## 3 Analysis of Cavity Asymmetry Effects on Beam-Dynamics
#### Coaxial Sliding Shorts Asymmetric Position  
The presented asymmetry and its effect on beam dynamics is compatible with the current use of the cyclotron. However, one of the major requests of the planned upgrade is to reduce as much as possible the beam losses and energy spread of the extracted beam. For this scope, a mitigation for the transverse electric field has been searched. We are presenting here a simple mitigation that does not require any mechanical changes on the accelerator machine. The solution here proposed is to introduce an asymmetry in the sliding shorts position that counteracts the effects of the electric field asymmetry introduced by the presence of the coupler and the trimmer. Starting from the symmetric shorts condition (length of 556 mm for 33.7416 MHz), the upper one has been extracted at 55 mm while the lower one has been inserted at 50 mm, as shown in Figure 14.  
Figure 12: Details of beam transverse displacement evolution, magnetic field and electric field components during the first part of the acceleration inside the cyclotron.  
Figure 13: Details of electric field components and transverse momentum provided to the beam for two consecutive RF cavities.  
The parasitic electric field component produced in the standard, symmetric sliding shorts, configuration was already presented in Figure 10. Same electric field analysis has been performed for the \"asymmetric shorts\" configuration. The electric field produced by the latter configuration is reported again in Figure 15 for easier comparison with the same color range scale of Figure 16 to simplify the comparison. Thanks to the applied asymmetry, it can be observed that the undesired electric field parasitic component corresponding with the center of RF cavity has been reduced to negligible values.  
It is worth noticing that the applied asymmetry keeps unaltered the accelerating field component \(E_{\parallel}\), parallel to the middle plane, as it is possible to see in Figure 17a,b.  
Figure 16: \(E_{\perp}\) produced by the asymmetric sliding short configuration at 33.7416 MHz  
Figure 14: Asymmetrical sliding shorts position to counteract the asymmetry introduced by the coupler.  
Figure 15: Magnitude of the electric field component \(E_{\perp}\) orthogonal to the median plane.  
With the tracking code, we can estimate the improvement in beam dynamics. The standard configuration produces 2 mm amplitude oscillation of the beam center for the first half of the acceleration time (Figure 11), with a beam size in the extraction region of about 10 mm. With the asymmetric sliding shorts configuration (Figure 14), the beam center amplitude oscillation is reduced by a factor of four and the beam size in the extraction region is reduced within range of 15-20% (see Figure 18) with respect to the standard operation condition, reported again in Figure 19 for easier comparison.  
Reducing the amplitude of the electric field component orthogonal to the middle plane, the amplitude of the oscillation, as visible in Figure 20, is consequently reduced, while the 0.5 us oscillation period is not modified because it is related to the beam inertia and the magnetic focusing effect, which remain unchanged. The reduction in beam oscillation amplitude results is correlated to the reduction of the transferred momentum (shown in Figure 21) by a factor 4.  
Figure 19: Beam transverse envelope for the standard operation condition.  
Figure 17: Electric field component \(E_{\parallel}\) parallel to the middle plane produced by the actual geometry (**a**) vs. the asymmetric sliding short configuration (**b**) at 33.7416 MHz.