## 3 Analysis of Cavity Asymmetry Effects on Beam-Dynamics  
Typically, any cyclotron RF cavity design has a vertical symmetry with equal half upper and lower structures. The symmetry of the cavity and the TEM fundamental mode operation ensures the best condition for electric field distribution in the beam region, parallel to the middle plane. However, the \"real\" machines--such the INFN-LNS compact cyclotron under investigation in this work--have some intrinsic limitations. The coupler is located on the bottom side of the Dee and the trimmer in the upper side. These elements break the symmetry of each RF cavity. Before of the present work this asymmetry was not quantitatively evaluated and the operation was driven by experimental evidence. In the following, to be able to evaluate the amplitude of the two components of interest, we used the expression \(E_{\parallel}\)=\(\sqrt{{{E_{x}^{r}}^{2}+{E_{x}^{i}}^{2}+{E_{y}^{r}}^{2}+{E_{y}^{j}}^{2}}}\) to extract the module of the component parallel to the middle plane, and the expression \(E_{\perp}\)=\(\sqrt{{{E_{z}^{r}}^{2}+{E_{z}^{r}}^{2}}}\) to extract the module of the component orthogonal to the middle plane. The superscript \(r\) and \(i\) identify the real and imaginary parts of the solution, respectively. The electric field component \(E_{\perp}\) orthogonal to the median plane, introduced by this asymmetry, was computed and presented in Figure 10. The color-scale of Figure 10 has been cut to 250 V/m to better identify the amplitude in the region of interest. Figure 9: Cavity unloaded quality factors \(Q_{0}\) vs. frequency. Blue curve refers to simulations while red curve reports measured values. Thanks to a beam dynamics code we are developing for the upgrade of our cyclotron, we can estimate the effects of this unwanted electric field component on the beam dynamics. The code has been developed in Matlab, importing the magnetic field and the RF field maps from comsol, then computing the beam trajectory using the relativistic Boris mover approach [24]. Before to use the electric field in the beam-dynamics code, its amplitude needs to be normalized in order to obtain the magnitude correspondent to the acceleration of the beam of interest. We consider the evolution of an oxygen beam with atomic mass of 18 and a charge state of 6+ that is accelerated with a resonant frequency, here considered to be 33.7416 MHz (second harmonic acceleration mode), up to an energy of 45 MeV/A. The dynamics are computed from an inner radius of 11 cm up to the extraction diameter of 86 cm. The beam envelope related to the vertical plane, shown in Figure 11, is characterized by a sinusoidal fluctuation of the beam size and center. This sinusoidal motion is typical of a harmonic oscillator; the restoring force is provided by the cyclotron magnetic field focusing strength, and the excitation force is the orthogonal electric field component. In Figure 12 a detailed beam dynamics analysis is carried out to reveal the correlation between the beam center transverse position and the transverse electric field component (\(E_{z}\)) strength encountered by the beam itself. The cyclotron magnetic field (\(B_{z}\)) is shown to easily identify when the beam travels across the RF cavity, corresponding to a lower magnetic field strength (the so-called \"valleys\"). The electric field component parallel to the middle plane (\(E_{xy}\)) is shown to precisely identify the acceleration gap cross. This correlation shows that the orthogonal electric field component increases with the distance from the middle plane. The effect of this component will growth with the foreseen upgrade which will increase the beam current and the beam size. Figure 11: Beam transverse envelope for the standard operation condition. Figure 10: Magnitude of the electric field component \(E_{\perp}\) orthogonal to the median plane. The analysis of the transverse electric field component, shown in Figure 13 along two RF cavities crossings, reveals a complex electric field structure with a sequence of alternated (up and down) kicks.