Guided exercises · 1 of 6

Find the Resonance

Goal. Sustained acceleration happens only when the RF frequency matches the magnetic field, and the matching frequency depends on the field and the ion, not on how fast the ions already are.

Who it is for. Anyone who has read How a Cyclotron Works and wants to run a machine rather than design one. One formula, no calculus. About fifteen minutes.

Predict first

The machine takes ions from slow, on their first turns near the centre, to fast, on their last turns near the collector. Will the best RF frequency be different for the slow first turns than for the fast last ones? Most people expect that a faster particle needs a faster clock. Write your answer down, together with the frequency you expect the beam to peak at, before you open the laboratory.

The frequency has a formula, and the laboratory's displayed settings give it a number. An ion of charge q and mass m in a uniform field B, at speeds well below light and with the RF on its fundamental, circles at

f = qB / 2πm.

The laboratory's reference setting shows a field of 0.596 T. For a proton (q = 1.602 × 10⁻¹⁹ C, m = 1.673 × 10⁻²⁷ kg):

f = (1.602 × 10⁻¹⁹ × 0.596) / (2π × 1.673 × 10⁻²⁷) = 9.08 × 10⁶ Hz, about 9.1 MHz.

Nothing in that line is a speed, a radius or an energy. That absence is the prediction to test.

The shape to expect

The expected shape, sketched: collected current against RF frequency, near zero almost everywhere and peaked at the frequency set by field and charge-to-mass ratio (dg-1436). The rule fixes where the peak sits; its width and the off-peak floor are the laboratory's own. The width exists because a small mismatch is tolerated for a while (dg-1809).

Open the laboratory

The link opens the laboratory at its working reference with the RF frequency sweep already set up, a few percent either side of the reference frequency; the sweep runs once the opening beam has appeared. The one control in play is RF frequency. Leave the field and the dee voltage where they are. The plot to watch is collected current against RF frequency.

Open the Virtual Cyclotron Laboratory (hosted; illustrative, not a calibrated replica)

What you should see

Over most of the sweep the collector reads close to nothing. At one frequency the current rises to a sharp peak, and that frequency sits near the number worked above. The ions in that peak were kept in step by one frequency from their first slow turn to their last fast one; none needed a faster clock as it sped up. Pull the frequency a few percent either way and the collected current falls away again. Those ions have not vanished: the laboratory counts them as lost inside the chamber, and on a real machine they land on the dees and walls.

If you see something different

The peak may sit a fraction of a percent from the worked number, and its height will differ by a few percent from one run to the next: every reading is computed from an ensemble of simulated ions, so runs scatter as repeated measurements do; both figures belong to the simulation alone. The laboratory is an illustrative machine shaped like the reference machine, not a calibrated replica of any real machine; its 0.596 T and its peak frequency are the settings of a simulated machine. A peak that sits consistently a little away from the formula is itself the subject of the last exercise, Weigh the ion.

The physics, cited

The resonance condition in the form a cyclotron textbook writes it: "ω_HF = k · ω_Zyk = k · v/r = k · (q/m) B mit k = 1; 3; 5; …", the RF angular frequency equal to an odd multiple of the cyclotron frequency (dg-1436). The ratio v/r collapses to (q/m)B: speed and radius grow together and cancel — which is the whole reason a cyclotron works. The same rule names the condition's limit, relativity: it notes that at a tenth of the speed of light the frequency is already about 0.5% low. For a proton that speed means several MeV, far above the laboratory's roughly 150 keV, where the shift is a few hundredths of a percent.

The peak has width because the match need not be exact. A weak-focusing cyclotron meets the cyclotron condition at only one point in the ion's flight from source to collector, and the source's test is that "this error is acceptable, as long as the overall integrated phase slippage is less than 90°" (dg-1809). A small mismatch is tolerated until its accumulated effect spends that budget, which is the next exercise.

Background: the resonance condition 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.