cannot avoid volume deformation completely. An efficient method is to compensate the frequency shifts from electric and magnetic concentrated areas. According to the Slater perturbation theory, the frequency shifts and volume deformation can be related as follows ^[16],

where U is the total stored energy of the cavity, and Δf is the frequency shift, with high magnetic fields contributing positively, and the high electric fields negatively. Thus, the cavity can be divided into two compensation domains (Fig. 5).

Fig. 5Full-screen View

The naked cavity half model: the magnetic (left) and electric (right) domains for frequency sensitivity analysis.

For two extreme conditions, including one beam pipe being fixed as the reference position and the other being free, the simulated df /dp are as given in Table 4. When the pipes fixed, the total frequency sensitivity is the summation of the respective frequency sensitivity of electric and magnetic area. Thus, the compensation method can be used under the condition of pipes fixed.

4.2 Stiffener and vessel design

In order to identify the location of the peak equivalent stress, the pressure safety was simulated with one atmosphere loading on the naked cavity. The peak equivalent stress occurred at the edge of beam caps, as shown in Fig. 6. With the simulation results, the stiffer ribs and vessel were designed (Fig.7).

Fig.6Full-screen View

Equivalent stress results under one atmosphere of naked cavity.

Fig.7Full-screen View

Schematic diagrams of HWR051 with the ribs and helium vessel.

With C shaped ribs, the absolute values of df/dp that contributes from the electric areas decreased from −15.1 to −7.2 Hz/mbar on the condition of pipe fixed. Two daisy slabs are added on the high magnetic field terminal domes to compensate the frequency shifts from the high electric areas and the df/dp reduces from 20.7 to 16.9 Hz/mbar. The total frequency sensitivity can be obtained at 5.6 and −32.5 Hz/mbar under the two extremely boundary condition with the helium vessel. Actually, the beam pipes are constrained by the mechanical tuner and the df/dp will be within the range of the two extremely values. The mechanical parameters and LFD coefficient are shown in Table 5.

The cavity tuning sensitivity is 185 KHz/mm (total displacement) for the displacement on the beam pipes. The cavity stiffness is 4.9 KN/mm (total displacement) with ribs attached for tuning force on beam pipes. It is easy to get the tuning range of 160 KHz when 4.5 KN force applied on beam pipes. The tuning range and the cavity stiffness can meet the request of operation.

Taken the worst case into consideration, the frequency shift will be less than 50 Hz with the helium pressure fluctuation about 1.5 mbar at 4.2 K. For CW operation, the accelerating gradient is stable at a constant value, so the shift of resonator frequency will not change by the Lorenz force. When need to change the accelerating gradient, the frequency changes induced by the LFD effect can be compensated easily by the mechanical tuner.