Cyclotron

A cyclotron repeatedly passes ions through an alternating electric field between two hollow electrodes called cyclotron dees. A nearly uniform magnetic field perpendicular to their plane bends the charged particles back toward the gap. Inside each cyclotron dee, ideal shielding suppresses the accelerating electric field, so each segment is a constant-speed circular motion. The next gap crossing increases the speed and the radius of curvature. The complete path consists of connected, progressively larger arcs, rather than one circle or a smooth spiral with continuous acceleration everywhere.

The striking feature is a timing cancellation: the longer arc traveled by a faster particle takes the same time. This cyclotron resonance condition lets one fixed radio frequency accelerate many successive gap crossings. The following construction and cyclotron energy and orbit radius example describe an ideal model; the Lawrence-Livingston cyclotron experiment explains what the builders actually measured.

A hollow D-shaped electrode that shields a circulating ion from the accelerating electric field during one half-turn. Two opposed cyclotron dees provide two shielded regions separated by a gap; their electric potential difference alternates. The vacuum reduces collisions with residual gas. An ion source supplies particles near the center, and an extraction system directs an outer orbit toward a particle detector or target. Useful electric current depends on retaining particles near the median plane and delivering them to the outlet, as well as on increasing their kinetic energy.

Take an ion of mass $m$ and electric charge $q\ne 0$, with velocity perpendicular to a uniform magnetic field of magnitude $B>0$. In the classical mechanics limit, the Lorentz force provides the centripetal acceleration:Here $r$ is the local radius of curvature and ${\omega }_c$ is the cyclotron frequency, expressed as an angular frequency. A half-turn takesIncreasing speed increases the path length in exactly the same proportion. Drive the gap at angular frequency ${\omega }_{\mathrm{RF}}={\omega }_c$, or ordinary frequency $f_{\mathrm{RF}}=|q|B/(2\pi m)$. In each half-turn the gap electric field reverses once, just as the particle reverses its direction of passage through the gap. The particle therefore receives two accelerating kicks per complete turn, using one oscillation per complete turn. This is the fundamental resonance; transit time through the gap and departures from a uniform magnetic field have been neglected.

Define $\Delta V={\Phi }_{\mathrm{entry}}-{\Phi }_{\mathrm{exit}}$ as the electric potential drop along the particle's passage. If the electric field changes negligibly during that short crossing, the work givesThe sign matters: a positive ion gains kinetic energy while going to lower electric potential; a negative ion needs the opposite drop. A fixed-sign field would accelerate on one crossing and decelerate on the next. The alternating field must reverse before the particle comes back.

For a fixed electric charge sign and circulation direction, absorb this alternating crossing direction into the definition of the arrival phase ${\varphi }_n$. If $V_0$ is the peak electric potential difference across the whole gap, rather than the amplitude of either electrode relative to ground, a thin-gap model isArrival at ${\varphi }_n=0$ gives the largest accelerating kick; $|{\varphi }_n|<\pi /2$ gives a positive kick. Arrival at ${\varphi }_n=\pi$ gives deceleration. For an ideal classical particle initially on crest, the cyclotron resonance condition keeps successive arrival phases equal. It does not automatically bring an initially mistimed particle onto crest.

Combining $r=mv/(|q|B)$ with the classical kinetic energy givesHere $E$ denotes kinetic energy, not relativistic total energy. At fixed magnetic field, twice the final radius allows four times the classical kinetic energy. At fixed final kinetic energy, a larger accelerating gap voltage reduces the number of crossings needed, but does not change the required orbit radius.

For an illustrative proton with $B=1.00\mathrm{T}$ and peak gap voltage $V_0=10.0\mathrm{kV}$, take negligible injection kinetic energy, on-crest kicks, a zero-width gap, and no losses. Each crossing adds $10.0\mathrm{keV}$, so $100$ crossings give $1.00\mathrm{MeV}$, using electronvolts. With $e=1.602176634\times 10^{-19}\mathrm{C}$ and $m_p=1.67262192595\times 10^{-27}\mathrm{kg}$,The $100$th kick arrives after $99$ intervening half-turns, about $3.247\mu \mathrm{s}$ after the first kick in this idealization. The final two curvature radii differ by $r(1\mathrm{MeV})[1-\sqrt{99/100}]\simeq 0.724\mathrm{mm}$. That difference is a curvature-radius increment, not a general formula for separation of real extracted turns: successive arc centers shift, and extraction also depends on focusing and the gap geometry. These chosen settings are a calculation, not historical measurements.

Ernest Orlando Lawrence and Milton Stanley Livingston reported repeated ion acceleration in 1932. Their evacuated brass chamber used one driven hollow cyclotron dee; the grounded box and slotted divider supplied the other electrode. Filament electrons ionized hydrogen near the gap. A perpendicular magnetic field bent the ions; a high-frequency oscillator drove the gap. Deflecting plates and slits selected particles for a Faraday cup, read by an electrometer.^[ref]

Scanning the magnetic field produced narrow resonance peaks. The optimum deflecting voltage tracked the kinetic energy predicted from field and orbit radius, supporting the reported $1.22\mathrm{MeV}$ protons. Source width and slit geometry limited this check; it was not a direct energy measurement of every particle.^[ref]

This combination distinguishes producing a circulating electric current from establishing its kinetic energy. The electrometer measures collected electric charge per unit time; it cannot by itself tell whether the collected particles are fast. The separate deflection selection supplies a speed-dependent test. Agreement of two consequences of the model, the cyclotron resonance condition and the cyclotron energy and orbit radius relation, makes the inference stronger than either a current signal or a predicted orbit alone.

Milton Stanley Livingston built the early cyclotrons with Ernest Orlando Lawrence; the 1939 Nobel Prize in Physics went to Ernest Orlando Lawrence. His lecture, The evolution of the cyclotron, was delivered on 11 December 1951.^[ref][ref]

At larger kinetic energy, special relativity changes the relation between momentum and velocity. With $m$ the invariant mass, the Lorentz factor is $\gamma =1+K/(m{c}^2)$, and a relativistic cyclotron orbit in a fixed uniform magnetic field satisfiesThe cyclotron frequency decreases as the kinetic energy increases. If the drive stays at the classical value ${\omega }_0=|q|B/m$, the half-turn time is $\pi \gamma /{\omega }_0$. During it the drive advances through $\pi \gamma$ rather than $\pi$ radians. Its excess phase per half-turn is thereforeEven a small fractional frequency mismatch accumulates across many turns. A particle eventually reaches a weak or decelerating gap phase, so the classical fixed-field, fixed-frequency design has a practical kinetic energy limit. The limit depends on the kick size and acceptable accumulated phase error, rather than one universal cutoff energy. Changing drive frequency or shaping the magnetic field can restore synchronism; the argument does not forbid relativistic circular accelerators.

For the illustrative proton at $1.00\mathrm{MeV}$, $\gamma -1\simeq 0.00106579$: the frequency is lower by about $0.1065\%$, and the next half-turn adds about $0.19184^{\circ }$ of drive phase excess. To see cumulative sensitivity, prescribe an ideal ramp $K_j=jϵ$ with $ϵ=10\mathrm{keV}$ and sum the errors of the first $N$ coasts after kicks:This gives ${\Phi }_{100}\simeq 9.688^{\circ }$ after the coast following kick $100$; the arrival at kick $100$ instead has ${\Phi }_{99}\simeq 9.496^{\circ }$. The prescribed ramp diagnoses detuning but does not feed the resulting phase error back into the kicks. A self-consistent thin-gap model uses the cyclotron gap energy gain with $\cos {\varphi }_n$, then advances the phase with the new kinetic energy during the following coast.