Beam Quality: What It Is and What Degrades It

A cyclotron beam is born with the best quality it will ever have, in the fraction of a millimetre of a source slit and the third of an RF cycle that captures it. Everything after that moment — gas scattering, field errors, phase slip — degrades the population; slits and scraping can clean up what survives, but only by throwing beam away. This page makes the quality parameters concrete at amateur scale: what emittance, energy spread, and beam extent actually are, which mechanism spoils each one, and which of them a builder controls in the drawings versus at the knobs. How to measure any of these numbers is the beam measurement page's territory; this page covers what the numbers mean and what sets them.

A beam is a population, not a trajectory

Orbit theory follows one ion. A real beam, even a picoampere one, is millions of ions per second, and no two of them start alike: each leaves the source at a slightly different position across the slit, at a slightly different angle, at a different point in the RF cycle, with a few electron-volts of random thermal energy from the plasma (measured plasma temperature ~35,000 K, a central starting energy near 4.5 eV — Forringer thesis; the initial-condition recipe orbit codes now use). Beam quality is the statistics of that population: how narrow its spreads are in transverse position and angle (emittance), in energy, and in arrival time. The sections below take them in that order, arrival time folded into the phase-window discussion since the same window sets both — and for a first machine, most reduce to the pass/fail checks collected near the end.

Two views of the same physics divide the labor here. The single-particle view — how one ion oscillates about its equilibrium orbit, what the field index does to the motion — belongs to the beam dynamics laboratory. This page takes the statistical view: the beam as a distribution whose spreads the machine inherits, transports, and degrades. Where the two meet (betatron amplitudes are the transverse spread), the concepts are used and cross-referenced, never rederived.

Emittance is decided at the source slit, on turn zero

Take a snapshot of every ion the instant it clears the source slit, and plot each one as a dot: horizontal axis, its radial position; vertical axis, the angle its velocity makes with the ideal orbit. The dots fill a region. Its area (divided by π, quoted in mm·mrad) is the radial emittance — that is the whole concept, though published values differ in bookkeeping (rms or full-contour, normalized or not), so comparing two numbers means checking conventions first. At the slit the cloud is tall and narrow: every ion is inside the 0.25 mm width of a typical PIG chimney slit, but the plasma's few-eV thermal energy sprays them over a large range of angles. The first published emittance measurement of a cold-cathode internal PIG cyclotron source — the claim is the dissertation's own — comes from exactly such a slit, on Forringer's MSU test stand: about 25 mm·mrad radial, roughly independent of arc current (Forringer thesis, pp. 77–91).

Two properties of this picture carry all the practical consequences. First, focusing only reshapes the area: lenses and field gradients rotate and shear the ellipse, trading a small spot for a large divergence and back, and can never shrink it — Liouville's theorem, the foundation the accelerator literature builds every transport calculation on (Barletta, USPAS lecture notes; glossary). Acceleration does flatten the plotted angles — the same transverse velocities matter less against more forward momentum — but at a fixed exchange rate: the momentum-scaled (“normalized”) emittance is the invariant, so the shrinkage is bookkeeping, never a free improvement. Whatever the slit emits is the best the machine will ever carry. Second, processes that are not smooth transport — collisions with residual gas above all — add area, and the addition is permanent. Emittance can only be spent, never earned back.

That makes the slit and its plasma the quality decision that outranks every other, settled before the first turn. Even the shape of the aperture matters: a slit gives an approximately flat plasma boundary and a converging beam, while a round hole of similar area gives a concave boundary, a diverging beam, roughly 50% larger normalized radial emittance, and half the luminosity at the same arc current (Forringer thesis, pp. 73–76 and 91–107; Smirnov's review draws the three possible boundary shapes). The ion sources page covers building these chimneys; here the point is only that their geometry is the beam's birth certificate.

The same snapshot in the axial (vertical) plane gives the axial emittance, with the slit's 5 mm height in place of its 0.25 mm width. The asymmetry is deliberate: a tall slit passes more current through the same radial emittance, which is why standard arc-source slits run several times taller than wide (Handbook of Ion Sources; design guide ion-source rules).

Energy spread is manufactured during acceleration, not at the source

At cyclotron scale the source contributes almost nothing. Ions leave a PIG discharge with 10–50 eV of energy spread, a hot-filament arc with 0.2–5 eV (Handbook of Ion Sources, Table 2.1) — tens of times below even a single keV-scale gap crossing at the worst, thousands of times below the final energy. The spread the beam ends up with is made later, by RF phase, in two steps.

First, the machine accepts a wide window of starting phases. An ion crossing the gap at phase φ gains qV cos φ, not qV; with an internal source and cosine RF the central region captures starting phases across roughly (−90°, +20°) — about a third of the cycle (Smirnov review, §5.5). Every phase in that window survives, but each accumulates a different history: the phase evolves with energy as the ion's revolution frequency drifts against the RF (sin φ grows by an integral of the frequency error over energy — Smirnov, eq. 5.11), so at any given radius the population spans a band of phases and therefore a band of energies. The phase budget is finite: tune the RF exactly to the central field of a radially falling magnet and the slip reaches 90° within a dozen or so turns — the worked number in one Houghton design study (King thesis, 2002) — and acceleration stops. The standard placement — RF slightly below the central-field frequency, above the edge-field frequency, so the error first grows and then unwinds — exists precisely to keep the whole population's phase histories inside the window (King thesis; design guide beam-dynamics rules).

Second, turns overlap. Turn spacing shrinks as 1/r while the radial beam width does not, so past the first turns any fixed radius contains ions on several different turns at once — the path length & turns calculator does the arithmetic. A collector, septum, or target placed there accepts the whole overlapping stack, and the standard estimate for the resulting spread of a multi-turn beam is about 2qV_dee — ~20 keV at a 10 kV dee, ~3 keV at a 1.5 kV dee (Baartman, JINST 2023) — with the exact width set by how many turns the aperture actually accepts. That number, not anything the source did, is the energy spread of a multi-turn beam. Extracting a single turn instead would need the RF phase width held below √(2/N) — a few degrees for a few hundred turns — plus field stability near 2 × 10⁻⁴ (Baartman), which is why single-turn machines are a professional discipline.

The failure mode at the end of the phase budget is distinctive: once an ion's accumulated slip passes 90° it gains nothing at the gap, then loses energy and spirals inward, so the beam population ends abruptly at whatever radius exhausts the budget (Morrow thesis). On a current-versus-radius log that reads as a sharp cliff — the beam measurement page uses it as a diagnostic signature; this is the mechanism behind it. The 86-inch's designers put the whole compromise in one sentence: frequency, field level, and field shape "must involve compromises to keep the ions within necessary boundaries in space and in phase" (ORNL-1196, p. 18).

Extent: what the beam spends its aperture budget on

The beam's physical size — radial width, axial height — is the quality parameter the chamber feels directly: axial extent must fit inside the dee aperture and radial extent decides what a probe or septum intercepts. Three contributions stack.

Betatron amplitude, the incoherent share, is the phase-space ellipse of the previous section seen in real space: each ion oscillates about its equilibrium orbit at f_axial = √n · f₀ and f_radial = √(1−n) · f₀, stable only while the field index stays between 0 and 1 — the weak-focusing relations of the classical machine, this page's scope throughout; sector-focused fields follow different tune rules (Livingston & Blewett, p. 161). The envelope is not constant with radius: measured on the MIT machine, the beam fills the dee aperture out to one-third or one-half of final radius, then narrows almost linearly — axial amplitudes damp as the focusing strengthens, 0.8 in at the center to ~0.1 in at the exit slit (Livingston & Blewett, pp. 163–167). The design consequence is stated in the same pages: the first third of the radius needs the generous vertical aperture, because that is where ions are lost.

The center is worse than weak — it is unfocused. By symmetry n = 0 at the machine center, so magnetic focusing vanishes exactly where the beam is largest; the first turns survive on the accelerating gap's electric-lens action alone, and mostly do not survive: the 86-inch design analysis put the central loss at about 90% of the starting ions, reduced by raising dee voltage so the ions clear the region in fewer turns (ORNL-1196, pp. 17–18). Purpose-designed central regions do better; the mechanism is the same. Even the 1949 small-machine cookbook knew where this lever was: "Better ion focussing can be obtained by installing a 'dummy' grounded dee edge symmetric to the insulated dee" (UCRL-476, Wouters, on the single-dee tank).

Coherent displacement is the share field errors add: the whole population's orbit centers pushed off the machine center together, so the beam sweeps a wider annulus than its emittance requires. The sensitivity is startling. A first-harmonic field error — one side of the machine slightly stronger than the other — displaces the equilibrium orbit by ε₁R/(ν_r²−1), and a bump of 10⁻⁴ of the main field (0.6 gauss in a 0.59 T field) can mean millimetres of displacement near ν_r = 1 (Botman & Hagedoorn, CERN Accelerator School) — an error invisible on a casual gaussmeter sweep can move the beam millimetres. The classic self-inflicted case: an ion-source mounting hole on one side of a pole, with no matching hole opposite, produced a first harmonic that grew radial oscillations to ~30 mm; adding a symmetric dummy hole on the opposite side cut the motion to under 3 mm (Antokhin et al., RuPAC 2006; design guide dg-050). Asymmetric dee drive does the same job electrically — voltage droop of up to 5% along a dee face has forced source offsets of more than 2 inches on large machines (Livingston & Blewett, p. 164). Extraction designers create exactly this displacement on purpose and size it to match the betatron amplitude; uncontrolled, it is pure quality loss.

Resonance growth is the share the field profile can add catastrophically. The 184-inch commissioning crew met it first: the machine was designed to accelerate to 85 inches radius, "but it was found that the beam disappeared at a radius of 82 inches" (MDDC-1092, 1947). Radioautographs showed the vertical spread staying under 2½ inches until, at the 82-inch radius, the beam blew up vertically into the dee — exactly where the field index passed 0.2 and the radial and axial oscillations came into 2:1 resonance. The memo's energy bookkeeping still stings: when radial oscillation energy converts to vertical, "the amplitude of vertical oscillation will be, at times, at least double that of the radial oscillations," and — the sentence every low-budget builder should read twice — "for systems having low accelerating voltages … the ions will rapidly increase the amplitude of their vertical oscillations at the point where n = 0.2" (MDDC-1092, pp. 1–2). Few turns near n = 0.2, or an aperture priced for double the radial amplitude: those are the options. The beam dynamics laboratory animates the motion itself; here it is one line in the extent budget.

Gas scattering is the degradation a pressure knob trades down

Every mechanism above is geometry and fields. Residual gas is different: it degrades all four quality parameters at once, and its knob — chamber pressure — turns mid-run, no metal cut. A large-angle collision or charge exchange removes the ion outright — intensity loss. The far more frequent small-angle collisions each add a random kick to the angle coordinate: in the phase-space picture, the ellipse's dots random-walk vertically, and the occupied area grows. That growth is the dashed ellipse in the figure above, and no downstream focusing undoes it.

The exposure is set by path length, which is why dee voltage appears again. A beam that needs N turns travels roughly N circumferences — hundreds of metres in a low-voltage machine (~300 m for a 2 MeV design at 1 kV per gap in a 2.7 T field) — and the working rule is a mean free path at least ten times the total flight distance, which still concedes roughly 10% single-collision loss and says nothing about the emittance the surviving ions pick up from small-angle scattering on the way (Dewan thesis, MIT 2007; design guide vacuum rules). The vacuum & beam survival calculator turns pressure, gas species, and path length into a surviving fraction — its worked example, a 15 m flight at a mean free path of 30 m, loses 39% of the beam to gas.

An internal-source machine cannot simply pump harder: the source needs gas to ionize. Houghton College mapped the resulting window on a 15 cm machine — that machine's operation confined to roughly 10⁻⁶ to 10⁻⁴ Torr of hydrogen, with the biggest current near the top of that range (Houghton, Cyclotrons 2013). The quality cost of running there is visible in the same theses: the machine's highest current to date, ~100 nA, came at ~10⁻⁴ Torr with resonance peaks "much broader due to the high pressure within the chamber," while scans at lower pressure gave smaller, well-defined peaks (Fuller thesis, pp. 47–52). And the cheapest quality upgrade on record: dropping the hydrogen partial pressure about seven-fold (with a gentler filament bias) raised the same machine's beam from 10 to 70 pA at zero additional RF watts (Yuly, Small Cyclotron Conference 2010) — less gas scattered away more than the source lost by starving.

Intensity and quality are one knob with two ends

Nearly every intensity decision in a classical cyclotron buys current with quality or quality with current — the exceptions below are worth knowing by name. The cleanest statement of the trade is the phase acceptance window itself.

The same trade repeats at every quality-setting element, with measured exchange rates:

• Slit width. Doubling a PIG chimney slit from 0.25 to 0.51 mm raised beam current ~4.4× (52 to 230 μA at 50 mA arc) and cost ~1.7× in radial emittance (Forringer thesis, pp. 66–69). Current buys faster than emittance sells — widen the slit until the machine's acceptance is full, then stop.
• Slit versus hole. Converting slit apertures to equal-area round holes on a Siemens Eclipse raised source-to-target transmission from 19% to 30% but cut target current from 120 to 40 μA (Potkins et al.): the hole's beam fits the machine better; the slit simply emits more. Which end wins depends on whether the machine is acceptance-starved or current-starved.
• Arc power. The one lever that is nearly free: raising arc current 50 to 450 mA raised beam current 52 to 227 μA with no measurable emittance change (Forringer thesis, pp. 77–78). The commonly quoted boundary below which space charge stays negligible is a few hundred microamperes (Smirnov review) — orders of magnitude above amateur beams, so at that scale quality is set by geometry and fields, not by current.
• Gas pressure. The Houghton trade of the previous section: peak current at 10⁻⁴ Torr with broadened resonance peaks, clean narrow peaks at lower pressure and lower current (Fuller thesis).
• Phase slits. The deliberate version of the window in the figure: a slit placed where the beam's radial size is largest, close to the center, cuts the accepted phase band down to a few RF degrees — cleaner turns, proportionally less current (Smirnov review, §5.5). The 1950s ANU machine ran beam-defining slits on turns 1–3 and reported 100% extraction efficiency, at low current and with dee voltage held to 0.5% (ORNL-2644, the 1958 world accelerator compendium, p. 27).

The accepted window also fixes the beam's time structure: a "continuous" cyclotron beam is really a bunch train, one bunch per RF cycle, as long as the captured phase band — the measured value on the Rutgers machine was about 40° of each cycle (Koeth, CAARI 2014), so average current there understates peak by roughly 9×. Any counting experiment downstream inherits that duty cycle.

"Good beam" at 100 keV is not "good beam" at 10 MeV

The four parameters never stop mattering, but which one binds depends on what the beam is for.

At amateur energies — tens to hundreds of keV, picoamps to nanoamps on an internal probe — good beam means three things: it survives to full radius (the current-versus-radius curve does not cliff early from phase slip or die of gas scattering), it fits (axial extent inside the dee aperture, no dee-lip burns), and it is identifiable (a resonance peak sharp enough to assign a species and harmonic). Emittance is rarely worth measuring — the machine is its own transport line, and the numbers worth logging are the humbler ones the beam measurement page lists: radial profile, vertical extent, energy spread, time structure. Even energy spread is mostly a bookkeeping entry: ~2qV_dee of turn-overlap spread on a 1–2 kV machine is a few keV on a few-hundred-keV beam — a few percent, invisible until an experiment needs a sharp reaction threshold.

At production scale the same numbers become economics. The Oak Ridge 86-inch circulated a milliampere at 23 MeV (ORNL-1196): at that current, emittance decides whether the beam lands on the target or on the beamline, energy spread decides whether turns separate at the septum — at 30 MeV and 100 keV per turn the spacing is 0.83 mm against a typical 4 mm beam width (Botman & Hagedoorn) — and space charge, negligible below a few hundred microamperes, is now a design driver (Smirnov review). The lesson for reading spec sheets runs in both directions: a professional machine's mm·mrad figures are survival requirements, not polish, and an amateur machine chasing those figures is solving a problem it does not have.

Some quality is welded in; the rest is at the knobs

Sorted by when the decision is made — the practical summary of everything above:

Fixed at design time (changing these means rebuilding):

• Source slit geometry — width, height, slit-versus-hole: sets emittance and the current/quality exchange rate (Forringer thesis).
• Dee voltage capability — the master quality variable. Turn count sets the phase budget, the gas-scattering exposure, the central-region loss, and the 2qV_dee energy-spread scale all at once (ORNL-1196, pp. 17–18; Livingston & Blewett, p. 172; design guide beam-dynamics rules).
• Field profile and symmetry — for a classical machine, n(r) inside (0, 1) with the n = 0.2 radius outside the working beam or crossed in few turns, and azimuthal symmetry watched at the gauss level near full radius: any asymmetric hole, slot, or chamber feature is a candidate first-harmonic generator (MDDC-1092; Botman & Hagedoorn; design guide dg-050). Field shaping practice lives in magnet design.
• Dee aperture — the axial budget, priced for the wide envelope of the first third of radius plus resonance headroom (Livingston & Blewett, pp. 163–167; MDDC-1092).
• Pumping and gas handling — base pressure and pumping speed set where in the 10⁻⁶–10⁻⁴ Torr window the machine can sit (Houghton, Cyclotrons 2013; vacuum calculator).
• Adjustability itself — a source mount movable by a few millimetres is a design decision that pays at tuning time (Livingston & Blewett, p. 164).

Adjustable at tuning time (session to session, run to run):

• RF frequency placement between edge-field and central-field frequency — the phase-history shape, and with it where the beam dies (King thesis, 2002).
• Magnet current — resonance itself, and via the trim of field against frequency, the phase budget's sign.
• Gas pressure — the intensity/peak-sharpness trade, tunable in minutes (Fuller thesis).
• Arc current and filament bias — current nearly free of quality cost at amateur scale (Forringer thesis; Yuly 2010).
• Source position — centering against the real, imperfect field and dee drive rather than the drawing (Livingston & Blewett, p. 164).
• Between-run hardware — shims and dummy holes: hours of work rather than a rebuild, and measured order-of-magnitude reductions in coherent oscillation (design guide dg-050).

Go deeper

• The design guide's beam-dynamics rules (209 rules), plus the vacuum and ion-source domains — each rule with formula, source, and page.
• Vacuum & beam survival and path length & turns — the two calculators behind this page's scattering and turn-overlap arithmetic.
• Beam measurement — how every number on this page is actually taken on a small machine.
• Hosted classics: the 184-inch commissioning memos (MDDC-1092: what field imperfections did to the biggest beam on Earth), ORNL-1196 (phase grouping and the 90% central loss, pp. 17–18), and UCRL-476 (Wouters' small-machine recommendations).
• Emittance, phase stability, betatron oscillation, phase slip, and turn separation in the glossary.
• E. R. Forringer's dissertation (the emittance measurements this page leans on) and the Houghton College theses — both in the library.

Sources

• D. C. Sewell, L. Henrich & J. Vale, “Some Operating Phenomena Associated with the 184-inch Cyclotron,” MDDC-1092 (1947) — hosted here; beam loss at 82 inches, n = 0.2 coupling, precession fine structure.
• M. S. Livingston et al., The Oak Ridge 86-Inch Cyclotron, ORNL-1196 (1952) — hosted here; fixed-frequency theory summary and central-region losses, pp. 17–18 (PDF pages).
• L. F. Wouters, General Recommendations for the Design of Small Cyclotrons, UCRL-476 (1949) — hosted here; dummy-dee focusing, operating pressure.
• M. S. Livingston & J. P. Blewett, Particle Accelerators, McGraw-Hill, 1962 — field-index stability and oscillation frequencies (p. 161), beam envelope (pp. 163–167), dee-face droop and source offset (p. 164), phase migration and dee voltage (pp. 166–172).
• E. R. Forringer, Phase Space Characterization of an Internal Ion Source for Cyclotrons, PhD dissertation, Michigan State University (2004), MSU library — slit/hole emittance measurements, plasma-boundary and plasma-temperature recipes.
• V. L. Smirnov, “The Cyclotron and Its Modeling,” Physics of Particles and Nuclei 52, 913 (2021), doi:10.1134/S106377962105004X — phase acceptance (§5.5), phase-history equation (5.11), phase slits, plasma boundary shapes, space-charge threshold.
• R. Baartman, “Cyclotrons: why/how are their dynamics different?,” JINST 18 T03005 (2023), doi:10.1088/1748-0221/18/03/T03005 — multi-turn energy spread 2qV, single-turn phase-width bound.
• J. I. M. Botman & H. L. Hagedoorn, “Extraction from Cyclotrons,” CERN Accelerator School (CAS) proceedings — turn separation, first-harmonic orbit displacement, coherent-amplitude criterion.
• Houghton College: N. A. Fuller thesis (2013) and M. Yuly et al., Small Cyclotron Conference 2010 / Cyclotrons 2013 papers — pressure window, peak broadening, the 10 → 70 pA result; S. I. Morrow thesis (2015) — phase-slip cutoff; B. T. King thesis (2002) — RF placement and the exact-tuning phase-slip example. See the library (search “Houghton”).
• D. Potkins et al. (D-Pace / Siemens Molecular Imaging), “Improvements to Siemens Eclipse PET Cyclotron Penning Ion Source” — slit-versus-hole transmission and current figures.
• F. T. Howard, Cyclotrons and High-Energy Accelerators — 1958, ORNL-2644, p. 27 (PDF page) — hosted here; ANU beam-defining slits and reported extraction efficiency.
• T. W. Koeth, Rutgers 12-inch cyclotron, CAARI 2014 — bunch-width measurement; see the library (search “Rutgers”).
• B. Wolf (ed.), Handbook of Ion Sources, CRC Press, 1995 — source energy-spread table, arc-source slit proportions.
• W. A. Barletta, Introduction to Accelerators, Lecture 4 (USPAS/MIT, 2010) — phase-space and emittance formalism; see the library.
• L. Dewan, Design and Construction of a Cyclotron Capable of Accelerating Protons to 2 MeV, B.S. thesis, MIT (2007) — mean free path versus flight distance rule.
• E. Antokhin et al., “Magnet System for PET Cyclotron Based on Permanent Magnets,” RuPAC 2006 — first-harmonic dummy-hole result.
• Design Guide rules cited by id (dg-050 first-harmonic dummy hole, and the beam-dynamics domain generally) — each carries its own verbatim quote and page citation in the design guide.