Guided exercises · 6 of 6
Weigh the Ion
Goal. The machine is a measuring instrument. The resonance frequency and the field together give the particle's charge-to-mass ratio, and with the charge known that is a weighing.
Who it is for. Readers who have done Find the resonance; this exercise repeats its sweep on purpose, as a measurement. About twenty minutes.
Predict first
Before sweeping, compute the mass the laboratory should report from its displayed field and the frequency you expect the peak at. Then: will the sweep's peak return exactly the proton mass? If not, by how much, and which way? Most people expect either exact agreement or a random error of no particular sign. Write your number and your expectation down.
The first exercise's formula, solved for the mass. It holds for a singly charged ion well below the speed of light, with the RF on its fundamental, and with B the field the orbit actually averages:
m = qB / 2πf.
The laboratory's reference setting shows 0.596 T. A proton (q = 1.602 × 10⁻¹⁹ C, m = 1.673 × 10⁻²⁷ kg) in exactly that field circles at 9.08 MHz, so a peak at exactly 9.08 MHz returns the proton mass. The sensitivity is one to one: a peak 0.5% above that frequency returns a mass 0.5% below the proton's. Read your peak off the sweep to the resolution the sweep allows, and compute. The displayed field is a setting at the centre; whether it is the field the orbit averages is the question the exercise ends on.
The shape to expect
Open the laboratory
The link opens the laboratory at its working reference with the RF frequency sweep already set up, as in the first exercise; it runs once the opening beam has appeared. The controls in play are the sweep and the field readout. Read the peak frequency off collected current against RF frequency (the Full workspace exports the sweep as a CSV). Optional second point: set a different field, re-centre the sweep on the frequency the formula predicts for it, and sweep again. The laboratory labels settings outside its verified domain, and the second field is one: treat that point as an illustration of the method only.
Open the Virtual Cyclotron Laboratory (hosted; illustrative, not a calibrated replica)
What you should see
A peak near 9.08 MHz, to the sweep's resolution, and from it a mass close to the proton's. Nothing on the screen looks like a particle being weighed — a frequency was read, a field was read, and the mass followed. With the second point, the two peaks should fall near one line through the origin: two points test the proportionality that one can only assume. Here the method weighs a simulated ion whose mass the laboratory was given; on a real machine it applies with more to pin down first, the charge state, the harmonic, and the field the orbit averages, which takes a field map.
If you see something different
On the laboratory's machine the computed mass comes out slightly low, by a fraction of a percent, with the same sign each run. That disagreement is real physics, not an error. The resonance frequency reports the field averaged over the orbit, centre to collector, while the laboratory displays the field at the centre; on this machine's field profile the orbit averages slightly more than the displayed value, and the computed mass comes out low by the same fraction. Reverse the reasoning and you have a magnetometer: with the mass known, the peak measures the average field. Other things move a peak a little too (the sweep's grid, the ensemble, which starting phases survive), so repeat the sweep before trusting the sign; ensemble scatter moves the peak's height more than its position. The laboratory is an illustrative machine shaped like the reference machine, not a calibrated replica of any real machine; the size of the fraction is this machine's field shape, and a real magnet has its own.
The physics, cited
Livingston and Blewett describe the running cyclotron as its own magnetometer: "the magnetic field can be determined with high precision … this resonance frequency represents an average value of the magnetic field from the center out to the exit radius … an error of the order of 0.5 per cent is possible" if that average is assigned to one radius (dg-019). The exercise runs the sentence backwards, field given and mass unknown, and the average is what makes the two disagree.
How a real machine knows its field independently of its beam: proton nuclear magnetic resonance. "The frequency for resonance can be measured and reduced to magnetic field through the relation B = (234.82 +/- 0.13)f where B is in gauss and f is in megacycles per second" (dg-511). A related resonance, a proton's spin precessing in the field where the probe sits, calibrates the magnet point by point; the beam's resonance is orbital, and averages over the whole orbit.
Background: measuring the real field in the magnet-design page, and energy: one formula and several cross-checks in beam measurement.
Closing question
In one sentence: what did you change, what happened, and why? There is no form; the sentence is for you. This is the last exercise in the series; the series index lists the other five.