How this instrument works
Send charged particles across a magnetic field and they wheel into circles. Cyclotron frequency counts how many of those laps finish each second. What earns the quantity a page of its own is what has been left out of it — speed does not appear, and neither does radius. Hand a particle more energy and its orbit widens in exact proportion, so it travels farther at higher speed and arrives back where it began on the same schedule. Charge-to-mass ratio and field strength set that pace between them, and nothing else does.
Ernest Lawrence noticed this isochronism in Berkeley's physics library in 1929, working through a German paper by Rolf Widerøe on drift-tube acceleration that he could barely read. If laps take equal time regardless of size, one radio-frequency voltage across one gap can kick a particle twice per turn, spiralling it outward through hundreds of modest pushes instead of one impossible enormous one. M. Stanley Livingston built the proof in 1931: a 4.5-inch brass chamber that drove hydrogen ions past 80 keV on roughly a kilovolt of drive. Nobel recognition followed in 1939, by which point Berkeley's machines had reached 60 inches across.
Three assumptions hold that simplicity in place. Mass must stay put, which fails once motion turns relativistic — divide by γ and fixed-frequency machines slide out of step with their own beams. Electrons pass 1% after just 5 keV, so classical cyclotrons were never built for them; microtrons and linacs took that job. Field must be uniform across an orbit, and particles must survive their laps without colliding, which is why magnetised plasma work lives in vacuum. Magnitudes cover many decades: 27.99 GHz for an electron per tesla, 15.25 MHz for a proton, near 870 kHz for magnetospheric electrons at 31 µT, and 148 GHz at ITER's 5.3 T magnetic axis — its 170 GHz heating gyrotrons resonate further inboard, where field runs higher.
- Set Particle charge, choosing C, mC, µC or nC. Elementary charge is 1.602176634 × 10⁻¹⁹ C; scale it by charge state for bare nuclei or multiply ionised atoms.
- Magnetic flux density (T) accepts tesla only. For scale: sunspot umbrae reach about 0.3 T, laboratory electromagnets 1 T, clinical MRI bores 3 T.
- Give Particle mass in kg or g. Atomic mass units want converting first — one u is 1.66053907 × 10⁻²⁷ kg, and an electron is 9.109 × 10⁻³¹ kg.
- Read Cyclotron frequency, moving its menu to kHz, MHz or GHz to keep digit counts sane. Want radians per second instead? Multiply by 2π yourself.
Worked example — an electron in a one-tesla field
One tesla is what stout laboratory electromagnets deliver, or low-field MRI bores. Particle charge 1.602176634 × 10⁻¹⁹ C, Magnetic flux density (T) 1, Particle mass 9.1093837015 × 10⁻³¹ kg. Charge times field, over 2π × 9.1093837015 × 10⁻³¹ = 5.7236 × 10⁻³⁰, comes to 27,992,489,872.3 Hz. Move that output menu to GHz and Cyclotron frequency reads 27.9925 GHz.
Which lands squarely in Ka band, and explains why electron spin resonance benches pair their magnets with microwave sources the way they do — an X-band klystron near 9.5 GHz asks for roughly 0.34 T. Spin turns out slightly more interesting than orbit here. A free electron precesses at 28.0250 GHz in that same one-tesla field, 0.116% quicker than it circles, and that small excess is g/2 − 1, its anomalous magnetic moment. Penning-trap experiments measure that beat to better than a part in 10¹², which makes this rare ground where hand arithmetic and the most precise number in physics share one formula.
Questions
Why does speed not appear in the formula?
Because orbit radius grows with speed and cancels it out. A particle moving at v traces r = mv/(|q|B), so a circumference of 2πmv/(|q|B) gets covered at v — leaving a period of 2πm/(|q|B) with no v anywhere in sight. Faster particles run bigger laps in equal time. Lawrence built an industry on that cancellation, and relativity eventually spoiled it: as γ climbs, effective mass rises and laps stretch out.
Is cyclotron frequency the same thing as Larmor frequency?
No, and confusing them costs a factor of nearly three. Cyclotron frequency describes orbital motion of a whole particle, |q|B/(2πm). Larmor frequency in NMR and MRI describes precession of a magnetic moment, set by spin structure rather than charge-to-mass ratio alone: a proton at 1 T orbits at 15.25 MHz yet precesses at 42.58 MHz. A third usage muddles matters further — classical Larmor precession of an orbiting charge in a weak field equals exactly half a cyclotron frequency. Establish which one an author means before trusting any published figure.
Should there be a 2π in the denominator or not?
Both forms are right, for different units. f_c = |q|B/(2πm) counts laps per second, in hertz. ω_c = |q|B/m counts radians per second, and plasma physics writes it that way almost exclusively — for electrons that value is 1.7588 × 10¹¹ rad/s per tesla, which is e/m exactly. This sheet reports hertz. Dropping a plasma textbook figure into a hertz field costs 6.283, an error big enough to send a resonance search hunting in altogether the wrong waveband.
When does the formula stop being true?
Three places, plus one for trap builders. Relativity: swap m for γm, so a 5 keV electron already runs 1% slow and a 9 MeV proton likewise. Collisions: a particle needs to finish its lap undisturbed, meaning ω_c τ must exceed 1, which air at atmospheric pressure will not allow. Field gradients: flux density varying across an orbit gives frequency varying with it, plus drift motion this expression never describes. Trapped-ion work adds electrostatic shifts, corrected out via the invariance theorem rather than ignored.
Does the sign of the charge change anything?
Not the rate, only the direction of travel. Absolute value sits in this formula deliberately: an electron and a positron in one field circle at identical frequencies in opposite senses. That opposition earns its keep — antiproton work at CERN compares an antiproton against a hydrogen ion inside the same magnet, testing charge-to-mass symmetry between matter and antimatter to roughly 16 parts per trillion.
How does this let anyone weigh an ion?
Rearrange to m = |q|B/(2πf_c). Frequency is the most precisely measurable quantity going, so an ion held in a well-mapped field yields a mass at accuracy no balance can approach. Fourier-transform ion cyclotron resonance mass spectrometry runs on exactly this: a singly charged ion of 1000 u in a 7 T magnet circles near 107 kHz, and a whole spectrum arrives at once as one frequency transform. Penning traps push further still, supplying atomic masses that nuclear binding-energy tables are assembled from.