Kleeven, W.J.G.M. JACoW Publishing Geneva, Switzerland 2673-5482 978-3-95450-212-7 10.18429/JACoW-CYCLOTRONS2022-THAO01 English 255-259 cyclotron resonance hadrontherapy focusing hadron Contribution to a conference proceedings 2023 2023-10 https://doi.org/10.18429/JACoW-CYCLOTRONS2022-THAO01 https://jacow.org/cyclotrons2022/papers/thao01.pdf Existing analytical models for transverse beam dynamics in isochronous cyclotrons are often not valid or not precise for relativistic energies. The main difficulty in developing such models lies in the fact that cross-terms between derivatives of the average magnetic field and the azimuthally varying components cannot be neglected at higher energies. Taking such cross-terms rigorously into account results in an even larger number of terms that need to be included in the equations. In this paper, a method is developed which is relativistically correct and which provides results that are practical and easy to use. We derive new formulas, graphs, and tables for the radial and vertical tunes in terms of the flutter, its radial derivatives, the spiral angle and the relativistic gamma. Using this method, we study the 2nur=N structural resonance (N is number of sectors) and provide formulas and graphs for its stopband. Combining those equations with the new equation for the vertical tune, we find the stability zone and the energy limit of compact isochronous cyclotrons for any value of N. We confront the new analytical method with closed orbit simulations of the IBA C400 cyclotron for hadron therapy.