The Magnet, and Staying in the Gap

The Mullins 8-inch home-built H-frame found its resonances with a cheap hand-held Hall meter and a swept magnet supply. Its peaks sat within a fraction of a percent of the field its RF frequency predicted (build log, “It works!”). Houghton and COLUMBUS ran on laboratory electromagnets with flat poles (Yuly (2013), p. 2-3; census).

If it does not work. If you cannot find a resonance peak, one easy thing to check is whether the magnetic field is approximately what the RF frequency predicts: check the meter’s range and zero, read it with the RF off because RF upsets a Hall meter’s reading, and confirm the supply holds a setting, since the resonance is only a few gauss wide. But a missing peak does not tell you the magnet is the problem. An ion source making too few ions, a dee without enough voltage, the wrong RF frequency, or a collector that is not seeing the beam all produce the same empty sweep (How do I find a first beam when all these settings affect one another? Does everything have to be perfect at once before anything happens?). A peak that appears only with the source off-center, or beam that falls away as the collector moves outward, points to a radial shape of the magnetic field or an orbit centering that is marginal. Houghton saw the second and attributed it to loss before full radius (Fuller (2013), p. 52-53). A radial line of field readings at a few azimuths, turned into the field index, how fast the field falls with radius, is the next step (Do I really need to map my magnet's field accurately? Is one reading in the middle enough, and which value goes into the frequency and energy calculations?). A peak that is broad and asymmetric, or two peaks where one is expected, can also come from gas pressure, source bias or the RF, so it does not point at the iron on its own (How do I find a first beam when all these settings affect one another? Does everything have to be perfect at once before anything happens?, I'm feeding hydrogen into the chamber. Does that mean the accelerated particles must be protons? Why might I see more than one resonance peak?).

To understand it. “Uniformity” is four requirements that bind differently. Strength and stability: the resonance is narrow, so the supply has to hold a setting. Livingston and Blewett ask for one part in a thousand (Livingston & Blewett (1962), p. 194), and Iowa State held ±4 G in 17 kG (McGuire (1961), p. 9). Both are matters of a regulated supply and a hysteresis routine (Zickler (2010), p. 26) rather than of the iron. The routine exists because the field iron gives at a given current depends on the currents it has been through before. Radial shape: classical machines settled on a total fall of the field, from center to full radius, of a few percent, 1.75 to 3.4 % in the 1958 census (Howard (1958), p. 164). A common design goal is to keep the field index n positive but below about 0.2 through the useful radius. Positive n gives vertical focusing, the push back toward the gap’s center plane that keeps ions off the dee; n = 0.2 is a coupling resonance that can cause serious losses (Livingston & Blewett (1962), p. 161, Koeth (2015), p. 3). At that index the ion’s radial wobble repeats exactly twice per vertical wobble, so an imperfection can pass radial motion into vertical motion. But 0.2 is not a wall. Whether particles survive crossing it depends on the imperfections present and how long they spend there (beam-dynamics page). Variation around a circle: 0.1 to 0.2 % is the bound operators of large machines agreed on near the exit slit, where the beam leaves the machine (Livingston & Blewett (1962), p. 196-197). It protects the orbit’s center, since a field stronger on one side of the circle pushes the whole orbit off center. On a small machine the tolerance follows from the displacement arithmetic against your own aperture, the room the dee and collector leave (How can my magnet's field have "harmonics" when the magnet runs on DC? I thought harmonics were an audio thing, like musical notes. Do I have to worry about them?). Material: low-carbon steel is the conventional choice for the flux path because carbon costs permeability most (Ramler & Parker (1959), p. 11, Tanabe (2005), p. 249-251), permeability being how easily steel magnetizes, and scrap steel has worked when oversized and verified by measurement (Baumgartner & Heuer (2010), p. 8). Flat poles often give the needed falloff; the actual profile has to be measured (Antaya (2010), p. 36-37).

Focusing makes the machine tolerant of realistic starting positions, angles and imperfections. The documented small machines got their confinement from unshaped or lightly tapered poles, faces cut so the gap widens toward the edge (Koeth (2006), p. 1, Heuer & Baumgartner (2009), p. 31).

If it does not work. The signs of weak confinement can be seen without a tracking code, a program that computes ion paths. Beam current that falls as the collector moves outward, which Houghton attributed to loss before full radius (Fuller (2013), p. 52-53). Marks where ions land. The Rutgers ion-source chimney, the small tube the ions leave through, carried a spiral track pitched downward at 4.3°, read as the signature of a parasitic vertical electric field at the source (Koeth (2007), p. 1). The 86-inch’s beam striking one dee’s periphery revealed an orbit center three inches off the magnet’s center (Howard (ed.), p. 7). A beam photographed on a phosphor probe, a plate that glows where the beam lands, visibly wobbling as it goes out (Ruisard (2013), p. 4). The first things to check: that the source and the collector sit on the magnetic midplane, which may not be the geometric one (How accurately do I need to line up the magnet, chamber, dee, ion source and collector? Which dimensions are really critical, and what should be adjustable?); that the dee aperture, the open height inside the dee, has margin; and then the radial field profile.

To understand it. The field index, n, measures how fast the field falls with radius: zero for a flat field, larger as the fall steepens. For n above zero there is a vertical restoring force, and the ion makes √n vertical wobbles per turn. For n below one there is also a radial restoring force. So 0 < n < 1 is the ordinary weak-focusing stability condition (Livingston & Blewett (1962), p. 161, Antaya (2010), p. 36-37). A small n means a slow wobble: at n = 0.03 the ion takes about six turns per vertical oscillation (Koeth (2006), p. 6). At n = 0.2 the radial oscillation frequency becomes exactly twice the vertical one. A push that repeats in step with a wobble adds up, and the 2:1 ratio keeps the radial wobble in step with the vertical one. Whenever an imperfection couples the two motions, radial motion can then feed vertical motion until particles strike the dee. How serious that is depends on the imperfections and on how many turns the particles spend near the condition. Classical machines therefore taper the pole so that n reaches 0.2 only near the last radius (Koeth (2015), p. 3, Koeth (2013), p. 2); for the same reason the 184-inch lost its beam at exactly the radius where its n reached 0.2 in 1947 (beam-dynamics page). A useful design target is a gently falling field with n below 0.2 through the useful radius. If a measured profile crosses 0.2 sooner, that deserves investigation; it does not automatically mean the machine cannot work.

The first turns are the exception. At the center the field is flat by symmetry, so n is zero and there is no magnetic focusing at all (Humphries (1986)). The first few turns survive on the electric field across the dee gap, which acts as a weak lens, and only for ions that cross while the RF voltage is falling (Livingston & Blewett (1962), p. 166-171). Professional machines add a small central field bump so the field falls from the center outward over the first turns (Smirnov (2021), p. 50-51), or grids across the dee aperture (Howard (1958), p. 27). Where an ion starts therefore matters as much as the field’s shape. The Rutgers 12-inch three-hole experiment launched ions 2.5 mm above and below the midplane. It lost one of the three beams to the dee lid, where a launch angle and an initial amplitude, the starting height off the midplane, added (Koeth (2007), p. 2). Before first beam, the vocabulary you need is the field index and “vertical focusing”. “Betatron oscillation” is the wobble, “tune” is wobbles per turn, and the resonances beyond n = 0.2 can wait until the beam reaches full radius.

The Mullins cyclotron used a cheap, uncalibrated Hall-effect meter (one never checked against a known field) to establish the approximate field, then varied the magnet current to find reproducible beam-current peaks; a second peak appeared at 427 mT (build log, “It works!”). The second peak, at twice the field, was the hydrogen molecular ion H₂⁺, a hydrogen molecule missing one electron and so twice the mass on the same charge.

If it does not work. Separate three jobs: measuring how the field varies with position; knowing its absolute magnitude; finding the resonance by watching the machine respond. For the first two, read the cheap meter with the RF off; the Mullins log made that a standing rule after one coil change (hosted instrumentation note). Keep the probe flat against a reference surface, since a Hall element reads only the part of the field perpendicular to its face and a tilted probe therefore reads low. Cycle the magnet to the same maximum each time so the field-against-current curve repeats (Zickler (2010), p. 26). If you suspect the field changes too much across the magnet, measure it at several positions between the poles. Start near the center and move the probe outward toward the edge, recording the field every centimetre or so; then repeat in another direction, toward twelve o’clock and toward three o’clock, say. That shows whether the field gets weaker toward the edge, as it should slightly for the vertical focusing, and whether one side of the magnet behaves differently from another. Houghton did this with a Hall probe on a rotating aluminium jig at 1 cm steps (Loucks (2007), p. 36-38), and a more detailed analysis of such readings reveals problems with particle focusing (magnet-design page). How the Mullins experiment was recorded is one way to do the sweep, not a requirement: a slow triangle ramp from a function generator into the magnet supply’s analog input, with an oscilloscope in XY persistence plotting field against electrometer current (hosted instrumentation note). The electrometer is the sensitive current meter on the collector. If a once-around pattern dominates a circle, the field high on one side and low opposite, check pole parallelism, since tilted poles produce exactly that, and the jig’s pivot before shimming anything, that is, before adding iron to correct the field (Livingston & Blewett (1962), p. 197-284). For the third job, the Mullins remedy for a resonance that was easy to miss, since the peak is narrow, was to set the field near the expected value, tune the RF frequency for maximum DC current at the collector, and only then sweep (hosted instrumentation note).

To understand it. The map a magnet designer wants has four parts. Field against current at the center. The radial profile and its index n(r). Circles at two or three radii, including the collector radius, to size the once-around variation (Livingston & Blewett (1962), p. 197). And the magnetic midplane, the plane about which the field is symmetric, found with a pair of opposed search coils; it sat half an inch off the geometric midplane at MIT (Livingston & Blewett (1962), p. 197). Measure with the chamber in place if its lids are steel, because steel lids become part of the poles. Houghton’s modelled magnetic lids moved the n = 0.2 radius outward by centimetres (Morrow (2015)). Rutgers found the normalized profile, the field at each radius divided by the field at the center, unchanged across 20, 30 and 40 A of magnet current on its 12-inch (Koeth (2006), p. 3), which is worth checking. A calibrated Hall probe can serve both the mapping and the absolute roles. For accuracy beyond it, the classical reference is proton NMR, whose resonance frequency is set by the field alone, at 234.82 G per MHz (Livingston & Blewett (1962), p. 286-287, McGuire (1961), p. 9-10). The species is which ion it is; the harmonic is whether the RF runs at the ion’s own frequency or an odd multiple of it. Once the beam’s species and harmonic are established, the resonance frequency constrains the orbit-averaged field more tightly than any meter (magnet-design page); until they are, a peak is a candidate. Artifacts: position error multiplies by the local gradient and dominates at the edge (Livingston & Howard (eds.), p. 31). Probe settling and backlash, play in the positioning mechanism, need handling (Koeth & Krutzler (2015), p. 3, Koeth & Krutzler (2015), p. 2). A feature that repeats at two excitations (two magnet currents) is likely the magnet’s, though a repeatable positioning or calibration error survives the same test (Alford (1961), p. 11).

Which number goes where. Into the frequency: the azimuthal average of the field at the radius in question, the average round the circle, which the ion samples whole on each turn, so the resonance is found by sweeping rather than computing (Livingston & Blewett (1962), p. 156). Into the energy: the field at the final orbit radius, inside the usable region about half a gap inside the pole edge (Livingston & Blewett (1962), p. 158, Livingston & Blewett (1962), p. 260). Into the focusing check: n(r). Into the centering check: the once-around at the collector radius.

If it does not work. A once-around error pushes the center of the orbit away from the center of the magnet. The ion curves more tightly where the field is stronger, so the circle it traces is no longer centered on the magnet. In the weak-focusing approximation for a first-harmonic error, the shift is roughly the error fraction, the once-around variation as a fraction of the field, times the radius divided by the field index, n (Botman & Hagedoorn (1996), p. 9). At small n the denominator is small and the shift is large. On a 10 cm machine at n = 0.05, a tenth of a percent gives about 2 mm (arithmetic). A displaced orbit also precesses, meaning the off-center circle itself slowly turns about the magnet’s center, which shows as regular pips of current on a probe, a collector moved in on an arm (Sewell (1947), p. 4). So the diagnostic is to measure the field around a circle at the same distance from the center as your beam collector, convert the once-around variation to a shift with the formula above, and compare it to the room your dee and collector leave. If the shift is a small fraction of that room, look elsewhere. If it is large, look for the cheap causes first. An ion-source hole on one side only: one machine’s radial oscillations fell from 3 cm to under 3 mm with a dummy hole opposite (Antokhin et al. (2006), p. 2-3). A yoke, the iron frame carrying flux between the poles, that is not symmetric. Or pole faces not quite parallel, which a dominant once-around sinusoid points to before anything else (Livingston & Blewett (1962), p. 197-284).

To understand it. The large classical machines held their once-around under 0.1 to 0.2 % near the exit slit, because the extraction channel left no room for a wandering orbit (Livingston & Blewett (1962), p. 196-197). Higher harmonics can drive other resonances and are not irrelevant. They are usually a lower priority for a modest first-beam objective, provided the measured errors and the available aperture support that judgment. They matter again for full radius or extraction. Three uses of the word. Field harmonics, above, are variations around a circle in a DC magnet. RF harmonics are multiples of the RF frequency in time. A push-pull amplifier’s third harmonic sat only 8 to 10 dB below its fundamental in one documented build (Buckler (2015), p. 3). A tank circuit, the coil and dee that resonate together at the RF frequency, can also resonate at a harmonic mode; on the Rutgers 9-inch that produced an unexpected beam peak at roughly half the expected field, traced to the tank and not to a different ion (Koeth, p. 8). Harmonic operation means running the RF at an odd multiple of the ion’s own frequency, which still accelerates it on every gap crossing, since between one crossing and the next the voltage has then reversed an odd number of times and points the right way again. At a third of the field the ion circles at a third of the rate, so a fixed RF is then its third harmonic. A magnet sweep therefore shows the same species again at a third and a fifth of its field (Yuly (2013), p. 4-5), as the Mullins low-field scan did (build log).

The Mullins log records one early lesson in reproducibility: a dee not fixed in position in the chamber “makes repeatable experiments more difficult” (build log, “It works!”).

If it does not work. Four functions, each with its own check. Geometry: the gap and the parallelism of the pole faces set the field error, roughly in proportion when the magnetic circuit is dominated by the gap and the iron is well below saturation (magnet-design page). In proportion means a given fraction of gap error gives about the same fraction of field error; gap-dominated means nearly all the coil’s drive is spent across the air gap, and saturation is where iron stops carrying more flux. A tilt is a once-around harmonic, and a circle of field readings finds it (How can my magnet's field have "harmonics" when the magnet runs on DC? I thought harmonics were an audio thing, like musical notes. Do I have to worry about them?). Alignment: the magnetic midplane holds particles centered vertically; in a real magnet it can sit off the geometric midplane, half an inch at MIT (Livingston & Blewett (1962), p. 197). A dished midplane, one that curves instead of lying flat, points to asymmetric iron or a shorted turn, which weakens the coil it is in (Livingston & Blewett (1962), p. 288). The 9-inch adjusted its chamber’s midplane to the magnetic one (Koeth, p. 2); machines with separately powered upper and lower coils move the magnetic midplane to the dee instead (Koeth (2015), p. 2). The dee’s centering is read from the beam. The 86-inch found its beam striking one dee’s periphery, worked out that the center of rotation was three inches off, and moved the dees half an inch (Howard (ed.), p. 7). Stability: things move after assembly. Flat lids bow under atmosphere, pressed inward by the air outside once the chamber is pumped down. The Mullins log records 0.200-inch steel lids designed for 0.030 in of center deflection and measured at 0.033 in (vacuum-chamber page). The Cyclotron Kids’ thin lids bowed enough to change the dee capacitance and lower the flashover voltage (Baumgartner (2013), p. 2). A lid closer to the dee changes the capacitance between them, which sets the RF tuning, and shortens the distance a spark has to jump. Steel lids are also pulled toward the poles by the magnetic field at a pressure comparable to atmosphere at these fields (vacuum-chamber page). The poles close up under magnetic load, by 0.05 to 0.10 mm on the Carnegie model (Creutz (1950), p. 9). Heat moves the tank coil and detunes the RF, moving the circuit’s resonant frequency off the ion’s (Koeth et al. (2010), p. 21). A dial indicator on the lid during pump-down and a field reading during ramp-up are the cheap checks. Adjustment: the source position is set empirically in every account that describes it. Dee voltage droop along the gap, the voltage being lower toward the far end of the dee than near its feed, shifts orbit centers, and large machines moved their sources more than two inches to compensate (Livingston & Blewett (1962), p. 164). Ohio State’s filament slid inside its housing and was adjusted from outside while the beam was watched (Hausman (1954), p. 6). The 86-inch gave its source a bracket fixing position, grounding and reproducibility at once (Howard (ed.), p. 10), and realigned it after every reinstall (Cyclotron Staff (1970), p. 26). The Rutgers group’s planned improvement was a four-axis adjuster usable while running (Koeth (2007), p. 6).

To understand it. The classical numbers, as scale. The Argonne 60-inch held gap variation at any radius under 0.13 mm on a 56 cm gap (Ramler & Parker (1959), p. 10). Classical practice machined pole faces parallel to about one part in 50,000 of the diameter (Livingston & Blewett (1962), p. 193), surface-grinding work at tabletop scale. COLUMBUS mills a recess in its lid so the upper pole drops into it and the chamber has one position in the magnet (Prechtl & Wolf (2020)). The Carnegie synchrocyclotron’s chamber split and withdrew without disturbing the shimmed pole tips (Creutz (1950), p. 15). Rutgers keeps its pole tips as removable inserts so the field can be reshaped without touching the forgings (Koeth (2015), p. 2). Members held in place by field symmetry sit in unstable equilibrium and creep (Wakerling & Guthrie (eds.)). A steel part pulled equally by both poles is pulled harder by whichever it moves toward.

A frequency-adjustable signal generator is not that by itself; the resonant circuit has to follow the frequency too (I keep seeing "cyclotron resonance", "RF resonance" and "impedance matching". Are these the same thing? What has to match what?).

If it does not work. A missing beam has all the usual causes (How do I find a first beam when all these settings affect one another? Does everything have to be perfect at once before anything happens?); two challenges are distinctive to the permanent-magnet choice, and a magnet-current sweep would have hidden both. The resonance is not where the design put it, because the assembled field is a few percent off the calculation. Analytic and 3-D estimates of average field have run a few percent optimistic in documented magnets (Antokhin et al. (2006), p. 2). Without a current to adjust, the fixes are RF frequency range; trim coils over a few percent, small windings that add or subtract a little field, as the ASTRID2 hybrid dipole uses (Bødker et al. (Danfysik / Aarhus University), p. 1); a movable iron shunt that diverts some flux away from the gap, as the Los Alamos dipoles use (Barlow (1993), p. 5); or magnet material installed with margin so it can be cut back, which is how the LBNL cyclotron was shimmed (Clark (1995), p. 3). The resonance also drifts with room temperature. NdFeB (neodymium-iron-boron) loses about 0.11 % of field per degree uncompensated in the grades Argonne measured (Kim & Doose (Argonne National Laboratory, p. 1). The proton’s resonance frequency follows the field at 15.2 MHz per tesla, so that field drift is the same fraction of frequency drift, which the RF must track or the magnet must be compensated against. The combined field-and-frequency operating point has to be stable, rather than the field alone. Passive nickel-iron shunts bring the field’s temperature coefficient, its fractional change per degree, to a few parts in a hundred thousand per degree near a design temperature. The compensated set point itself shifts over the first thermal cycles (Kim & Doose (Argonne National Laboratory, p. 1, Kim & Doose (Argonne National Laboratory, p. 4, Kim & Doose (Argonne National Laboratory, p. 4).

To understand it. Halbach’s 1987 argument gives permanent magnets a fundamental scaling advantage as machines get smaller. Shrinking an electromagnet at fixed field forces the coil current density up until cooling forbids it, because the ampere-turns a gap needs shrink with the gap while the room for the coil shrinks with its area; a permanent-magnet system, whose field depends on the shape of its material rather than its size, scales to any size with no loss of field (Halbach (Lawrence Berkeley Laboratory), p. 3). Whether that makes it the better choice for a particular cyclotron still depends on field strength, adjustment range, temperature stability, cost and what is to hand. Where both can reach the field, the choice is cost and convenience (Halbach (Lawrence Berkeley Laboratory), p. 3). The precedent, the LBNL cyclotron mass spectrometer, put 1 T on 15 cm-radius iron poles from about 6,000 cm³ of SmCo (samarium-cobalt) in two hundred glued blocks across a 1.6 cm gap, with no coils, no supplies and no cooling (Li (1992), p. 5). The iron pole was the precision element and the blocks behind it coarse (Clark (1995), p. 3), and rotatable magnet rods along chords served as field tuners (Li (1992), p. 5). The Berkeley Cyclotrino team, weighing the same choice, took an electromagnet “because the field could be tuned” (Cressman (2006), p. 16). The costs of going without coils. Every millimetre of gap must be driven by magnet material (Li (1992), p. 4), where an electromagnet would spend more current. Rare-earth material is linear until its knee, the point beyond which part of its magnetization is lost for good, and then recoils permanently, and nothing in an assembled magnet can remagnetize it (Halbach (Lawrence Berkeley Laboratory), p. 2). SmCo is steadier and more radiation-resistant than NdFeB at lower remanence (less field from each block) and roughly twice the mass, in the grades ASTRID2 compared (Bødker et al. (Danfysik / Aarhus University), p. 3). Ferrite ages by fractions of a percent over years (Brown et al. (Fermi National Accelerator Laboratory), p. 3). And it is never off. A built PM cyclotron magnet measured hundreds of gauss at its zero setting (safety page). Assembly forces run to hundreds of kilograms or more (Hsu (2025), p. 2), and opening the ASTRID2 dipole took 90 kN (Bødker et al. (Danfysik / Aarhus University), p. 2). The amateur data point is the Takanashi apartment cyclotron, whose permanent magnet was about half its budget. The builder states he got the machine running and exhibited it, and has published no beam, field or frequency (census).

If it does not work, or before you buy. Start from the magnet’s field-versus-gap curve, measured or from the maker, rather than its nameplate maximum (Dewan (2007), p. 7-8). The gap must take the chamber: Houghton’s 15 cm poles end up 3.81 cm apart with the chamber in place (Yuly (2013), p. 2-3), and the field falls as the gap opens (Baumgartner (2013), p. 1-2). The usable radius ends about half a gap inside the pole edge (Livingston & Blewett (1962), p. 260). On flat poles the one-percent-uniform region ended at about 80 % of the pole radius in one measured case (King (2002), p. 22-23). From gap, usable radius and the field there, the energy ceiling and the RF frequency follow (How do the particle, the energy, the magnet size, the field and the RF frequency fit together? Which of them can I actually choose?). At 1.2 T that is 18 MHz; at 2 T, 30 MHz. The weak-focusing falloff, the gentle radial fall, has to exist, from the flat poles’ natural profile (Antaya (2010), p. 36-37) or from new pole tips, which Rutgers made for its NMR magnet and keeps interchangeable (Koeth (2015), p. 2). Check the supply and the cooling, which may not come with the iron; Houghton’s field was capped by its chiller, the coil cooler, not its magnet (Loucks (2007)). The magnet’s original field characterization no longer describes it once you have changed its pole tips and gap, though its iron circuit and material still matter: assess the field for the new geometry.

The record, and the two alternatives. The Rutgers 9-inch ran on a Varian NMR electromagnet turned on its side with new 9-inch pole tips, at 0.889 T (census). Houghton runs a commercial 15 cm laboratory electromagnet (Yuly (2013), p. 2-3). COLUMBUS runs a donated Bruker laboratory magnet (census). A 1958 Philadelphia high-school build had a donated 6-ton magnet with 7-inch poles and no documented beam. The Knox College build designed around a 2 T NMR magnet documented no beam either, and the record is silent on why (census). An MIT thesis design around a 2.7 T Varian magnet was not completed (census, Dewan (2007), p. 7-8).

MRI magnets. A clinical MRI magnet is a superconducting solenoid. A solenoid is a coil wound round an open bore, with its field running along the bore; a superconducting winding carries its current without resistance only while held at cryogenic temperature. It produces a highly uniform axial field inside a large bore, rather than a field across iron pole faces that can be shaped and that falls gently with radius. Its field cannot be conveniently swept during commissioning. Its shimming, which exists, is designed for imaging uniformity rather than cyclotron focusing. It can be ramped down or quenched, but only as a specialized and hazardous operation by qualified people, not as a commissioning adjustment. A quench is the winding losing its superconductivity all at once and dumping its stored energy as heat. Its cryogenic, mechanical and stray-field hazards need facility infrastructure. The census records no machine built on one. The same applies to a superconducting NMR solenoid, so “NMR magnet” has to be read carefully: the usable surplus item is the iron-pole electromagnet, not the cryostat.

Transformer cores. Microwave-oven transformer cores are small laminated E-I stacks (core pieces shaped like an E and an I), a few centimetres on a side with a window, the opening for the windings, to match. A cyclotron pole of six to nine inches (Wouters (1949), p. 11-12) needs a gap area and a return cross-section, iron to carry the flux back round from one pole to the other, dozens of times larger. Assembling that from many small cores is cumbersome: every joint is a parasitic air gap in series with the real one (Wouters (1949), p. 3), costing ampere-turns as the real gap does, and the structure has none of the stiffness the pull between the poles demands. Laminations are not the problem in themselves; they reduce eddy currents, the currents a changing magnetic field induces in solid iron, and are used in production magnets for reproducibility (Tanabe (2005), p. 4-5). The documented scrounger’s route is welded plate and bar. Niell’s C-frame, a yoke shaped like a C, was welded 5-inch soft-steel bar (Niell (2010), p. 11-17); the Cyclotron Kids’ scrap-steel yoke was sized so its flux density fell to 1.2 T, a bigger cross-section spreading the same flux thinner (Baumgartner & Heuer (2010), p. 8); the backyard Cyclotron Project’s yoke came from discarded steel (census). The record shows steel by the kilogram working when it is oversized and then checked with approximate field measurements and a demonstrated resonance; a detailed map is for diagnosing and improving the magnet afterward. The general warning stands over every option: a machine built fast from adapted parts can end up with limits none of which can readily be corrected (Howard (ed.), p. 20).