Cyclotron Energy Calculator
The three headline numbers of any classical cyclotron — beam energy, magnetic field, and extraction radius — are locked together by one relation. Pick a particle, enter the field and the radius at which the beam is extracted, and this calculator returns the kinetic energy, the RF resonance frequency, the magnetic rigidity, and the particle speed, plus an estimate of how much the relativistic correction matters.
Results
| Kinetic energy (non-relativistic) | — |
| Cyclotron resonance frequency f | — |
| Magnetic rigidity Bρ | — |
| Final speed v | — |
| β = v/c | — |
| Lorentz factor γ (exact) | — |
| Relativistic kinetic energy | — |
| Non-relativistic overestimate | — |
| Field / radius in customary units | — |
Note: the relativistic correction here exceeds 1%. A fixed-frequency classical cyclotron loses resonance as γ grows; energies this high call for isochronous-field or synchrocyclotron treatment beyond this calculator.
The math
A particle of charge q and mass m moving at speed v perpendicular to a uniform field B feels a centripetal force qvB = mv2/r, so its momentum is
Non-relativistically, T = p2/2m, giving the classical cyclotron energy relation:
The orbital period is independent of radius — the fact that makes a cyclotron work — so the RF drive sits at the cyclotron resonance frequency:
Magnetic rigidity is momentum per unit charge, Bρ = p/q; for a particle on its extraction orbit it is simply Bρ = B·r. Speed follows from v = p/m and β = v/c.
When relativity matters
The momentum p = qBr is exact; only the energy–momentum conversion changes. The exact kinetic energy is
The true resonance frequency is f/γ. The calculator reports γ and the percentage by which the non-relativistic energy overshoots the exact one. Below a few MeV for protons the correction is a fraction of a percent; a fixed-frequency machine typically tolerates roughly a 1–2% frequency slip before phase excursions kill the beam (Livingood 1961, ch. 6).
Assumptions and limits
- Uniform, flat field — no radial field falloff, no azimuthal (AVF) flutter.
- The particle reaches the stated radius; extraction efficiency is not modeled.
- No sanity checks are applied — unphysical inputs give unphysical outputs.
Worked check
Protons at B = 0.582 T: f = 8.873 MHz. At r = 0.104 m the same field gives T = 175.5 keV, Bρ = 0.0605 T·m, β = 0.0193, γ = 1.000187 — a relativistic correction of only 0.009%, comfortably classical.
Sources
- J. J. Livingood, Principles of Cyclic Particle Accelerators, Van Nostrand, 1961 — chs. 5–6 (classical cyclotron relations, resonance limits).
- M. S. Livingston & J. P. Blewett, Particle Accelerators, McGraw-Hill, 1962 — ch. 6.
- Physical constants: CODATA 2018 recommended values.