Beam Extraction

Every subsystem page on this site ends with beam circulating inside a chamber. This one asks whether it has to leave — and shows that the answer is arithmetic before it is hardware: the radial gap between the last two turns, measured against the beam's own radial width, decides which extraction schemes are open. The decision runs in a fixed order. Default to an internal target, which is what every machine in the builds census did. Test extraction by pure acceleration, one line of algebra on the magnet geometry. If that fails, the deflector and septum — helped, where spacing falls short, by precession — is the classical route. And H⁻ stripping bypasses the whole problem, if the vacuum and the ion source can be paid for.

Scope: this page ends where the beam leaves the magnetic field. Beam transport after extraction — lenses, beamlines, targets in air — is out of scope, as are the regenerative and magnetic-channel schemes of MeV-class machines, which are named below only where the sources rank them. H⁻ source construction belongs to ion sources; the probes that measure turn structure belong to beam measurement.

The default is no extraction at all

Of the seventeen documented amateur and student machines in the builds census, none is documented to have run an external beam. The best-documented of them, the Rutgers 12-inch, ran 800 keV protons for years with the beam ending inside the chamber; its electrostatic deflection channel served as an energy diagnostic, not a beamline (Ponter et al., Small Cyclotron Conference 2010; see beam measurement). One successor project, the University of Maryland 19-inch, is being built to extract — intention, not yet operation. The default has a long pedigree: the machines that did extract were professional. The Argonne 60-inch delivered 200 μA of deflected deuterons through a 47-inch-long electrostatic channel at 75 kV, with an oil-immersed supply in its own “deflector vault” (ANL-5907, pp. 16–18).

The internal target is not a consolation prize. A target on the radial probe collects the circulating beam that survives to its radius, at an energy selected by radius through T = (qBr)²/2m (calculator) — while a classical electrostatic deflector delivered on the order of a quarter of the circulating beam: “up to 25 per cent… under optimum conditions” at MIT, and 25–40% nominal acceptance on the Argonne machine at the strong end (Livingston & Blewett, Particle Accelerators, p. 166; ANL-5907, p. 18). A builder whose internal current is nanoamperes should expect the extracted remainder to be a fraction of a nanoamp. Extraction buys geometry — a beam in a defined direction, clear of the dees, available for targets the chamber cannot hold — and charges for it in current and in high-voltage engineering. It also changes the radiological situation, since beam can now reach material outside the shielded gap (safety).

Turn separation is the constraint everything else serves

A cyclotron beam spirals outward by a little less each turn. From acceleration alone, the radius gain per turn is

Δr ≈ R · (ΔE/E) · γ/(γ+1) · 1/ν_r²

with ΔE the energy gain per turn, E the kinetic energy at radius R, γ the relativistic Lorentz factor (barely above 1 at amateur energies), and ν_r the radial tune (Kleeven & Zaremba, CAS 2015, p. 44). At sub-MeV energies γ/(γ+1) is ½ and ν_r² = 1 − n is near 1, so the rule of thumb is that “the relative radial increase is only half the relative energy increase” (Kleeven & Zaremba, p. 44). Spacing therefore shrinks as energy grows — turn separation is generous at the center and worst exactly where extraction must happen. And at fixed radius the spacing is proportional to dee voltage: doubling the dee voltage doubles it (Kleeven & Zaremba, p. 44).

The professional numbers explain why extraction chapters exist. For 30 MeV protons gaining 100 keV per turn at R = 0.5 m, the spacing is 0.83 mm — “a rather small number, e.g. when compared with a radial beam width of for instance 4 mm” (Botman & Hagedoorn, CAS, p. 6). Successive turns overlap; no blade can sit between them. A small machine is better off fractionally, because its per-turn energy gain is a far larger share of its final energy: a 155 keV machine gaining 2.6 keV per turn at r = 9.7 cm spaces its last turns by about 0.8 mm, and the same geometry with a 10 kV dee spaces them by several millimetres — the arithmetic the path length & turns calculator runs for any machine, including the septum verdict for the final pair of turns.

Two refinements matter before trusting that verdict. The energy gain per turn is not the full dee voltage unless the geometry cooperates: a dee of angular width θ delivers 2V₀ sin(hθ/2) per pass through the dee at optimum phase, with h the harmonic number (h = 1 and the full 2V₀ for 180° dees on the fundamental) — the ORIC designers noted that shaving even a 15° wedge off a dee lip would cost third-harmonic ions 30% of their gain per turn (ORNL-2648, p. 95). And the turn pattern is a thing to measure, not assume: a differential radial probe with fingers about 2 mm apart resolves individual turns and the radial oscillation near the edge (Botman & Hagedoorn, pp. 11–12), provided everything ahead of the collecting surface is thinner than the spacing being measured — on the Rutgers machine, spacing near 4 inches radius was 0.04 inch at 10 kV peak-to-peak (Koeth, ion source studies, 2006).

When turns do overlap, extraction still works — it just takes several turns' worth of beam at once. The cost of that multi-turn compromise is energy spread: at least the gain separating adjacent turns — of order 2qV_dee in Baartman's convention — and proportionally more when several turns straddle the septum. True single-turn extraction, for the few-hundred-turn case Baartman analyzes, demands an RF phase width of a few degrees and field stability near 2 × 10⁻⁴ — the phase acceptance shrinks as √(2/N) with turn count N (JINST 18 T03005, 2023). A machine running hundreds of turns should not chase it.

Extraction by acceleration: one line of algebra before any hardware

The simplest scheme adds no hardware at all — and usually fails the arithmetic on small magnets. Push the energy gain per turn high enough and the beam runs out of field before it runs out of turns: the fringe region past the pole edge stops bending effectively, the orbit opens, and the beam leaves on its own. Baumgarten derived the condition: the turn count must satisfy N_t ≤ (R/g)²/(πN_hγ(γ+1)), with g the pole half-gap at extraction and N_h the harmonic number — “it is mostly the squared ratio of extraction radius and pole gap at extraction which determines the maximal number of turns” (arXiv:2205.04124, pp. 5–6). Wide-gap tabletop machines usually fail it: at R ≈ 10 cm with a 12.7 mm half-gap the budget is roughly nine turns, meaning ~17 keV per turn to reach 155 keV — borderline for the 5–10 kV dee systems small machines actually run. Narrowing the gap at the pole edge stretches the budget toward thirty turns. The formula costs nothing to evaluate and settles immediately whether the no-hardware option exists for a given magnet.

The deflector and septum: kilovolts, knife edges, and honesty about losses

The classical answer is a DC electric field applied to the last turn only. A thin grounded blade — the septum — separates the final turn from its predecessors; outside it, a negative electrode — the deflector — pulls the beam outward along a channel that follows the orbit for tens of degrees of azimuth, giving a kick of typically 50 to 100 mrad, enough for the orbit to walk out of the field (Botman & Hagedoorn, p. 14).

The deflector voltage follows, to first order, from geometry. For a channel of gap d peeling the orbit from radius R to R + ΔR, the required potential is roughly V_d = (2T/q) · d·ΔR/(R(R+ΔR)), with ΔR = 0.15R the typical compromise — a slower peel needs less voltage but a longer channel, a faster one risks breakdown (Livingston & Blewett, pp. 163–166). The voltage scales with kinetic energy, which is the small builder's one structural advantage — with a second lever in how fast the peel must happen. MIT's 42-inch needed 47 to 87 kV to peel 16 MeV deuterons across a 0.3 inch gap over a long, gentle channel; the Rutgers 12-inch needed 32.5 kV to bend 472 keV protons from 4 to 4.5 inches radius within just 43° of arc, forcing the beam onto a 7-inch curvature (Ponter et al., 2010); and a 350 keV machine content with a 100 mrad kick spread over 10 cm of arc needs about 7 kV/cm — 3.5 kV across a 5 mm gap, from the kick relation θ = EL/(2T/q) (Botman & Hagedoorn, p. 14). Kilovolts, not tens of kilovolts, if the geometry is patient.

The septum is where physics meets machining. Its entrance edge is a knife: modern practice runs 0.1 to a few tenths of a millimetre at the entrance, tapering to several millimetres downstream, with a V-shaped entrance slot to spread the heat of the turns that inevitably land on it (Botman & Hagedoorn, p. 14). At production currents that heat is the design driver: Argonne destroyed uncooled and water-cooled splitters of tungsten, carbon, and copper before settling on the water-cooled V-notch copper design that then ran for thousands of hours (ANL-5907, pp. 16–17). A nanoampere machine has the opposite problem — a 350 keV beam of 1 nA deposits ~0.4 mW even if all of it hits the septum, so cooling is irrelevant and the blade can be as thin as it can be made straight. Both septum and deflector should be adjustable in place: “such calculations and designs can only be approximate,” and MIT built its deflector to be trimmed empirically for maximum beam in operation (Livingston & Blewett, pp. 181–183).

The high-voltage engineering has its own rules, all inherited from machines that arced first. The empirical Smith–Grunder guideline for clean, conditioned vacuum electrodes is V·E < 1.5 × 10⁴ kV²/cm, derated a further 20–30% because the deflector sits in a magnetic field (Botman & Hagedoorn, p. 14) — a kilovolt-class channel sits far below it, which is margin, not a guarantee: it is a design guideline for well-prepared electrodes, not a breakdown bound. In practice edges and feedthroughs spark long before bulk gaps do. Rutgers estimated its edge fields with E_max = 0.9V/(r ln(a/r)) for an edge of radius r facing a gap a, rounded every deflector edge to 3/16 inch, and held peak fields to 170 kV/inch against the ~290 kV/inch breakdown allowable the team assigned to aluminum (Ponter et al., 2010) — surface finish and conditioning set the real margin, which is why the allowable is not a number to run close to. The same team arced a 30 kV-rated feedthrough run at 35 kV and rebuilt with the feedthrough inside the vacuum — the supply and cable store 0.1–0.4 J at 30 kV, enough to pit electrodes, so the working rules are a series resistor at the feedthrough, a short HV cable, and a feedthrough rated two to three times the operating voltage (Ponter et al., 2010). A deflector supply in the tens of kilovolts is lethal to people long before it pits copper; the HV practices in safety apply to this subsystem in full.

Expectations, finally, should come from the record for positive-ion machines (stripping plays by different rules, below): about 10% extracted for early synchrocyclotron precessional schemes, up to 25% for a well-tuned classical deflector, 75–80% for IBA's self-extracting design — a shaped-field scheme, not a septum — and above 90% for modern machines with well-centered beams and refined precessional schemes (Livingston & Blewett, p. 166; Botman & Hagedoorn, p. 2; Jongen, Cyclotrons 2004). A small deflector that delivers a tenth of the internal beam is working as its ancestors did.

Precession: turn spacing bought with a deliberately off-center beam

When acceleration alone cannot open a millimetre for the septum, the standard trick makes the orbit center move. A beam given a coherent radial oscillation of amplitude x has an orbit center that circulates — precesses — at the frequency (1 − √1−n) ω₀, so on successive turns the oscillation adds to the radius on a slowly rotating azimuth. Placed correctly, it contributes up to 2π|ν_r−1|·x of extra separation per turn on top of the acceleration term: a 3 mm amplitude carried to ν_r = 0.8 buys 3.8 mm (Botman & Hagedoorn, p. 7). Compare the brute-force alternative — bending the whole beam outward with a non-resonant azimuthal field asymmetry — where, in Heikkinen's typical conventional-cyclotron case, a 1 gauss first harmonic in a 1.7 T field gains only ~0.2 mm per turn (CAS 1994, p. 14). Precession is the cheap lever. The orbit-center motion itself runs live in the beam dynamics laboratory.

The phenomenon was measured before it was exploited. In 1947, Berkeley's crew photographed recurring “pips” in the 184-inch probe current, added a second probe 155° away, and proved the fine structure was the beam bundle precessing past the probes (MDDC-987); the companion analysis showed the pip frequency equals (1 − √1−n) ω₀, making pip-counting a measurement of the field index (MDDC-1092; both hosted with the 184-inch commissioning papers).

Creating the coherent amplitude differs by machine class. Isochronous machines designed for precessional extraction cross ν_r = 1 near the outside and let a deliberate first-harmonic bump drive the resonance for its ~10-turn effective width, then extract where ν_r has fallen to about 0.8 (Heikkinen, p. 14); the orbit-center response to a bump is ε₁R/(ν_r²−1), so close to the resonance a bump of one part in 10⁴ of the main field — gauss-scale — displaces the orbit by millimetres in Botman and Hagedoorn's R = 1 m example (p. 9); the displacement scales with radius. A classical weak-focusing machine never has ν_r > 1 (ν_r = √1−n with 0 < n < 1), so the amplitude must be created outright — most simply by off-centering the ion source, a displacement builders discover empirically anyway when beam peaks with the source shifted (Livingston & Blewett, p. 164). Off-centering at turn one has a cost: over the following 30–100 turns, RF phase mixing smears the direction of the oscillation even though its amplitude survives, which is why a trim bump applied late, near the extraction region, gives a cleaner extracted beam than a source offset (Botman & Hagedoorn, p. 9).

The trick has a wall, and every classical machine hits it in the same place. The amplitude should be no larger than the beam's incoherent (emittance) amplitude — “a coherent radial oscillation amplitude of the same size as the incoherent amplitude, is a good criterion for efficient extraction” (Botman & Hagedoorn, p. 7) — because large radial amplitudes feed the coupling resonance at n = 0.2 (ν_r = 2ν_z, the Walkinshaw resonance), which converts radial oscillation into vertical at double the amplitude or more (MDDC-1092; Heikkinen, p. 18). The ORIC designers called the axial-stability loss during deflection “a very serious danger in this deflection system” (ORNL-2648, p. 88), and the 184-inch supplied the canonical demonstration: its beam spread vertically and died at about 81½ inches radius, closely matching the measured n = 0.2 point (MDDC-984). The fringe region that provides ν_r ≈ 0.8 for precession sits next to the resonance that eats the beam — the working rule is to keep the induced amplitude to a few millimetres and cross n = 0.2 in as few turns as possible, which argues for high energy gain per turn on the last turns (Heikkinen, p. 18).

Stripping: extraction without a septum, paid for in vacuum

Accelerate H⁻ instead of protons and extraction stops being a separation problem. A stripping foil — pyrolytic graphite, 50 to 200 μg/cm² — removes both electrons in one pass; the particle's charge flips sign, its orbit curves the other way, and it exits the field in a fraction of a turn. Virtually every ion that reaches the foil is converted and extracted — near-100% in current, though at sub-MeV energies the foil itself taxes beam quality, costing energy and adding multiple scattering on the way through. The “device” is a foil on an arm with reported lifetimes above 2 × 10⁴ μAh (microampere-hours — two-plus years of continuous operation at a full microampere), extracted energy is selected by foil radius, and no septum-clearance arithmetic applies (Botman & Hagedoorn, pp. 3–4) — though where turns overlap, the radial beam width at the foil still sets the extracted energy spread (ΔT/T = 2Δr/r; Botman & Hagedoorn). Baartman's summary is the right frame: stripping simply evades the septum-survival and phase-acceptance logic that governs everything above (JINST 18 T03005). Near-total extraction by stripping is a large part of why H⁻ machines dominate the compact PET-isotope segment of the commercial industry (Kleeven & Zaremba, §4).

The price is exactly where a small machine is weakest. The extra electron that makes stripping possible is bound by only 0.754 eV (Calvo et al.), and collisions with residual gas remove it early: the H⁻ gas-stripping cross-section “is maximal for the energy range (0.1–300) keV” — a band that covers all of a 150 keV machine's spiral and much of a 500 keV one's (Calvo et al., PRAB 24, 090101, 2021). Survival falls as exp(−nσL) — gas density times cross-section times path length — and cyclotron paths are long: the compact AMIT design loses (14.6 ± 1.5)% of its beam to residual gas at ~10⁻⁴ hPa, i.e. 10⁻⁴ mbar (7.5 × 10⁻⁵ Torr) central pressure (Calvo et al.). Applying peak-band cross-sections to a tabletop spiral of a few tens of metres (path length calculator) gives an indicative screen, nothing more: losses of rough order 5–10% at 10⁻⁶ Torr, and most of the beam gone at 10⁻⁵ Torr. The honest number is the energy-weighted path integral of σ(E) over the actual pressure profile — the calculation Calvo et al. implement — and any serious H⁻ design should run it. Vacuum, more than any beam-optics consideration, decides whether the H⁻ option exists — Botman and Hagedoorn name stringent vacuum, alongside the H⁻ source itself, as the price of the scheme (pp. 3–4). The other famous H⁻ loss channel can be ignored: magnetic (Lorentz) stripping, which constrains fast H⁻ beams — the governing parameter is the rest-frame electric field γβcB, velocity times field, not field alone — has a lifetime that becomes astronomically long below a few MeV even at 4 T (Calvo et al.). The remaining cost is the ion source itself — making useful H⁻ current is a harder source problem than making protons, and it belongs to the ion sources page.

Where the documentation goes quiet

Between “first internal beam” and “working external beamline” the amateur record is nearly silent: one deflection channel used as a diagnostic (Rutgers), one extraction-capable machine under construction (Maryland 19-inch), and no documented amateur external beam. The site's own literature holdings showed the same gap from the other side, and closing it was the explicit goal of two 2026 acquisition rounds: the extraction domain of the design guide has gone from 21 rules to 71 as five hosted reports on deflectors, pulsers, regenerators and RF extraction were mined — a measured electrode-materials study and four more spanning 1948 to 1963. What that literature is about has not changed, though. Build reports document magnets, RF, and ion sources in loving detail, then stop at first beam; the machines that extracted were professional, and their extraction literature — the newly hosted reports included, along with the CERN Accelerator School lectures cited throughout this page — assumes MeV energies, isochronous fields, and engineering staff. The gap is now better documented rather than closed.

Being addressed. The collection has since acquired a primary source aimed squarely at this gap: UCRL-10654, Smith and Grunder’s 1963 report on the Berkeley 88-Inch deflector, hosted in full. It supplies what this page has had to infer — electrode materials ranked by measured spark damage, the VE relationship between gap voltage and cathode gradient, a stated design derating (1.5 × 10⁴ against 2.25 × 10⁴ achieved on the test model), and a septum-material trade with numbers on it. Its rules have not yet been extracted into the design guide, so the 21-rule count above still stands; this page will be revised when they are.

For a builder, the silence has two practical readings. Designing an extraction system for a small machine means working from professional sources scaled by the formulas above, with no peer build to copy — so over-provision adjustability (Livingston & Blewett, pp. 181–183) and measure the turn structure with a differential probe before committing to a septum position (Botman & Hagedoorn, pp. 11–12). And a documented result either way — a working small-machine deflector, or a careful account of why one failed — would be a real contribution to a record that currently lacks any.

Go deeper

• The design guide's extraction rules — all 21, each with formula, source, and page citation — and the beam-dynamics rules behind the tune and resonance claims.
• Path length & turns calculator — turn count, spacing of the last turns, and the septum verdict for any machine; energy, field & radius for the internal-target energy at each probe radius.
• The two CAS extraction tutorials this page leans on: Botman & Hagedoorn, Extraction from Cyclotrons (the backbone), and Heikkinen, Injection and Extraction for Cyclotrons; Kleeven & Zaremba's CAS 2015 review, extraction chapter, for the illustrated industrial view.
• Hosted classics: ANL-5907 (a working deflector system described end to end), ORNL-2648 (deflection studies and the axial-stability warning), and the 184-inch commissioning papers (precession, observed and explained, 1947).
• Glossary: extraction, deflector, septum, turn separation, precession, stripping foil, H−.
• Beam measurement — the differential probe and the deflected-spot energy check that turn extraction hardware into diagnostics.

Sources

• J. I. M. Botman & H. L. Hagedoorn, “Extraction from Cyclotrons,” CERN Accelerator School CERN-1996-002, pp. 169–190, doi:10.5170/CERN-1996-002.169 — turn-separation equation, precession, deflector and septum practice, field limits.
• P. Heikkinen, “Injection and Extraction for Cyclotrons,” CERN Accelerator School CERN-1994-001, doi:10.5170/CERN-1994-001.819 — first-harmonic and precessional extraction, resonance-crossing rules.
• W. Kleeven & S. Zaremba, “Cyclotrons: Magnetic Design and Beam Dynamics,” CAS 2015 proceedings, arXiv:1804.08961 — turn-separation scaling (§4, p. 44), extraction survey.
• C. Baumgarten, “Cyclotron Beam Extraction by Acceleration,” arXiv:2205.04124 — the (R/g)² turn-count condition.
• R. Baartman, “Cyclotrons: why/how are their dynamics different?,” JINST 18 T03005 (2023) — single- vs multi-turn extraction, phase acceptance, stripping as evasion.
• P. Calvo et al., “Beam stripping interactions in compact cyclotrons,” Phys. Rev. Accel. Beams 24, 090101 (2021), arXiv:2103.15583 — H⁻ gas- and Lorentz-stripping losses, AMIT numbers.
• The Argonne 60-Inch Cyclotron, ANL-5907 (1959) — hosted here; deflector system pp. 16–18 (PDF pages).
• The Oak Ridge Relativistic Isochronous Cyclotron, ORNL-2648 (1958) — hosted here; dee-width energy gain p. 95, deflection studies and axial-stability warning pp. 85–88 (printed pages).
• 184-inch commissioning memos: F. W. Yeater, MDDC-987 (two-probe precession proof); MDDC-1092 (precession frequency, n = 0.2 coupling); J. Vale, MDDC-984 (beam loss at n = 0.2) — all hosted here.
• M. S. Livingston & J. P. Blewett, Particle Accelerators, McGraw-Hill, 1962 — deflector sizing pp. 163–166, extraction efficiency p. 166, septum adjustability pp. 181–183, source off-centering p. 164.
• Rutgers 12-inch cyclotron documents: T. Ponter et al., Small Cyclotron Conference 2010 (deflector design, edge radii, feedthrough failures); T. W. Koeth, Ion Source Studies Part I (2006) (turn spacing at the probe) — see the library (search “Rutgers”).
• Y. Jongen, “New Cyclotron Developments at IBA,” Cyclotrons 2004 — self-extraction efficiency figures.