Why is the cyclotron frequency independent of particle speed?

Faster particles have larger orbits but travel proportionally longer paths. The increased circumference exactly compensates for the higher speed, keeping the period constant. This breaks down at relativistic speeds where mass increases with velocity.

What is the Larmor radius?

The Larmor radius (gyroradius) r_L = mv/(qB) is the radius of the circular orbit. It increases with speed and decreases with field strength. In a cyclotron, the Larmor radius grows as the particle gains energy, creating a spiral path.

How does this relate to MRI?

In MRI, protons in body tissue precess at the Larmor frequency: f = γB/(2π) where γ is the gyromagnetic ratio (42.577 MHz/T for protons). A 1.5T scanner operates at 63.87 MHz; a 3T scanner at 127.74 MHz.

What limits the energy of a classical cyclotron?

At relativistic speeds, the particle mass increases (γm), reducing the cyclotron frequency. The particle falls out of sync with the fixed-frequency RF. This limits proton cyclotrons to about 20 MeV. Synchrocyclotrons vary the RF frequency; synchrotrons vary both RF and magnetic field.

What is magnetic rigidity?

Magnetic rigidity Bρ = p/q (in units of T·m) determines how much a magnetic field bends a particle. Higher momentum or lower charge = higher rigidity = larger bending radius. Used to design beam transport and spectrometers.

How does the electron cyclotron frequency compare to the proton?

Since f_c ∝ q/m and the electron is 1836× lighter than the proton (same |q|), the electron cyclotron frequency is 1836× higher. At 1T: electron = 27.99 GHz, proton = 15.25 MHz.

Cyclotron Frequency Calculator

Calculate cyclotron (gyro) frequency, Larmor radius, and relativistic corrections for charged particles in magnetic fields.

Particle Source

Particle

Magnetic Field (B)

Kinetic Energy

Cyclotron Frequency

22.8684 MHz

ω_c = 143,686,476.1449 rad/s · T = 43.7284 ns

Relativistic Frequency

22.6234 MHz

γ = 1.010832 · β = 0.146001

Larmor Radius

304.6221 mm

r = mv/(qB) at KE = 10 MeV

Particle Velocity

43,770,077.4723 m/s

β = v/c = 0.146001 (14.6001% of c)

Magnetic Rigidity (Bρ)

0.4619 T·m

Bρ = p/q — determines bending in dipole magnets

Revolutions to Energy

100

At V_dee = 50 kV, gain = 100.0 keV/rev

Relativistic Assessment

14.60% of c

Highly relativistic — synchrocyclotron or synchrotron required

Frequency vs Magnetic Field

B (T)	f_cyclotron	Larmor Radius
0.01	152.456 kHz	45.693 m
0.1	1.525 MHz	4.569 m
0.5	7.623 MHz	913.866 mm
1	15.246 MHz	456.933 mm
1.5	22.868 MHz	304.622 mm
2	30.491 MHz	228.467 mm
3	45.737 MHz	152.311 mm
5	76.228 MHz	91.387 mm
10	152.456 MHz	45.693 mm

Particle Comparison at B = 1.5 T

Particle	m (kg)	q (e)	f_c	q/m (C/kg)
e⁻ Electron	9.109e-31	1	41.989 GHz	1.759e+11
p⁺ Proton	1.673e-27	1	22.868 MHz	9.579e+7
d⁺ Deuteron	3.344e-27	1	11.440 MHz	4.792e+7
α Alpha (He²⁺)	6.645e-27	2	11.513 MHz	4.822e+7
t⁺ Triton	5.007e-27	1	7.639 MHz	3.200e+7
µ⁻ Muon (µ⁻)	1.884e-28	1	203.078 MHz	8.507e+8
C⁶⁺ Carbon-12 (C⁶⁺)	1.993e-26	6	11.518 MHz	4.824e+7

Planning notes, formulas, and examples

About the Cyclotron Frequency Calculator

The cyclotron frequency (gyrofrequency) is the frequency at which a charged particle orbits in a uniform magnetic field: f_c = qB/(2πm). This frequency depends only on the charge-to-mass ratio q/m and the field strength B — remarkably, it is independent of the particle's speed (in the non-relativistic limit).

This property is the basis of the cyclotron particle accelerator, invented by Ernest Lawrence in 1932. Particles spiral outward in a fixed-frequency oscillating electric field, gaining energy each revolution while maintaining constant orbital frequency. The proton cyclotron frequency in a 1.5T field is 63.86 MHz — this is also the Larmor precession frequency used in MRI.

This calculator computes cyclotron frequency, Larmor radius, relativistic corrections, and particle comparison for any charged particle in any magnetic field. It supports multiple particle types including electrons, protons, deuterons, alpha particles, and custom ions. It is especially useful when you want to compare how the same magnetic field behaves for different charge-to-mass ratios without doing repeated unit conversions by hand.

When This Page Helps

The cyclotron frequency connects fundamental particle properties to observable frequencies in magnetic fields. Calculating it requires precise particle masses, charge states, and unit conversions between Tesla, Gauss, eV, and SI units.

It covers any particle and field configuration, with relativistic corrections that become important for electrons above a few hundred keV and protons above about 100 MeV. The particle comparison table is useful for accelerator design, MRI, and mass spectrometry.

How to Use the Inputs

Select a particle from the database or enter custom mass and charge.
Enter the magnetic field strength with units (Tesla, mT, or Gauss).
Enter the kinetic energy to calculate the Larmor radius and velocity.
Use presets for common scenarios (NMR, accelerators, geophysics).
Review cyclotron frequency, relativistic correction, and Larmor radius.
Compare particles and field strengths using the reference tables.

Formula used

f_c = qB/(2πm). ω_c = qB/m. Larmor radius: r_L = mv/(qB). Relativistic: f_rel = f_c/γ where γ = 1/√(1−β²). Magnetic rigidity: Bρ = p/q.

Example Calculation

Result: f_c = 22.84 MHz, r_L = 0.304 m

ω_c = (1.602×10⁻¹⁹ × 1.5)/(1.673×10⁻²⁷) = 1.435×10⁸ rad/s. f_c = ω_c/(2π) = 22.84 MHz. At 10 MeV: v = √(2×10×1.602×10⁻¹³/1.673×10⁻²⁷) = 4.38×10⁷ m/s. r_L = mv/(qB) = 0.304 m.

Tips & Best Practices

For MRI: proton Larmor frequency = 42.577 MHz × B(Tesla). At 3T, this gives 127.73 MHz — in the VHF radio band.
The electron cyclotron frequency at Earth surface (~50 µT) is about 1.4 MHz — in the AM radio band. It is observed in ionospheric physics.
In plasma physics, the electron cyclotron frequency must exceed the plasma frequency for electromagnetic wave propagation along the magnetic field.
Cyclotron radiation power ∝ q⁴B²γ²/m² — electrons radiate ~10¹³ times more than protons at the same energy.
Modern medical cyclotrons (for PET isotope production) accelerate protons to 11-18 MeV in a 1.0-1.5T field.

The Cyclotron: Concept and History

Ernest O. Lawrence conceived the cyclotron in 1929 and built the first working model (4.5 inches diameter) in 1932. The key insight was that charged particles in a magnetic field orbit at a frequency independent of their energy, so a single RF oscillator can continuously accelerate them as they spiral outward.

The particle enters near the center, is accelerated by the electric field between two D-shaped "dee" electrodes, curves in the magnetic field, and re-enters the accelerating gap half a period later — now moving faster. Each revolution adds energy equal to 2qV_dee (two gap crossings). After hundreds of revolutions, the particle reaches the outer edge at maximum energy and is extracted.

Relativistic Limitations and Solutions

The non-relativistic cyclotron frequency ω_c = qB/m is constant. But as v → c, the relativistic mass γm increases, reducing the actual orbital frequency: ω = qB/(γm). The particle arrives later to each gap crossing, eventually falling out of phase with the RF.

Solutions: (1) The **synchrocyclotron** decreases the RF frequency as the particle accelerates, matching the declining orbital frequency. (2) The **synchrotron** uses a toroidal vacuum chamber and increases the magnetic field synchronously with particle energy, keeping the orbit radius constant. (3) The **isochronous cyclotron** uses azimuthally varying magnetic fields (sector focusing) to compensate for the relativistic mass increase — modern compact medical cyclotrons use this design.

Applications Beyond Accelerators

The cyclotron frequency appears in many contexts: MRI scanners (Larmor precession of nuclear spins), plasma confinement in tokamaks (electron and ion gyration around magnetic field lines), mass spectrometry (separating isotopes by cyclotron frequency in a Penning trap — the most precise mass measurement technique), and astrophysics (cyclotron radiation from electrons in stellar magnetospheres and pulsars).

Sources & Methodology

Last updated: January 15, 2025

Frequently Asked Questions

Faster particles have larger orbits but travel proportionally longer paths. The increased circumference exactly compensates for the higher speed, keeping the period constant. This breaks down at relativistic speeds where mass increases with velocity.