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 per ion (non-relativistic) | — |
|---|---|
| Cyclotron frequency f = qB/2πm (classical) | — |
| Revolution frequency f/γ (relativistic) | — |
| Magnetic rigidity Bρ | — |
| Speed v = p/γm (exact) | — |
| β = v/c | — |
| Lorentz factor γ (exact) | — |
| Relativistic kinetic energy per ion | — |
| Non-relativistic overestimate | — |
| Field / radius in customary units | — |
Note: the classical energy estimate differs from the relativistic value by more than 1% here. This calculator does not determine whether a beam stays in accelerating phase — that depends on the field profile, dee voltage, RF harmonic, and turn count (see phase).
Enter a positive value for every input — charge, mass, field, radius — to calculate. The calculator takes positive ions; for a negative ion use the charge magnitude (the energy and frequency are unchanged; only the orbit's sense reverses).
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 — relativistically the same balance reads qvB = γmv2/r, and either way the momentum is
Non-relativistically, T = p2/2m, giving the classical cyclotron energy relation:
In this classical limit 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 magnitude, Bρ = p/|q|; for a particle on its extraction orbit it is simply Bρ = B·r. Speed follows from v = p/γm, which reduces to the familiar p/m when γ ≈ 1, and β = v/c.
When relativity matters
The momentum p = qBr holds at any speed in this uniform-field model, with r the orbit's curvature radius; the energy–momentum conversion, the speed, and the revolution frequency all change. The exact kinetic energy is
The revolution frequency falls to f/γ, so a fixed-frequency drive slips phase against the beam as γ grows. The calculator reports γ and the percentage by which the classical energy overshoots the exact one — a gauge of the approximation, not a verdict on whether a machine accelerates: that depends on the accumulated phase over all turns, hence on field profile, dee voltage, harmonic, and turn count (Kleeven & Zaremba, §1.1; the phase section works the budget arithmetic). Below a few MeV for protons the correction is a fraction of a percent.
Assumptions and limits
- Uniform, flat field — no radial field falloff, no azimuthal (AVF) flutter.
- RF at the fundamental (harmonic h = 1): the drive frequency equals the revolution frequency.
- The particle reaches the stated radius; extraction efficiency is not modeled.
- Inputs must all be positive (anything else shows dashes); beyond that no sanity checks are applied, and extreme inputs give extreme 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 — the classical energy formula is exact to 0.009% here.
Sources
- J. J. Livingood, Principles of Cyclic Particle Accelerators, Van Nostrand, 1961 — chs. 5–6 (classical cyclotron relations, resonance limits).
- W. Kleeven & S. Zaremba, “Cyclotrons: Magnetic Design and Beam Dynamics”, CERN Accelerator School proceedings, arXiv:1804.08961 — §1.1 (turn-by-turn energy/phase evolution; retrieved September 2026).
- M. S. Livingston & J. P. Blewett, Particle Accelerators, McGraw-Hill, 1962 — ch. 6 (library entry).
- Physical constants: CODATA 2018 recommended values (NIST).
Educational reference, not an operating procedure: results reflect the stated model and assumptions. Verify anything safety-critical against primary sources, and read the safety fundamentals before applying numbers to real hardware. Last reviewed: · Report a correction.