Making Ions, and the First Few Turns

One optional bench test gets closer, where the dee can be biased: put DC on it, no RF, and collect the ions. It shows that ions can be drawn out of the source; it does not say which ions, or rule out leakage and other currents on the meter, and the documented first beams were found without it. Leakage is current reaching the meter through imperfect insulation or the supply rather than through ions.

For a first beam: known, controllable emission, set on the meter rather than by the glow; the gas step, the rise in emission when gas is admitted, seen on the emission meter; and a source geometry that delivers ions into the accelerating region, which a glow cannot tell you (Where exactly should the filament or the source opening go? Is "at the center" good enough, and how do the first orbits avoid hitting the source?). Refining the chimney and slit, and optimizing the bias, can wait.

If it does not work. Read the gas step carefully. Rising emission when gas arrives is consistent with the electrons finding something to ionize along their path; adding gas can change the discharge and the current collection in other ways too. To ionize a molecule is to knock an electron off it, leaving it charged; an ion is any charged atom or molecule. It does not say the ions are born where the dee’s RF electric field can pull them out, or how many reach the gap. With gas present the electrons sustain a discharge in it, an arc; the arc voltage is the bias supply’s voltage across it, and the arc current is the emission current through the gas. A creeping arc voltage at constant current is, in the hot-cathode sources the handbook describes, read as gas running short rather than as a filament dying (Wolf (ed.), p. 83); it is the first thing to check, not a verdict. The closest test is the one the Rutgers group used to characterize its source: DC-bias the dee negative and collect ion current across pressures and arc currents, with the RF off (Koeth (2013), p. 2). It measures collected current; reading it as ions still means checking that the reading responds to gas and emission and not to leakage or the bias supply, and it says nothing about which ion species is present. Species means which kind of ion the source is delivering.

Two knobs get blamed too early. Bias sets the electrons’ energy, the voltage each one falls through after leaving the wire; on the Houghton machine it made no measurable difference to beam over the range tested, around −90 V, while pressure and dee voltage did (Fuller (2013), p. 48-49); a later change on the same machine that cut the bias to −6 V along with the pressure and the frequency did raise the beam, several variables at once (Yuly et al. (2010), p. 17-18; What pressure should I aim for when the cyclotron is running? Don't I need more gas to make ions, but less gas to let them spiral around?), so the null holds for the range tested, not for bias in general. Tune it, but not first. Heater current is the steepest knob and the hardest on the filament: tungsten evaporates fast at the top of its range (the handbook tables give a 1 mm wire thousands of hours at 2, 500 K and tens of hours at 2, 900 K, Wolf (ed.), p. 41-42), and extra heater power warms the chamber and spoils the vacuum (Prechtl & Wolf (2020)). Where the trade between emission and filament life sits depends on the wire’s diameter, shape and material and on the source geometry. Choose the working point from the emission meter and the filament’s life. One symptom to recognize: above about 5 × 10⁻⁵ torr with more than 1 mA of emission, the 9-inch showed a dramatic cathode ray, a visible stream of electrons, running from the filament down the magnet’s field lines to the chamber floor (Koeth, p. 6). The ray is a discharge.

To understand it. Thermionic emission, the escape of electrons from a hot metal surface, rises steeply with temperature. For a tungsten filament near 2,500 K, Richardson’s law gives roughly an order of magnitude more emission for a 10 % rise in temperature (arithmetic from the law with tungsten’s work function); the sensitivity grows at lower temperatures, and the ion-source handbook quotes ten- to a hundred-fold for the cathodes it covers (Wolf (ed.), p. 38-39). On the Rutgers 9-inch a test filament at −300 V of bias gave essentially no emission below 4.5 A of heater current and about 3 mA of emission at 5.0 A (Koeth, p. 6). The numbers belong to that filament; the knee, the sharp turn-on, is general. The Mullins bench test found the same shape: emission turning on sharply between 1.8 and 2.0 A of heater current at 101 V of bias, and the machine has run just above that knee since (hosted rf-dee-source note, §6.3; /library/amateur/#mullins). The Rutgers 12-inch sequence, for its filament and bias supply, ramps the heater current until emission reaches about 10 mA at −200 V of bias, then admits hydrogen slowly until emission rises (Koeth (2006), p. 5).

Some amateur machines managed that with little more than a bare filament; others use a housing with a small opening toward the gap. Every documented amateur machine started there, most of them with something cruder than the drawings in the books. The Mullins button rests on an insulator sheet on the chamber floor, just inside the dee’s straight edge, in a chamber whose interior is only an inch high, so the hairpin stands within that inch of the midplane, with “no source housing, chimney, puller or extraction geometry” (hosted rf-dee-source note, §6). Houghton’s is an electron-microscope filament mounted between the dee and the dummy dee, the grounded electrode that faces a single dee across the gap (census). The Rutgers 9-inch mounted its source on the dummy dee, which doubled as the central mounting surface (Koeth, p. 2). Niell’s second machine made ions with a nichrome heater coil wound on a glass pipette (census). All of these found beam. In each, the ions were born near the machine’s axis, near the midplane (the plane halfway between the pole faces), and within reach of the dee’s electric field.

The ion returns on something close to a circle whose size depends on how hard it was pulled. The radius grows as the square root of the voltage and shrinks as the magnetic field rises (Kamiguchi (2019), p. 1, the scaling). COLUMBUS computed that its protons clear the source from 200 V of dee amplitude and its H₂⁺, a hydrogen molecule missing one electron, from 400 V, against a 20 mm chimney region, the housing tube around the filament (Prechtl & Wolf (2020), p. 42-43). The heavier ion needs twice the voltage because, at the same RF frequency, an ion of twice the mass and the same charge needs twice the dee voltage to make a first circle of the same size (Prechtl & Wolf (2020), p. 41-42). The Rutgers 12-inch calculated 165 W of RF power to clear its chimney (the dee voltage rises with the RF power fed in) and watched the beam drop abruptly to zero at 170 W as the power was reduced (Koeth (2006), p. 4). Treat the first-orbit radius as a way to judge how much room you have, not as a dimension the source must fit inside. The source’s shape, position and direction, the dee voltage and the ions’ starting directions all enter, and the first orbit is not a circle centred exactly on the source.

If it does not work. Beam that fades and then stops abruptly as the dee voltage is reduced is consistent with the first turn striking the source structure (Koeth (2006), p. 4). It is not proof. Less voltage means a smaller first circle, and below some voltage that circle no longer clears the housing. A weak RF system also ends beam abruptly when it drops out (stops oscillating), so check that the dee is still driven before blaming the geometry. Beam that peaks with the source moved off the geometric center is normal, not a fault. On the documented machines whose sources could be moved, the position was settled by watching the beam. A fixed, roughly centred source has also worked; the Mullins filament was never moved on purpose (hosted rf-dee-source note, §6.2). Ohio State slid its filament inside the housing from outside the vacuum while the beam was monitored at the target (Hausman (1954), p. 6). The 63-inch found source alignment “extremely critical” and added external adjustments (Howard (ed.), p. 10). COLUMBUS, with ten turns or fewer, deliberately made its source movable toward the gap, because with so few turns the best starting point need not be the centred one (Frank (2013), p. 1). If your source is fixed, the slower equivalent is a series of pump-down cycles with the source shifted a millimetre or two each time. Beam that is lost in the first few turns up or down is the vertical version. Near the center the magnet does not hold the beam vertically at all. The field index, the measure of how fast the magnetic field weakens with radius and so of the vertical holding it gives, is zero there by symmetry, and only the electric field of the gap does the work, as a lens (Humphries (1986)). So a source sitting a millimetre above or below the midplane, or a dee sitting a millimetre off it, launches ions into a part of the gap’s electric field that points up or down rather than across. They strike the dee lips, the edges of the dee along the gap, within a few turns. A millimetre of vertical source offset costs a large fraction of the collected current while the emission readout, the meter showing the filament’s electron current, does not move (guided exercise “Where the ions are born”). Burn marks and heat-tint patterns on the dee lips and chamber floor tell you where ions are going. The 86-inch read a three-inch orbit-center offset from where its beam struck a dee (Howard (ed.), p. 7). Look before moving anything, because ions lost for arriving out of step with the RF (phase) and ions clipped by a narrow opening (aperture) produce similar marks.

The three “centers” matter only when they disagree. The mechanical center is the axis of the poles and the chamber. The magnetic center and midplane are where the magnetic field is actually symmetric, which can sit off the geometry. MIT found its magnetic midplane half an inch below the geometric one (Livingston & Blewett (1962), p. 197), and the 9-inch adjusted its chamber’s plane to the magnetic one (Koeth, p. 2). The orbit center is where the circles the ions actually trace are centered. It is set by where the ions start and how they are kicked on the first crossings. A radial offset of the source simply displaces the orbit by the same amount at first (Koeth (2013), p. 2). Uneven dee voltage along the gap also shifts the orbit center, and on large machines that has needed source displacements of more than two inches from the geometric center to find the best beam (Livingston & Blewett (1962), p. 164). For a first beam, put the source at the mechanical center and let the beam tell you where the other two are.

To understand it. In the central region timing and geometry are entangled, and the amateur machines did the least engineering there. Ions are accepted only from part of the RF cycle, and the gap’s electric focusing works only for part of it (Livingston & Blewett (1962), p. 166-171; Smirnov (2021), p. 56; My ion source is on all the time, but the RF keeps reversing. How do the ions start at the right moment? Do I have to synchronize the source with the RF?). One modeled Rutgers geometry captured best from a point 8 mm above the midplane, because, as the report reads its model, an unintended vertical field component there pulled ions down (Koeth (2007), p. 3). The same group measured a 0.32 mm vertical offset of its dee-mounted puller plates, the electrodes that draw ions out of the source, and found that enough to dominate where the beam went (Koeth (2007), p. 5). Those numbers belong to that machine; the lesson that transfers, and that group’s own conclusion, is that adjustability beats machining in the central region. The ion-sources chapter’s section on position covers the hooded sources, the ones with a housing and an exit slit (puller and slit geometry; the breakdown limits on the gap; and why extraction at these currents is a geometry problem; What is "space charge", and when would it matter in a small cyclotron? Is it a problem in the source, during acceleration, or both?), and the beam-dynamics chapter covers what an off-center orbit does later in the spiral (Koeth (2013), p. 2).

The rest of what the source makes is never a candidate. The ion-sources chapter gives the window in degrees. The accepted ions then travel as a train of short bunches, one bunch per RF cycle, since the ions let in during one window travel together. The Rutgers 12-inch measured its beam occupying about 40 degrees of each 360-degree cycle (Koeth (2015), p. 9), so the current an electrometer averages (the collector’s current meter) is a small fraction of the peak.

You control how long each accepted ion stays in step. The magnet, the RF frequency and the ion’s mass set a resonance, the condition in which the ion comes back to the gap in step with the RF every turn; nothing holds the ion at it. If the frequency is a little off, the ion arrives at each successive gap crossing a little earlier or later in the cycle. The error accumulates turn after turn until the electric field it meets at the gap is opposing it. The budget is about a quarter of a cycle of accumulated slip, 90 degrees, before an ion stops gaining energy (Gonski (2013), p. 1; the how-it-works chapter). A half-percent frequency error slips the phase, the point in the RF cycle at which the ion reaches the gap, by 1.8 degrees a turn and spends the budget in about fifty turns (guided exercise “Watch an ion fall out of step”). The beam peak is therefore narrow in field or frequency, and a detuned beam, one with the magnetic field or the frequency moved off the matching value, is lost by falling out of step. A higher dee voltage does not shrink the error added on each turn; for a given mismatch, every turn adds the same slip. A higher voltage does give the ion more energy per crossing. It cuts the number of turns needed to reach the collector, so less error has accumulated by the time the ion gets there (Livingston & Blewett (1962), p. 172; Loucks (2007), p. 24-26; How many turns should a proton make before it reaches my target energy? Fewer turns sounds easier, but isn't the whole point of a cyclotron to accelerate over lots of turns?).

If it does not work. Three things to look at, in order of cost. First, the resonance itself. A missing or narrow peak of collector current is the ordinary sign that magnetic field and RF frequency do not match for the ion you expect. The magnet sweep with the RF fixed, stepping the magnet current while watching the collector, is the standard way to find it (Do I really need to map my magnet's field accurately? Is one reading in the middle enough, and which value goes into the frequency and energy calculations?). A detuned machine loses most of its beam. Second, the filament bias. The bias is the negative voltage the filament is held at, which drives its electrons out into the gas. On the Mullins machine, changing the bias also moved the magnetic field at which the proton resonance peaked, by a few percent. The note records that no model of the effect was made (hosted rf-dee-source note, §6.5). Houghton found no measurable effect of bias on beam over the range it tested (Fuller (2013), p. 48-49). Both are observations about one machine each, and neither establishes a mechanism. Re-find the peak after you move the bias. Third, the dee voltage. If the beam appears near the center and dies farther out, the ions may be running out of phase budget before they reach the collector. More volts per turn is the literature’s first remedy (Livingston & Blewett (1962), p. 172; I can detect acceleration near the centre, but not farther out. Is that a partly working cyclotron? How do I find where the particles are being lost?). One caution from the 86-inch: the surviving beam self-selects its phase, so a detuned machine can hide part of the detuning from you. A field error that should have shifted the phase by 45 degrees shifted the measured phase by 10. The ions at the old phase were lost and ions of a more favorable phase took over (Cohen (1952), p. 9). Current on the collector does not by itself say that the machine is tuned where you think it is.

To understand it. A classical fixed-frequency cyclotron has no phase stability in the synchrotron sense. There is no restoring force that gathers early and late ions back toward a synchronous phase, a common arrival point in the cycle. An ion that drifts out of the window is lost. The ±90° figure is a budget for the whole trip, not a boundary the ion can bounce off (Gonski (2013), p. 1; the how-it-works chapter calls it the machine’s phase budget). Books and lectures written for synchrotrons describe phase focusing, bunch oscillations and synchrotron frequency. None of that applies to the machine on your bench. The classical machine has instead a phase history. The magnetic field falls with radius for vertical focusing. A weaker field turns an ion more slowly, so the revolution frequency falls with radius, and the ion’s phase first drifts one way and then the other. The standard trick is to set the RF slightly below the frequency the central field implies. Near the center the ion then turns faster than the RF; farther out, in the weaker field, slower. The slip then first grows and then unwinds, and the whole population’s histories stay inside the window. The Rutgers group describes the same trick as starting in a field that is too high (Gonski (2013), p. 1); the beam-quality chapter’s energy-spread section gives the worked case. The window is lopsided, about 90 degrees of RF phase on one side of the crest, the moment the gap voltage peaks, and 20 on the other. One reason lies in the first turns: the electric field of the gap focuses vertically only for ions crossing while the voltage is falling, and the central region has no magnetic focusing to spare them (Livingston & Blewett (1962), p. 166-171, Humphries (1986)). The window itself, in Smirnov’s review, runs from about 90 degrees before the voltage crest to 20 degrees after it for an internal source and cosine RF (Smirnov (2021), p. 56). The chapters cover the 1961 Iowa State phase-window calculation that predicts an intensity-versus-field curve (Mueller (1961), p. 5-8), phase slits that trade most of the current for a few degrees of accepted phase (Howard (1958), p. 27; beam-quality chapter), and the energy spread of a multi-turn beam, of order the energy gained per turn (Baartman (2023), p. 10).

Momentum is charge times magnetic field times radius, and the dee voltage does not appear in that relation (Prechtl & Wolf (2020), p. 31); the voltage decides how many pushes it takes to get there. The general form of the turn count, with the phase factor, is in dg-272.

Fewer turns are generally an advantage, all else being equal, and the “all else” is real: more dee voltage costs RF power, invites sparking across the gap, and does nothing for a wrong frequency or an unsuitable magnetic field shape (Livingston & Blewett (1962), p. 172). Fewer turns means a shorter path through the residual gas, so fewer collisions with gas molecules, which can remove ions from the beam, and the beam survives the trip in a worse vacuum. COLUMBUS’s design criterion is stated as a path-length condition, the mean free path at least the total spiral length, so lower dee voltage demands lower pressure (Prechtl & Wolf (2020), p. 54). The mean free path is the average distance an ion travels between collisions. Houghton measured beam current rising with dee voltage at fixed field and pressure (Fuller (2013), p. 49-50). Fewer turns means less accumulated phase slip against the same 90-degree budget, so the resonance is wider and the magnetic field profile has to be less perfect (Gonski (2013), p. 1; Livingston & Blewett (1962), p. 172; My ion source is on all the time, but the RF keeps reversing. How do the ions start at the right moment? Do I have to synchronize the source with the RF?). Fewer turns also means less time in the central region, where the losses are. The 86-inch lost “some 90 % of the initial supply of ions” to the dee surfaces near the center and cut that by raising the dee voltage (Livingston & Boch (1952), p. 18, one machine’s figure). The amateur machines ran low voltages and took the turns that followed, and the documented range is wide. COLUMBUS’s 2013 design table set 6 to 8 revolutions at 2 to 3 kV between the dees, a pre-beam figure; its later papers ran at lower voltages toward lower energies (Wolf (2013), p. 1, Wolf & Prechtl (2022), p. 2). The Rutgers 12-inch runs at about 8 kV peak (Ruisard (2013), p. 2). The Mullins machine’s dee is “~800 V nominal, uncalibrated”, so its computed turn counts “inherit this uncertainty in full” (hosted performance note, §6). The path-length calculator with that voltage, two ideal crossings and the machine’s field and collector radius returns a count of order a hundred. The note says that no such number is a measurement. Real pushes are smaller than the peak voltage, because most ions cross the gap away from the voltage crest, so real machines take more turns than the ideal count (the calculator’s assumptions list).

If it does not work. Beam near the center that fades before the collector radius says the ions are not surviving the trip. Three causes scale with the number of turns: gas scattering, accumulated phase slip, and the central-region losses. Which to look at first depends on what you have already measured and what you can change reliably; there is no fixed order. Pressure is quick to change, and the Houghton pressure window shows what a scan looks like (ion-sources chapter, gas-feed section). A frequency a fraction of a percent off spends the phase budget in tens of turns, and the magnet sweep tells you where the peak actually sits (My ion source is on all the time, but the RF keeps reversing. How do the ions start at the right moment? Do I have to synchronize the source with the RF?). More dee voltage shortens the path and the accumulated slip and, by widening the first orbit, can clear some central-region hardware. It does not correct a field shape or remove every central loss, and it costs RF power and sparking (Livingston & Blewett (1962), p. 172, Morrow (2015), p. 27-28). Livingston and Blewett observe that most machines settle for a slightly smaller radius and energy to keep intensity (Livingston & Blewett (1962), p. 172). Turn structure you cannot see on a radial scan, a probe stepped inward through the beam, is normal. Successive orbits lie close together, closer as the radius grows, and the measured profile of the inner turns on COLUMBUS was nearly continuous rather than a comb of peaks (Wolf (2016), p. 3). A collector whose shield is thicker than the turn spacing intercepts the beam before it reaches the sensitive surface. Turn spacing is the radial step between one turn and the next; the beam can step over a shield edge thinner than that and lands on a thicker one. The Rutgers 12-inch found this and replaced a shielded cup with a bare aluminum block, moving the RF rejection, the filtering that keeps RF pickup out of the current reading, outside the vacuum (Koeth (2006), p. 4). The path-length calculator reports the turn spacing at the collector radius; compare it with whatever projects in front of your collector.

To understand it. Because energy grows linearly with turns and radius grows as the square root of energy, the orbits crowd together toward the outside. The spacing per turn near the final radius is the radius divided by twice the turn count (the calculator’s derivation; Antaya (2010), p. 43). Extraction is therefore hard on a many-turn machine, and extraction programs chase dee voltage: a septum, the thin partition at the mouth of the extraction channel, has to fit between the last two turns (Livingston & Blewett (1962), p. 181-183; Antaya (2010), p. 43; the beam-extraction chapter). For the same reason a multi-turn beam’s energy spread is of order the per-turn gain, since a collector at fixed radius accepts ions from several overlapping turns (Baartman (2023), p. 10; beam-quality chapter). Relativity adds a per-turn slip that grows with energy and caps a fixed-frequency machine. By relativity the ion’s mass rises with its speed, so it turns a little more slowly each turn. At amateur energies the per-turn effect is tiny, but it still accumulates with turn count, the scoped statement already in lit-which-rules-apply (Antaya (2010), p. 51, Humphries (1986), p. 530-531). With one set of pole tips the Rutgers group met the opposite case: a field profile that produced too much phase slip at the normal 8 kV never carried ions to the deflector, the extraction electrode at the outer radius, and simulation said 20 kV would have (Ruisard (2013), p. 2, Koeth (2011), p. 9). The RF system was the limit.

Whether it matters at all depends on how many charges are packed into how small a volume, how fast they move, how tightly they are bunched and how strongly they are focused. The Rutgers 12-inch after its source upgrade reached tens of microamperes (census). The current at the collector is far smaller than the current in the source or in the first turns, and any estimate belongs to the bunch rather than the average (both below). The beam-quality chapter’s statement is the same: at amateur currents, beam quality is set by geometry and fields, not by current.

Inside the ion source many more charged particles are close together, and their electric fields affect how ions leave the source, how much usable current comes out, and where it goes. Once a discharge runs in the source gas, the gas is a plasma, a mixture of free electrons and positive ions. The bulk of a discharge plasma is close to electrically neutral, electrons and positive ions together. The strong fields and the charge separation lie in thin layers, the sheaths, next to the electrodes and at the opening where ions are drawn out. In vacuum, an arc source’s filament is throttled by its own electron cloud (the electrons already emitted repel the ones behind them) to milliamperes per square centimetre at a modest voltage. With the gas discharge struck, positive ions reaching the filament cancel that electron space charge and amperes flow, in the calutron arithmetic (Yonts (1954), p. 11). At the exit opening, the shape of the plasma boundary and the pulling electric field together decide how much current can leave through a given hole. The idealized ceiling is the Child–Langmuir law: the current density that can be drawn across a gap goes as the voltage to the three-halves power over the gap squared. For protons at 10 kV across 5 mm that is about 220 mA per square centimetre, thousands of times a small machine’s whole beam (Wolf (ed.), p. 376-377; the ion-sources chapter works the formula). It is a limit, not a model of your source. Real extraction also depends on the plasma density and on the shape its boundary takes, so the handbook pairs the formula with the instruction to match plasma density to the pulling field by tuning the arc (Wolf (ed.), p. 379). The Rutgers 12-inch used the law as a scale estimate when puller plates on the dee face raised the electric field at the plasma edge by a large factor: the law predicted a 130-fold increase in available current, and about two orders of magnitude were measured (Koeth (2007), p. 4). Space charge acts in the source and at its mouth routinely, which is what the slit and puller rules address. In the spiral it is unlikely to be the limit at the currents a first machine collects, the expectation stated above rather than a proof.

Do not read your source’s condition from your collector. The current that leaves a working source is three to five orders of magnitude (a thousand to a hundred thousand times) above what a small cyclotron captures (the ion-sources chapter, citing Forringer). The losses between them are timing, position and aperture, not repulsion (My ion source is on all the time, but the RF keeps reversing. How do the ions start at the right moment? Do I have to synchronize the source with the RF?, Where exactly should the filament or the source opening go? Is "at the center" good enough, and how do the first orbits avoid hitting the source?). A weak reading at the collector therefore says almost nothing about how dense the plasma is or whether space charge is active at the slit. The hosted Mullins note shows the simple diode picture (a hot emitter and a collector, with the current between them capped by its own space charge, the three-halves law above) failing at the bench: a filament-to-collector test the builder tabulated against the three-halves law rose as roughly the 3.6 power of the bias instead, so that collection was not a simple space-charge-limited diode (hosted rf-dee-source note, §6.3).

If it does not work. Space charge is rarely the first suspect for a missing or weak first beam. Exclude the ordinary causes: the source is making ions (My filament is glowing. How do I know it's actually making ions? And what are the filament current, the bias voltage and the emission current each telling me?); they start in the right place (Where exactly should the filament or the source opening go? Is "at the center" good enough, and how do the first orbits avoid hitting the source?); the magnetic field and frequency match (Do I really need to map my magnet's field accurately? Is one reading in the middle enough, and which value goes into the frequency and energy calculations?); the pressure is in the window (What pressure should I aim for when the cyclotron is running? Don't I need more gas to make ions, but less gas to let them spiral around?); the collector can see what arrives (How do I know my beam-current detector could detect the beam I'm looking for, before I use it to troubleshoot the cyclotron?). Two source-side symptoms do involve the plasma. Pushing the arc or emission harder and getting less beam was seen on the calutron sources (the arc is the discharge current through the source gas), where rising ion density at the exit defocused the beam. The optimum arc was specific to the electrode geometry and the pulling voltages (Yonts (1954), p. 36). On a filament source the equivalent is the emission knob. The Mullins log’s working filament currents sit just above the emission knee, the filament current at which electron emission starts to rise steeply, and more was not better (hosted rf-dee-source note, §6.4). A beam that changes as the gas is turned up points first at pressure-dependent collisions and at the source’s own behavior, both of which move with gas. The observation alone does not identify the mechanism. The Houghton pressure window is the reference for what a pressure scan looks like (ion-sources chapter).

To understand it. The bunches matter more than the average. The accepted ions travel in short bunches, about 40 degrees of each RF cycle on the Rutgers machine. The peak current inside a bunch is then roughly nine times (360 divided by 40) the average an electrometer reads (Koeth (2015), p. 9). Any space-charge estimate for the circulating beam should use the bunch current at the lowest energy, in the first turns, where the ions are slowest and closest together and where the effect would appear first (dg-580’s note). Three things should make you reconsider: average currents approaching tens of microamperes on an internal probe; an external or high-current source feeding a central region designed for a weak one; an extraction program, where beam size and turn separation start to count. The professional machines that run milliamperes design their vertical focusing around space charge. VECC set its vertical tune, the number of up-and-down oscillations an ion makes per turn, above 0.5 for a 5 mA average current (Pandit (2007), p. 1), a requirement that belongs to that current. In the source, the plasma edge’s shape and the slit-versus-hole choice set the beam’s emittance, its spread in position and direction, before the first turn. The beam-quality chapter treats that as the quality decision that outranks all the others (Forringer’s test-stand results, Forringer (2004)). Wolf’s handbook gives the extraction-optics rule of thumb, an aperture-radius-to-gap ratio near one half, and then matching the plasma density to the pulling electric field by tuning the arc (Wolf (ed.), p. 379). All of that is Level 3 for a machine chasing nanoamperes, and Level 1 for anyone building a hooded source for microamperes.