Cyclotron Magnet Design

A cyclotron magnet is designed backward from the field the beam needs, forward through an iron circuit that can carry the flux and coils that can drive it, and finished by a measure-and-shim loop that closes the gap between the field calculated and the field actually built. This page covers that whole chain for the classical small cyclotron. Why the field must have this shape — weak focusing, the field index, resonance — is derived in How a Cyclotron Works; superconducting magnets belong to the industrial machines. The 1940s–50s construction classics appear throughout because they published what scales down — proportions, model methods, tolerances — and the closing table anchors the same quantities at benchtop size.

The field is the specification

The beam imposes three requirements, and every later choice traces to one of them. First, the magnitude: it sets the RF frequency — f = qB/2πm, or 15.2 MHz per tesla for protons on the fundamental harmonic — and, with the usable radius, the energy:

T ≈ 48 (Z R B)² / A

with T in MeV, R in meters, B in tesla at the outer orbit, and Z, A the ion's charge and mass numbers — non-relativistic, which is fine below ~20 MeV (Humphries 1986). At 1.2 T and 10 cm of usable radius, protons reach about 0.7 MeV. Second, the radial profile: the field must fall with radius — by roughly 3 to 4 percent total in a small machine, less in larger ones — so the field index n stays between 0 and 1 and the beam is focused toward the median plane (Livingston & Blewett 1962, p. 161). Third, azimuthal uniformity: on every circle of constant radius the field should vary by less than 0.1 to 0.2 percent, the bound most operators of classical machines agreed on (Livingston & Blewett 1962, pp. 196–197).

One definition does silent damage in budgeting: R in the energy formula is the radius the last orbit actually reaches, and that orbit must sit inside the usable field — field within about 2 percent of the central value — not merely inside the iron. The usable region ends about half a gap-length inside the pole edge: 0.6g inside for a bare pole, 0.45g with edge shims (Livingston & Blewett 1962, pp. 158, 260). A 30 cm (12 in) pole with a 10 cm gap therefore offers roughly 9–10 cm of usable orbit radius, and the energy budget must be built on that smaller number.

The rest of the page follows the design sequence, which is still UCRL-476's 1949 "Summary of Initial Steps" with the model-and-measure loop appended (UCRL-476, p. 5): fix field and usable radius from the beam; budget the gap stack; compute ampere-turns with leakage; choose conductor and cooling to match the available supply; design the iron around gap and coils; then model, measure, and shim until the field built matches the field specified.

The iron circuit as a flux budget

Below saturation the gap field follows the air-gap formula, B ≈ μ₀ NI / g, because unsaturated iron consumes only 1–2 percent of the coil's magnetomotive force — essentially the entire NI drops across the gap (Zickler, CAS 2010). The relation is linear up to roughly 1 T (10 kG); at 1.8 T (18 kG), on the classical magnets Livingston & Blewett surveyed, the delivered field was only about 73 percent of the linear prediction, the balance lost to iron reluctance and leakage (Livingston & Blewett 1962, pp. 258–260).

The iron's job is to gather that flux and return it. Not all of the flux crosses the gap where the beam is: some fringes around the gap periphery and some bypasses the gap entirely through the space the coils occupy. UCRL-476 quantifies the penalty with a multiplying factor on gap flux, set by the gap proportions: a gap-height-to-diameter ratio of 1/2 doubles the flux the yoke must carry, 1/4 gives a factor 1.5, and 1/10 — "the region of small cyclotrons" — gives 1.2 (UCRL-476, p. 3). The yoke and pole cores are sized for the multiplied flux, and Figure 1 shows where it flows.

Saturation is the budget's hard ceiling. Ordinary soft irons begin saturating near 1.6 T (16 kG), some usable to 2.1 T (UCRL-476, p. 3); low-carbon 1010 steel goes strongly nonlinear above 1.5 T and is essentially saturated near 2 T (Tanabe 2005, SLAC-R-754). The steel is chosen for this: plain low-carbon steel for every flux-path part, because carbon is the impurity that costs permeability the most — and because B-H behavior also shifts heat-to-heat and with rolling direction, from consistent stock with a specified analysis (Tanabe 2005). The Argonne 60-inch forgings analyzed at 0.12 % C (ANL-5907, p. 9); ORIC specified 0.15 % C maximum (ORNL-2648, p. 112). The proportions follow from it: give the return yoke at least 25 percent more total cross-section than the pole cores, so that if anything saturates it is the cores, whose MMF (magnetomotive force — the ampere-turns) drop the coils can still overcome (UCRL-476, p. 4). The rule is a default for conventional proportions, and a model test settles it for any particular geometry — on the 86-inch's unusual layout, growing the yoke from 68 to 100 percent of core area bought almost nothing (ORNL-1196, p. 29). So do the joints: machine every yoke-to-yoke and yoke-to-core contact surface flush, because an unplanned air film in the return path is a parasitic gap in series with the real one (UCRL-476, p. 4).

Yoke topology is a trade documented since the CAS lectures: a C-frame gives open access to the chamber but is less rigid and needs asymmetry shims; an H-frame is symmetric and stiff at the cost of access (Marks, CAS 2004). Amateur builds have used both — the yoke is usually the single heaviest procurement, and scrap steel has worked when oversized and then verified by measurement: the Cyclotron Kids ran 1.6 T poles into a scrap-steel yoke sized so its flux density fell to 1.2 T (CYCLOTRONS 2010).

Gap and pole geometry

The pole gap is the most expensive dimension in the machine. Ampere-turns grow linearly with it and excitation power roughly as its square (Livingston & Blewett 1962, p. 194), and a wider gap also pushes the usable-field boundary further inside the pole edge. But the gap must house the vacuum chamber, the dees with their voltage clearances, any liners, and the beam's vertical excursions — and a very tight gap leaves no room for probes, source, or pumping, and amplifies the effect of every mechanical error (Zaremba, CAS 2005; Beeckman 2009). Classical practice settled near a gap of 1/8 the pole diameter for energy-oriented machines: 13–15 cm gaps on 107 cm (42 in) poles, 20–23 cm on 152 cm (60 in) (Livingston & Blewett 1962, p. 194). Small machines run relatively wider gaps; the Rutgers 12-inch uses a 10 cm gap on 30 cm poles.

The Argonne 60-inch shows what the gap actually buys — and that "gap" must always be qualified as iron-to-iron or beam aperture. Its deliberately roomy 56 cm (22 in) pole gap shrinks to a 30 cm working space once the two 13 cm-thick chamber lids are in, and to a 23 cm acceleration aperture after Rose rings and RF liners (ANL-5907, p. 8). Budgeting the gap means budgeting that whole stack — chamber walls, dee and liner clearances, beam aperture — and the beam is the smallest item in it.

Pole shaping does the rest of the geometric work. A taper from a wider base to the pole face keeps flux density roughly constant along the pole as leakage joins the flow — an 18 kG design wants a pole base about 24 percent larger in diameter than the face (Livingston & Blewett 1962, p. 193). A chamfer or rolled edge (45° works; the Rogowski profile is the analytic ideal) keeps the corners from saturating and softens the fringe (Heuer & Baumgartner 2009; Marks, CAS 2004). And the falling radial profile itself can be machined in: the 2 MeV design's slight convex face taper, 0.5 mm (0.020 in) from center to edge on each face of a 5.4 cm gap, gives the roughly 2 percent droop that weak focusing needs at that scale (Heuer & Baumgartner 2009). The Rutgers machine bolts all of this into removable 2.5 cm pole-tip inserts, so field-shaping experiments never touch the main forgings — a practice worth copying at any scale.

Coils: ampere-turns and watts

The excitation follows directly from the gap. In SI form NI = Bg/μ₀; UCRL-476's design sheet states the same thing in shop units, ampere-turns = 2.02 × gauss × inches of gap (UCRL-476, p. 14). So 1.5 T across a 5 cm gap needs about 60,000 ampere-turns; the Cyclotron Kids' 2 MeV design put 720 turns at 110 A (79 kA-turns) across a 5.4 cm gap for 1.6 T, a sensible ~15 percent margin over the air-gap minimum (Heuer & Baumgartner 2009). The magnet power calculator runs these numbers with the iron and leakage corrections.

Splitting NI between turns and amperes is a power-supply decision: many turns at low current means cheap thin wire but high, more dangerous voltage; few turns at high current means busbar-class conductors and a welder-class supply (Marks, CAS 2004). Neither option is the safe one: a low-voltage, high-current supply still carries arc-flash, burn, and fire hazards, plus the coil's stored inductive energy. UCRL-476 leaned toward large conductors and high current because it simplifies insulation and winding (p. 4). Either way the coil wants an approximately square cross-section hugging the pole — "either too flat or too tall a coil intercepts more leakage flux and thus wastes turns" (UCRL-476, p. 4; Zickler gives the same 1:1 to 1:2 aspect rule with a 0.6–0.8 packing factor).

The binding constraint is dissipation, and cooling sets the allowed current density. Close-wound coils in still air: about 1.2 A/mm² (750 A/in²) continuously, 1.55 A/mm² intermittently (UCRL-476, p. 4); Tanabe's modern rule is the same 1.5–2 A/mm² air-cooled ceiling. Thin copper cooling plates between pancake windings plus a fan buy about 2 A/mm² (UCRL-476, p. 5). Water-cooled hollow conductor runs 5–10 A/mm² (Tanabe 2005; Smirnov 2021). These densities are starting points for sizing, and no substitute for thermal design — insulation temperature class, duty cycle, flow verification, and thermal interlocks set the real limit (the coil rules collect them). A modest water-cooled example: 60 kA-turns at 5 A/mm² with a 1.2 m mean turn dissipates about 6 kW, whatever the turns-versus-current split, since P = ρj × NI × l_turn. UCRL-476's own 6-inch machine is the standing caution: its 6000-turn #13-wire coils saturated the iron at under 10 A — and overheated in under an hour (p. 5). At the next scale up, the Iowa State 1.5 MeV machine held 1.7 T across its gap with about 20 kW of water-cooled DC (McGuire 1961). The coil geometry calculator turns a candidate winding into length, resistance, and mass.

Two operational rules belong in the design, not the manual. Regulate the excitation to about 1 part in 1000 or better — a drifting field walks the machine off resonance — and the standard approach is precision current regulation on top of a reproducible magnetization cycle: a precision shunt against a reference held Iowa State's 17,000 gauss to ±4 gauss (Livingston & Blewett 1962, p. 194; McGuire 1961). And never open the coil circuit at current without a surge path across the coil; the stored energy will arc across whatever opens first. UCRL-476 specified a surge path — thyrite resistor or electrolytic dump tank — in 1949 (p. 6); the modern equivalent is a properly rated dump resistor or diode network permanently across the coil.

Model first, iron later

The construction classics agree on method more than on any number: prove the magnet on a cheap model before cutting full-size iron. The Carnegie Tech synchrocyclotron report is the best-documented case. The team started with a rough 6-inch model in lab-scrap steel just "to orient ourselves," then built a 2-inch model that "could be changed quickly and cheaply to give rough data over a wide range of parameters" (NYO-780, p. 2). From the 2-inch sweeps they extracted the coil height minimizing combined power-plus-steel cost, the pole taper's effect on efficiency — defined as the fraction of magnetic potential dropped in the gap — and the yoke-to-pole area ratio worth paying for. Two 9-inch models followed: the first, designed for 16 kG, proved too long when oscillator progress made 18–20 kG attractive; the second proved shimming workable at 20 kG. An 8.667-inch model with the optimized proportions then hosted the detailed shim studies, down to a measurement method good to 0.1 percent along a radius. The final model poles were machined from the very forgings destined for the full-scale magnet; when they produced 96.7 percent of central field at 96.5 percent of pole radius, "the design was frozen" (NYO-780, pp. 3–5). The 1500-ton magnet built from that model chain shimmed successfully at 20–21 kG on the first campaign (p. 6).

Oak Ridge ran the same discipline at both ends of the era. The 86-inch conversion was settled by tests on a 1/16-scale model — including the finding that growing the yoke from 68 to 100 percent of core area bought almost nothing (ORNL-1196, p. 29). Six years later the ORIC designers wrote the method's plainest statement: "many of the design parameters can only be determined by model magnet tests… the model magnet test program played the central role in the design of the magnet" (ORNL-2648, p. 23). Their 1/8-scale models were mapped on a 6 mm (1/4 in) grid with a rotating-coil fluxmeter to about 0.6 percent RMS point accuracy (pp. 24–26).

A 2-D field code now does what the 2-inch model did: cheap parameter sweeps before any metal moves. FEMM and POISSON solve the axisymmetric pole-and-gap cross-section on a laptop — yokes and features that break the symmetry need a 3-D solver (see modeling tools) — and industry runs the loop as hand calculation → 2-D code → 3-D code, with a good 3-D model agreeing with measurement to better than 3 percent in IBA's practice (Zaremba, CAS 2005). Two caveats from practitioners: expect calculated average fields to come out a few percent optimistic — one compact-magnet team measured 5 percent below calculation and had to cut the valley gap to recover it — and trust codes for differences between two designs more than for absolute values (Beeckman 2009). What the code does not replace is the other half of the model discipline: measurement of the thing actually built.

Measuring the real field

Nothing in the shim loop works without a trustworthy map. The classical instruments were the search coil on a pivoted arm feeding an integrating fluxmeter — a full-circle sweep must integrate back to zero, which doubles as the drift check (Livingston & Blewett 1962, pp. 283–285) — and the rotating-coil fluxmeter that gridded the ORIC models. The modern amateur splits the job in two: a Hall gaussmeter for mapping, and an absolute reference for setting the operating point, because an uncalibrated Hall probe alone is not accurate enough to establish the resonance condition. Iowa State used an NMR magnetometer readable to 1 gauss for the absolute number and reserved the Hall probe for relative work (McGuire 1961). The running machine is itself the final magnetometer: at resonance, the RF frequency gives the orbit-averaged field to high precision — but only the average (Livingston & Blewett 1962).

The Rutgers 12-inch project published the fullest amateur-scale mapping recipe: a stepper-driven stage at ~800 steps per inch, readings taken only while moving forward to cancel backlash, probe dwell time chosen empirically on the steepest gradient, and — before trusting any of it — a repeatability proof of 1600 moves returning within the resolution of a dial indicator (Koeth & Krutzler 2015). The professional literature adds the operating discipline: hold current regulation to 0.3 percent or better for the duration of a map — tighter when the errors being chased are smaller than that (ORNL-2648, p. 26); cycle the magnet to maximum current before settling at the operating point, so hysteresis is reproducible run to run (Zickler, CAS 2010); and find the magnetic median plane with a pair of opposed search coils — it can sit well off the geometric midplane, a half inch in one documented case (Livingston & Blewett 1962, p. 197).

Shimming closes the loop

Classical practice assumed no magnet would be machined straight to its final field: every published machine was mapped and shimmed after machining. The design plans for a correction stage; shims are the precision trim on deliberately approximate iron (ORNL-1196, p. 35). The corrections divide by what they fix, as in Figure 2 — and what an uncorrected field error does to the beam runs live in the beam dynamics laboratory.

For the radial profile, the standard tool is a flat pyramidal stack of thin iron discs of graded diameter in the shimming gap between chamber and pole: MIT used four 0.5 mm (0.020 in) discs of graded diameter; Argonne stacked twelve 1/16-inch discs from 15 cm up to 100 cm diameter (Livingston & Blewett 1962, p. 195; ANL-5907, p. 9). Thin sheets are the rule — Lawrence's shims ran 6 mm (0.25 in) or less, because a thick shim makes the field change too abruptly at its own edge (Morrow 2015). For the fringe region, a raised iron ring near the pole periphery — the Rose ring — holds the field up at large radius; Argonne's sat at 84–97 percent of pole radius. Size rings cautiously: an oversized ring, or a correct one run at lower field, produces a local field minimum that defocuses (Livingston & Blewett 1962, p. 196). Azimuthal errors are the least forgivable, since they drive orbit distortions: sector- and wedge-shaped shims after mapping brought classical machines under 0.1 percent, and the 86-inch's machined contour shims held azimuthal variation below 0.2 percent (Livingston & Blewett 1962, pp. 196–197; ORNL-1196, p. 35).

The 86-inch commissioning shows the loop's honest texture. First came patchwork: 5 cm steel discs carrying up to ten layers of 0.4 mm shim stock, rearranged through repeated field plotting until the pattern worked — then the whole pattern was machined into solid 5.7 cm plate, because the layered stacks outgassed at every pumpdown (ORNL-1196, pp. 100–101). Iterate in cheap, reversible iron; freeze in machined iron. Copper corrections come last and stay small: auxiliary pole coils to steer the beam onto the magnetic median plane (the 86-inch wound 65 turns per pole; Rutgers powers upper and lower main coils from separate supplies for the same effect), and trim coils for fine radial profile — but measurements at Houghton found bucking coils could move the n = 0.2 radius only ~2 mm while costing 20 percent of peak field, so copper does not rescue a wrong pole contour (ORNL-1196, p. 35; Morrow 2015).

Forces and tolerances

The field the beam rides is also a mechanical load. Magnetic pressure is B²/2μ₀ — about 1.0 MPa (150 psi, ten atmospheres) at 1.6 T (Tanabe 2005). Across a 30 cm (12 in) pole that is roughly 74 kN (16,700 lbf) of attraction between the pole faces, and UCRL-476's shop-unit version of the same formula — pounds = (kilogauss)² × in² / 1.735 — gives the identical number (p. 14). Fastenings, pole mounts, and any structure bridging the gap are designed against this force with normal structural margins on top of the static estimate — and it dwarfs atmospheric load: ORIC's structure was designed for 4.7 MN (1,055,000 lb) of magnetic force against 0.27 MN (60,000 lb) of vacuum load (ORNL-2648, p. 113). The load also moves the iron: Carnegie Tech measured its model poles deflecting 0.05–0.10 mm under magnetic load, at model scale (NYO-780, p. 4).

Tolerances translate directly into field errors: to first order, below saturation, gap-height error is field error at the same fractional size. The working figures from machines that ran: gap variation under 0.13 mm (0.005 in) at any radius on the Argonne 60-inch (ANL-5907, p. 8), pole faces parallel to about 1 part in 50,000 of pole diameter in classical practice (Livingston & Blewett 1962, p. 193), and pole-tip machining held to ±0.06 mm (±0.0025 in) at Carnegie Tech (NYO-780, p. 6). At tabletop scale the first two are careful tool-room work; parallelism at 1 part in 50,000 — 6 μm across a 30 cm pole — calls for surface grinding and real metrology. Either way, the figures belong on the drawings and get checked, and the magnet's currents, stored energy, and crane-weight iron deserve the same respect as the machine's other hazards.

Proven numbers at small scale

Documented small machines cluster tightly, and their parameters make useful sanity anchors for any new design. A dash means the report did not state the value. Row sources: UCRL-476 p. 5; Yuly (Cyclotrons 2013); McGuire 1961; Koeth 2015; Heuer & Baumgartner 2009 with the Cyclotron Kids' CYCLOTRONS 2010 paper — each also carried, with formulas and page numbers, in the design guide.

Go deeper

• Design guide: magnet domain — 322 sourced rules — and the 103 coil rules, each with formula and page citation.
• Magnet power & amp-turns and coil geometry calculators.
• NYO-780 — the model-magnet chapter is the best short course in this method; UCRL-476 is the small-machine recipe it distills to.
• Free modeling tools — FEMM and friends for the 2-D magnet cross-section.
• Why the field must fall with radius — the physics this page takes as given.

Sources

• E. Creutz (ed.), Design and Construction of Synchro-Cyclotron, NYO-780, Carnegie Institute of Technology, 1950 — model-magnet program, shim studies, engineering design. Hosted copy.
• L. Wouters, General Recommendations for Design of Small Cyclotrons, UCRL-476, 1949 — iron and coil recipes, leakage factors, design sheet. Hosted copy.
• The Argonne 60-Inch Cyclotron, ANL-5907, 1959 — as-built tolerances, gap stack, Rose rings, steel analysis. Hosted copy.
• The Oak Ridge 86-Inch Cyclotron, ORNL-1196, 1952 — 1/16-scale model, patchwork-to-machined shim campaign. Hosted copy.
• H. G. Blosser et al., The Oak Ridge Relativistic Isochronous Cyclotron, ORNL-2648, 1958 — model-magnet method, measurement discipline, magnetic forces. Hosted copy.
• M. S. Livingston & J. P. Blewett, Particle Accelerators, McGraw-Hill, 1962, ch. 8 — the classical-cyclotron magnet chapter behind most of the design guide's magnet rules.
• J. Tanabe, Iron Dominated Electromagnets, SLAC-R-754, 2005 — steel properties, current densities, forces. OSTI 878409.
• Th. Zickler, "Basic design and engineering of normal-conducting, iron-dominated electromagnets," CERN Accelerator School, 2010. arXiv:1103.1119.
• S. Zaremba, "Magnets for Cyclotrons," CAS 2005 (CERN-2006-012, p. 253); W. Kleeven & S. Zaremba, "Cyclotrons: Magnetic Design and Beam Dynamics," CAS 2015. arXiv:1804.08961.
• N. Marks, "Conventional Magnets I & II," CAS Baden 2004 — yoke topologies, coil sizing.
• W. Beeckman, "Cyclotron Magnets," ECPM37 lecture, Groningen, 2009 — gap trade-offs, iron-first shaping, field-code practice.
• T. W. Koeth & J. E. Krutzler, "Field Mapping in Cyclotron Magnets," Rutgers 12-Inch Cyclotron project, 2015 — amateur-scale mapping hardware and verification.
• Machine reports cited in the table via the design guide: McGuire, Iowa State 1.5 MeV (1961); Heuer & Baumgartner, Design of a 2 MeV Cyclotron (2009) and the Cyclotron Kids CYCLOTRONS papers (2010, 2013); Yuly, Houghton College (2013) and Morrow, Focusing in the Houghton College Cyclotron (2015); Koeth, Rutgers 12-inch (2015); Smirnov, The Cyclotron and Its Modeling, Phys. Part. Nuclei 52 (2021).