Guided exercises · 2 of 6
Watch an Ion Fall Out of Step
Goal. A slightly wrong frequency does not simply trim the beam a little. The error accumulates, turn by turn, until the pushes stop helping and the ions stop gaining energy.
Who it is for. Readers of How a Cyclotron Works who have done Find the resonance. The arithmetic is a fraction times 360 degrees. About twenty minutes.
Predict first
Detune the RF frequency by 0.5% from the peak you found in the first exercise. Does each ion now gain 0.5% less energy per push, or something else? And after how many turns is a push no longer helping? Most people expect "slightly wrong, slightly worse." Write both answers down before you open the laboratory.
Rough arithmetic for the second question, with the RF on its fundamental. An RF cycle that runs 0.5% fast gains on the circling ion by 0.5% of a cycle every revolution: 360° × 0.005 = 1.8° per turn. An ion that started on the crest reaches the gap as the voltage crosses zero, 90° away, after about 50 turns; beyond that, each crossing takes energy away. The pushes shrink long before they stop, though: each is reduced by the cosine of the accumulated slip, so by the fiftieth turn they have long been dwindling.
Now count the turns. At the laboratory's sensitive-machine setting (2 kV on the dee), a singly charged ion receiving the full displayed voltage at both crossings of every turn would gain 4 keV a turn and need roughly forty turns to reach the laboratory's roughly 150 keV. Detuned, each of those forty pushes is smaller than the last, and their sum leaves the ion well short of that energy by the time its phase reaches the zero crossing. An ion that started a little off the crest on the side the slip is heading for runs out sooner; one that started on the other side is carried through the crest first and does better. The detune selects as well as slows.
The shape to expect
Open the laboratory
The link opens the laboratory at its sensitive 2 kV setting, already detuned by half a percent from that setting's own resonance peak. The controls in play are RF frequency and path inspection: click any drawn path, or use "Next path", and the inspection panel shows that ion's phase at each gap crossing and its energy after each crossing. In the Full workspace the same panel sits beside the RF-acceptance view.
Open the Virtual Cyclotron Laboratory (hosted; illustrative, not a calibrated replica)
What you should see
The phase dots walk. Crossing after crossing — and the panel shows every one — they move steadily in one direction instead of staying near the crest; whether they walk early or late depends on which side of the peak the frequency sits. The energy staircase climbs with steps of nearly full height at first, then the steps shrink, then they reverse: the ion arrives when the gap voltage opposes it. Meanwhile the collected current has fallen by far more than half a percent. The beam is lost by losing step, not by losing strength.
If you see something different
How far the staircase climbs before it turns over depends on the detune and on the dee voltage: more volts per turn means fewer turns to the collector and less slip accumulated on the way, so at a 5 kV setting an ion may reach the collector before the staircase bends. Use the lower-voltage setting, or a larger detune, and the bend returns. Every reading comes from an ensemble of simulated ions, so the collected current scatters by a few percent from run to run; the scatter is the ensemble's own. The laboratory is an illustrative machine shaped like the reference machine, not a calibrated replica of any real machine.
The physics, cited
The accounting is per turn, summed. A lecture on cyclotron limits writes it for the relativistic case, "dphi/dn=360 [gamma-1] -> 8 deg. An ion on peak phase is lost in 11 revolutions" (dg-273): a few degrees of slip a turn, and a budget of 90° spent in a countable number of revolutions. The cause there is relativity, here a detuned oscillator, and the bookkeeping is the same; the rule's note states the design check as the summed slip across all turns, checked against the ±90° window.
The budget itself: "this error is acceptable, as long as the overall integrated phase slippage is less than 90°" (dg-1809). The budget is a boundary, where a push stops adding energy; inside it the ion still gains, but by less each turn, and whether it reaches the collector at the energy wanted depends on the sum of those shrinking pushes. The same rule records that raising the dee voltage reduces the number of turns and so the accumulated slippage.
Fewer turns, less slip: "higher potential on the dees results in fewer orbits and a shorter time of acceleration, allowing for a higher maximum kinetic energy" (dg-309). That is why a lower-voltage setting, with more turns to the collector, is the more sensitive to a detune.
Background: staying in step in How a Cyclotron Works.
Closing question
In one sentence: what did you change, what happened, and why? There is no form; the sentence is for you.