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 for this energy estimate below ~20 MeV (Humphries 1986); the same 1–2 percent relativistic mass shift matters to RF resonance far sooner, and is what caps fixed-frequency machines near that energy. 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 — within about 2 percent of the intended profile, the criterion Livingston & Blewett use to locate where the fringe takes over (distinct from the deliberate weak-focusing droop inside it) — 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 (the safety page carries the thresholds and controls). 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.

The permanent-magnet alternative

Everything the coil section budgets (ampere-turns, watts, cooling water, the regulated supply) can be deleted at once: rare-earth permanent magnets can drive the same iron circuit with no power at all. Halbach's 1987 scaling argument says where the trade favors them. Shrink an electromagnet at fixed field and the coil current density must rise as the inverse of the linear size, until cooling or conductor limits force the field down. A permanent-magnet system scales to any size with no loss of field. So below some critical size, which depends on field, configuration and material, “a permanent magnet will always produce higher fields than an electromagnet” (Halbach 1987, dg-2122). Where either technology reaches the field, the choice turns on cost and convenience, and favors permanent magnets less as the smallest magnetically relevant dimension grows (dg-2123). A small cyclotron's magnetically significant dimensions are a gap of a few centimetres and a pole of tens of centimetres, which is why the idea keeps returning at this scale. What the deletion costs is the main-field current knob: nothing turns a PM field up, down, or off, and what can be bought back (trim, shunts, tuners, compensation) is priced below. The Berkeley Cyclotrino team, weighing exactly this choice, took an electromagnet “because the field could be tuned” (Cressman 2006). The team that went the other way was at LBNL. Their compact cyclotron mass spectrometer put 1 T on 30 cm-diameter iron poles from SmCo blocks alone, with no coils, no supplies and no magnet cooling, and they judged the resulting instrument portable enough for hospitals, trucks, or airplanes (Clark et al. 1995, dg-1557). Los Alamos, building variable-field PM beamline dipoles in 1993, reported fabrication about 20 percent cheaper than comparable electromagnets with their supplies and coolant systems, and noted that the water-flow, over-temperature, over-voltage and ground-fault interlocks went with them (Barlow et al. 1993, dg-2121): one data point for small dipoles, and the stored field keeps hazards of its own. Both cyclotron records are design decisions rather than beam results, and the section's closing part returns to that distinction.

Material: the linear region and its limits

Material choice is settled, and the reason is the shape of the demagnetization curve. Rare-earth material (NdFeB, Sm₂Co₁₇) has a straight-line B–H characteristic through the first quadrant and much of the second, with slope close to 1 (Halbach measured 1.04 on the SmCo of 1980), so the working point moves reversibly along that line as the gap changes. Push it into the knee and the material recoils on a lower parallel line; the original magnetization returns “only by driving the material very far into the first quadrant” (Halbach 1980, dg-2130). That makes partial demagnetization permanent for an assembled magnet, since nothing in the assembly produces a magnetizing field of that size. Alnico has no such straight line: each deeper excursion — a widened gap, a removed magnet, steel tools across the gap — walks its operating point down a new minor loop and permanently costs strength (Leupold & Potenziani 1996). Halbach's whole design method rests on the straight line. He calls the material “analytical”: permeability near 1 plus equivalent current or charge sheets on the block surfaces, which lets vacuum fields superpose block by block and lets a hybrid system be designed with a linked set of analytical formulas rather than a general field solver (Halbach 1987, dg-2125; dg-2131). The one standing assumption is that no block is ever driven beyond the knee (dg-2124). Low-coercivity NdFeB grades run hot can still be driven irreversible. The check is every block's worst-case working point, including blocks in reverse-field regions near the gap and pole edges, against the supplier's curves over the full temperature range and under assembly and fault conditions. The 1987 remanence range of 0.8 to 1.2 T predates today's grades; the supplier's curve for the actual grade governs.

Four practical limits ride on the same curve. Blocks come thin: in 1980 manufacturing capped a rare-earth block's smallest dimension at a few centimetres, so a pole-driving volume is assembled from many pieces, and Halbach counted the labour of making and handling them as part of the cost (Halbach 1980, dg-2134). Price per volume depends on grade, piece size and shape, dimensional and magnetic tolerances, and order volume (dg-2132), which is the economic case for the pole-filter principle below: a design whose iron pole averages block errors can buy looser, cheaper blocks. SmCo can be taken to 200 °C and higher with precautions, but the temperature “should always be fairly uniform and, for that reason, should not be changed too rapidly” (Halbach 1980, dg-2133). NdFeB grades rate lower, grade by grade, and a temperature gradient across an assembly distorts the field rather than merely shifting it. And an iron-free rare-earth magnet has a hard ceiling: for given geometry the obtainable field is limited by the material (dg-2136). The hybrid escapes that ceiling in principle, because replacing all the magnet material inside a closed scalar-potential surface with high-permeability steel leaves the performance unchanged, and Halbach judged this an attractive design for a dipole whose working volume is much wider than it is high (Halbach 1980, dg-2135). A cyclotron gap is that dipole. What limits first in a given hybrid — pole or yoke saturation, the recoil line, or leakage — comes from a circuit calculation or a field solver, early.

Two screening numbers size the idea before any modeling. A simple open PM assembly reaches gap fields up to roughly half the material remanence, about 0.6 T from 1.3 T NdFeB, a feasibility screen rather than a hard ceiling. Past it, a closed yoke, cladding, or a Halbach-class geometry buys more far more efficiently than added material in the same arrangement (Leupold & Potenziani 1996). And in the ideal relation the amount of material follows the particle, not the pole. Required PM volume scales with (B_gR)², that is, with energy, times gap height, independent of pole radius, and the magnet is used most efficiently at the maximum-energy-product point B_m = B_r/2 (Antokhin et al. 2006). Antokhin's 10 MeV design spent 900 kg of material across a 20 mm hill gap; the relation's ideal arithmetic prices a 1 MeV machine at the same gap near 90 kg, before leakage, pole margin, trim provision and block packing raise it. The one built cyclotron magnet gives a measured ratio to set beside the scaling. LBL energized 1 T across a 1.6 cm gap on 15 cm-radius poles with about 200 SmCo magnets totalling about 6000 cm³, roughly five times the gap volume (Li et al. 1992, dg-2111), a ratio that belongs to that geometry's leakage paths and is not a constant.

The circuit that works: iron poles, magnets outside the vacuum

The circuit that works at cyclotron geometry keeps the iron and replaces only the excitation. In the LBL design (the magnet concept was Klaus Halbach's), the PM material per pole splits into a “crown” above the pole, driving flux axially down into pole and gap, and a “barrel” outside the pole rim, driving flux radially inward. A cylindrical yoke closes the loop; blocks on one pole are magnetized toward its face, on the other away from it (Clark et al. 1995, dg-1548; Young et al. 1993). The iron pole stays the precision element of the pole-geometry section: its face is the surface the gap sees, and it carries the high-accuracy machining. The magnet blocks behind it can be ordinary stock, coarsely arranged, because the intervening pole filters the short-wavelength part of their block-to-block variation (dg-1549, dg-1550). Halbach stated the general principle for his hybrid quadrupole: the field distribution is controlled by the iron surface and is very insensitive to magnet-material properties as long as the iron permeability stays large compared with 1 (Halbach 1987, dg-2126). A built NdFeB dipole confirmed it at 1 T. The ASTRID2 “Green Magnet” at Aarhus reached homogeneity below 0.6×10⁻³ against a 1×10⁻³ specification, its authors writing that “the pole shape will determine the field homogeneity” (Bødker et al. 2014, dg-2137). Low-order errors still reach the midplane: a remanence spread around the azimuth, a displaced block, a gap or yoke asymmetry. The sorting-and-shimming part of this section is about them.

How the blocks were actually bought, cut and placed is in the 1992 design report. LBL bought SmCo as larger blocks, cut and ground them to size, and glued them to the flux-return yoke for permanent mounting (Li et al. 1992, dg-2111). The annulus between the round pole and the cylindrical yoke was filled with rectangular stock blocks rather than custom arc segments. Seen from above they form an 18-sided polygon, inner corners touching, small wedge-shaped gaps at the outer corners, with ledges machined in the yoke to align them (dg-2112). Coarse field level was set by leaving blocks out of the crown layers, the pole smoothing the voids (dg-2113); after any such omission the mean field and the harmonics get re-verified, by map or by 3-D calculation. NSRRC assembled large blocks by gluing smaller magnetized units together, with dimensional and per-block flux acceptance set from a simulation of their field effect (Hsu et al. 2025). The gap was made as short as the chamber allowed, because every millimetre of it is paid for in magnet material. The inner pole faces form part of the vacuum wall, and an aluminium spacer ring between them sets the 1.6 cm gap and carries the radial ports (dg-2109). The magnet material itself, whose vacuum properties the authors judged good, was still kept outside the vacuum by O-ring seals in the spacer ring and by welding the beam-entrance and pump-out tubes to the pole faces (dg-2110). A separate chamber with its own lids inside the gap costs the lid thickness twice in gap height; the pole-as-lid design trades that for seals at the pole faces and the outgassing of pole steel.

The design sequence is the one this page already prescribes, with three substitutions. The analytic first pass is a flux budget: LBL's program calculated the direct and indirect fluxes of candidate configurations to fix the dimensions of magnets, poles and yoke, and POISSON then verified and optimized the survivor (Clark et al. 1995, dg-1551). Leupold's permeance method is the same calculation by hand, trusted to about 2 percent on total flux but only ~30 percent on local field (Leupold & Potenziani 1996). The 2-D verification pass needs a solver that handles permanent-magnet materials (PANDIRA in the POISSON family; see modeling tools), and for the absolute field it needs a 3-D check or a measured correction, because the material cannot be turned up after assembly. The Los Alamos dipole, with poles about 10 cm long, is the cautionary number: its 2-D FLUX2D model, lacking the third-dimension fringe loss, overestimated the central field by about 40 percent, so the 2-D results were normalized to measurement and the final design modelled in TOSCA (Barlow et al. 1993, dg-2118). A round cyclotron pole has a different 2-D/3-D relationship, which is the point: it has to be checked for that pole. A fast fixed-magnetisation model, blocks as sheet currents with an effective remanence calibrated once against a full nonlinear solve, is the tool Brookhaven used for shim optimisation (Brooks et al. 2017, dg-2169) and is the natural third pass for a cyclotron whose iron stays unsaturated. LBL took its magnet to fabrication on a 2-D prediction of ±2 parts in 10⁴ (Young et al. 1993), a prediction of the nominal design; the as-built number appears under tuning below.

Two physics cautions ride along. LBL could run a deliberately flat field only because electrostatic focusing at RF harmonic 15 replaced the weak-focusing droop at its very low energy (dg-1546). A proton machine on the fundamental still needs the falling profile of Figure 2, which a PM-fed pole can be contoured for exactly like a coil-fed one. And a magnet with no free currents obeys a constraint a coil-driven one does not. By Ampère's law the line integral of H along the whole axis is identically zero, so for a hybrid dipole with a hole along its centre line the on-axis field must reverse somewhere outside the gap (Halbach 1987, dg-2129). A PM cyclotron with an axial-injection bore through pole, magnet layer and yoke is that case, and the injection optics belong to the full on-axis field including its reversals. There is also a middle option. Permanent-magnet material set on the poles of an electromagnet, oriented against the coil-driven flux, reduces the flux density in the pole, letting the magnet reach higher aperture fields or be made smaller while the coil keeps the adjustability (Halbach 1987, dg-2128). Halbach notes the resulting saturation curve is asymmetric, the gain lying in the direction the pre-load assists.

Setting the field without a current knob

Correction without a current knob is subtraction, shunting and rotation instead of dialing, and the literature now holds five mechanisms, each with a measured or designed range. The first is margin in the removable direction. LBL installed its barrel magnets slightly oversize on purpose, planning to cut them back after mapping; the as-built field measured flat to 7 parts in 10⁴ over the 5–12 cm acceleration region against a 2 parts in 10⁴ design target, the excess anticipated and waiting for trim (Clark et al. 1995). The maps were taken across four midplane diameters so an azimuthal asymmetry of the assembly would show (dg-1553), and the margin was planned from the start (dg-1552). It is the shim loop of the shimming section below, with the margin deliberately built in the removable direction. The second is the rotating tuner, the mechanism the 1992 design report planned for fine adjustment. Rectangular magnets with radially directed magnetization sit in rotatable tubes lying along chords in the top and bottom magnet volumes, in place of ordinary blocks; turning a rod sets its field parallel, antiparallel or at any angle to the main field and raises or lowers the midplane field with no coil current (Li et al. 1992, dg-2108). The report says the tuning magnets “significantly improve the operational requirements” over an electromagnet instrument (dg-2115). The tuning range the rods achieve is not reported, so the trim authority for another magnet is computed or measured; and a tuner rod feels torque from the main field, which its drive must hold.

The third is the moving iron shunt. The Los Alamos variable-field dipole (a SmCo block between iron poles, a 2.8 cm-high by 8 cm-wide aperture, poles about 10 cm long) varies its central field continuously from 0.07 to 0.3 T by moving an iron shunt across the back of the magnet. A flat shunt gave a steep, non-linear tuning curve, little flux diverted until the shunt gap approached the dipole gap; a C-shaped shunt with tapered tips that saturate linearized the curve, extended the travel, relaxed the positioning precision and reduced the force on the shunt (Barlow et al. 1993, dg-2116). The drive is part of the design: a lead screw from a small stepper through a 100:1 reduction, preloaded against the magnetic pull, an electric brake to lock the setting, microswitch limits doubling as the position reference, and step counting for readout (dg-2119). For another movable element the force is calculated or measured over the whole travel, because it can vanish or reverse, and step counting is a home datum that slip or a power loss can lose. Each dipole was then calibrated with an NMR probe, central field against shunt position, and reproduced its field to better than 0.1 percent (dg-2120): the per-unit calibration table that turns a repeatable mechanical setting into a known field. The fourth is a small trim coil on a PM main field. The ASTRID2 dipole adjusts ±3 percent with air-cooled trim coils on a 20 A supply, with no cooling water, saving at least 99 percent of the 4 kW the equivalent electromagnet would draw (Bødker et al. 2014, dg-2138). The NSRRC hybrid dipole shows the same buy-back on one measured prototype (Hsu et al. 2025): a 42-turn trim coil worth ±1.25 percent of field over ±15 A, at 0.086 percent per ampere, and movable outer iron plates, set with aluminum spacers, as a zero-power coarse adjust worth +1.85 percent closed and −0.14 percent opened. The intended workflow pre-adjusts each magnet on the bench and leaves only fine trim to the coil. Halbach's hybrid undulator adds the fifth, local rather than global: threaded steel tuning studs through a steel backing plate behind the blocks (Halbach 1987, dg-2127), whose field change per turn is calibrated by measurement.

Antokhin's design even bought back the off switch: equal PM volumes on a rotatable rotor cancel the gap field when turned 180°, at the price of a drive built for 1800 N·m of torque against the field. The measured residual at that zero setting was up to 560 gauss, which its builders called enough for easy disassembly (Antokhin et al. 2006); the safety page reads the same number the other way around, as a field that still snatches tools and is never an off state. What any of these mechanisms must cover is set by the machine: the resonance search at commissioning, the drifts of the next part, and whatever first beam reveals. The Fermilab Recycler's 484 permanent magnets all met a production specification of central field within 1 part in 10³ of ideal and higher harmonics below 1 part in 10⁴ of it (Volk 2012, dg-2157). The ring still needed small skew-quadrupole and sextupole correctors that appeared only in operation (dg-2163), so a PM cyclotron reserves mounting points for trim it has not yet identified. What a fixed-field machine's resonance search looks like, sweeping RF frequency against a field that will not move, can be watched on the site's Virtual Cyclotron Laboratory (hosted; illustrative, not a calibrated replica of any machine).

Temperature: the drift the regulated supply used to hide

Antokhin's team measured −0.07 percent per °C on their built magnet and judged it acceptable for normal cyclotron work (Antokhin et al. 2006). The material itself is worse. Kim and Doose's Argonne test dipoles, sintered NdFeB blocks driving soft-steel poles and a return yoke, measured −1.1×10⁻³ per °C uncompensated, a coefficient the authors attribute to NdFeB's low Curie temperature, about 300 °C (Kim & Doose 1997, dg-2145). Brookhaven's uncompensated Halbach magnets showed strength errors of a few parts in 10³ that the authors called expected from the same coefficient with no temperature control of magnet or measurement area (Brooks et al. 2017, dg-2168). Strontium ferrite is worse again: a Fermilab Recycler quadrupole built without compensator, frozen for 48 hours and warmed slowly over four days, fitted 0.18 percent per °C (Brown et al. 1999, dg-2152). With fixed RF the resonance condition turns field drift into frequency error one for one, so an uncompensated NdFeB circuit with the Argonne coefficient moves the cyclotron frequency by about 1.1 kHz per °C for every megahertz of RF. Over the 10 °C seasonal swing of an unconditioned workshop even Antokhin's milder figure is 0.7 percent of field: about thirty times the ±2.4×10⁻⁴ Iowa State held by current regulation, and far more than the resonance tolerates without retuning.

The passive countermeasure is a parallel flux shunt of low-Curie-point alloy. As temperature rises the shunt's permeability falls and it diverts less flux from the gap, offsetting the magnet's falling remanence; for sensitivity the Curie point should sit close to ambient, which Kim and Doose found in 30–32 percent Ni-Fe (dg-2146). Their recipe: 25 mm-wide strips of 30 percent Ni-Fe, Curie temperature 60 °C, inserted between the blocks at an optimum strip-to-NdFeB volume ratio of 20 percent (dg-2147), which brought the coefficient below 2×10⁻⁵ per °C over ±6 °C (dg-2145). ASTRID2 compensated its NdFeB below 30 ppm per °C over 15–35 °C against a 50 ppm specification (Bødker et al. 2014, dg-2139). On the NSRRC prototype, every 2 mm of Ni₃₀Fe₇₀ plate cost 0.4 percent of static field, and 4 mm cut the drift from 0.043 to 0.027 percent per °C, a 37 percent reduction bought with a known field loss (Hsu et al. 2025). Three cautions come with the strips. Compensation is flat only near a design temperature. Because the strips are nonlinear (relative permeability in the magnet calculated below 5), the Argonne curve has a turning point: held to its 2×10⁻⁵ over 31 ± 6 °C and, as the text states it, below 1×10⁻⁴ per °C over the whole 25–45 °C cycle (dg-2148). The design question for a cyclotron is whether its operating band, room swing plus RF and filament heat in an enclosure, stays on the plateau. The set point moves in the first cycles: the compensated field rose about 1×10⁻⁴ over the first three slow thermal cycles while the compensation temperature fell about 1 °C, then held over seven more (dg-2149). So the finished magnet is cycled through its range before the final map, and re-measured after. And the alloy's Curie point varies batch to batch with chemistry, heat treatment and handling, which is why the Recycler builders mixed strips from different batches at random and set each magnet's strip count empirically. The nominal Curie point of the same Ni(30%)Fe(70%) differs between accounts: 45–50 °C in Brown et al. 1999, 40–45 in Volk, 60 in Kim and Doose (dg-2153).

How much compensator, and how to prove it, has a production answer. Volk states that for the Recycler ferrite the degree of compensation is linear in the amount of compensator, so a 20-fold reduction, from 0.2 to 0.01 percent per °C, needs the amount set to 1 part in 20, which “poses no difficulties for production” (Volk 2012, dg-2158). Fine tuning was done by adjusting compensator at the magnet ends, like strength trimming. The same bookkeeping sizes the job for a rare-earth magnet (the wanted coefficient divided by the raw one is the fractional accuracy the compensator amount needs), with the caveat that Kim and Doose's strips were nonlinear, so the relation is a first estimate. The acceptance test was two points: every Recycler magnet was frozen to 0 °C during manufacture, its field measured at 0 and 22 °C, and the compensator steel adjusted until the change sat within the 0.01 percent tolerance, the fleet averaging about 0.004 percent per °C rms (dg-2159). Since compensated curves bend, the two points should bracket the band the machine will run in. Where even that is too loose, regulate the temperature instead. The Recycler's high-beta quadrupoles, with a requirement ten times tighter, could not be held by strips against daily and seasonal tunnel variation, so electric heating blankets under feedback hold them at 32 ± 1 °C (Brown et al. 1999, dg-2154). Each was pre-warmed overnight to that temperature before its final adjustment, so the field was set where it would run. The measurement itself has to be matched to the level sought. Kim and Doose used Hall probes calibrated to ±10⁻⁵ of reading per °C, temperature sensing accurate to 0.5 °C, and ramped their incubator at about 0.8 °C per hour so the magnet stayed isothermal (dg-2150). ASTRID2's seven-month stability record, a weighted mean deviation of 212 ± 12 ppm, was taken with a probe whose own coefficient of −80 ppm per K was never corrected and exceeds the compensated magnet's drift (dg-2142). A PM field number means little without its temperature, the way NSRRC quotes 0.7545 T at 20 °C (Hsu et al. 2025).

Ageing and radiation

A permanent magnet also loses strength on its own, slowly. On the Recycler's strontium-ferrite gradient magnets the loss after magnetisation followed a logarithmic law, Δ(M₂/M₁) = 9.0×10⁻⁴ log(t₂/t₁), which Brown et al. extrapolate to a 0.35 percent loss over a 20-year life; the figure reproduces only with a base-10 logarithm from a one-day origin, although the text calls it the natural log, one of several small inconsistencies the design guide brackets in these two Fermilab reports (Brown et al. 1999, dg-2151). Volk's record of the first production magnet, measured weekly for a year, then monthly, then every six months for ten years, is reported as 0.35 percent lost in the first 3 years and stable since (Volk 2012, dg-2160). That statement is at odds with the 1999 fit's 20 years and with the plotted data, which were still falling a decade on. What transfers is the measure-over-log-time method, and the consequence: in a fixed-frequency machine the loss is a resonance shift, so the field is measured after initial ageing and periodically after, and the drift carried in the operating calibration or trimmed out. An NdFeB cyclotron magnet needs its own record; the Recycler coefficient is strontium ferrite in a ring dipole.

Radiation is the other slow loss, and the ASTRID2 group measured what protects against it. Irradiating N35 and N35M samples under a controlled reverse bias with 100 MeV electrons, they found the fluence for 1 percent demagnetization correlates exponentially with the normalized work-point margin (H_ci − H_d)/H_ci (Bødker et al. 2014, dg-2143). A high work point in an iron-dominated circuit, a low internal demagnetizing field, buys robustness. No proton or mixed-field tolerance follows from an electron test, and the thermal margin is a separate check against H_ci at the worst-case block temperature. The same authors put samarium-cobalt's radiation resistance about three orders of magnitude above NdFeB's in their test context, at the price of about 25 percent lower remanence and about twice the magnet mass, and judge it mostly relevant to low-field magnets (dg-2144). A cyclotron's choice between them weighs mass and cost against radiation and temperature robustness, and the lifetime estimate needs particle- and grade-specific data.

Block sorting, the first harmonic, and shims computed from a map

The pole filters short-wavelength block errors. A coherent error passes straight through. Brown et al. note that in a permanent magnet strength is controlled by integrated brick strength, where an electromagnet's is controlled by the gap. Brick count and placement alone, the authors say, could produce longitudinal nonuniformity of as much as 20 percent, on top of a brick-to-brick strength spread of about 1 percent; the Recycler controlled it by sorting bricks on measured strength and adjusting total volume with pieces as small as 0.5 × 0.5 × 6 inches (Brown et al. 1999, dg-2155). Around a cyclotron pole the same spread maps into azimuthal variation, including a first harmonic, before the iron smooths it, so blocks are sorted by measured strength before placement. Designing around one block size helps the sorting as much as the buying: ASTRID2 used a single N48M block with dedicated force tooling, which let one person mount nearly 500 magnets in a week (Bødker et al. 2014, dg-2140). The measurement that follows reduces the map to what the beam feels. The Recycler scanned each magnet's longitudinal field with an automated Hall probe, computed first and second moments about the mechanical centre, and used the first moment to set each magnet's installation position (dg-2156). A cyclotron's counterpart is a polar map reduced to its first harmonic about the mechanical axis. A first harmonic can come from a translated axis or from intrinsic errors with different radial dependence, so the map is combined with an orbit calculation before anything is moved.

Then the shim is computed, machined and measured, in that order. Each Recycler gradient magnet was measured with a rotating coil up to the 12-pole; a program computed an end-shim shape to adjust the quadrupole, skew quadrupole, sextupole and octupole terms and wrote the wire-EDM code to cut it. Shims were ready within 24 hours; one iteration usually sufficed and rarely two were needed (Volk 2012, dg-2161). On the Recycler quadrupoles, whose 1008-steel poles were cold-extruded to a hyperbola with no further machining, steel washers in the corners adjusted gradient and the low harmonics, and washers on the pole faces removed the 12-pole (dg-2162): small removable steel at defined places, an adjustment path with no power. Brookhaven took the same loop furthest, for magnets built without precision machining. An iron wire in a locally uniform field magnetises, for ideal iron and in the authors' notation, to M = 2B₀ (μ₀M in tesla) and acts as a small induced dipole whose strength scales with the wire's cross-section (Brooks et al. 2017, dg-2164). The team measured each assembled Halbach magnet on a rotating coil, added the observed errors to the 2-D field model, and optimised the radii of 32 to 64 iron wires just inside the bore to drive the harmonics back to zero (dg-2165). The solver is a regularised SVD pseudo-inverse of the response matrix of harmonics to small shim changes, so that noisy measured errors do not demand large shims (dg-2166). The wires were cut from a few stock gauges to lengths holding the same volume as the ideal-radius wire, and the magnets cost $1,070 and $779 apiece in 2017, almost all of it block material (dg-2170). One iteration cut the error multipoles by about a factor of 4 on average, to 1.22–3.41 units of 10⁻⁴ at 10 mm for the quadrupoles; the abstract's “roughly 1 unit” is the method's aim, approached on one prototype only after a second pass (dg-2167). Two limits on what shimming does. It is radius-dependent: the same magnets scored 5.9–13.8 units at 20 mm, so the radius at which a cyclotron field is specified matters as much as the figure. And harmonic shimming does not fix overall strength; Brookhaven's mean gradients still sat a few parts in 10³ off design after shimming, and the paper's own readings show strength moving with the shims (dg-2168), so mean field and harmonics are separate targets in one response model. The same linear-response method would take an isochronous or weak-focusing radial profile as its goal, an extension the source does not test.

Forces and access: no routine off-state

Assembly is where the technology bites hardest, because the field is on before the parts meet and stays on; the safety page treats a PM machine as permanently energized because it is. Attraction during NSRRC's assembly could exceed several hundred kilograms. The documented answer is engineered handling: custom fixtures with mechanical guides, magnetic shielding, and locking mechanisms for staged installation. Inserting the magnets last, or applying a force-reducing reverse field, are the recorded alternatives, the reverse field only well inside the magnets' intrinsic coercivity at worst-case temperature, from grade data rather than a nominal room-temperature figure (Hsu et al. 2025). Access after assembly is the same problem in reverse. Separating the ASTRID2 magnet halves to install the vacuum chamber meant overcoming 90 kN, done with four removable assembly bolts mounted on the magnet sides and kept for later component replacement (Bødker et al. 2014, dg-2141). A PM cyclotron has no routine off-state for chamber access unless a flux shunt, bucking coil, rotor or removable-magnet provision is designed in, so the opening fixture is part of the magnet design. The force arithmetic of B²/2μ₀ gives the scale, and only the scale: a part-built PM assembly produces local forces and torques the uniform-gap formula does not capture, so rated fixtures, mechanical stops, and modeled worst-case alignments do the protecting rather than the estimate.

The record at small scale

At amateur scale the idea has been prototyped, and the record's state should be read precisely. The Houghton program's 2002 design study built and measured a permanent-magnet gap at 0.493 T on 15 cm poles, uniform to 1 percent over 80 percent of pole radius, the bare-pole curve of Figure 2 (King 2002). The same thesis computed that closing its 3.8 cm gap to 1.3 cm would buy about 0.75 T where the ideal inverse-gap law promised 1.4 T, leakage and the magnets' moving operating point eating the difference. The Los Alamos dipole offers one datum for pole overhang at this scale: an 8 cm-wide pole met 1 part in 1000 uniformity across the 2.8 cm beam-pipe width without edge shims, about 0.9 gap-heights of overhang per side (Barlow et al. 1993, dg-2117), a ratio for that straight dipole and a field calculation for any round pole. None of the permanent-magnet cyclotrons in this site's library (surveyed September 2026) records a demonstrated beam: LBL described its magnet in 1992 as an instrument under construction (Li et al. 1992), fabricated and field-verified it by 1995 with beam demonstration left “continuing at a low level, limited by funding” (Clark et al. 1995), Antokhin's magnet was designed, manufactured, and measured with the RF still at design stage (Antokhin et al. 2006), and every working machine in the table below runs an electromagnet. The magnet technology is demonstrated at 1 T on a 30 cm pole; the machine record is still open. The design guide's permanent-magnet rules carry the numbers above with their formulas and page citations.

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: the machines in these reports were all 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.

Reported numbers at small scale

Documented small machines cluster tightly, and their parameters make useful sanity anchors for any new design. The rows mix as-built machines with one design study, labeled as such. 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 — 548 sourced rules — and the 131 coil rules, each with formula and page citation.
• Permanent-magnet rules — 78 rules from the ten permanent-magnet sources, from Halbach's material model to the Recycler's production tests.
• 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.
• D. J. Clark, K. Halbach, W. B. Kunkel, K. N. Leung, C. Y. Li & A. T. Young, "A Compact Permanent Magnet Cyclotron for Accelerator Mass Spectrometry," Proc. Cyclotrons '95 — the crown-and-barrel PM circuit, pole-as-precision-element, oversize-and-cut-back shimming, as-built field. JACoW.
• A. T. Young et al., "Development of a Compact Permanent Magnet Cyclotron for Accelerator Mass Spectrometry," Proc. PAC 1993 — the same machine's design paper: POISSON field prediction, block polarities, multicusp source. JACoW.
• Y. Y. Hsu, J. C. Jan, Y. L. Chu & F. Y. Lin, "Integrating Permanent Magnets and Electromagnets: A Hybrid Dipole Magnet Design," Proc. IPAC 2025, WEBD3 — trim coil, movable outer plates, NiFe thermal compensation, glued-block construction, assembly fixtures. JACoW.
• E. Antokhin, T. Fujisawa, M. Kumada, T. Matsumoto & S. Wakasa, "Magnet System for PET Cyclotron Based on Permanent Magnets," Proc. RuPAC 2006 — PM volume scaling, measured temperature coefficient, the flux-engine off switch. JACoW.
• C. Y. Li, M. Cooper, K. Halbach, W. B. Kunkel, K. N. Leung, R. P. Wells & A. T. Young, "A Permanent Magnet System for a Cyclotron Used as a Mass Spectrometer," LBL-32037, Proc. Cyclotrons 1992 — the same machine's magnet design report: rotating field tuners, blocks left out for coarse level, the 18-gon of rectangular blocks, pole faces as vacuum wall, magnets outside the vacuum. OSTI 6996533.
• K. Halbach, Use of Permanent Magnets in Accelerator Technology: Present and Future, LBL-23556, 1987 — the scaling argument, the "analytical" material model, hybrid insensitivity to material, the zero line integral, PM-assisted electromagnets. OSTI 6183769.
• K. Halbach, Physical and Optical Properties of Rare Earth Cobalt Magnets, LBL-11393, 1980 — the straight-line B–H model, the knee and irreversible recoil, block size, price variables, temperature uniformity, the steel-substitution theorem. OSTI 5068077.
• D. B. Barlow, R. H. Kraus Jr. & R. E. Meyer, Variable-Field Permanent-Magnet Dipole, LA-UR-93-3237, 1993 — the moving iron shunt with tapered tips, its preloaded drive, NMR calibration, the 2-D/3-D overestimate, the cost comparison. OSTI 10192222.
• F. Bødker et al., "Experience with a NdFeB Based 1 Tm Dipole," Proc. IPAC 2014, TUPRO080 — the ASTRID2 Green Magnet: pole-shaped homogeneity, ±3 percent trim coils, passive compensation, the 90 kN opening force, radiation tolerance versus work point. JACoW.
• S. H. Kim & C. Doose, "Temperature Compensation of NdFeB Permanent Magnets," Proc. PAC 1997 — the Ni-Fe flux-shunt recipe, the flat region and its first-cycle shift, measurement rigor at the 10⁻⁵ level. OSTI 541882.
• B. C. Brown et al., "Time Dependence and Temperature Stability of the Permanent Magnets for the Fermilab Antiproton Recycler Ring," FERMILAB-Conf-99/267, 1999 — logarithmic ageing, the uncompensated ferrite coefficient, batch mixing, heater regulation, brick sorting, first-moment installation. OSTI 12569.
• J. T. Volk, Experiences with Permanent Magnets at the Fermilab Recycler Ring, FERMILAB-TM-2497, 2012 — production tolerances, linear compensator bookkeeping, the two-point test, the ten-year ageing record, EDM end shims, washer trims, correctors found in operation. arXiv:1209.4324.
• S. J. Brooks, G. Mahler, J. Cintorino & A. K. Jain, "Production of Low Cost, High Field Quality Halbach Magnets," Proc. IPAC 2017, THPIK007 — iron-wire shims sized by SVD from a measured map, the fixed-magnetisation block model, what shimming does not fix. JACoW.
• H. A. Leupold & E. Potenziani II, A Permanent Magnet Circuit Design Primer, ARL-TR-946, 1996 — permeance method, material comparison, open-source feasibility bound. Internet Archive.
• B. T. King, A Preliminary Design for a Small Permanent Magnet Cyclotron, Houghton College B.S. thesis, 2002 — the measured amateur-scale PM gap behind Figure 2's bare-pole curve. Houghton repository; M. J. Cressman, The Design and Construction of a Small Cyclotron, Houghton College thesis, 2006 — the Cyclotrino team's magnet choice. Houghton repository.
• 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).