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Why do classical cyclotrons have a limit on the energy they can give protons?

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Diagram of a linear proton resonance accelerator, or linac, showing protons entering from a source, passing through a series of labeled drift tubes powered by a radio-frequency source, and exiting as a beam through a cavity. — particle accelerator The accelerator is a large-diameter tube within which an electric field oscillates at a high radio frequency. Within the accelerator tube are smaller diameter metallic drift tubes, which are carefully sized and spaced to shield the protons from decelerating oscillations of the electric field. In the spaces between the drift tubes, the electric field is oriented properly to accelerate the protons in their direction of travel.

In a classical cyclotron, the energy imparted to protons is limited by the relativistic increase in mass that occurs as a particle's speed approaches the speed of light.12

Because the operation of a classical cyclotron relies on the isochronicity of particle orbits—meaning the time taken to complete a circuit remains constant—the device requires a fixed frequency for the accelerating voltage. As a proton gains energy and its speed increases, the relativistic effect causes its mass to rise, which in turn decreases its orbital frequency. Eventually, the particle falls out of phase with the alternating voltage, causing it to cross the gap at times when the electric field decelerates rather than accelerates it. Consequently, classical cyclotrons are generally limited to proton energies of less than 25 MeV.12