loading...
running...
How to use
Pick a preset, then press Pause or Space to freeze the run and Reset (or R) to restart. Watch the central ion (white) spiral outwards: it gains the same energy at every gap crossing while the green or red arrows in the gap show the field direction. The blue waveform shows the gap voltage with orange dots where ions crossed.
Try: Relativistic stall and watch the crossing phase in the second graph slide through the red lines; then switch Frequency programme to the synchrocyclotron or isochronous and compare. Raise the phase spread to see how the bunch smears in energy, or detune the RF by 1-2% to watch the machine lose synchronism. Space pauses, R resets.
Drag to orbit, wheel to zoom. The upper yoke is lifted so you can see the dees; untick the cutaway to see the closed machine. The dee that is positive at this instant glows green and the negative one glows red.
The science
A charged particle of charge q and mass m in a uniform magnetic field B moves on a circle. The magnetic force does no work, so the speed is constant while the field only turns the velocity:
q v B = γ m v² / r ⇒ r = p / (qB), ωc = qB / (γ m), T = 2πγm / (qB)
Non-relativistically (γ = 1) the period does not depend on the speed or radius. That is the whole idea of Lawrence's cyclotron: a fixed radio-frequency voltage f = qB / (2πm) across the gap between two hollow D-shaped electrodes (the dees) pushes the ion forward every time it crosses, and the ion's circle grows while the timing stays right. Inside a dee there is no electric field, so the ion only circles.
ΔE = |q| V0 cos φ · Tg, Tg = sin θ / θ, θ = ωrf g / (2 v) (transit-time factor of a gap of width g)
Two gap crossings per turn give an energy gain of 2|q|V0 cos φ per revolution. The ion leaves at the dee radius R, so the final energy is limited by the magnet, not by the voltage:
Emax = (q B R)² / (2 m) (non-relativistic), E = √((pc)² + (mc²)²) − mc², p = qBR (exact)
Relativistic limit. As the energy approaches the rest energy, γ grows (1.1% at 10 MeV for protons, 11% at 100 MeV). The orbit takes longer than the RF half-period, so every crossing arrives a little later in the RF cycle. The phase slip per half-turn is π(γ − 1); the ion stops gaining energy when the accumulated slip reaches 90 degrees and then returns the energy it gained. The simulation shows this as the crossing phase drifting through the red lines in the phase graph. Two cures exist:
synchrocyclotron: f(t) = qB0 / (2π γ(t) m), isochronous (AVF) cyclotron: B(r) = B0 γ(r)
The synchrocyclotron lowers the RF frequency while a bunch is accelerated (low duty cycle, high energy); the isochronous cyclotron raises the field with radius so that the revolution time stays constant, which works with a fixed frequency but needs azimuthal field variation (Thomas focusing) for vertical stability, ignored here. Electrons reach γ = 2 at just 0.5 MeV, which is why they are accelerated in synchrotrons and linacs instead.
The model. The orbit is integrated exactly, not by small Euler steps: during a step the momentum rotates by an exact angle and the position advances along the exact circular arc. When an arc crosses the gap plane x = 0 the crossing time is found by bisection and the energy kick is applied there, with the gap voltage V0 cos Φ(t) at that instant. The bunch of 33 ions is injected over the chosen phase spread; the central ion is the reference. Dee radius, gap width and field profile are free parameters; the field profile B(r) = B0[1 + k (r/R)²] is evaluated at the current radius of the ion.
Numerical checks run in the page. (1) The classical extraction energy lies between (qB(R − 2r1))²/2m and (qBR)²/2m + 2qV (the spiral is centred on the ion source, which costs a little radius). (2) Every half-turn is a semicircle: the chord between consecutive crossings equals 2p/(qB). (3) The turn period equals 2πm/(qB) for slow and fast ions in the classical case. (4) The relativistic period of a 150 MeV proton equals 2πγm/(qB). (5) The magnetic field alone conserves kinetic energy over 60 turns. (6) The energy after 150 crossings with a fixed frequency agrees with the independent two-line phase-slip map E' = E + qV cosφ, φ' = φ + π(γ − 1). (7) The isochronous machine extracts at the energy given by v = qB0R/m.
Simplifications. The gap is a thin impulsive kick (the transit-time factor accounts for the finite width); there are no space-charge forces, no vertical motion, no edge focusing, no fringe field of the dees, no residual gas scattering; the extraction is an idealised radius cut and the stripped ion beam is not steered. The electric field is uniform across the gap and independent of y.