Cyclotrons and mass spectrometers

Balance the magnetic force against the centripetal requirement for a charge q of mass m crossing a uniform field at right angles: qvB = mv²/r, so r = mv/(qB), and a faster ion simply swings wider. Now ask how long one lap takes. T = 2πr/v = 2πm/(qB), and v has cancelled — the ion moving twice as fast covers twice the circumference in the same time. The frequency f_c = qB/(2πm) is therefore a property of the particle and the magnet alone, not of the beam's energy. For a proton in 1.00 T it is (1.602 × 10⁻¹⁹ × 1.00)/(2π × 1.673 × 10⁻²⁷) = 1.52 × 10⁷ Hz = 15.2 MHz; for an electron in the same field, lighter by 1836, it is 28.0 GHz. That cancellation is the fact the rest of this lesson is built on, and it is also the first thing to fail.

A cyclotron is two hollow D-shaped electrodes facing each other across a narrow gap, inside the field of one large magnet. There is no electric field inside a dee, so the ion coasts along a half circle; at the gap it meets a potential difference V₀ and gains qV₀. Half a period later it reaches the gap again, and because the oscillator has reversed in exactly that time, it is accelerated rather than slowed. Two crossings per turn means K = 2NqV₀ after N turns, so N = K/(2qV₀). Notice what V₀ does and does not control. It fixes the energy step, and hence the number of turns and the time the beam spends inside the machine; it does not fix the final energy. The beam leaves when its radius reaches the pole radius R, and putting r = R into r = √(2mK)/(qB) gives Kₘₐₓ = q²B²R²/(2m). A stronger field or a wider magnet raises the energy; a larger dee voltage only gets you there sooner.

Everything above assumed m is fixed. Relativistically the momentum is γmv, so the orbit period is 2πγm/(qB) and the revolution frequency falls as the ion speeds up: ω = qB/(γm), with γ = 1 + K/(mc²). A fixed-frequency oscillator therefore meets a beam that is steadily later. On turn n the ion is behind by a fraction γₙ − 1 = Kₙ/(mc²) of a cycle, and those errors accumulate. Summing Kₙ = 2nqV₀ over N turns gives a total slip of about K²/(4qV₀mc²) cycles, and acceleration fails once that reaches roughly a quarter of a cycle, so a plain cyclotron tops out near K ≈ √(qV₀ mc²). For protons at V₀ = 100 kV that is √(0.100 × 938) = 9.7 MeV, and raising the dee voltage to 500 kV only lifts it to 21.7 MeV. The two cures are to sweep the oscillator frequency down during the acceleration, giving a synchrocyclotron that accelerates pulses rather than a steady beam, or to let B rise with radius so that qB/(γm) stays constant, giving an isochronous cyclotron.

A magnetic analyser measures m/q, and how it does so depends on what fixed the ion's speed. Accelerate singly charged ions from rest through a p.d. V_acc and qV_acc = ½mv², so v = √(2qV_acc/m) and the heavier ions arrive slower. Substituting into r = mv/(qB) gives r = √(2V_acc m/q)/B, so the radius grows as √m, and the measurement inverts to m/q = B²r²/(2V_acc). A ²⁴Mg⁺ ion at V_acc = 3.00 kV in B = 0.450 T leaves the gun at 1.55 × 10⁵ m s⁻¹ and turns on r = 8.59 cm; ²⁶Mg⁺ sits at 8.94 cm, wider by a factor √(26/24) = 1.041. The other standard arrangement puts a velocity selector in front, so every ion enters at the same speed whatever its mass. There r = mv/(qB) is linear in m instead. Same magnet, different power of m: which one applies is decided upstream, not in the analyser.

In a 180° instrument the ion enters through a slit, turns a half circle and lands on the plate a distance D = 2r away, so the plate reads the radius directly. Since m ∝ r² at fixed V_acc and B, differentiating gives Δm/m = 2Δr/r, and two species arrive separated by ΔD = 2Δr = r Δm/m. They are resolved when that separation exceeds the width w of the entrance slit imaged on the plate, which gives a resolving power m/Δm = r/w. The number is unforgiving. With r = 8.59 cm and w = 0.50 mm the instrument reaches m/Δm ≈ 172, ample for telling ²⁴Mg⁺ from ²⁵Mg⁺ and useless against the classic mass-28 pair: N₂⁺ at 28.0061 u and CO⁺ at 27.9949 u differ by 0.0112 u and need m/Δm ≈ 2500. Getting there means a far larger radius, a much narrower slit, or optics that refocus a beam that is slightly divergent and slightly spread in energy.

Nothing in m/q = B²r²/(2V_acc) separates a mass from a charge, so every peak is a mass-to-charge ratio, and a doubly charged ion of mass 2m lands exactly where a singly charged ion of mass m does. ¹²C²⁺ arrives at m/q = 6.00 u per elementary charge, within 0.015 u of ⁶Li⁺ at 6.015, so telling those two apart needs m/Δm ≈ 400. The same blindness is what makes a cyclotron versatile, because f_c = qB/(2πm) and Kₘₐₓ = q²B²R²/(2m) also depend on the ion only through q/m. An alpha particle has half the charge-to-mass ratio of a proton, so in the same magnet it needs half the oscillator frequency; and writing Kₘₐₓ = (q/m)² mB²R²/2, its quartered (q/m)² cancels against its fourfold mass, so it leaves with the same total kinetic energy — a quarter as much per nucleon.