Cyclotron Radius & Frequency Calculator

A charged particle in a magnetic field moves in a circle, or a helix if it also moves along the field. This calculator gives the size of that circle and how often the particle goes around it, for electrons, protons, or any ion at any energy, from electronvolts to GeV. Use it to size a bending magnet or an energy analyzer, to check whether a field confines a beam or a plasma, or to find the ion cyclotron resonance (ICR) frequency of an m/z in a Penning trap.

Key terms
Cyclotron (Larmor) radius
The radius of the circle a charged particle follows in a uniform magnetic field, also called the gyroradius. It grows with the particle’s momentum and shrinks as the field gets stronger: a 1 keV electron in a 10 mT field circles with a radius of about 1 cm.
Cyclotron frequency
How many times per second the particle goes around. At low energy it depends only on the charge-to-mass ratio and the field, not on the energy: 28 GHz per tesla for an electron and 15.2 MHz per tesla for a proton.
Magnetic rigidity (Bρ)
The particle’s momentum divided by its charge, in tesla-meters. It is the field times bending radius a magnet must provide: a beam of rigidity 0.46 T·m (10 MeV protons) bends on a 0.46 m radius in a 1 T dipole.
Pitch angle
The angle between the particle’s velocity and the magnetic field. At 90° the particle moves in a flat circle; at smaller angles it spirals along the field, and only the velocity across the field sets the radius.
Lorentz factor
γ, the ratio of a particle’s total energy to its rest energy. Relativity makes a fast particle behave as if heavier by this factor, so its orbit is larger and its cyclotron frequency lower than the classical formulas give.
Ion cyclotron resonance (ICR)
Measuring an ion’s cyclotron frequency in a strong, very uniform magnetic field to find its m/z. FT-ICR mass spectrometers reach the highest mass resolution of any type, because a frequency can be measured very precisely.

Inputs

Particle

Fills in the mass and charge. For an ion not in the list, choose Custom and enter its mass; for an m/z value, enter it as the mass with charge 1.

u
e

The sign of the charge sets only the direction of rotation (positive particles circle clockwise when the field points at you).

Motion and field

The particle’s total kinetic energy (not per nucleon). An ion accelerated through U volts has K = |z|·U eV; a thermal particle at temperature T has about 3kT/2 (0.04 eV at room temperature).

1 T = 10 kG = 10 000 G. The Earth’s field is about 50 µT; a strong permanent magnet gives 0.2–1 T near its face; FT-ICR and NMR magnets run at 7–21 T.

deg

The angle between the velocity and the field: 90° for motion straight across the field (a flat circle), smaller for a helix along it.

An ICR frequency read from a Penning trap or FT-ICR spectrum. The result gives the m/z it belongs to in the field B above.

Results

Cyclotron (Larmor) radiusrc—
Cyclotron frequencyfc—
Angular frequencyωc = 2πfc—
PeriodTc—
ICR frequency (low-energy limit)f0 = |q|B/2πm—
Magnetic rigidityBρ = p/|q|—The field × bending radius needed to bend the whole momentum, whatever the pitch angle.
Momentumpc—
Speedv—
Lorentz factorγ—
Helix pitch (advance per turn)—
m/z for the measured frequency—

Radius vs. kinetic energy

How it is calculated

The magnetic force \(q\,\mathbf v \times \mathbf B\) is always perpendicular to the velocity, so it changes the particle’s direction but never its speed or energy. Across the field the particle moves on a circle; along it, it coasts. For mass \(m\), charge \(q = ze\), and kinetic energy \(K\):

\[ \begin{gathered} \gamma = 1 + \frac{K}{mc^2}, \qquad pc = \sqrt{K^2 + 2Kmc^2}, \qquad v = \frac{pc^2}{\gamma m c^2}, \\[8pt] r_c = \frac{p \sin\alpha}{|q| B}, \qquad \omega_c = 2\pi f_c = \frac{|q| B}{\gamma m}, \qquad B\rho = \frac{p}{|q|}. \end{gathered} \]

The helix advances \(v\cos\alpha \cdot T_c\) along the field per turn. Classically \(\gamma = 1\) and \(p = \sqrt{2mK}\), which gives the familiar \(r_c = mv_\perp/|q|B\) and an orbit frequency \(f_0 = |q|B/2\pi m\) that does not depend on energy. That independence is why a classical cyclotron can accelerate with a fixed radio frequency, and why ICR measures m/z from a frequency alone:

\[ \frac{m}{z} = \frac{e B}{2\pi f_m}\ \ \text{(in kg; divide by 1 u for u/e)}, \qquad f_0 \approx 15.356\ \text{MHz} \times \frac{B/\text{T}}{(m/z)/\text{u}}. \]

Relativistically, the orbit frequency falls as \(1/\gamma\). For 10 MeV protons γ = 1.011, so a fixed-frequency cyclotron would slip out of step after a few hundred turns: the reason isochronous cyclotrons shape their field to rise with radius, and synchrocyclotrons sweep the frequency.

In practical units, \(B\rho\,[\mathrm{T\,m}] = 3.3356\, pc\,[\mathrm{GeV}]/|z|\) (the factor is \(10^9/c\)), and the electron cyclotron frequency is 27.992 GHz per tesla (2.80 MHz per gauss).

Typical values

From the formulas above, motion straight across the field (α = 90°):

Particle and energyBRadiusFrequency
Electron, 1 keV (an electron gun)10 mT10.67 mm279.4 MHz
Electron, 1 MeV (beta particle)0.1 T47.4 mm946.7 MHz
⁴⁰Ar⁺, 2 keV (ion beam)0.5 T81.4 mm192.1 kHz
Proton, 10 MeV (medical-isotope cyclotron)1.5 T305.4 mm22.63 MHz
Ion of m/z 500, at rest (FT-ICR)7 T—215.0 kHz

Worked example

The default inputs: a 10 MeV proton (m = 1.007276 u, mc² = 938.272 MeV) in a 1.5 T field, moving across it.

Assumptions and limits

References

Primary sources and standards

  1. J. D. Jackson, Classical Electrodynamics, 3rd ed., Wiley (1999), §12.2 (relativistic motion in a uniform static magnetic field: gyration frequency ωB = ecB/γmc² in Gaussian units, gyration radius, and helical motion).
  2. A. G. Marshall, C. L. Hendrickson, and G. S. Jackson, “Fourier transform ion cyclotron resonance mass spectrometry: A primer,” Mass Spectrom. Rev. 17, 1–35 (1998), doi:10.1002/(SICI)1098-2787(1998)17:1<1::AID-MAS1>3.0.CO;2-K (the ICR frequency, f = 1.535611 × 10⁷ B/(m/z), and its trap shifts).
  3. P. J. Mohr, D. B. Newell, B. N. Taylor, and E. Tiesinga, “CODATA recommended values of the fundamental physical constants: 2022,” Rev. Mod. Phys. 97, 025002 (2025), doi:10.1103/RevModPhys.97.025002 (e, u, the electron, proton, deuteron, and alpha-particle masses, and c).
  4. M. Wang, W. J. Huang, F. G. Kondev, G. Audi, and S. Naimi, “The AME 2020 atomic mass evaluation (II). Tables, graphs and references,” Chin. Phys. C 45, 030003 (2021), doi:10.1088/1674-1137/abddaf (the atomic masses of the preset ions).

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