Cyclotron Frequency Calculator
Enter the particle charge, magnetic field strength, and particle mass to find the cyclotron frequency - the rate at which a charged particle orbits in a uniform magnetic field. Choose from built-in presets for protons, electrons, alpha particles, and more, or enter a custom charge and mass. Results include cyclotron frequency, angular frequency, orbital period, and a chart showing how frequency varies with field strength.
What is cyclotron frequency?
Cyclotron frequency (also called the cyclotron resonance frequency or Larmor frequency) is the rate at which a charged particle completes circular orbits in a uniform magnetic field. When a charged particle moves perpendicular to a magnetic field, the Lorentz force acts as a centripetal force, bending the trajectory into a circle. The remarkable property of this motion is that the orbital frequency depends only on the particle's charge-to-mass ratio and the field strength, not on the particle's speed or the radius of the orbit. This isochronous property is the physical principle that makes cyclotron particle accelerators possible.
How cyclotron frequency is calculated
The derivation starts by equating the Lorentz magnetic force to the centripetal force required for circular motion. For a particle of charge q and mass m moving at speed v in a field of strength B, the Lorentz force is F = qvB. The centripetal force needed is F = mv2/r, where r is the orbit radius. Setting these equal gives qvB = mv2/r, which simplifies to r = mv / (qB). The orbital period T is the circumference divided by speed: T = 2*pi*r / v = 2*pi*m / (qB). Cyclotron frequency f = 1/T = qB / (2*pi*m). Angular frequency omega = 2*pi*f = qB/m. Notice that neither v nor r appears in the final frequency formula - only the charge-to-mass ratio q/m and the field B.
Particle presets and charge-to-mass ratios
Because cyclotron frequency depends on the charge-to-mass ratio (q/m), particles with higher q/m orbit faster. The electron has the highest q/m ratio of any stable particle (about 1.76e11 C/kg), so it has an extremely high cyclotron frequency - about 28 GHz per tesla. The proton q/m is about 9.58e7 C/kg, giving about 15.26 MHz per tesla. The alpha particle carries twice the proton charge but is four times heavier, so its q/m is roughly half the proton value and its cyclotron frequency is about 7.68 MHz per tesla at 1 T. Use the particle preset dropdown to explore these differences automatically.
Relativistic effects and practical limits
The formula f = qB/(2*pi*m) assumes the rest mass m is constant. At high kinetic energies the particle's relativistic mass increases, causing the cyclotron frequency to decrease. This makes conventional cyclotrons impractical for proton energies above about 50 MeV without isochronous corrections (a sector-focused or spiral-ridge cyclotron) or a switch to a synchrotron design where the RF frequency is varied to match the changing revolution period. For protons, relativistic corrections are smaller than 1% below about 9.4 MeV kinetic energy, and smaller than 5% below about 47 MeV. For electrons, relativistic effects are important even at a few hundred keV.
Cyclotron frequency of common particles in a 1 T field
| Particle | Charge (C) | Mass (kg) | f at 1 T (MHz) |
|---|---|---|---|
| Electron (e-) | 1.602e-19 | 9.109e-31 | 28,024 |
| Proton (p+) | 1.602e-19 | 1.673e-27 | 15.26 |
| Muon (mu-) | 1.602e-19 | 1.884e-28 | 135.5 |
| Deuteron (2H+) | 1.602e-19 | 3.344e-27 | 7.63 |
| Alpha particle (4He2+) | 3.204e-19 | 6.645e-27 | 7.68 |
All values computed from f = qB / (2*pi*m) at B = 1 T (non-relativistic).
Frequently asked questions
What is the cyclotron frequency formula?
The cyclotron frequency is f = qB / (2*pi*m), where q is the particle charge in coulombs, B is the magnetic field strength in tesla, and m is the particle rest mass in kilograms. The angular frequency version is omega = qB / m. The orbital period is T = 2*pi*m / (qB). Crucially, the formula contains no speed or radius term, so the frequency is the same for all particle speeds (in the non-relativistic limit).
Does cyclotron frequency depend on speed?
No, not in the non-relativistic limit. This is the isochronous property that makes cyclotrons work: slow particles follow smaller circles and fast particles follow larger ones, but both complete each orbit in exactly the same time. The RF cavities can therefore be driven at a fixed frequency and will always be in phase with the orbiting particles. At relativistic speeds this breaks down because the particle effective mass increases with kinetic energy.
What is the difference between cyclotron frequency and Larmor frequency?
Larmor frequency and cyclotron frequency refer to the same physical quantity - the orbital frequency of a charged particle in a magnetic field. The term "Larmor frequency" is more common in NMR and plasma physics, while "cyclotron frequency" is used in accelerator physics. Some texts define the Larmor frequency as omega = qB/m (the angular version) while cyclotron frequency means f = qB/(2*pi*m); others use them interchangeably. This calculator shows both.
What is the cyclotron frequency of a proton in a 1 T field?
Using f = qB/(2*pi*m) with q = 1.602e-19 C, B = 1 T, and m = 1.673e-27 kg: f = (1.602e-19 x 1) / (2 x pi x 1.673e-27) = 15.26 MHz. The orbital period is 1/15.26 MHz = 65.5 ns. This is in the standard RF range for proton cyclotrons used in medical isotope production.
Why do cyclotrons use a fixed RF frequency?
Because the cyclotron frequency is independent of particle speed (isochronous property), a fixed-frequency RF accelerating voltage stays in sync with the spiraling particle across all orbits. Each time the particle crosses the dee gap it receives a kick in phase with the RF. This design simplicity is why cyclotrons can be compact and efficient for moderate energies. Synchrotrons, by contrast, ramp both the magnetic field and RF frequency to track relativistic particles to much higher energies.
How does magnetic field strength affect cyclotron frequency?
Cyclotron frequency is directly proportional to the magnetic field B. Doubling the field doubles the frequency. A proton in a 1 T field orbits at about 15.26 MHz; in a 2 T field it orbits at about 30.52 MHz. This linear relationship is why high-field superconducting magnets allow compact cyclotrons to reach higher beam energies in a smaller radius.