How a Cyclotron Works
The physics of the classical cyclotron, from the Lorentz force to the relativistic ceiling — written for a motivated general reader, with enough precision to be useful to an undergraduate or a builder. Every formula here is available interactively in the calculators.
The problem it solves
To probe an atomic nucleus you must throw a charged particle at it with an energy of millions of electron-volts (MeV). The direct approach — accelerate the particle across a single gap held at millions of volts — runs straight into the limits of insulation: somewhere around a few MV, everything arcs. In 1928 Rolf Wideröe demonstrated the way around this: use a modest alternating voltage many times, timed so the particle always arrives when the push is in its direction. Wideröe's machine was a linear accelerator, and it got long fast — each successive push happens at higher speed, so each drift section must be longer than the last.
Ernest Lawrence's insight of 1929 was that a magnetic field can fold that long line into a compact spiral. Bend the particle in a circle, and it can cross the same accelerating gap over and over. Better still — and this is the fact the whole machine hangs on — the time per lap turns out not to depend on how fast the particle is going.
A charged particle in a magnetic field
A particle of charge q moving with velocity v through a magnetic field B feels the Lorentz force,
which is always perpendicular to the velocity. A force that never points along the motion does no work — it changes the particle's direction but not its speed. In a uniform field, a particle moving in the plane perpendicular to B therefore travels in a circle, with the magnetic force supplying the centripetal force:
The orbit radius grows in proportion to momentum. A slow proton circles near the center of the magnet; a fast one sweeps a wide arc near the edge. Read backwards, this is also the machine's energy gauge: the momentum of a particle on its outermost orbit is p = qBr, fixed entirely by the field and the radius. For a proton in a 1 T field reaching a radius of 0.5 m, that works out to about 12 MeV — the energy calculator does this arithmetic for any particle, field, and radius.
The resonance condition
How long does one lap take? The circumference is 2πr and the speed is v, so the period is T = 2πr/v = 2πm/qB — the radius and speed cancel. Every lap takes the same time, no matter how fast the particle is going, because faster particles travel proportionally larger circles. The orbital frequency, called the cyclotron frequency, is
For protons this is 15.2 MHz per tesla of field — squarely in the shortwave radio band for the fields ordinary iron magnets produce. That constancy is the cyclotron's central trick: an alternating voltage at this one fixed frequency stays in step with the particle from its first lap to its last. The particle and the radio wave are in resonance.
Dees and the accelerating gap
The classical cyclotron places two hollow, D-shaped copper electrodes — the dees — face to face inside a vacuum chamber between the magnet poles, with a narrow gap between them. An RF oscillator drives the dees in antiphase at the cyclotron frequency, so an electric field appears across the gap, reversing direction every half-cycle. Inside a dee the particle coasts in a field-free half-circle; each time it crosses the gap, it arrives just as the field points its way, and picks up energy qV, where V is the gap voltage at crossing.
Two crossings per lap, thousands of laps: a dee voltage of a few kV builds to hundreds of keV or MeV, and the orbit spirals outward as the momentum grows. The number of turns required is roughly the final energy divided by the energy gained per turn (2qV at best), and the total path length can run to hundreds of meters — which is why the vacuum must be good enough that the particle survives the trip without scattering off residual gas. The path-length calculator and the vacuum calculator quantify both sides of that bargain.
Staying in step: phase
Resonance is never perfect. If the RF frequency, the field, or the particle's mass drifts, the particle arrives at the gap slightly early or late — its phase slips. A particle that slips too far arrives when the field opposes it and is decelerated; the beam is lost. A classical fixed-frequency cyclotron therefore has a phase budget: the total slip accumulated over all turns must stay within roughly ±90°, and in practice a frequency or field error of order 1–2% is the most such a machine tolerates before the beam dies (Livingood 1961, ch. 6). Fewer turns at higher dee voltage spend the budget more slowly — one reason builders chase dee voltage.
In 1944–45 Veksler and McMillan independently discovered phase stability: if the RF frequency is made to follow the particles rather than stay fixed, ions near a stable phase are automatically herded back toward it, oscillating gently about the synchronous phase instead of drifting off resonance. That principle rescued the cyclotron from its relativistic limit (below) and underlies the synchrocyclotron and every synchrotron since.
Weak focusing: the field index
A beam also has to be held together vertically. A particle drifting above the magnet midplane must feel a force pushing it back down, or it will strike a dee within a few turns. In a classical cyclotron this axial focusing comes from making the field decrease slightly with radius: the field lines then bow outward, and their radial component pushes off-plane particles back toward the midplane. The amount of decrease is captured by the field index,
Vertical focusing requires n > 0 (field falling with radius); radial orbit stability requires n < 1. Classical machines run with a small positive index — typically a few hundredths to ~0.2 near extraction — produced by shimming the pole faces. This is weak focusing: gentle restoring forces, large vacuum chambers, and a beam that occupies real vertical space. The design guide collects the practical shimming rules with their sources.
Why classical cyclotrons top out
The resonance condition contains the particle's mass — and relativity makes mass effectively grow with energy. At kinetic energy T, the orbital frequency falls to f/γ, where γ = 1 + T/mc2. For a 20 MeV proton, γ ≈ 1.021: the particle circulates 2% slower than the RF expects, and 2% is already about the whole phase budget. Worse, the two fixes conflict — restoring resonance would require a field that rises with radius, exactly what weak focusing forbids. Bethe and Rose pointed out this ceiling in 1937; in practice, fixed-frequency classical cyclotrons top out around 10–25 MeV for protons, the exact figure set by how much dee voltage the builder can afford (Livingston & Blewett 1962, ch. 6). Two escape routes exist, and both were taken:
The synchrocyclotron gives up on fixed frequency. Sweeping the RF downward in step with the accelerating bunch keeps it in resonance to arbitrary γ, and phase stability holds the bunch together during the sweep. The price is duty cycle: only one bunch rides each frequency sweep, so the beam arrives in pulses at the modulation rate (tens to hundreds of Hz) and the average current drops by orders of magnitude compared with a CW machine. The Berkeley 184-inch and its postwar siblings reached hundreds of MeV this way.
The isochronous (AVF) cyclotron gives up on a radially falling field. Instead the average field is made to rise with radius exactly as γ does, keeping the orbital period truly constant at all energies — and the vertical defocusing that rising field causes is beaten by azimuthal field variation: the poles carry ridges ("hills") and grooves ("valleys"), so the orbit scallops through alternating strong and weak field sectors, which produces a net vertical restoring force (L. H. Thomas worked this out in 1938; spiral-shaped sectors, suggested by Kerst in 1955, strengthen it further). Isochronous machines deliver continuous beam at high current and are the design of essentially every cyclotron built since the 1960s, from 12 MeV medical machines to the 520 MeV separated-sector giant at TRIUMF.
The cyclotron among accelerators
Every circular accelerator is a different answer to the same two questions: what do you do with the magnetic field as the particle gains energy, and what do you do with the RF frequency?