Beam Dynamics: An Interactive Laboratory

How a Cyclotron Works shows weak focusing as a still picture. This page runs it: two simulations integrate the betatron-oscillation equations in the browser, with the field index on a slider and the classic field imperfections on two more. The finale replays a documented 1947 beam loss from the Berkeley 184-inch cyclotron against the memoranda that diagnosed it.

Stability is two numbers

An ion on the equilibrium orbit of a weak-focusing cyclotron is in balance: magnetic bending exactly matches its momentum. Displace it outward and, because the field falls with radius more slowly than 1/r when n < 1, it meets more bending than its new radius needs and is pushed back. Displace it above the midplane and the outward-bowing field lines push it back down, provided n > 0. For small displacements both restoring forces are linear — a field whose fall-off is even approximately linear in radius gives simple-harmonic axial motion (dg-138) — so the ion executes two independent harmonic betatron oscillations. Measuring azimuth θ in radians, with x and z the radial and axial displacements:

The oscillation counts per turn — the tunes ν_r = √(1 − n) and ν_z = √n — are the two numbers of the title. Both restoring forces are positive only for 0 < n < 1 — at either endpoint one tune reaches zero and that motion, left with no restoring force, drifts away unchecked — which is the whole content of the classical stability rule (Livingston & Blewett 1962, p. 161, dg-003; modern notation dg-561). The same pair of frequencies appears, derivation and all, in Figure 4 of MDDC-1092 — the 1947 memo this page ends on. The two tunes are also chained to each other: ν_r² + ν_z² = 1. A classical field buys axial stiffness only by selling radial stiffness. The sliders below make that trade felt.

Panel 1 — the healthy machine

One control: the field index. The top view looks down the magnet bore at the orbit; the side elevation watches the ion's height above the median plane; the tune diagram plots the working point. Drag n toward 0 and the axial wobble goes limp — ν_z → 0, one axial oscillation stretching over many turns — while the radial motion tightens. Drag toward 1 and the roles swap. Working weak-focusing machines settled near the soft end: in the 1958 world accelerator census (ORNL-2644, a questionnaire survey with a data sheet per machine), cyclotrons shimmed for a total field drop of only 2–4% from center to full radius — Copenhagen 1.75%, Canberra 2%, Rochester 3–4% (dg-702) — placing n at a few hundredths over most of the pole and letting it climb toward 0.2 only at the working edge (dg-637, dg-152). Why 0.2 is the fence is panel 2's subject.

The tune diagram maps the trouble

The tune diagram condenses the machine to a single point at (ν_r, ν_z). In the linear model above, every weak-focusing working point lies on the quarter-circle ν_r² + ν_z² = 1; the slider just moves the point along it (a real machine sits near the arc, not exactly on it — electric focusing, acceleration, and field imperfections all shift the tunes slightly). The danger is the dashed lines. A field error fixed to the magnet repeats in the ion's experience exactly once per turn, so if an oscillation also repeats in step with the turns, the error's kicks land at the same oscillation phase every time and add coherently instead of averaging away. ν_r = 1 is the worst case — the integer resonance, driven by a first-harmonic field error, sitting at the n = 0 end of the arc. In Smirnov's assessment, its passage near the center goes without noticeable loss only because it is crossed in one to three turns with no large first harmonic present (dg-562). The second line, ν_r = 2ν_z, is the Walkinshaw coupling resonance (dg-563): radial oscillation pumps axial oscillation when the radial frequency is exactly twice the axial. On the weak-focusing arc, that happens at one place only — 1 − n = 4n, so n = 0.2. This single point is why the weak-focusing shimming guides tell builders to taper the field so n = 0.2 arrives only at the final ion radius (dg-152, dg-136): a resonance crossing parked inside the acceleration region eats the beam, as a deliberately mis-tapered pole has demonstrated (dg-153).

Panel 2 — the imperfection lab

Real magnets are imperfect, and the two classic imperfections get a slider each. A first-harmonic error ε₁ = B₁/B₀ — the field a little stronger at one azimuth θ₁ than opposite it — comes from an off-center ion-source hole or an asymmetric yoke (dg-050). An electric asymmetry — dee voltage drooping along the gap — displaces the orbit center too, by its own mechanism rather than through the field, and machines have off-set their ion sources by inches to compensate (dg-244); the slider below is the magnetic error only. A first harmonic displaces the closed orbit by

which is already millimeters for a part-per-thousand error and diverges as ν_r → 1: the extraction literature's worked example puts a 5 mm shift at ε₁ = 10⁻⁴ — roughly one gauss of error in a 1 T (10 kG) field — for a 1 m orbit near the integer line (dg-587). Drag n toward 0 with ε₁ nonzero and watch the displacement readout diverge; that is the integer resonance announcing itself. A displaced ion also precesses: its orbit center circles the equilibrium center — the shifted one, when a first harmonic is present — at (1 − √(1 − n)) revolutions per turn, the warm trail in the top view. That precession is measurable on a real machine as multiple "pips" per beam pulse on a probe — the diagnostic MDDC-987 nailed down with a second probe 155° away, and MDDC-1092 turned into a formula (dg-696; the measurement side of the story is in beam measurement).

The gradient coupling g = R·dn/dr is subtler. If n grows with radius — as it does at the fringing edge of a conventional pole (dg-637) — an ion oscillating radially samples a different axial stiffness on each half-swing: z″ + (n + gx/R)z = 0 (the simulation measures x in units of R, which absorbs the 1/R). A restoring force modulated at frequency ν_r pumps an oscillator of frequency ν_z hardest when ν_r = 2ν_z — the Walkinshaw condition again, now as an equation you can run. The model keeps the coupling's energy-conserving pair of terms, so the transfer saturates: when all the radial oscillation energy has moved to the axial motion, the axial amplitude is √((1−n)/n) times the radial — double at n = 0.2. That cap is a result of this model — lowest-order coupling, a coasting beam, no nonlinear detuning, and an aperture that has not yet intercepted the growth — not a law; the memoranda's own statements bracket the same number, MDDC-1092 as the bound "at least double" and the modern summary as "twice" (dg-563). The one imperfection this panel does not model is a top/bottom field asymmetry, which shifts the magnetic median plane itself; machines steer it back with unequal coil currents or auxiliary pole coils (dg-151, dg-174).

Recreate 1947

The Berkeley 184-inch synchrocyclotron was designed to accelerate to an 85-inch radius, where its field index reached 1 and the energy maxed out. The beam vanished at 82 inches. The theory group predicted the culprit — vertical oscillations blowing up where n crossed 0.2 — and in February 1947 the crew tested it with U-shaped copper targets, slots 2½ to 4½ inches wide, bombarded one at a time and laid on film. MDDC-984 reports the verdict in one sentence: "The autographs indicate a rapid spreading vertically of the beam at about 81 1/2 inches. This agrees quite closely with the point at which n = 0.2 from magnetic measurements." Four months later MDDC-1092 supplied the theory — the ν_r = 2ν_z energy transfer, with the conclusion that "the amplitude of vertical oscillation will be, at times, at least double that of the radial oscillations" — and the operational lesson: machines with low accelerating voltage spend many turns near the resonance and build the amplitude fast. Berkeley chose not to fight it: acceleration past that radius "has been postponed, since the available energy of the ions at this radius is within 5 per cent of the maximum of the system" (MDDC-1092, dg-693).

The preset below dials panel 2 into that event: orbit radius 2.08 m (82 in), a radial oscillation of about 2½ inches, an axial seed, and a slow upward ramp of n standing in for the orbit's outward march through the field's edge gradient. The n = 0.2 crossing radius and the factor-two amplitude transfer are documented (MDDC-984, MDDC-1092); the first-harmonic amplitude and the gradient are representative values chosen to reproduce the documented outcome, since the memoranda record neither. The loss line is a modeling choice too: it sits at |z| = 3.5% of the orbit radius, 2.9 inches at 82 inches, because the memoranda never state the dee aperture itself. The geometry they do record is the targets' 2½-to-4½-inch slots — full openings, so 1¼ to 2¼ inches each side of the median plane — which the simulation's line deliberately errs wide of. Watch the side view: the beam rides quietly, meets the crossing, and walks off the median plane onto the dee — the same failure mode the radioautographs caught on film.

The same lines on an AVF map

Isochronous (AVF) cyclotrons abandon the falling field — their average field rises with radius to hold the orbital period constant — and buy axial focusing from sector flutter and spiral instead, so their tunes leave the quarter-circle locus entirely (ν_z² gains flutter and spiral terms, dg-154, dg-156). The Walkinshaw line follows them off the arc — and the N-fold sector periodicity adds danger lines of its own, the structure resonances lν_r + mν_z = pN, so an AVF tune diagram carries more lines than panel 1's, not fewer. The ORNL isochronous-cyclotron design study tracked its operating points against ν_r = 2ν_z explicitly, listed the structure resonance (N−1)ν_r + 2ν_z = N right beside it, and warned, of extraction schemes that deliberately grow radial amplitude: "Whenever large radial amplitudes are present certain nonlinear coupling resonances become important and may lead to a disastrous growth in the axial oscillations" (ORNL-2648, p. 88; operating-point proximity pp. 46, 71). What 1947 discovered as a wall, the 1958 designers carried as a line on the map.

Go deeper

• Design guide — beam dynamics — the 209 sourced rules behind this page: the n-window, tune, resonance, and first-harmonic families.
• How a cyclotron works — the static introduction this page animates.
• Magnet design — shimming — producing a field whose n(r) stays out of panel 2's trouble.
• Beam measurement — probes and pips: how the oscillations on this page are seen on real hardware.
• The 184-inch commissioning papers — the six 1946–47 memoranda, hosted, with a reading guide.
• Glossary — betatron oscillation, tune, field index, resonance crossing.
• Beam extraction — precession — where these resonances are used on purpose: orbit-center motion turned into septum clearance.
• Beam quality — the statistical view: the beam as a population whose transverse spread these oscillation amplitudes set.

Sources

• J. Vale, "184-inch Cyclotron Vertical Beam Oscillations in the Region of 82-inch Radius," MDDC-984, UC Radiation Laboratory, Feb. 1947 — hosted PDF; the radioautograph experiment.
• F. W. Yeater, Jr., "184 Cyclotron: Synchroscope Beam Pictures on Two Probes," MDDC-987, Feb. 1947 — hosted PDF; pips proven to be precession.
• D. C. Sewell, L. Henrich & J. Vale, "Some Operating Phenomena Associated with the 184-inch Cyclotron," MDDC-1092, June 1947 — hosted PDF; the n = 0.2 mechanism, the 2× amplitude bound, the precession formula.
• The Oak Ridge Relativistic Isochronous Cyclotron, ORNL-2648, 1958 — hosted PDF; coupling resonances in AVF design (pp. 46, 71, 88).
• F. T. Howard, Cyclotrons and High-Energy Accelerators — 1958, ORNL-2644 — questionnaire census of the world's accelerators, one data sheet per machine; the field drop-off figures quoted in panel 1.
• M. S. Livingston & J. P. Blewett, Particle Accelerators, McGraw-Hill, 1962 — the 0 < n < 1 window (p. 161) and orbit-center effects of azimuthal voltage/field asymmetry (p. 164).
• V. L. Smirnov, "The Cyclotron and Its Modeling," Phys. Part. Nuclei 52, 913 (2021) — modern tune formalism, resonance orders, the Walkinshaw resonance by name.
• J. I. M. Botman & H. L. Hagedoorn, "Extraction from Cyclotrons," CERN Accelerator School proceedings — the first-harmonic displacement law and precession amplitudes.