In a uniform field B a particle of charge q and momentum p moves on a circle of radius r = p/(qB) with angular frequency w = qB/(gamma m). A cyclotron exploits that the period does not depend on speed (for gamma = 1): an RF voltage across the gap between two dees flips sign every half turn, so each crossing adds q V cos(phi) and the orbit spirals outward.

Relativity breaks the synchronism: gamma grows with energy, the gyro-period lengthens, the ion arrives later and later at the gap, the phase slips and acceleration stops (around 10-30 MeV for protons with a fixed RF). The orange line is the ideal gain N q V, the blue line the real energy. Detune the RF and the slip comes earlier.

Method: relativistic Boris pusher at 200 steps per period, a 6 mm gap, injection energy 0.5 MeV, extraction radius 0.5 m. Simplifications: no space charge, no vertical focusing, uniform B (no isochronous profile), idealised dees. The check uses a square-wave RF at tuned frequency (gain exactly qV per crossing) and a separate fine time step.