Tools / Physics
100% client-side · CODATA 2018 constants · non-relativistic modelCyclotron Frequency Calculator
A charged particle spiraling in a uniform magnetic field orbits at the cyclotron frequency. Compute it from charge, field and mass — or invert the formula to design the magnet field or find the mass-to-charge ratio.
Mode 1 — Cyclotron frequency from q, B, m
Sign is ignored — the orbit frequency uses |q|. Electron preset fills q = −1 automatically.
| Angular frequency ω = qB/m | — |
|---|---|
| Period T = 1/f | — |
Relativistic check at radius r
| v/c (β) | — |
|---|---|
| Lorentz factor γ | — |
| Kinetic energy | — |
Mode 2 — Magnet field B from target frequency
Inverting the formula: B = 2πmf / q. Useful when designing a cyclotron or ICR cell for a desired resonance.
Mode 3 — Mass-to-charge ratio m/q from f and B
m/q = B / (2πf). The workhorse of mass spectrometry: measured resonance frequency in a known field identifies the ion.
also —
| Ion mass (m = (m/q)·Z·e) | — |
|---|
Reference values (CODATA 2018)
Particle data used
| Particle | Charge | Mass |
|---|---|---|
| Electron | −e | 0.00054857990946 u |
| Proton | +e | 1.007276466621 u |
| Deuteron | +e | 2.013553212745 u |
| Alpha particle | +2e | 4.001506179127 u |
e = 1.602176634×10⁻¹⁹ C (exact), u = 1.66053906660×10⁻²⁷ kg, c = 299,792,458 m/s (exact). Source: CODATA 2018.
What is the cyclotron frequency?
The cyclotron frequency is the orbital frequency of a charged particle circling in a uniform magnetic field: f = qB / (2πm), where q is the charge, B the magnetic field and m the particle mass. For example, a proton in a 1 T field orbits at 15.245186 MHz - about 15.2 million revolutions per second.
What is the formula for cyclotron frequency?
f = qB / (2πm) in hertz, with q in coulombs, B in tesla and m in kilograms. The angular frequency is ω = qB/m (rad/s) and the orbital period is T = 1/f. Example: an electron in 1 T gives f = 27.992490 GHz with a period of only 35.723868 ps.
Is the proton cyclotron frequency 42.58 MHz per tesla?
No - that is a common mix-up. 42.57747892 MHz/T is the proton spin-flip (NMR) frequency, which comes from the magnetic moment, not the orbital motion. The orbital cyclotron frequency of a proton at 1 T is 15.24518646 MHz. Different physics, different number - this calculator uses the orbital one.
How do I choose the magnetic field for a target frequency?
Invert the formula: B = 2πmf / q. Enter the target frequency in Mode 2 and the calculator returns the required field. Example: a 10 MHz proton resonance needs B = 0.65594475 T. Results are shown in T, mT and G.
What is the mass-to-charge mode for?
Mode 3 computes m/q = B / (2πf) from a measured resonance frequency in a known field - the workhorse relation of mass spectrometry (FT-ICR). Example: f = 15.24518646 MHz at B = 1 T gives m/q = 1.04396849×10⁻⁸ kg/C, i.e. 1.00727647 u/e (a proton). If you also enter the charge state Z, the ion mass is solved as m = (m/q)·Z·e.
Is there a quick Z/A rule of thumb?
Yes: f ≈ 15.356118 × (Z/A) × B MHz, with B in tesla, Z the charge state and A the mass number. It is approximate because A is the nominal mass number: for a deuteron (Z/A = 1/2) at 1 T the rule gives 7.67805877 MHz while the true value with the real deuteron mass (2.013553212745 u) is 7.62637794 MHz - about 0.7% lower.
When do relativistic effects matter?
The formula is non-relativistic. Enter an orbit radius and the calculator evaluates β = v/c with v = qBr/m: if β > 0.1 you get a warning with the corrected resonance f_rel = f/γ. Example: a proton at B = 1 T and r = 0.5 m reaches β = 0.15975774, so γ = 1.01301086, the true frequency is 15.04938111 MHz (not 15.24518646 MHz) and KE = 12.20772395 MeV. At r = 0.05 m, β = 0.01597577 - safely non-relativistic.
Does the sign of the charge matter?
No. The orbit frequency depends on |q|: an electron (q = −e) in 1 T circles at the same 27.992490 GHz as a hypothetical +e particle of electron mass. Only the sense of rotation (clockwise vs counter-clockwise) flips with the sign.
What units can I use?
Magnetic field: T, mT or G (1 T = 1000 mT = 10,000 G). Frequency: Hz, kHz, MHz or GHz. Orbit radius: m or cm. Custom ions: charge in units of e, mass in atomic mass units u. All results are also shown converted across every unit.
What is the difference between this calculator and a frequency converter?
This page computes physics - the resonance frequency from charge, magnetic field and mass (f = qB/2πm). A frequency converter only changes units (Hz to kHz, MHz, rpm, rad/s) without any physics. For unit conversion, wavelength, Nyquist rate and band lookup, use the Frequency Converter hub instead.
What does the calculator assume?
A uniform, static magnetic field in vacuum and the non-relativistic regime (with an explicit warning when β > 0.1). Real machines have fringe fields, energy loss to synchrotron radiation, and space-charge effects - treat results as design reference, not a substitute for beam-dynamics simulation.
Where is the cyclotron frequency used?
Cyclotron and synchrotron design (setting the RF to match the orbit frequency), Fourier-transform ion cyclotron resonance (FT-ICR) mass spectrometry, electron cyclotron resonance (ECR) ion sources and plasma heating, and astrophysics - e.g. cyclotron emission lines from magnetized neutron stars.
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