Cyclotron

The core idea

A cyclotron speeds a charged particle up to high energy using two simple tools working together: a steady magnetic field that bends the particle into circles, and an alternating voltage that gives it a small push every half-turn. Neither tool alone would do much — but combined, they let a particle be accelerated thousands of times by a modest voltage, all inside a compact magnet.

Picture two hollow, D-shaped metal boxes (the "dees") sitting flat between the poles of a big electromagnet, with a thin gap between their straight edges. A charged particle is injected at the center. The magnetic field curves it into a half-circle inside one dee; when it crosses the gap, the voltage gives it an electric kick that speeds it up; the field then curves it through a larger half-circle inside the other dee. Speed up, circle wider, kick, repeat. The path is an outward spiral, and after enough laps the particle is flung out the edge at high energy.

How it works, step by step

• Inject at the center. A source drips low-energy ions into the gap at the middle of the magnet.
• The field bends, the metal shields. Inside a dee, the metal walls shield the particle from any electric field, so it feels only the magnetic force and coasts in a perfect half-circle.
• The gap accelerates. The only place an electric field exists is the gap between the dees. The alternating voltage is timed so that whenever the particle is in the gap, the field points the way that speeds it up.
• Resonance keeps the timing. Crucially, each half-circle takes the same time regardless of speed — so a single fixed-frequency oscillator stays in step the whole way out.
• Spiral outward. Each kick raises the speed, which raises the orbit radius, so the particle traces an ever-widening spiral.
• Extract at the rim. When the spiral reaches the edge of the magnet, a deflector plate peels the beam off toward a target.

The governing physics

Start with the magnetic force on a moving charge — the Lorentz force with no electric field:

F = q·v × B

This force is always perpendicular to the velocity, so it does no work — it only turns the particle. Setting it equal to the centripetal force m·v²/r needed for circular motion:

q·v·B = m·v² / r    →    r = m·v / (q·B)

So the radius grows in direct proportion to speed. Now find how long one revolution takes. The period is T = 2πr / v, and substituting r:

T = 2π·m / (q·B)    →    f = q·B / (2π·m)

The speed v has dropped out entirely. The cyclotron frequency f (also written f_c) depends only on the charge-to-mass ratio and the field strength — not on how fast the particle is going or how big its orbit is. The angular version is the cyclotron angular frequency:

ω_c = q·B / m

This is the magic of the cyclotron: drive the gap voltage at this one frequency and the particle stays in step for its entire outward spiral.

Maximum energy

The particle keeps spiraling until its orbit radius reaches the physical edge of the magnet, R. At that point r = R, so its speed is v_max = qBR/m. The kinetic energy ½mv² becomes:

E_max = q²·B²·R² / (2m)

Two things jump out. First, the final energy doesn't depend on the gap voltage — a bigger voltage just means fewer laps to get there. Second, energy scales with the square of both the field strength B and the magnet radius R. Want four times the energy? Double the field, or double the radius. This is exactly why higher-energy cyclotrons need dramatically bigger and stronger (and more expensive) magnets.

Each crossing of the gap adds energy qV, where V is the peak gap voltage. To reach E_max the particle needs about N = E_max / (2qV) full turns (two gap crossings per turn). A 50 kV gap driving a 16 MeV proton means roughly 160 turns before extraction.

Worked numbers

Quantity	Formula	Example value
Proton cyclotron frequency at B = 1.5 T	f = qB/(2πm)	≈ 22.9 MHz
Electron cyclotron frequency at B = 1.5 T	f = qB/(2πm)	≈ 42 GHz (1836× higher — tiny mass)
Orbit radius of a 16 MeV proton at 1.5 T	r = mv/(qB)	≈ 0.39 m
Max proton energy, B = 1.5 T, R = 0.5 m	E = q²B²R²/(2m)	≈ 26 MeV
Turns to reach 16 MeV at 50 kV gap	N ≈ E/(2qV)	≈ 160 turns
Total path length over those turns	Σ 2πr	tens of metres inside a ~1 m machine

Note how the electron frequency lands in the microwave/radar band — the same physics underlies cyclotron resonance and electron-cyclotron-resonance plasma sources.

The relativistic limit and its fixes

The whole scheme rests on f = qB/(2πm) being constant. But m here is the relativistic mass, m = γm₀, and as the particle nears light speed γ climbs above 1. The cyclotron frequency drifts downward, the particle arrives at the gap a little late each time, and eventually it slips out of phase with the fixed-frequency voltage and stops gaining energy. For protons this becomes a real problem around 20–25 MeV. Two redesigns get around it:

Machine	Trick	Trade-off
Classic cyclotron	Fixed B, fixed RF frequency	Capped near ~25 MeV (protons) by relativity
Synchrocyclotron	Slowly sweep the RF frequency down to track γ	Reaches ~100s of MeV, but only pulsed (bunches), low average current
Isochronous cyclotron	Shape B to rise with radius so f stays constant	Steady (CW) beam at high current; needs azimuthally varying field for focusing
Synchrotron (different family)	Ramp B and frequency, fixed-radius ring	Reaches GeV–TeV, but huge and not compact

Where cyclotrons show up

• PET-scan isotopes. The dominant use. Hospital cyclotrons fire ~16 MeV protons at targets to make fluorine-18 (110-minute half-life), carbon-11 (20 min), nitrogen-13, and oxygen-15. The half-lives are so short that the isotope must be made on-site or nearby, often the same morning it's injected.
• Proton-beam cancer therapy. Larger cyclotrons (~230–250 MeV) produce proton beams that deposit their dose in a sharp Bragg peak inside a tumor, sparing tissue beyond it.
• Making new elements. Cyclotrons produced technetium (1937) — the first artificially made element — plus astatine, and contributions to many transuranics.
• Nuclear and materials research. Probing nuclei, producing exotic isotopes, and ion-beam analysis of materials.
• Radiocarbon and trace dating. Cyclotron-based accelerator mass spectrometry counts rare isotopes atom by atom.
• The same resonance in nature. Charged particles in planetary magnetospheres and the solar wind gyrate at their local cyclotron frequency, radiating cyclotron/synchrotron emission radio astronomers detect.

Common misconceptions and edge cases

• "The magnet accelerates the particle." It doesn't. The magnetic force is perpendicular to velocity and does zero work — it only bends. All energy comes from the electric field in the gap.
• "You need a higher voltage for higher energy." No — the final energy is set by B and R, not by the gap voltage. A bigger voltage just reduces the number of laps; the particle reaches the same E_max faster.
• "The frequency must track the particle's speed." For a non-relativistic classic cyclotron, the frequency is constant on purpose, because the period is speed-independent. It's only the relativistic drift that forces a sweep (synchrocyclotron) or a shaped field (isochronous cyclotron).
• "Electrons work great in a cyclotron." Barely. Electrons are so light that they go relativistic at low energies, so the constant-frequency trick fails almost immediately — cyclotrons are practical mainly for protons and heavier ions.
• "The spiral is uniform." The spacing between turns shrinks as the particle speeds up — radius goes as √(energy), so each equal-energy kick adds a smaller and smaller increment of radius near the rim.
• "Cyclotron and synchrotron are the same." A cyclotron has a fixed radius magnet and a spiraling orbit at fixed field; a synchrotron keeps a fixed-radius ring and ramps the field as the beam accelerates. Cyclotrons are compact; synchrotrons scale to far higher energies.