Exploring the Limits of Perturbation Theory for TESLA RF Cavities to Evaluate Various Geometric and Electromagnetic Values
Description
Particle accelerators use radio-frequency (RF) cavities to accelerate charged particles, and accurate cavity modeling is essential for optimizing beam acceleration and minimizing energy loss. While pillbox cavities admit analytic electromagnetic solutions, cavities with complicated geometries like the TESLA cavity require numerical solvers, obscuring direct mathematical relationships between the geometry and field properties. In this work, a semi-analytic approach based on first-order perturbation theory is developed by applying small geometric deformations to a pillbox cavity to approximate the TESLA cavity. Two deformation profiles were tested and applied to a nine-cell coupled cavity model in π phase. Compared to a pillbox approximation, perturbation theory reduces the resonant frequency error from 14.5% to below 1% while the ”R-upon-Q” value is reduced to around 36% error. The electromagnetic field behavior also improved, with the error in the ratio of peak electric field to accelerating field reduced to about 20%. Larger discrepancies remain in quantities dependent on power dissipation, such as the geometry and quality factors. These results demonstrate that first-order perturbation theory, combined with Slater’s formula, provides a promising framework for predicting RF cavity frequency shifts and field behavior without full numerical simulations. The software uploaded was created to implement the equations derived in the paper where intial conditions were based on the accelerating electric field of the TESLA RF cavity.