## 2 Rf Model: Simulation Results vs. Experimental Measurements  
### Model Geometry, Materials, Mesh, and Boundary Conditions  
The RF design of the INFN-LNS superconducting cyclotron, presented elsewhere [20; 21], is based on three vertical half-wave (\(\lambda/2\)) coaxial resonators, placed \(120^{\circ}\) apart along the median plane (i.e., the area where particle acceleration occurs) and operating in the frequency range of 15-48 MHz. In the median plane, an accelerating high voltage electrode, named \"Dee\" for its shape, is connected to the shorted inner conductor lines (\"Dee-stem\"). The grounded outer conductor, called \"Liner\", is the wall of the vacuum chamber region. In the gap between the RF oscillating high-voltage Dee and the grounded Liner, an accelerating electric field is formed with an intensity inversely proportional to the distance between facing surfaces. For the sake of simplicity, only one of the three identical RF cavities is considered in our model. A picture of the accelerating vacuum chamber is shown in Figure 1. It is composed by the upper and lower liner (only the inferior one is shown) and the inner part of the cryostat. The RF cavities including coaxial stem and Dees are installed in the valley of the liner. The 3D geometry of the actual RF cavity adopted in our model is shown in Figure 2. It is fully comprehensive of all the RF details and it has been derived from the mechanical drawings, including the variations done along the years.  
The simulated RF cavity geometry includes the inner coaxial, the sliding shorts, the Dee-stems, the Dee, the RF power coupler, the trimmer, and the Liner internal region here represented as a transparent blue object. The model also includes three alumina insulating parts which are designed to provide the separation between air and vacuum pressure side: two of them are placed between the outer and the inner coaxial Dee-stem, and the other one is placed in the coupler. Both the Comsol and CST models have been designed resembling the real machine capability to move the position of the sliding shorts and trimmer for frequency tuning and of the coupler for the impedance matching. In such a way it was also possible to have an extensive comparison between measurements and model results as shown in the next sections both in terms of range of operating frequency and power coupling condition. The cavity model has been simulated with same settings in CST and COMSOL: the internal volume material was set as \"vacuum\" while a proper dielectric constant was used for the three alumina volumes. The copper conducibility was used for metal boundary conditions (\"impedance boundary condition\" [18]) on the conductive parts of the model. A representation of the employed mesh is visible in Figure 3 for the cavity full model. A detailed view of the Dee and the coupler are also shown. It is worth noting that the mesh properly follows the different size-scales of the geometries. A fine-mesh region was added in the Dee and coupler region where it is expected high gradient and intense electric fields will occur.  
Figure 1: Accelerating vacuum chamber. It is composed by the upper and lower liner (only the inferior one is visible) and the inner part of the cryostat. The RF cavity including the coaxial stem and Dees are installed in the valley of the liner.  
Figure 3: Mesh of the model. Top right: Dee supported by inner coaxial Dee-stem. Bottom right: external part of the coupler.  
Figure 2: Superconducting cyclotron RF cavity model. Main components are indicated. NOTE: ceramic isolators between inner and outer coaxial are not visible.