Guided exercises · 5 of 6
What the Leftover Gas Costs
Goal. Vacuum is a beam-transport requirement with an exponential price. Each ion either survives the whole spiral or is lost from the beam, and the price depends on how long the spiral is.
Who it is for. Readers of How a Cyclotron Works. One exponential, one logarithm. About fifteen minutes.
Predict first
The sweep starts at a pressure where nearly all the beam survives. At one hundred times that pressure, how much of the beam is lost: about 1%, about a quarter, or almost all of it? A common answer is the middle one, or the reasoning that a hundred times a small loss is still a smallish loss. Write your answer down before you open the laboratory.
The method matters more than any number. An ion crossing a path of length L through gas of number density n, with a cross-section σ for being lost, survives with probability
S = exp(−nσL).
The exponent is proportional to pressure, because n is, so inside this model one reading fixes the whole curve. In the Full workspace the accounting panel reports the share of the beam lost to gas at the current setting; at the sweep's start that share is small. Call the surviving fraction there S₀. At ten times the pressure the exponent is ten times larger and the survival is S₀¹⁰; at a hundred times, S₀¹⁰⁰. If S₀ were 0.95, those would be 0.60 and 0.006: a twentieth gone at the start, four tenths at ten times, nearly all at a hundred. Work the same lines from the fraction the laboratory shows you.
The shape to expect
Open the laboratory
The link opens the laboratory at the sweep's starting pressure with the pressure sweep already set up, logarithmically spaced from a pressure where nearly all the beam survives to one where almost none does; it runs once the opening beam has appeared. The one control in play is the hydrogen pressure (under "More controls"). Watch collected current against pressure and, in the Full workspace, the accounting panel's gas line.
Open the Virtual Cyclotron Laboratory (hosted; illustrative, not a calibrated replica)
What you should see
Across the first decade of pressure the current barely moves. Then the fall steepens — and by the top of the sweep little of the beam remains. On the logarithmic axis the curve is flat where the exponent is small and drops away once it is not; against pressure itself, each added increment of pressure removes the same fraction of whatever beam remains, so the curve bends toward zero without reaching it. The drawn paths do not change: the laboratory attenuates the statistical weight of each simulated ion along its path, standing in for the loss of some ions by charge exchange (a real ion is lost whole or not at all), so the same paths carry less and less of the beam and the loss appears in the accounting as gas attenuation.
If you see something different
Where the bend falls depends on the length of the path. At the laboratory's reference setting, 5 kV on the dee, the ideal turn count worked in More voltage, same energy is about sixteen, a spiral of a few metres, so the exponent stays small at pressures that would cost a lower-voltage, many-turn machine most of its beam (dg-1398). Nothing else that high pressure does to a real machine, a dee voltage breaking down, a source misbehaving, is modelled. Every reading comes from an ensemble of simulated ions and scatters by a few percent from run to run, a figure that belongs to the simulated ensemble alone. The laboratory is an illustrative machine shaped like the reference machine, not a calibrated replica of any real machine.
The physics, cited
The exponential form comes from a loss rate per metre. A compilation of cross-sections for hydrogen ions in hydrogen defines the "cross section for 'destruction' or loss of ions or fast neutrals in m2"; times the gas density it is a loss rate per metre, and integrated along the orbit it gives the surviving fraction exp(−n Σ Qd(Ek) Lk) over the turns (dg-1909). The rule's note adds that the exponential form matters once losses exceed about 10%, where a linear estimate overstates the loss.
A teaching-cyclotron textbook states its vacuum criterion as a path-length condition: "the mean free path must be >= the total path length" of the spiral, so lower dee voltage, meaning more turns, demands lower pressure (dg-1398). Its worked trade: "16 accelerations need 2.76 m of path, barely reached with 3 m" (dg-1399). The criterion is weak, and the rule says so: a mean free path equal to the path means about 37% survival, so a real specification starts from an allowable loss.
Gas costs twice. A design review gives the vacuum a second job, in its own reasoning: "to ensure the stability of operation of electrical components of the accelerator… by increasing the breakdown voltage" (dg-577); how voltage holding behaves with pressure is the vacuum page's subject. The laboratory shows only the first cost.
Background: the pressure target comes from the beam in the vacuum page, and the vacuum and beam-survival calculator.
Closing question
In one sentence: what did you change, what happened, and why? There is no form; the sentence is for you.