Cyclotron Principle and Working: How Particles Fly

A cyclotron is a particle accelerator that uses a magnetic field to bend charged particles into a spiral and an alternating electric field to give them a push on every half-turn. Two hollow D-shaped electrodes, the dees, sit inside a vacuum chamber between the poles of a large magnet.

Its importance is practical: cyclotrons produce energetic protons and ions used in medical isotope production and research. Its clever trick — that revolution frequency stays constant as the particle speeds up — is what makes simple, repeated acceleration possible.

Key Concepts

Dees: hollow semicircular electrodes; the particle is shielded from the electric field inside a dee and is accelerated only in the gap. Magnetic field B: perpendicular to the plane, forces circular motion, unit tesla (T). Oscillator: alternating voltage applied to the dees at the cyclotron frequency. Resonance condition: oscillator frequency equals the particle's revolution frequency. Extraction: at maximum radius the beam is deflected out onto a target.

Formulas

Cyclotron frequency: f = qB/(2πm), where q in coulomb (C), B in T, m in kilogram (kg); f in hertz (Hz) and independent of speed and radius. Maximum kinetic energy: Kmax = q²B²R²/(2m), where R is the dee radius in metre (m). Each gap crossing adds energy qV, where V is the dee voltage. Assumption: non-relativistic motion and uniform B. Limit: at relativistic speeds the effective mass rises, the frequency falls out of step, and a simple cyclotron loses synchronism.

Worked Example

A proton (q = 1.6 × 10⁻¹⁹ C, m = 1.67 × 10⁻²⁷ kg) in B = 1.0 T with dee radius R = 0.50 m. Find f and Kmax.

1. f = qB/(2πm) = (1.6 × 10⁻¹⁹ × 1.0)/(2π × 1.67 × 10⁻²⁷) ≈ 1.5 × 10⁷ Hz = 15 MHz.
2. Kmax = q²B²R²/(2m) = (1.6 × 10⁻¹⁹)² × 1 × 0.25/(2 × 1.67 × 10⁻²⁷) ≈ 1.9 × 10⁻¹² J.
3. In electron-volts: 1.9 × 10⁻¹²/(1.6 × 10⁻¹⁹) ≈ 1.2 × 10⁷ eV = 12 MeV.

The oscillator runs at about 15 MHz and protons leave with roughly 12 MeV energy, fixed by B and R rather than by the dee voltage.

Common Misconceptions

• Higher dee voltage gives higher final energy: It only reduces the number of turns needed; final energy is set by B and the exit radius.
• The particle spirals because the field weakens: Radius grows simply because speed grows at constant frequency, r = mv/(qB).
• Electrons work just as well: Electrons become relativistic at low energies and fall out of synchronism quickly, so cyclotrons mainly accelerate protons and ions.

Real-World Applications

• Production of medical radioisotopes for PET scans and cancer therapy.
• Nuclear physics research and materials irradiation.
• Proton therapy centres treating tumours with precisely targeted beams.

The cyclotron exploits constant revolution frequency in a magnetic field to push particles repeatedly in a widening spiral, delivering mega-electron-volt beams from a compact machine.