Coupling impedances of a gap in vacuum chamber

A gap in the vacuum chamber stands between a beam and the outside world, and the theoretical elucidation of the interaction mechanism between the gap and the beam is of great importance to understand the interaction of any device with the beam. In this paper, we will present the formulas for the longitudinal and transverse impedances due to a gap in the beam chamber. In this process, we will derive the complete solutions of electromagnetic fields effective in the entire region, including the inside and the outside of the chamber, in a form that they can be easily numerically evaluated. The newly developed technique can provide new methods of solutions of electromagnetic fields also for a rather broad class of structures such as cavities. The numerical results of impedances are consistent with the ABCI results and their behavior in high frequency agrees well with the prediction of the diffraction theory. Our theory can also accurately reproduce the behavior of the impedance near and above the cutoff frequencies. In addition, our theory is applicable even to the impedances for nonrelativistic beams. We found that the broadband impedance of the small cavitylike structure can be estimated from the gap size and the chamber radius only, regardless of the exact shape of the structure. We also found that the transverse impedance of a gap has a large resonance peak at the frequency where the wavelength is equal to the chamber circumference. This resonance peak appears around 1–2 GHz in most of the cases, and we should be careful to design a ceramic break so that this transverse mode will not leak out to interact with nearby devices