Cyclotron beam dynamics design rules

Protons: T (Mev) = 3.12 x 10-4 B2R2 ... the radius R applies to the extent of the uniform magnetic field; the physical radius of pole faces must be larger by about one-half the gap length.

Livingston & Blewett, Particle Accelerators (1962) — p. 158

Editorial note, tabletop extrapolation: For 8-in poles at 5.9 kG with the reference machine's 1.42-in gap, the flat field ends near R = 3.2 in, predicting ~110 keV - below what the machine demonstrates, because its cup collects further out, in the fringe. Read the formula as the energy the UNIFORM field alone buys; a wider pole or smaller gap moves that number as B^2R^2.

for particle oscillations about an equilibrium orbit to be stable for both axial and radial coordinates, the value of n must be in the range 0 < n < 1.

Livingston & Blewett, Particle Accelerators (1962) — p. 161

Editorial note, tabletop extrapolation: Map n(r) on the 8-in poles; any region where the field rises with radius (n < 0) is axially defocusing, and the longer the beam spends there the less of it survives - shim such regions out rather than reasoning about how much defocusing is tolerable.

Cited in: Beam Dynamics: An Interactive Laboratory

the total decrease below the value of the central field out to the exit slit is about 2 per cent. The radial decrease can be larger (3 to 4 per cent) in small machines in which D voltage is relatively high.

Livingston & Blewett, Particle Accelerators (1962) — p. 161

Editorial note, tabletop extrapolation: The reference machine is the small, few-turn case the quote names, and the census machines converged on the same few-percent smooth droop (dg-702). Aim for a smooth 3-4%-class fall-off shaped against the machine's own n(r) requirement (dg-003), with the number as the historical anchor rather than the spec.

The n value rises almost linearly from zero at the center to 0.02 at 15 in. (where fringing effects start), then increases rapidly to 0.40 at 18.75 in. (exit-slit location) and to 1.0 at 19.25 in.

Livingston & Blewett, Particle Accelerators (1962) — p. 161-183

Editorial note, tabletop extrapolation: One documented profile, useful as a shape target rather than a scaling law: fringe onset and width depend on gap-to-pole ratio, edge shape and shims, so map n(r) on the actual 8-in poles (FEMM, then measurement) and place extraction where the MEASURED n has climbed toward ~0.4 - before the n = 1 radial-stability edge.

the magnet generates 2.7 T of magnetic field with a 1.25 inch pole separation... capable of accelerating protons to a maximum kinetic energy of 1.96 MeV

Dewan, Design and Construction of a Cyclotron Capable of Accelerating Protons to 2 MeV — MIT thesis (2007) — p. 7-8

Editorial note, tabletop extrapolation: The surplus-NMR-magnet route to MeV energies: small radius is fully compensated by high B (energy ~ B^2*r^2), so a 6-inch 2.7 T machine beats a 12-inch 1 T machine.

implement a .02'' convex taper from the center of the pole to the edge, to create sufficient bending of the magnetic field lines.

Heuer & Baumgartner, Design of a 2 MeV Cyclotron (2009) — p. 31

Editorial note, tabletop extrapolation: The taper's size does not transfer: the gradient it produces depends on gap, pole radius, saturation and yoke geometry. Choose a target field index n = -(r/B)dB/dr for the reference machine, get the contour from magnetostatic modeling (FEMM), and finalize by field mapping and shimming - the source's 0.02 in is one machine's value, not a starting spec.

at a dee radius of 6'' the field is 1.57 T, a .08 T drop off from 1.64 T directly at the center.

Heuer & Baumgartner, Design of a 2 MeV Cyclotron (2009) — p. 31

Editorial note, tabletop extrapolation: One machine's by-design droop as a sanity anchor: a few percent total is the common class (dg-702's census clustering). What YOUR field may droop is set by the phase-slip budget and the n(r) requirement - verify with Poisson/FEMM against those, not against 5%.

To minimize weight and size of magnet system the average magnetic field value has to be high, limited by iron saturation ... average magnetic field value was chosen as 1.4 T provided of 2.3 T and 0.5 T of hill and valley region fields

Antokhin et al., Magnet System for PET Cyclotron Based on Permanent Magnets (2006) — p. 1

Editorial note, tabletop extrapolation: A worked AVF datapoint, not a scaling law: hill/valley ratio, sector count and sector angle set flutter and tunes in a geometry-dependent way, so an 8-12 inch pole set re-derives them (FEMM plus a tune calculation) rather than copying 4.6:1 and 45 degrees. What does transfer is the design order: average field as high as iron saturation allows, then focusing from the hill/valley split - iron, not coil power, is the ceiling on a PM machine.

As hill gap providing enough beam intensity was chosen as 20 mm and then corresponding valley gap is 100 mm.

Antokhin et al., Magnet System for PET Cyclotron Based on Permanent Magnets (2006) — p. 1

Editorial note, tabletop extrapolation: One worked ratio for a first AVF pole-tip sketch: hill gap from beam-aperture needs, valley several times deeper - re-derived for the actual field contrast and the RF/pumping geometry rather than copied. A deep valley is indeed where an amateur's Dee and pumping naturally live.

Note that radial oscillations for this conditions and measured field is large enough as about 3 cm, that may lead to increasing axial oscillations through the nonlinear resonance Qr-2⋅Qz. The reason for increased radial oscillations is big first harmonic of magnetic field, which caused by non-symmetric structure of central part of cyclotron magnet ... radial oscillations now does not exceed 3 mm and axial ones does not exceed 2 mm ... to make symmetric central magnet part by setup second hole on opposite side with respect to ion source hole.

Antokhin et al., Magnet System for PET Cyclotron Based on Permanent Magnets (2006) — p. 2-3

Editorial note, tabletop extrapolation: A ten-fold reduction in orbit wander at the source machine for the cost of a symmetry-restoring second hole. For the reference machine, treat any asymmetric central feature as a first-harmonic suspect - but measure the harmonic (field mapping or orbit calculation) and choose the compensating geometry from the data; a mirror feature is not guaranteed to cancel a given perturbation.

Cited in: Beam Dynamics: An Interactive Laboratory · Beam Quality: What It Is and What Degrades It

the fringe field is longer at the center of the magnet and drops off near the edges. This distribution is approximately quadratic and the integrated multipole field looks like a sextupole field. ... a removable insert with a machined chamfer installed on the SPEAR3 prototype gradient magnet. ... The shape of the chamfer depth was determined empirically and was approximately parabolic. It was designed to reduce the integrated sextupole field.

Tanabe, Iron Dominated Electromagnets, Lecture 10: Forces, Stored Energy, Fringe Fields, End Chamfering (2005) — p. 20-21

Editorial note, tabletop extrapolation: Mostly relevant if the builder adds edge shaping for extraction: expect the field falloff at the pole rim to vary azimuthally with any non-axisymmetric pole feature, and fix it empirically with removable machined inserts.

Shimming of pole edges or shims based on equation: dB(r)/B(r) = gamma(r)^2 * df_p(r)/f_p(r)

Zaremba, Magnets for Cyclotrons (2005) — p. 10

Editorial note, tabletop extrapolation: For protons from ~150 keV to 1 MeV, gamma is about 1.00016-1.00107. Relate the reference machine's measured phase slip to a local orbital-frequency error first, then use the formula for the required field correction, and get shim thickness from simulation or measured shim sensitivity - relativistic effects are small at these energies but not automatically subdominant to mechanical field errors.

The maximum kinetic energy T determines magnetic rigidity: B*rho = sqrt(T^2+2T*E0)/(300*Z)

Zaremba, Magnets for Cyclotrons (2005) — p. 19

Editorial note, tabletop extrapolation: For 1 MeV protons B*rho = 0.145 T*m: at 1 T that is a 14.5 cm final orbit radius, which immediately sizes the next machine's pole diameter (with overhang and fringe allowances added).

coil cooling estimation: dT(C) = 60*P(kW)/(4.19*N(l/min))

Zaremba, Magnets for Cyclotrons (2005) — p. 30-32

Editorial note, tabletop extrapolation: The cooling formula is immediately usable: a next machine's 5 kW coil at 4 L/min runs ~18 C water rise; the flutter formulas matter only if the builder adds sector (AVF) pole faces.

For best RF efficiency, k=0.5 BUT to decrease machine dimensions k >0.5 (more hill, thus more field) CHOICE : k=0.67 (60 deg hills)... CHOICE : nu_z = 0.2

Zaremba, Magnets for Cyclotrons (2005) — p. 32-33

Editorial note, tabletop extrapolation: If a next machine goes AVF to escape the weak-focusing energy ceiling, IBA's documented choices are a starting point, not proven tabletop values: k between 0.5 (best RF room) and 0.67 (compactness), and a modest vertical-tune target like their nu_z = 0.2 - each re-derived for the actual geometry, since a 60-degree hill only gives k = 0.67 in their sector periodicity, and sector count and valley usage carry their own trades.

A general rule of thumb is that the length of the fringe field beyond the edge of the steel pole tip is = h, = h/2, or = h/3, for the dipole, quadrupole or sextupole

Tanabe, Iron Dominated Electromagnets: Design, Fabrication, Assembly and Measurements — SLAC-R-754 (2005) — p. 252-253

Editorial note, tabletop extrapolation: Tells the builder where usable field really stops on an 8-inch pole: the fringe extends about one half-gap BEYOND the steel edge before dying away, while the flat, usable region ends somewhat inside the pole radius as the falloff begins. Map B(r) (FEMM, then Hall probe) to place the maximum stable orbit; the h rule sizes how much radial real estate the fringe transition consumes.

the radial sector poletips have the greatest average falloff - so great that it amounts to be an unusable field. The spiral sector AVF field is a compromise between the two.

Koeth & Krutzler, Field Mapping in Cyclotron Magnets (2015) — p. 10

Editorial note, tabletop extrapolation: Direct guidance for a next machine's pole-tip upgrade at the 8-12 inch scale, as a measured comparison among these candidates rather than a ban: radial-sector AVF machines exist, but making one work takes sector-angle and profile design these candidates did not carry. Also warns that narrow spiral vanes saturate at large radius - the measured field fell below simulation there.

this analysis approach can be used to quickly assess a field during design, relegating the laborious task of phase space mapping and determining the limits of stability to the few the final contenders.

Koeth & Krutzler, Field Mapping in Cyclotron Magnets (2015) — p. 10-11

Editorial note, tabletop extrapolation: A cheap, quantitative design filter that works from measured maps of a home-built magnet, no orbit code required.

we choose a reference circle to have a radius half that of the maximum ion radius ... the reference circle is swept to maximize the 4th harmonic, while minimizing the second, third, and fifth.

Koeth & Krutzler, Field Mapping in Cyclotron Magnets (2015) — p. 6-7

Editorial note, tabletop extrapolation: Applies if a next machine moves to sectored pole tips; on a 12-inch machine the whole analysis is a spreadsheet/Octave job on the map you already took.

The vertical field magnitude decreases away from the magnet over a scale length comparable to the gap width.

Humphries, Principles of Charged Particle Acceleration (1986) — p. 140, 526

Editorial note, tabletop extrapolation: Rule of thumb for a next machine's layout: expect roughly one gap-height of field transition at the pole edge - how much of it is actually unusable depends on the field tolerance, so map it (dg-098) - and shield any deflector with a grounded septum as designed practice.

fx = (gamma mo vz/qBo)/tan beta = rgo/tan beta.

Humphries, Principles of Charged Particle Acceleration (1986) — p. 141

Editorial note, tabletop extrapolation: If a next machine ever extracts a beam, angling the magnet exit edge can focus the diverging beam without any extra magnet - check the sign convention for the actual bend geometry and verify the full extracted-beamline optics rather than trusting the thin-lens number.

The bending magnets were shaped to produce a field with index in the range 0 < n < 1.

Humphries, Principles of Charged Particle Acceleration (1986) — p. 159, 521

Editorial note, tabletop extrapolation: The outer bound that pairs with Koeth's n<0.2 refinement: the reference machine's field must fall (n>0), but slowly, all the way to full radius.

Tmax = 48 (Z RB)2/A, where Tmax is given in MeV, R in meters, and B in tesla.

Humphries, Principles of Charged Particle Acceleration (1986) — p. 524

Editorial note, tabletop extrapolation: The master sizing formula: the reference machine's 0.582 T at r ~ 0.10 m gives ~163 keV, which is its best demonstrated run; 1 MeV needs (R*B) ~ 0.144 T-m, e.g. 1.2 T at 12 cm.

Have real sinusoidal solutions for 0<n<1; this condition is true in a classical cyclotron

Antaya, Cyclotron Basics — MIT 8.277, Unit 10, Lecture 14 (2010) — p. 36-37

Editorial note, tabletop extrapolation: The reference machine's flat-pole H-frame gets its weak focusing from natural radial falloff - but a flat pole is nearly uniform over much of its radius and falls mainly near the edge, so map n(r) rather than assuming it: the design task is confirming where n is usefully positive, then controlling how fast it rises.

Almost (but not quite) spherical: Efficient cyclotron magnetic circuits include more iron laterally than axially

Antaya, Cyclotron Basics — MIT 8.277, Unit 10, Lecture 14 (2010) — p. 48-50

Editorial note, tabletop extrapolation: The B^2 energy leverage argues for raising a next machine's field before enlarging poles: doubling B quadruples energy at fixed radius. The 1/B^3 mass saving holds only while the whole magnet scales geometrically - gap included - and the iron's own saturation (dg-096) caps how far the argument runs.

A radially decreasing field can be described as Bz = B0(r0/r)^n for n >= 0, where n = 0 implies a uniform field and n > 0 implies a restoring force.

Chun, The Cyclotron Magnet and RF Oscillator (2003) — p. 7-9

Editorial note, tabletop extrapolation: Exactly the reference machine's problem class and size: machine a gentle crown or stepped 'lump' into the 8-inch poles (or shim equivalently), aiming for the few-percent center-to-edge fall-off of the source's worked case - and verify the result against the mapped n(r) (dg-003, dg-561) rather than the geometric recipe, since the actual profile depends on gap and permeability.

For educational applications a six to nine-inch pole piece is an economical range; for inducing light element reactions ... a somewhat larger machine, say, 12 to 15 inches

Wouters, General Recommendations for Design of Small Cyclotrons — UCRL-476 (1949) — p. 11-12

Editorial note, tabletop extrapolation: Frames the next machine's decision the way the source does: 8-inch-class poles sit in the educational range, and light-element reaction goals argue for the 12-15 inch class. Pole diameter is a proxy - field and species matter as much - and small does not mean neutron-incapable: deuteron operation makes neutrons at any energy via D(d,n)3He, which is a hazard question before it is a capability one (see the safety rules).

Slight taper on pole applies a corrective Lorenz force to the beam. Made with freeware! Poisson Superfish

Baumgartner & Heuer, The Cyclotron Kids 14-Inch Accelerator (2010) — p. 8

Editorial note, tabletop extrapolation: The documented amateur approach at the reference machine's scale - Cyclotron Kids here, with the pole-shaping rules (dg-119, dg-152) carrying the design math: put the field index into the pole profile deliberately, designed in the field code before cutting steel, rather than relying on accidental fringing.

the field index value n=0.2 must not occur inside the maximum ion orbit radius to avoid coupled resonances

Morrow, Focusing in the Houghton College Cyclotron — Houghton College thesis (2015) — p. 2, 39

Editorial note, tabletop extrapolation: One necessary check for weak-focusing pole shaping on a 100 keV-1 MeV tabletop machine, and computable from a measured B(r) curve - necessary, not sufficient: axial focusing margin (n > 0), radial stability (n < 1), phase slip, aperture and orbit clearance all still have to be verified against the actual B(r). Where the contour cannot be pushed out to the final radius (dg-152), the design question becomes how few turns the beam spends near it, not whether the machine can work at all.

Cited in: Beam Dynamics: An Interactive Laboratory

weak magnetic focusing can be achieved by producing a magnetic field in which Bz linearly decreases.

Morrow, Focusing in the Houghton College Cyclotron — Houghton College thesis (2015) — p. 26-27

Editorial note, tabletop extrapolation: For a shimming attempt on the reference machine's 8-inch poles the plottable acceptance test is n(r) from the measured B(r), by finite differences, kept inside its stable range over the whole used radius - not a straight-line fit to B(r). A field profile that passes that test still has to be checked for isochronism and phase slip (dg-1328), the n = 0.2 contour (dg-136, dg-152) and radial stability; "no orbit code needed" was an overreach, though a simple n(r) plot does reject a bad shim before any tracking is run.

Cited in: Beam Dynamics: An Interactive Laboratory

the magnetic field actually increases in magnitude out to around r = 5 cm at which point it begins decreasing again. This is problematic because it yields a negative field index

Morrow, Focusing in the Houghton College Cyclotron — Houghton College thesis (2015) — p. 51-53

Editorial note, tabletop extrapolation: A concrete trap when adding iron or coils near the center of an 8-inch pole: check the sign of dB/dr over the whole usable orbit range, not just at the edge - the 5 cm crossover and the -0.1 index are that geometry's numbers, not general thresholds.

the difference in radius is minimal - about 0.2 cm - and comes at the steep cost of a ~20% reduction in maximum magnetic field from 1.27 T to 1.07 T. As such, this modification was considered insufficient.

Morrow, Focusing in the Houghton College Cyclotron — Houghton College thesis (2015) — p. 53-54

Editorial note, tabletop extrapolation: Saves a next machine's builder from spending months on one class of trim-coil fix inside a small gap (they also steal gap height) - but this is one modelled geometry: evaluate any other trim-coil design from its full B(r) map and orbit dynamics.

Pole face: 150mm, Pole gap: 39mm, Current: 70A ;NOTE: this is a slice down the middle of the magnet

Morrow, Focusing in the Houghton College Cyclotron — Houghton College thesis (2015) — p. 60-68

Editorial note, tabletop extrapolation: Low-cost simulation path for a hobbyist - Superfish is free, SIMION is paid but widespread, and FEMM plus the playbook's Python tracker is the all-free equivalent; the appendix geometry file is a working starting template for an 8-15 cm pole magnet.

much of the beam current is being lost by the time the beam reaches larger radii... This could be done by adding shims of ferromagnetic material between the chamber and pole faces.

Fuller, Exploring the Capabilities of the Houghton College Cyclotron — Houghton College thesis (2013) — p. 52-53

Editorial note, tabletop extrapolation: Predicts the current-vs-radius profile the builder should measure. If a next machine loses beam before full radius, diagnose first - map B(r) and n(r) and identify the loss mechanism - then shim, and re-verify field and transmitted current; a shim can worsen the index if misshaped.

the value of n for the cyclotron must be between 0 and 1; it has been determined empirically the index should increase with r roughly linearly between 0 and 1

Loucks, Initial Results from the Houghton College Cyclotron — Houghton College thesis (2007) — p. 21-23

Editorial note, tabletop extrapolation: Source-specific weak-focusing guidance, and not a target to shim toward: 0 < n < 1 is the stability condition, but the empirical 0-to-1 ramp is Loucks' description of one machine's profile, not an instruction to drive n as high as possible. n = 0.2 is the Walkinshaw coupling resonance (dg-136, dg-152, dg-563, dg-694), and in a many-turn classical cyclotron it can constrain the usable orbit long before n approaches 1. Map B(r) with a Hall probe, compute n(r) by finite differences, and shape the profile with that contour in mind. [Note added 2026-08-22: the resonance cross-reference was missing; read in isolation the rule invited shimming toward n = 1.]

Cited in: Beam Dynamics: An Interactive Laboratory

The magnet's upper and lower coils are independently energized enabling an intentional axial field imbalance so as to vertically shift the accelerating plane.

Koeth, Undergraduate Education with the Rutgers 12-Inch Cyclotron (2015) — p. 2

Editorial note, tabletop extrapolation: Cheap beam-height trim for a next machine: two supplies, or a properly rated current-trim circuit on one coil, instead of re-machining anything - verify the result with a field or beam measurement, since unequal excitation perturbs the midplane symmetry it exploits.

Cited in: Beam Dynamics: An Interactive Laboratory

if n = 0.2 is to be avoided (vx=2vz), then the rate at which the vertical field decreases must be moderated such that n=0.2 occurs at the final ion radius.

Koeth, Undergraduate Education with the Rutgers 12-Inch Cyclotron (2015) — p. 3

Editorial note, tabletop extrapolation: The quantitative pole-taper design rule for a next machine: map n(r) from the field profile and keep 0 < n < 0.2 out to full beam radius.

Cited in: Beam Dynamics: An Interactive Laboratory

We have built a set of poletips designed to intentionally drive a destructive axial resonance; we refer to these as the 'bad' weak focusing poles tips. The n=0.2 location occurs near r=3.5 inches, well within the 5 inch DEE radius, so as to allow the ion displacement to grow. Since the n=0.2 is a difference resonance the axial peak-to-peak amplitude is bounded by the initial radial offset. A displacement of the chamber's center of about 3mm with respect to the magnet center was necessary to seed the resonant axial blow up

Koeth, Undergraduate Education with the Rutgers 12-Inch Cyclotron (2015) — p. 5

Editorial note, tabletop extrapolation: Shows how little margin there is between good and bad tapers on an 8-12 inch machine; motivates measuring n(r), not guessing it.

Cited in: Beam Dynamics: An Interactive Laboratory

The axial tune... can be summarized by: vz2 = -k + F(1+tan2xi) and the radial tune is written as: vr2 = 1+k

Koeth, Undergraduate Education with the Rutgers 12-Inch Cyclotron (2015) — p. 6

Editorial note, tabletop extrapolation: If a next machine gets sector pole tips (allowing a rising average field), these two lines are the first-order SCREEN - in the source's conventions: check the sign convention for k and the flutter definition before substituting (dg-156's lesson) - with resonance avoidance and then tracking completing the design.

Cited in: Beam Dynamics: An Interactive Laboratory

F = ((<B^2> - <B>^2)/<B>^2) is called the flutter and represents the hill to valley field difference

Beeckman, Cyclotron Magnets — ECPM37 lecture, Groningen (2009) — p. 25-26

Editorial note, tabletop extrapolation: The complete design equation set for an AVF follow-on build; every term is measurable from a 2-D Hall-probe map of the built magnet.

Cited in: Beam Dynamics: An Interactive Laboratory

N large: high maximum energy, F small and quasi circular orbits -> spiral compulsory

Beeckman, Cyclotron Magnets — ECPM37 lecture, Groningen (2009) — p. 26, 28

Editorial note, tabletop extrapolation: Explains when the extra machining pain of spiral tips pays: when flutter is small and orbits quasi-circular - the quoted regime, where 'spiral compulsory'. Whether an 8-12 inch N = 4 design wants spiral or more hill/valley contrast is a computed comparison (the (1+2tan^2 xi) factor against achievable flutter), not a default.

It implies that N must be larger than 2 (lower limit of the pi stop-band) and there is an energy limit for every N value T = (N/2 - 1)E0

Beeckman, Cyclotron Magnets — ECPM37 lecture, Groningen (2009) — p. 27-28

Editorial note, tabletop extrapolation: Rules out 2-sector 'butterfly' pole tips that look easy to machine; N=3 or 4 is the practical amateur choice and neither limits sub-MeV protons.

For best RF efficiency, k=0.5 BUT to decrease machine dimensions k >0.5 (more hill, thus more field) CHOICE : k=0.67 (60 deg hills)

Beeckman, Cyclotron Magnets — ECPM37 lecture, Groningen (2009) — p. 69

Editorial note, tabletop extrapolation: The arithmetic to go from a required <B> to hill/valley fields and sector angle - first-order and reusable at any scale, with fringe and gradient effects refining it in the field code.

CHOICE : nu_z = 0.2. Flutter and spiral not too large. Field gradient can be strong. Spiral angle of pole completely determined since n, N, F and nu_z are known

Beeckman, Cyclotron Magnets — ECPM37 lecture, Groningen (2009) — p. 69

Editorial note, tabletop extrapolation: Gives a numeric focusing target to design toward instead of 'as much focusing as possible'. On a classical weak-focusing machine at the reference machine's energies, nu_z = 0.2 means n = 0.04 - modest and achievable from pole-face falloff. The caveat belongs to AVF designs: there the isochronous average field RISES with radius (vertically defocusing on its own), and the flutter/spiral term must supply the whole tune, which is exactly why the source treats nu_z as a choice that determines the spiral angle.

The radial decrease in field strength is at a rate of one percent in 13 inches out to a radius of 20 inches ... These shims reduce azimuthal variations to less than 0.2%.

Livingston & Boch, The Oak Ridge 86-Inch Cyclotron — ORNL-1196, OSTI 4357145 (1952) — p. 15, 35

Editorial note, tabletop extrapolation: The fractional pattern transfers, not the inches: a smooth, monotonic few-percent center-to-edge fall-off with azimuthal ripple shimmed to the few-per-mille level is what the 86-inch exemplifies. The right numbers for an 8-inch pole come from its own n(r) stability requirement (dg-003, dg-119), not from this machine's profile.

By means of auxiliary coils wound on the pole pieces it is possible to control the position of the beam with respect to the median plane of the tank. The coils consist of 65 turns of #6 wire wound on each pole piece. A dc power supply provides up to 75 amperes

Livingston & Boch, The Oak Ridge 86-Inch Cyclotron — ORNL-1196, OSTI 4357145 (1952) — p. 35

Editorial note, tabletop extrapolation: Cheap and direct for a next machine: an auxiliary winding on the poles gives a vertical-centering knob instead of mechanical re-shimming - size its ampere-turns from the field asymmetry the orbit calculation asks for (the 86-inch used up to ~4900 A-turns; a small machine needs proportionately less, but compute it), with a reversible supply and thermal check.

Cited in: Beam Dynamics: An Interactive Laboratory

Increasing the dee voltage, however, requires increasing the required voltage breakdown gap and thus the magnet hill gap, so that some compromise must be reached.

Livingston & Howard (eds.), The Oak Ridge Relativistic Isochronous Cyclotron — ORNL-2648, OSTI 4275955 (1958) — p. 65

Editorial note, tabletop extrapolation: The coupled optimization transfers: pick a next machine's dee voltage and magnet gap together, since dee clearance ultimately costs ampere-turns and field.

the design of the beam deflection system will be worked out simultaneously with the design of the magnet ... all the problems which arise from trying to obtain deflected beams after the machine is built would be avoided.

Livingston & Howard (eds.), The Oak Ridge Relativistic Isochronous Cyclotron — ORNL-2648, OSTI 4275955 (1958) — p. 85-86

Editorial note, tabletop extrapolation: Directly applicable lesson for a next machine: if an extracted beam is ever wanted, reserve the azimuthal slot, field-edge profile, and feedthrough ports now, even if the deflector comes later.

an input RF power level of 50 watts is too low, and 500 watts should be sufficient. The first ions are expected to clear the chimney at approximately 200 watts.

Koeth, Rutgers 12 Inch Cyclotron Ion Source Studies: Part I (2006) — p. 5

Editorial note, tabletop extrapolation: A geometry trap for a next machine: any chimney or source structure must be smaller than the first half-turn diameter set by the dee voltage, or beam dies before the first gap crossing.

Protons: f (megacycles) = 1.52B (kilogauss) ... The actual technique used to control resonance in a cyclotron is to vary the magnetic field, with the applied frequency held constant.

Livingston & Blewett, Particle Accelerators (1962) — p. 156

Editorial note, tabletop extrapolation: The reference machine's 0.59 T (5.9 kG) gives 8.97 MHz, confirming their ~9 MHz choice; for a next machine pick B first, then f = 1.52*B.

there will be a somewhat lower potential at the ends of the D faces nearest the lines ... measured in some cyclotrons to be as great as 5 per cent ... a displacement of the ion source of over 2 in. has been necessary.

Livingston & Blewett, Particle Accelerators (1962) — p. 164

Editorial note, tabletop extrapolation: Make the source mount adjustable in both directions and tune position for beam, not for geometric center. Size the travel from RF-field and orbit modelling for the actual dee geometry - the large machines needed over 2 inches; what a tabletop machine needs is its own calculation, and generous commissioning range is cheap.

Cited in: Beam Dynamics: An Interactive Laboratory

the practical maximum migration in phase will be from zero to -pi/2 and back to zero, a total phase migration of pi radians or one half-cycle.

Livingston & Blewett, Particle Accelerators (1962) — p. 166-171

Editorial note, tabletop extrapolation: With a 3-4% field droop and 160 turns-scale acceleration, the reference machine's dee voltage sets how much phase slip they can afford: higher V = fewer turns = more field-shape tolerance.

Increasing the D voltage requires fewer turns for acceleration to maximum energy and will compensate for a larger phase shift. However, D voltage is usually limited by ... power and spark breakdown.

Livingston & Blewett, Particle Accelerators (1962) — p. 172

Editorial note, tabletop extrapolation: At ~1.3 kV and ~150 keV the reference machine's ions make ~60 turns; doubling dee voltage halves turns and dramatically relaxes both field-uniformity and vacuum (scattering) requirements.

apply a radio frequency oscillating voltage to the electrode that is slightly less than the cyclotron frequency given at the center of the field, but greater than [that] near the edges.

King, A Preliminary Design for a Small Permanent Magnet Cyclotron — Houghton College thesis (2002) — p. 20

Editorial note, tabletop extrapolation: A concrete tuning rule for the builder: do not tune RF to the central field alone - place it inside the quoted window and find the best point empirically by beam current; the phase-history reasoning is the theory behind the knob, not a substitute for turning it.

It would only take a few cycles, on the order of 12, for most cyclotrons to have reached this velocity.

King, A Preliminary Design for a Small Permanent Magnet Cyclotron — Houghton College thesis (2002) — p. 20

Editorial note, tabletop extrapolation: Explains failed runs where beam dies at small radius, and quantifies how little phase budget a mistuned machine has - for the actual machine, integrate the slip turn by turn from the measured B(r) and dee voltage rather than using 12 turns as a threshold.

This value is about 10% for 120 uamp of deflected deuterons, increasing to 15% for a 200 uamp beam. About 30% of the internal beam at the exit radius is extracted.

Ramler & Parker, The Argonne 60-Inch Cyclotron — ANL-5907 (1959) — p. 19

Editorial note, tabletop extrapolation: Sets expectations if a next machine attempts a deflector: capturing a third of the circulating beam was a mature machine's result, so plan around numbers of that order - and account for where the rest goes (septum heating, sputtering, and at higher energies activation), rather than booking the loss as free.

fo = qBo/2pi mi = (1.52x10^7) Bo(tesla)/A

Humphries, Principles of Charged Particle Acceleration (1986) — p. 524

Editorial note, tabletop extrapolation: One-line check of the reference machine's operating point: 0.59 T -> ~9.0 MHz for protons; sets the next machine's RF band for any target field.

the final kinetic energy is maximized by taking Vo large... a high gap voltage accelerates particles in fewer revolutions so that there is less opportunity... to get out of synchronization.

Humphries, Principles of Charged Particle Acceleration (1986) — p. 530-531

Editorial note, tabletop extrapolation: At sub-MeV this limit is distant - by this formula a 10 kV dee puts the proton ceiling near 7 MeV - but the same physics governs field-flatness tolerance: fewer turns forgives more field error.

Low energy proton in 1 T field: 15.23 MHz

Antaya, Cyclotron Basics — MIT 8.277, Unit 10, Lecture 14 (2010) — p. 29

Editorial note, tabletop extrapolation: The single most-used number in the reference machine's notebook: 0.59 T -> 9.0 MHz; a 1.2 T higher-field successor -> 18.3 MHz, still comfortable amateur-radio-technique territory.

250 MeV protons; 17 KeV/turn: N~15,000... 250 MeV protons r=0.3m: dr/dN ~ 20 microns!

Antaya, Cyclotron Basics — MIT 8.277, Unit 10, Lecture 14 (2010) — p. 43

Editorial note, tabletop extrapolation: For the builder: 1 MeV at 2 kV per gap (2 gaps) is ~250 turns with final-orbit spacing ~0.25 mm at r = 12 cm - which is why higher dee voltage directly eases both extraction and vacuum requirements.

dphi/dn=360 [gamma-1] -> 8 deg. An ion on peak phase is lost in 11 revolutions. Only solution- very high energy gain per turn - 360kV

Antaya, Cyclotron Basics — MIT 8.277, Unit 10, Lecture 14 (2010) — p. 51

Editorial note, tabletop extrapolation: At 1 MeV the instantaneous slip is only ~0.4 deg/turn, but slip accumulates over every turn, so what that buys depends on volts per turn: a machine gaining a few kV per turn spends thousands of turns getting to 1 MeV and can run out of phase well below the textbook ceiling. The design check is the summed slip across all turns against the +/-90 deg window, not the per-turn number.

the 'single-dee' construction; this has many advantages ... Better ion focussing can be obtained by installing a 'dummy' grounded dee edge symmetric to the insulated dee, but this is a refinement

Wouters, General Recommendations for Design of Small Cyclotrons — UCRL-476 (1949) — p. 8

Editorial note, tabletop extrapolation: Exactly the reference machine's architecture. The dummy-dee edge is the source's named refinement for better ion focusing - a natural next-machine upgrade once the basic machine works, which is the sequencing the source itself implies ('but this is a refinement').

No peaks for magnetic fields larger than H2+ at 0.5 T -> 35 keV; 44 orbits at 800 Vpp

Yuly et al., Modifications on the Houghton College Cyclotron (2010) — p. 21

Editorial note, tabletop extrapolation: Quantifies why the reference machine's dee-voltage upgrade matters: at 1.3 kV their protons need ~hundreds of turns to reach interesting energies, and ~44 turns was already the survival ceiling at Houghton's pressures.

The axial component of the magnetic field can be decreased at larger radii in order to increase the radial (focusing) component, provided the ions only have a few revolutions left once they reach this portion of the field.

Morrow, Focusing in the Houghton College Cyclotron — Houghton College thesis (2015) — p. 27-28

Editorial note, tabletop extrapolation: One candidate for the reference machine's next big win: at ~150 keV on a low Dee voltage the turn count is large, and cutting it relaxes both the phase budget and gas-scattering exposure. Whether Dee voltage or field shaping pays more on a given machine is a diagnosis - measure what actually limits the beam first; the quote's own condition is narrower: late-radius focusing tricks need few turns remaining.

It can be seen that in general, an increase in dee voltage results in a higher beam current.

Fuller, Exploring the Capabilities of the Houghton College Cyclotron — Houghton College thesis (2013) — p. 49-50

Editorial note, tabletop extrapolation: For a fill-gas machine, dee volts are a strong current knob - the measured trend here: fewer turns, less path, fewer collisions. Whether they are THE binding knob depends on what limits the machine that day: source output, pressure, phase acceptance and detuning all compete (dg-359, dg-525). Measure before spending.

higher potential on the dees results in fewer orbits and a shorter time of acceleration, allowing for a higher maximum kinetic energy... gives a maximum of 15 MeV for protons with 50 kV peak-to-peak

Loucks, Initial Results from the Houghton College Cyclotron — Houghton College thesis (2007) — p. 24-26

Editorial note, tabletop extrapolation: At the reference machine's ~150 keV the relativistic shift is small (gamma-1 ~ 0.02%), but phase slip accumulates over the whole turn count, so low volts-per-turn can still spend the +/-90 deg budget well below the textbook ceiling (dg-273's summed-slip check). This rule sets the fixed-frequency ceiling for any future MeV-class ambition.

It is possible to calculate the various effects quantitatively and to predict the minimum dee voltage required to obtain a given energy in a particular cyclotron.

Livingston & Boch, The Oak Ridge 86-Inch Cyclotron — ORNL-1196, OSTI 4357145 (1952) — p. 17-19

Editorial note, tabletop extrapolation: Directly applicable design step for a next machine: compute threshold voltage for the target energy and field taper before freezing the RF chain power budget.

As measured, efficiency tends to increase with dee-to-dee potential and with beam power.

Livingston & Boch, The Oak Ridge 86-Inch Cyclotron — ORNL-1196, OSTI 4357145 (1952) — p. 24-26

Editorial note, tabletop extrapolation: The order of magnitude transfers as expectation-setting: most RF power goes to resonator and ion-loading losses, so size a next machine's RF from resonator dissipation (dg-313), not from beam power.

the maximum voltage gain/dee is Vd = 2*V0 sin(theta/2); for particles rotating on subharmonics of the dee frequency the angular width of the dee is n*theta to the particle

Livingston & Howard (eds.), The Oak Ridge Relativistic Isochronous Cyclotron — ORNL-2648, OSTI 4275955 (1958) — p. 100

Editorial note, tabletop extrapolation: Directly applicable when trimming a next machine's dee for probe or source clearance: keep the dee close to 180 degrees or compute the sin(N*theta/2) penalty for the harmonic in use. Fundamental-mode trims are gentle - 15 degrees off costs about 1% at N = 1 - but the same trim costs more at higher harmonics.

the beam width was found to be limited by the internal aperture of the D's out to about one-third of the final radius and then to narrow in a nearly linear fashion out to the exit slit.

Livingston & Blewett, Particle Accelerators (1962) — p. 163-167

Editorial note, tabletop extrapolation: Give the first third of radius generous vertical aperture - that is where the envelope filled the dee aperture on MIT's machine - and let the outer region run tighter, which also helps RF economy. Confirm on the actual machine (witness strips, dg-695) rather than assuming the same profile.

There is no vertical magnetic focusing at the center of the magnet. By a fortunate coincidence, electrostatic focusing by the accelerating fields is effective for low-energy ions.

Humphries, Principles of Charged Particle Acceleration (1986) — p. 524, 526

Editorial note, tabletop extrapolation: Explains why source-to-dee geometry (chimney position, puller gap, aperture height) dominates beam capture on small machines: at the center magnetic vertical focusing vanishes and only builds as n grows off zero with radius, so the electric gap lens is what the first turn or two get. Central-region electrode design is where capture is won on the documented machines.

some 90% of the initial supply of ions are lost to the dee surfaces. The loss may be reduced by increasing the dee voltage, thus reducing the number of turns an ion makes

Livingston & Boch, The Oak Ridge 86-Inch Cyclotron — ORNL-1196, OSTI 4357145 (1952) — p. 18

Editorial note, tabletop extrapolation: Directly applicable: at 1.3 kV the reference machine's protons make many turns, and the quoted machine cut its central losses with more dee volts. A strong transmission lever - alongside central-region geometry (dg-348), which shapes what the first turns even see; measure which binds before spending (dg-303).

In more practical units, this equation can be rewritten: j[mA/cm2] = 1.72 * sqrt(q*/u) * phi[kV]^(3/2) / d[mm]^2.

Wolf (ed.), Handbook of Ion Sources (1995) — p. 376-377

Editorial note, tabletop extrapolation: Confirms the reference machine's nA beams are nowhere near space-charge limits; if extraction is weak the problem is geometry/plasma matching, not the Child-Langmuir ceiling.

For the cylindrically symmetric case the maximum current can be estimated from References 11 and 12 and the assumption of a certain aspect ratio (aperture radius to electrode separation). A good aspect ratio is on the order of S = 0.5.

Wolf (ed.), Handbook of Ion Sources (1995) — p. 379

Editorial note, tabletop extrapolation: For the puller gap in a next machine, carry over the shape of the rule - aperture dimension about half the extraction gap, beam parallelism tuned by matching plasma density - but the cited numbers are for round apertures: model the actual chimney slit electrostatically or by simulation rather than substituting the slit half-width, and expect to adjust both arc density and geometry.

The electrode closest to the source was the suppressor and was located 2.5 cm from the cathode faceplate. A target electrode was placed 7.6 cm from the suppressor. During high-voltage operation, the suppressor and target were biased negative with respect to the ion source cathode (i.e., ground). ... at a pressure of 0.4 mTorr and 10.5 W PIG source power, a continuous 1 mA positive hydrogen ion beam has been focused onto the target and accelerator voltages up to -30 kV have been investigated.

Rovey, Ruzic & Houlahan, Simple Penning Ion Source for Laboratory Research and Development Applications (2007) — p. 3

Editorial note, tabletop extrapolation: Template for a bench extraction test stand to characterize the next machine's source before it goes into the magnet.

The chimney with the larger slit produces a beam with a larger emittance. However, the beam is also of higher intensity.

Forringer, Phase Space Characterization of an Internal Ion Source for Cyclotrons — MSU dissertation (2004) — p. 66-69

Editorial note, tabletop extrapolation: Suggests slit width is a powerful knob worth sweeping on a next machine: expect more current and more emittance from a wider slit, and stop widening when the machine acceptance is filled - but the 4.4x/1.7x ratios are that source's numbers; do your own aperture sweep and acceptance analysis before extrapolating.

an approximately flat plasma boundary provides the best match to the experimental beams emerging from the 'slit' style chimneys... while a concave plasma boundary... for the 'hole' style chimney ... The size of the hole in the chimney is 0.047 (1.19 mm) with a sixty degree chamfer. At 50 mA of arc current, the normalized luminosity of the beam which made it to the wire probe was 129 A/(mm2-sr), about half of that for the slit chimney with the same arc current (264 A/(mm2-sr)).

Forringer, Phase Space Characterization of an Internal Ion Source for Cyclotrons — MSU dissertation (2004) — p. 73-76, 91-107

Editorial note, tabletop extrapolation: Decides a next machine's chimney aperture style within the tested regime: cut a tall narrow slit rather than drilling a hole if beam brightness and predictable optics matter - and re-verify on the actual source, since the ranking comes from these apertures and operating points.

the general trend of increasing arc current producing increased beam current as expected... there was no noticeable change in the emittance of the beam for different currents

Forringer, Phase Space Characterization of an Internal Ion Source for Cyclotrons — MSU dissertation (2004) — p. 77-78, 81

Editorial note, tabletop extrapolation: Cranking arc power buys current with diminishing returns, and in this scan it did not spoil measured emittance. Do not read that as space charge being negligible in general: generalized perveance rises steeply at low velocity, so keep space charge in extraction and first-turn models until a sensitivity check shows it is negligible at your energy and current.

the plasma temperature that provides the best match for experimental beams is approximately 35,000 K (resulting in a central starting energy of 4.5 eV).

Forringer, Phase Space Characterization of an Internal Ion Source for Cyclotrons — MSU dissertation (2004) — p. 91-107

Editorial note, tabletop extrapolation: A starting calibration for any first-turn simulation of a next machine's central region: begin near 35,000 K / 4.5 eV, then sweep the initial temperature and meniscus shape and validate against measured emittance or beam profiles - the value is a fitted effective parameter, not a universal plasma property.

An acceptable vacuum would allow for a mean free path an order of magnitude larger than the expected flight distance. For protons accelerated by a 2.7 T magnetic field and a voltage difference of 1000 V between the electrodes, the expected flight distance is approximately 300 meters.

Dewan, Design and Construction of a Cyclotron Capable of Accelerating Protons to 2 MeV — MIT thesis (2007) — p. 9-11

Editorial note, tabletop extrapolation: The quantitative vacuum spec for a next machine: turns = final energy / energy-per-turn, and total path is turns times the average orbit circumference (about two-thirds of the final one for r proportional to sqrt(E)); halving dee voltage doubles the path and tightens the pressure requirement proportionally. Evaluate lambda with sigma(E) from ORNL-6086 along the orbit (the vacuum calculator's orbit mode does this); a 150 keV, 1 kV-per-dee proton machine needs ~1e-5 torr for 10% loss, not 1e-3.

Cited in: Beam Quality: What It Is and What Degrades It · The Vacuum Budget of a Cyclotron

A typical figure, used in the MIT cyclotron, is a dR of 0.15R. The deflector gap is usually tapered ... Spacings as small as 1/8 in. can be used at the entry slit, opening to 1/2 in. or greater at the exit.

Livingston & Blewett, Particle Accelerators (1962) — p. 180-181

Editorial note, tabletop extrapolation: Scaled to ~150 keV the same normalized geometry needs only ~500-900 V on the deflector - an easy supply. Entry-slit width is set against the local turn separation and beam width together (dg-495), not by a fixed prescription.

It is seen that particles are detected at about 70 kv. and the numbers increase more rapidly with increase of bombarding energy than with the lithium film.

Oliphant & Rutherford, Experiments on the Transmutation of Elements by Protons (1933) — p. 266-270

Editorial note, tabletop extrapolation: Proof that p-B11 alphas are observable far below the 675 keV resonance: the 1933 apparatus saw them at 60-70 kV using tens of microamps and large solid angle, so a lower-current machine compensates with integration time and geometry. Their alphas stopped in under 5 cm of air, hence the vacuum path to a PIPS detector.

Cited in: Experiments by Energy Band

the ion revolution turn spacing near r = 4 inches, in a B-field of 1.0T will be just 0.04 inches, which is smaller than the 0.06 inch RF shield of the tip

Koeth, Rutgers 12 Inch Cyclotron Ion Source Studies: Part I (2006) — p. 1

Editorial note, tabletop extrapolation: At the reference machine's ~1.3 kV the turn spacing is smaller still, so geometry matters doubly: a thick tip costs single-turn radial resolution first, and a shield mounted AHEAD of the collector can shadow it into reading zero while beam exists - the quoted case. A bare, grooved copper collector is the safe default until turn-resolved measurements are wanted.

The limit at the entry is set by the dr between successive turns at this radius ... it is desirable to have adjustable controls on deflector spacing and location which can be trimmed empirically.

Livingston & Blewett, Particle Accelerators (1962) — p. 181-183

Editorial note, tabletop extrapolation: With ~2.6 keV total gain per turn at ~150 keV, the reference machine's turn spacing at extraction is ~0.9% of r - about 0.7 mm at r = 3.2 in - so build the septum mount with millimetre-scale radial adjustment.

Emergent beam intensities up to 25 per cent of the resonant beam intensity have been obtained under optimum conditions ... practical operating intensities would in this case be limited to 80 or 100 ua.

Livingston & Blewett, Particle Accelerators (1962) — p. 182

Editorial note, tabletop extrapolation: Judge a next machine first on internal-probe current at full radius: documented machines commonly ran internal currents several times their extracted beam (MIT's optimum was 4:1), so a gap of that order is precedented rather than failure. The rung-by-rung extraction picture is dg-595's; the census cross-checks are dg-260's.

allows most of the resonant beam to pass into the deflector channel without striking the channel walls. In the MIT cyclotron the septum is formed of two strips of tungsten, 0.020 in. thick and 12 in. long and with the edges spaced 1/8 in. apart. Each strip is silver-soldered to a copper bar bent to the correct curvature, with copper tubing also soldered to the bar for cooling. Other designers use a long V slot in a tungsten-strip septum, so the heat is distributed over an extended surface

Livingston & Blewett, Particle Accelerators (1962) — p. 184

Editorial note, tabletop extrapolation: At the reference machine's beam power the thermal load is small but not zero - intercepted power is loss current times energy per charge (1 uA of 500 keV beam is 0.5 W into a very small spot) - so compute it, and keep the slotted geometry: it maximizes transmitted current into the channel either way.

once the particle has been accelerated to 10 MeV the mass has been changed by about 1%, which has a frequency shift of 1%.

King, A Preliminary Design for a Small Permanent Magnet Cyclotron — Houghton College thesis (2002) — p. 12, 21

Editorial note, tabletop extrapolation: At the reference machine's 100 keV-1 MeV scale the instantaneous shift is ~0.01-0.1%. Whether it can be ignored is a turn-count question: with hundreds of volts to kilovolts per turn a sub-MeV machine has phase budget to spare, but the check is the accumulated slip against the +/-90 deg window (dg-273), not the per-turn number.

The separation for non-relativistic ions is dR = (R/2) (2qVo sin phi_s/T)... Eq. (15.3) implies that dR = 0.44 cm.

Humphries, Principles of Charged Particle Acceleration (1986) — p. 527

Editorial note, tabletop extrapolation: Lets the builder compute whether a probe or future septum can distinguish final turns: at ~150 keV, r ~ 9.6 cm and the reference machine's 2.6 keV total gain per turn, dR = (r/2)*(2.6/150) ~ 0.8 mm - tight for a probe, and doubling volts-per-turn doubles it.

0.15 0.91 +/- 0.015... 0.65 218.42 +/- 0.55

Spraker et al., The 11B(p,α)8Be → α+α and the 11B(α,α)11B Reactions at Energies Below 5.4 MeV (2012) — p. 360

Editorial note, tabletop extrapolation: The master rate table for the reference machine's PIPS window (150-675 keV) - as fitted A0 coefficients: a count-rate prediction folds in the angular terms, solid angle, target thickness and integration time (the experiments-by-energy arithmetic). What the table quantifies exactly is what reaching the resonance is worth: ~240x in A0 from 150 to 650 keV.

Cited in: Experiments by Energy Band

Our parameters: B=.976 T ... T=.472 MeV ... d=.291 inches ... Combining yields: Vd/d = (2T/R)(dR/(R+dR)) ... Vd = 32.531 kV

Ponter, Beam Energy Measurements with a New HV Deflection System and Ion Source Upgrades on the Rutgers 12-Inch Cyclotron (2010) — p. 6 (R and dR on 7)

Editorial note, tabletop extrapolation: Gives the builder the extraction-voltage scale for a next machine: deflector voltage scales linearly with beam energy at fixed geometry, so a ~100 keV beam needs about a fifth of a 472 keV machine's figure in the same channel. Given the source's formula/number discrepancy (see formula field), size from the formula and confirm by measurement.

H2+/9 H+/7 H+/5 H+/3 H+ H2+

Yuly et al., Modifications on the Houghton College Cyclotron (2010) — p. 15-18

Editorial note, tabletop extrapolation: Prevents misidentifying beam in the reference machine's B-field sweeps: a peak at one-third the expected field is a CANDIDATE for the same ion on the 3rd harmonic - confirm by species diagnostics or scaling tests, since another species/harmonic combination can land at the same field.

Voltage -kV dc / Current milliamps / Pressure millitorr / Neutrons cpm: 16, 11, 18, 10 ... 25, 8.1, 14, 60 ... 31, 10.8, 13, 130. Figure 25 - Neutron readings versus other chamber parameters.

Kovalchick, Deuterium Fusion Using Inertial Electrostatic Confinement (2012) — p. PDF p.46 = printed p.46 (Kovalchick, 'Experiment 7 - Observations', Figure 25)

Editorial note, tabletop extrapolation: The same lesson as the p-B11 cross-section curves: sub-barrier yield rises steeply with particle energy, so extra beam energy buys far more counts than the same fractional increase in current. The specific sweep numbers are one fusor's; the steepness is the physics.

Cited in: Experiments by Energy Band

It was limited to low level output by its 22kv internal supply. 103 n/sec... a 33 kilovolt supply. 105 n/sec... currently produces in excess of 600,000 neutrons per second

Hull, Fusor: An Easy to Construct Fusion Reactor Based on Inertial Electrostatic Confinement (2009) — p. 36-39

Editorial note, tabletop extrapolation: Reinforces energy-over-current for the builder: sub-Coulomb-barrier reaction rates reward every extra keV steeply - though not by a fixed orders-per-10-kV law; Hull's own steps differ between jumps.

Cited in: Experiments by Energy Band

If many ions in the beam fall out of phase before reaching maximum Dee radius, the beam current will drop suddenly to near zero beyond whatever radius the ions tend to reach

Morrow, Focusing in the Houghton College Cyclotron — Houghton College thesis (2015) — p. 28, 57

Editorial note, tabletop extrapolation: Diagnostic direction, not verdict: a sharp cutoff in the radial current profile is CONSISTENT with phase slip - and also with aperture interception, wall collisions or vertical-envelope loss - so discriminate by what moves it: RF frequency and dee-voltage changes shift a phase-slip radius, mechanical interception does not, and field trim tells its own story.

for a fixed frequency f, resonances will occur for lower magnetic fields, e.g. B/3 and B/5, corresponding to an odd multiple of a lower frequency

Yuly, The Houghton College Cyclotron: a Tool for Educating Undergraduates — Cyclotrons 2013, WE1PB01 (2013) — p. 4-5

Editorial note, tabletop extrapolation: Practical commissioning technique: a magnet-current sweep plus an electrometer assigns candidate species/harmonic pairs to each peak. Confirming that a peak is really protons (and clean) still needs field calibration and, where purity matters, an independent species check.

Frame-by-frame analysis... revealed that ions nominally populate 40 degrees of the 360 degree RF cycle in our cyclotron.

Koeth, Undergraduate Education with the Rutgers 12-Inch Cyclotron (2015) — p. 9

Editorial note, tabletop extrapolation: Sets expectations for fast diagnostics and duty-factor arithmetic on any machine: measure your own bunch width (capacitive pickup, gated counting) and use it - 40 degrees is one measured machine's figure, not a constant.

The axial focusing, as shown above, takes place for any positive values of the field decay exponent. Therefore, orbital stability in both directions takes place only for 0 < n < 1.

Smirnov, The Cyclotron and Its Modeling — Phys. Part. Nuclei 52 (2021) — p. 18-20

Editorial note, tabletop extrapolation: The governing stability rule for the weak-focusing reference machine: check the FEMM-derived B(r) for 0 < n < 1 over the working radii. Two refinements: n tends to zero at the machine center by symmetry, so the requirement bites from the first working orbits outward; and the value n takes is the designer's shaping choice - weak-focusing machines run it small at inner radii, rising toward extraction.

Cited in: Beam Dynamics: An Interactive Laboratory

its passage without noticeable losses of particles becomes possible only due to the fact that the beam crosses it for 1-3 revolutions, and the first harmonic of the magnetic field with a large amplitude is absent

Smirnov, The Cyclotron and Its Modeling — Phys. Part. Nuclei 52 (2021) — p. 37

Editorial note, tabletop extrapolation: In the reference machine Qr = sqrt(1-n) sits just below 1 everywhere, so first-harmonic field symmetry is the load-bearing tolerance: a coherent distortion driven by B1 grows while the resonance condition holds, and the crossing survives when B1 is small (the quote's condition) and the crossing fast. How small is computed for the actual machine - the beam-dynamics laboratory's imperfection tools do it; pole tilt and off-center coils are the usual B1 sources.

Cited in: Beam Dynamics: An Interactive Laboratory

When transferring the energy of radial betatron oscillations into axial oscillations, the amplitude of the latter turns out to be twice the amplitude of radial oscillations.

Smirnov, The Cyclotron and Its Modeling — Phys. Part. Nuclei 52 (2021) — p. 39

Editorial note, tabletop extrapolation: Very concrete for the reference machine: if the edge-field falloff pushes n through 0.2 near the last turns, dwelling ions grow vertically as far as the coupling perturbation drives them - into the dee aperture if allowed. Keep n below ~0.2 out to the extraction radius, or cross the resonance fast (dg-152, dg-694).

Cited in: Beam Dynamics: An Interactive Laboratory

if the field is formed such that the deviation from the isochronous one for all operating radii is no more than 5 G, then this corresponds to a deviation of the RF phase... by no more than 5 degrees

Smirnov, The Cyclotron and Its Modeling — Phys. Part. Nuclei 52 (2021) — p. 50

Editorial note, tabletop extrapolation: The 5 G <-> 5 deg pairing is that machine's arithmetic, not a portable spec: phase slip accumulates with turn number, harmonic and energy gain per turn, so integrate it turn by turn from the measured B(r) and RF parameters, and set the reference machine's shimming tolerance from the resulting phase-acceptance budget.

Then the RF phase shifts to the values at which the particles cross the accelerating gaps with the optimal phase. Thus, conditions are created for the additional focusing of particles in the axial direction at the first revolutions by a high-frequency electric field. Depending on the configuration of the central region of the cyclotron, the level of the magnetic field in the center is raised to an amount of a few tens to a few hundred gauss

Smirnov, The Cyclotron and Its Modeling — Phys. Part. Nuclei 52 (2021) — p. 50-51

Editorial note, tabletop extrapolation: Usable on a next machine: shim a small central cone so B falls gently from center outward, and set the central-region phase so the electric focusing helps rather than hurts - then verify the resulting field index and phase history by model; RF electric focusing means n~0 first turns are not wholly unfocused even before the bump.

The presence of the resonance makes it possible to use the first harmonic of the field with a small amplitude (usually a few gauss) to obtain a significant increase in radial amplitudes.

Smirnov, The Cyclotron and Its Modeling — Phys. Part. Nuclei 52 (2021) — p. 64-65

Editorial note, tabletop extrapolation: The dR formula tells the builder exactly what turn spacing a ~kV energy gain buys at 4-inch radius (fractions of a mm), i.e. whether a septum/foil extraction is geometrically feasible for a next machine.

With a practically realizable energy set today, the final energy is limited to 10-15 MeV for protons when one or two [dees are used] ... By selecting the value of the generator frequency, it is possible to achieve that the point of changing the direction of the phase motion is near -90 degrees.

Smirnov, The Cyclotron and Its Modeling — Phys. Part. Nuclei 52 (2021) — p. 22

Editorial note, tabletop extrapolation: At 100 keV-1 MeV the reference machine is far from the ceiling. Setting the oscillator slightly below the central-field frequency is the source's strategy for spending the phase budget symmetrically - the same lever helps a machine whose field profile is imperfect, but the check remains the summed slip (dg-273), not the detuning itself.

the optimal case from the viewpoint of minimizing the radial beam losses is a mode of operation in which the value of the injection energy is less than the amplitude of the accelerating voltage across the dees

Smirnov, The Cyclotron and Its Modeling — Phys. Part. Nuclei 52 (2021) — p. 55-56

Editorial note, tabletop extrapolation: If a next machine ever moves to an external source and axial injection: the cited criterion (injection energy per charge below the dee amplitude) minimizes radial losses by letting the first gap crossings enlarge the orbit fast - set the actual injection energy jointly with inflector acceptance, transport and RF capture, not from the radial-loss criterion alone.

the phase acceptance of the center, as a rule, contains the particles, the initial RF phases of which do not go beyond the range of (-90; 20)

Smirnov, The Cyclotron and Its Modeling — Phys. Part. Nuclei 52 (2021) — p. 56

Editorial note, tabletop extrapolation: Explains why a large fraction of source output never accelerates: only starting phases inside a ~110-degree window of the full 360 are candidates at all - a uniform-emission estimate makes that a ~30% ceiling, before radial and axial acceptance cut further; it is not a measured capture efficiency. The builder tool should launch macroparticles across this window rather than a single reference phase.

The number of lost particles dN at each time step dt can be estimated by the formula dN = sigma nN v dt

Smirnov, The Cyclotron and Its Modeling — Phys. Part. Nuclei 52 (2021) — p. 70

Editorial note, tabletop extrapolation: Lets the builder tool convert a gauge reading and total path length (hundreds of turns) into a survival fraction - with the composition stated, the gauge's gas-sensitivity factor applied, and the cross sections taken at the right energies. An assumed oxygen-like composition is a labeled assumption, not a guaranteed worst case: water and hydrocarbons can exceed it for the processes that matter.

Cited in: The Vacuum Budget of a Cyclotron

The slit is most functional if it is installed in the place of the largest radial size of the beam... The closer to the center the device is installed, the more efficient it is, and the less radiation losses thereon. ... If there are several slits, then it is advisable to place them at different revolutions and azimuthally with a difference of half the period of the system, e.g., in a hill and a valley. ... Elements should be installed away from accelerating gaps, e.g., along the center lines of the space between the dees

Smirnov, The Cyclotron and Its Modeling — Phys. Part. Nuclei 52 (2021) — p. 56

Editorial note, tabletop extrapolation: Practical placement rules if the builder adds a beam-defining post or slit to clean up phase spread and improve turn separation at extraction radius.

Most often, the boundary of the transition to high intensities is determined at the level of a few hundred microamperes.

Smirnov, The Cyclotron and Its Modeling — Phys. Part. Nuclei 52 (2021) — p. 40

Editorial note, tabletop extrapolation: The reference machine's nA-uA beams sit far below the cited boundary, so omitting space-charge solvers is a reasonable default for the builder tool - but earn it with one calculation: peak (bunched) current through the low-energy first turns is where space charge bites first, so run the perveance estimate there before declaring it negligible.

the relative radial increase is only half the relative energy increase. However, for a given cyclotron, the turn separation dr0 will double when the dee voltage is doubled.

Kleeven & Zaremba, Cyclotrons: Magnetic Design and Beam Dynamics — CAS 2015, arXiv:1804.08961 (2018) — p. 44

Editorial note, tabletop extrapolation: The reference machine (~150 keV, 2.6 keV/turn, r ~ 9.6 cm) gets ~0.8 mm/turn. A 10 kV dee at the same radius scales it by the ratio of per-turn energy gains - computed from the actual voltage convention and gap count: 10 kV peak with two crossings at good phase is ~20 keV/turn, ~6 mm; one effective crossing or poor phase halves it or worse.

for a final energy of T = 30 MeV, dT = 100 keV, and an extraction radius of 0.5 m, we find dr = 0.83 mm. This is a rather small number, e.g. when compared with a radial beam width of for instance 4 mm.

Botman & Hagedoorn, Extraction from Cyclotrons — CAS, CERN 96-02 (1996) — p. 6

Editorial note, tabletop extrapolation: Small machines fare better because dr/R scales as dT/T: a 350 keV next machine at 10-20 keV per turn carries a fractional turn separation 9-17x this 30 MeV machine's. Its beam width does not shrink in proportion, though - so the separation-vs-width comparison still needs the machine's own numbers (dg-495).

when a coherent oscillation amplitude x of 3 mm has been built up ... and acceleration takes place until vr = 0.8, the maximum turn separation due to precession is 3.8 mm.

Botman & Hagedoorn, Extraction from Cyclotrons — CAS, CERN 96-02 (1996) — p. 7

Editorial note, tabletop extrapolation: A deliberate few-mm coherent amplitude (source off-centering is one way to seed it), plus letting nu_r fall toward 0.8 in the fringe, can multiply turn spacing severalfold - IF the precession phase is arranged so the separation appears at the septum azimuth. It is a designed, tracked orbit-dynamics move, not a free effect.

Accelerating the beam far into the fringe field often means passing the vr = 2 vz coupling resonance. Energy can be exchanged from the radial to the vertical motion, blowing up the beam vertically and leading to beam loss. If the radial oscillation amplitude is not too large, and if the resonance is passed in only a few revolutions, vertical amplitude increase is avoided. In practice, a coherent radial oscillation amplitude of the same size as the incoherent amplitude, is a good criterion for efficient extraction. Another reason for requiring not too large radial oscillation is avoiding strong non linear effects.

Botman & Hagedoorn, Extraction from Cyclotrons — CAS, CERN 96-02 (1996) — p. 7

Editorial note, tabletop extrapolation: If a next machine's radial beam half-width is ~2-3 mm, start near a ~2-3 mm coherent amplitude and set the acceptable ceiling by tracking through the extraction field - the equality criterion is the source's practical starting point, not a hard limit.

one has to limit the radial amplitude, induced from the v = 1 resonance, to a few mm.

Heikkinen, Injection and Extraction for Cyclotrons — CAS, CERN 94-01 (1994) — p. 18

Editorial note, tabletop extrapolation: In a weak-focusing field the last turns sweep the field index upward toward these resonances: compute the actual tune curves from the measured field map, and keep energy gain per turn high through any crossing so that tracking predicts acceptable vertical growth - speed of crossing, not a fixed turn count, is the criterion.

Cited in: Beam Dynamics: An Interactive Laboratory

taking eps1 = 10-4, R = 1 m and vr - 1 = 0.01, one finds an orbit centre shift, i.e. a radial oscillation amplitude, of dx = 5 mm.

Botman & Hagedoorn, Extraction from Cyclotrons — CAS, CERN 96-02 (1996) — p. 9

Editorial note, tabletop extrapolation: Gauss-level azimuthal asymmetry matters at 0.59-0.89 T NEAR nu_r = 1: the (nu_r^2 - 1) denominator is what turns the quoted 0.6 G into 5 mm, and the sensitivity falls away from the resonance and shrinks with radius. It is both the knob (a deliberate shim or coil bump) and the hazard (uncontrolled bumps de-center the beam) - dg-562's tolerance computation is the same physics from the defensive side.

Cited in: Beam Dynamics: An Interactive Laboratory

n_eff is the effective duration of the resonance (typically around ten revolutions). ... Typically the extraction takes place near v = 0.8.

Heikkinen, Injection and Extraction for Cyclotrons — CAS, CERN 94-01 (1994) — p. 14

Editorial note, tabletop extrapolation: A smooth azimuthally symmetric weak-focusing machine approaches nu_r=1 from below and never crosses it, so create the amplitude by ion-source off-centering instead (a different mechanism than resonant buildup - verify what it delivers by tracking) and use the fringe region where nu_r has fallen toward the source's typical ~0.8 for precession, with the actual tune taken from the measured field map.

the spreading of orbit centres for different HF phases due to HF mixing, is small for an originally well centreed beam, as in general the number of turns from the vr = 1 resonance till extraction is not so large.

Botman & Hagedoorn, Extraction from Cyclotrons — CAS, CERN 96-02 (1996) — p. 9

Editorial note, tabletop extrapolation: If a next machine uses source off-centering (amplitude created at turn 1, necessarily - it cannot be placed late), evaluate the phase-mixing integral from tracked particles in the actual field map before assuming the coherent centroid survives: individual amplitudes persist while the ensemble centroid can smear. A trim bump near the extraction region, where late placement IS possible, is the cleaner tool.

PIG ion source 10-50 [energy spread, V] 5-500 [typical ion current, mA] (Table 2.1)

Wolf (ed.), Handbook of Ion Sources (1995) — p. 51

Editorial note, tabletop extrapolation: Against a few-keV effective first-gap gain a 10-50 eV spread is a ~1 percent perturbation, so source simplicity is worth keeping - confirm with the machine's own capture/acceptance estimate, since capture depends on RF phase and central-region geometry, not the gap voltage alone; the filament-arc comparison needs its own source.

We observe that an approximately flat plasma boundary provides the best match to the experimental beams emerging from the 'slit' style chimneys in our study, while a concave plasma boundary (curving toward the source axis) provides a better match for the beam that emerges from the 'hole' style chimney. In all cases, the plasma temperature that provides the best match for experimental beams is approximately 35,000 K (resulting in a central starting energy of 4.5 eV). Using the methods presented in this dissertation, the orbit tracking code Z3CYCLONE is able to predict the beam produced by a cold cathode PIG ion source with adequate accuracy such that construction of actual cyclotrons can proceed with reasonably prudent confidence that the cyclotron will perform as predicted.

Forringer, Phase Space Characterization of an Internal Ion Source for Cyclotrons — MSU dissertation (2004) — p. 3

Editorial note, tabletop extrapolation: Drop-in starting condition for the reference machine's central-region orbit models: start protons from a flat sheet across the slit with the source's 4.5 eV central energy, not from rest at a point - and sweep the parameters against measured beams per dg-423.

Post-to-foil transmission increased dramatically (from 19% to 30%) but the total target current decreased from 120 uA to 40 uA

Potkins et al., Improvements to Siemens Eclipse PET Cyclotron Penning Ion Source (2017) — p. 3-4

Editorial note, tabletop extrapolation: For a machine starved of axial acceptance a hole source may waste less injected beam, while total current favored the tall slit in the Eclipse test. The reference machine's 1.42 in physical gap suggests but does not establish generous DYNAMIC acceptance - pick slit vs hole from central-region tracking or a measured acceptance/delivered-current comparison, not the gap dimension.

with 20,000 gauss at the center, produced a field of 96.7 percent of this value at 96.5 percent of the total radius (corresponding to the magnetic index n = .2), with an approximately linear decrease in field from center to edge.

Creutz, Design and Construction of Synchro-Cyclotron — NYO-780 (1950) — p. 9-10

Editorial note, tabletop extrapolation: The transferable practice is the method: measure H(r), compute n(r) from its local slope, and place the working radius where n stays in the focusing band - do not set a shim target from endpoint percentages, and do not adopt n = 0.2 as a goal without orbit, phase-slip and extraction analysis.

Cited in: Beam Dynamics: An Interactive Laboratory

The autographs indicate a rapid spreading vertically of the beam at about 81 1/2 inches. This agrees quite closely with the point at which n = 0.2 from magnetic measurements.

Vale, 184-inch Cyclotron Vertical Beam Oscillations in the Region of 82-inch Radius — MDDC-984 (1947) — p. 3

Editorial note, tabletop extrapolation: Fully applicable as a diagnostic: map B(r) on the bench, compute n(r) and the tunes, and if the reference machine's beam stalls early, the n = 0.2 crossing is suspect number one - but confirm with tracking and check field-error resonances and aperture before moving the target radius.

The usual beam pattern of two to three pips was obtained at several probe radii; namely, 22", 28 1/2" and 35". ... the regular probe radius was 28 1/2".

Yeater, 184″ Cyclotron: Synchroscope Beam Pictures on Two Probes — MDDC-987 (1947) — p. 3 (printed "- 1 -")

Editorial note, tabletop extrapolation: The two-azimuth comparison transfers to any machine: it is a test, not a verdict - accept the precession reading when the measured inter-probe phase agrees with an orbit model and controls exclude RF pickup and coherent source modulation (which can also arrive with a fixed offset).

Approximately 60 kv could be held on the high voltage electrode of this deflector.

Sewell, 184″ Cyclotron: Vertical D.C. Electrostatic Deflector — MDDC-1051 (1947) — p. 2

Editorial note, tabletop extrapolation: Compute the next machine's required deflector field from beam rigidity, channel length and allowed interception - then design insulation, clearances and stored-energy limiting for that voltage in its own right. The series spark-limiting resistor is worth copying; the assumption that deflector HV is low-risk is not.

accelerating particles past the radius where n = 0.2 in the 184-inch cyclotron has been postponed, since the available energy of the ions at this radius is within 5 per cent of the maximum of the system

Sewell, Henrich & Vale, Some Operating Phenomena Associated with the 184-inch Cyclotron — MDDC-1092 (1947) — p. 4

Editorial note, tabletop extrapolation: Budget a next machine's energy at the n = 0.2 radius, not the pole edge - and where shims can push the n = 0.2 contour outward, that buys usable energy more surely than chasing radius into the fringe (dg-152's taper rule is the design form of the same point).

Cited in: Beam Dynamics: An Interactive Laboratory

It must be kept in mind for systems having low accelerating voltages similar to the 184-inch cyclotron, that the ions will rapidly increase the amplitude of their vertical oscillations at the point where n = 0.2.

Sewell, Henrich & Vale, Some Operating Phenomena Associated with the 184-inch Cyclotron — MDDC-1092 (1947) — p. 4

Editorial note, tabletop extrapolation: The reference machine's few-kV dee means many turns near any resonance radius - the slow-crossing regime the quote warns about. Keep n below 0.2 over the whole usable radius (the mapped check, dg-138), and give the dee aperture real margin over the expected radial oscillation amplitude.

Cited in: Beam Dynamics: An Interactive Laboratory

The frequency of the minor pulses in each beam pulse agrees quite well with the calculated frequency of precession of the center of rotation of the ions about the magnetic center of the system

Sewell, Henrich & Vale, Some Operating Phenomena Associated with the 184-inch Cyclotron — MDDC-1092 (1947) — p. 4

Editorial note, tabletop extrapolation: Transfers with caveats: on a CW fixed-frequency machine you need a pulsed source or fast probe electronics to see the structure, but a pulsed-arc run makes precession directly visible on a scope - treat the inverted n as an approximate effective-tune diagnostic, cross-checked against the field map.

Cited in: Beam Dynamics: An Interactive Laboratory

Field drop-off 3-4 %

Howard, Cyclotrons and High-Energy Accelerators, 1958 — ORNL-2644 (1958) — p. 164

Editorial note, tabletop extrapolation: Directly transferable as a SHAPE target for the reference machine's field: the fixed-frequency population converged on a smooth, monotonic few-percent total drop. The total constrains the average only - the stability check remains the local n(r) map (n > 0 throughout, staying clear of 0.2; dg-003, dg-138), which the same total drop can satisfy or violate depending on where the fall concentrates.

Cited in: Beam Dynamics: An Interactive Laboratory

Internal Beam, Stable, ua 1000-2000 ... External Beam, Stable, 800 ua; 100 ua focused on target 15 ft from machine

Howard, Cyclotrons and High-Energy Accelerators, 1958 — ORNL-2644 (1958) — p. 107

Editorial note, tabletop extrapolation: If the reference machine sees nA, the gap to the historical uA-mA norm lives in source output and center-region transmission, not physics limits - and extraction cost most census machines most of their beam, so budget a next machine's external current pessimistically.

Beam defining slits used on 1, 2, and 3rd revolutions to define center of beam rotation; 3rd turn slit is 1/2 mm wide. 100% extraction efficiency with low beams, requires better than 1/2 % stabilization of dee volts.

Howard, Cyclotrons and High-Energy Accelerators, 1958 — ORNL-2644 (1958) — p. 27

Editorial note, tabletop extrapolation: A historically successful, mechanically simple extraction aid: slits in the center region plus tight dee-amplitude regulation. Evaluate it for a next machine by comparing its interception losses and centering benefit against the calculated turn separation and deflector tolerances - slits select phase space by throwing beam away, so they complement, not replace, deflector design.

Electric focusing with carbon grids on the dees successful on first four revolutions

Howard, Cyclotrons and High-Energy Accelerators, 1958 — ORNL-2644 (1958) — p. 27

Editorial note, tabletop extrapolation: First-turn loss at low dee voltage is a classic tabletop failure mode, and the ANU carbon-grid result makes grid focusing worth testing - model the electric fields first, note a slit plate is not the same as a transparent grid, and check interception, RF loading, heating and outgassing at low current before adopting it.

It minimizes electric defocusing, which is ordinarily a major cause of beam loss, and actually provides some focusing during the usually defocusing part of the phase excursion.

Goodman, A Square-Wave Cyclotron Oscillator — ORNL-2403 (1958) — p. 10-11

Editorial note, tabletop extrapolation: Electric defocusing on the first turns is a plausible and testable contributor to the reference machine's losses - not an established attribution; a flat-topped dee is likely too much RF plumbing for a next machine, but the mechanism explains why phase excursion and gap-crossing timing deserve modeling attention in any small machine.

the same bunching occurs even if a third harmonic is added to the r-f wave form to square the wave

Goodman, A Square-Wave Cyclotron Oscillator — ORNL-2403 (1958) — p. 10

Editorial note, tabletop extrapolation: Reassurance that waveform shaping and center-region bunching are separable problems in the source's treatment; any claimed voltage benefit depends on harmonic amplitude/phase and what is held fixed (peak voltage vs RF power), so quantify longitudinal acceptance by calculation before banking on it. The bunching mechanism itself (Cohen) is what sets which ions survive the center region. OCR note - theta prints as (c) in these appendix equations.

the radial displacement amplitude is equivalent to about twenty turns, or to about two kilovolts, or about 1% spread in energy at the exit radius

Goodman, A Square-Wave Cyclotron Oscillator — ORNL-2403 (1958) — p. 12

Editorial note, tabletop extrapolation: Budget this effect for any future extraction work by evaluating D and Delta-r with the machine's own V, omega, gap and phase distribution and tracking the offsets to extraction - the ORNL equivalences are that machine's numbers, not a floor. Flat-topping reduces the phase-dependent part; neither AVF nor flat-topping removes the displacement wholesale.

A major improvement was effected when an off-center source was installed which injected azimuthally into one of the dees.

Heusinkveld et al., Studies with a Three-Dee Three-Phase Proton Cyclotron — UCRL-3187 (1955) — p. 12

Editorial note, tabletop extrapolation: For the reference machine's filament source, radial position and slot azimuth are cheap, high-leverage experiment variables (directly relevant to the planned source-species test) - scan them, normalized to the first-orbit geometry. Expect improvement mechanisms to be entangled as they were historically; measure, don't assume a factor of two.

the high-voltage electrode should have the minimum possible surface area. This minimizes the amount of sparking required to bake out the cathode spots and reduces the amount of electrode contamination.

Smith & Grunder, Electrical Design of Electrostatic Deflectors for Sector-Focused Cyclotrons — UCRL-10654 (1963) — p. 16

Editorial note, tabletop extrapolation: Measure or track the next machine's actual beam envelope at extraction radius - vertical oscillations, alignment and median-plane shift included - set the field window from that worst case plus explicit margin, and keep the HV bar as small and short as the trajectory allows; the 88-inch's 2x is their outcome, not a sizing law.

the increase in radius of the burst of ions per rf cycle is approximately 0.1 inches and the deflector bars are spaced one inch apart, the pulse must occur within +/- 5 rf cycles.

Kerns et al., High Voltage Pulser for 184-inch Cyclotron Electric Deflector — UCRL-95 (1948) — p. 8

Editorial note, tabletop extrapolation: Synchrocyclotron-specific hardware (a CW deflector needs no pulse), but the requirements chain - turn separation sets element aperture sets timing budget - is the template for sizing ANY extraction element, including a next machine's septum entrance.

The extraction requirements, simply stated, are: (a) The precession must be arrested (b) Sufficient gain per turn must be obtained (c) Losses owing to axial blowup must be minimized.

Stubbins, Extraction of Synchrocyclotron Beams Near the Maximum Energy — UCRL-3476 (1956) — p. 7

Editorial note, tabletop extrapolation: The cleanest checklist in this collection for what a next machine's precessional-assist extraction must accomplish - phase-lock the precession to place orbit maxima at the septum azimuth, then count gain-per-turn against septum thickness. Machine-class independent.

The synchronous radius suitable for deflection in the cyclotron is just inside the radius at which normal beam destruction occurs.

Stubbins, Extraction of Synchrocyclotron Beams Near the Maximum Energy — UCRL-3476 (1956) — p. 7

Editorial note, tabletop extrapolation: The siting logic transfers, the threshold does not: put a next machine's septum or regenerator equivalent just inside where its OWN analysis says the beam dies - measured field map, tune calculation and tracking, not a universal n = 0.2 wall (linear radial stability formally extends to n = 1, and real loss radii are set by resonances, apertures and field errors). And every mm inward is extracted energy given away.

The determination of the required perturbation for extracting the beam of a synchrocyclotron is made in seven steps.

Stubbins, Extraction of Synchrocyclotron Beams Near the Maximum Energy — UCRL-3476 (1956) — p. 6

Editorial note, tabletop extrapolation: The workflow (measured field -> amplitude-dependent tunes -> impulse-matrix tracking -> element strength) is exactly the CYCLOPS-lite pipeline planned for a next machine; the peeler-regenerator field shapes themselves are synchrocyclotron machinery and need not transfer. Treat the sine-ratio gain coefficient as branch- and model-specific once the re-read pins its definitions - it is singular near its denominator zeros, so no monotone smaller-interval-more-gain rule survives unqualified.

For a 1-in. axial amplitude this disturbance is about twice as strong as that occurring in the radial motion from the same field perturbation.

Stubbins, Extraction of Synchrocyclotron Beams Near the Maximum Energy — UCRL-3476 (1956) — p. 18

Editorial note, tabletop extrapolation: The transferable warning: any radial-field-gradient extraction element has an off-midplane Br ~ z*dBz/dr whose vertical effect can focus or defocus depending on gradient sign and trajectory - include the deflector fringe and any field bump in the next machine's 3-D tracking rather than assuming the sign or which particles go first.

It is an advantage to choose the smallest value of l, since the choice allows the least sensitivity in matching the perturbing frequency to the particle motion.

Stubbins, Radiofrequency System for Extracting Particles from a Cyclotron — UCRL-8578 (1958) — p. 7

Editorial note, tabletop extrapolation: A candidate extraction assist worth a TRACKER experiment before hardware: note that for l = 1 with nu_r near 1 the drive lands near TWICE the revolution frequency, and the required gradient, electrode voltage, bandwidth against tune spread, and isolation from the main RF are exactly what the tracking study must produce before 'an electrode pair and a small oscillator' can be promised.

f_z = f - f_0, where f_z equals (sqrt n) f_0 ... and n is the conventional cyclotron magnetic field parameter. The relation f = f_0 ((sqrt n) + 1) is one required condition for this process to occur

Stubbins, Radiofrequency System for Extracting Particles from a Cyclotron — UCRL-8578 (1958) — p. PDF p.5 = printed p.-3- (UCRL-8578, Sec. I Introduction)

Editorial note, tabletop extrapolation: A real design caution at any scale: an E_z gradient of the right symmetry near a nu_z resonance can dump the beam. Note dee misalignment gives mostly a dipole-like midplane E_z, not the z-proportional gradient this parametric resonance needs - but asymmetric liners and gap geometry can supply the gradient term, so keep the dee/dummy-dee vertically symmetric and check nu_z against strong rf harmonics at operating field.

because of the precession of orbits all particles are ultimately perturbed to larger radial oscillation amplitudes.

Stubbins, Radiofrequency System for Extracting Particles from a Cyclotron — UCRL-8578 (1958) — p. 9

Editorial note, tabletop extrapolation: Sets expectations for any resonant/precessional scheme on a next machine: the growth is episodic over many turns, so judge schemes in the tracker by turns-to-extraction and septum-hit fraction rather than single-pass kick size - the detailed evolution is deterministic and scheme-dependent, so track your own.

The increase in proton energy resulted from relocation of the ion source 1 1/2" upward; the source is now effectively only one inch below the magnetic center.

Howard (ed.), Electromagnetic Research Division Quarterly, period ending 31 March 1952 — ORNL-1339 (1952) — p. 8

Editorial note, tabletop extrapolation: On the reference machine, treat filament/chimney height relative to the MAGNETIC median plane (find it by measurement - it need not match the mechanical midplane) as a tuned parameter worth systematic scans. What the height buys is centering, vertical transmission and usable radius; at a fixed field and radius the energy is p = qBr regardless, so measure where the gain actually comes from rather than expecting a fixed percentage.

the beam striking the periphery of the south dee. This condition resulted from the center of rotation of the beam being offset to the south by a distance of approximately three inches. ... The dees were moved 1/2 in south, measured at the horizontal center line of the dees.

Howard (ed.), Electromagnetic Research Division Quarterly, period ending 31 March 1952 — ORNL-1339 (1952) — p. 7

Editorial note, tabletop extrapolation: Witness marks on the reference machine's dee edges are a free orbit-centering CLUE - corroborate with radial probe scans and the field map before moving anything, since phase, axial focusing and apertures make similar marks; then correct at the source or dees once the cause is identified.

most of the loss of protons to the dees occurs during early revolutions. Only a small percentage of the beam is lost beyond one-half of maximum radius, Figure 5. ... [Figure 5:] Z-WISE BEAM DISTRIBUTION on Each of Five Segments

Howard (ed.), Electromagnetic Research Division Quarterly, period ending 31 March 1952 — ORNL-1339 (1952) — p. 16

Editorial note, tabletop extrapolation: Both the finding and the instrument transfer as guidance: stack 3-5 insulated foils as a segmented z-probe on the reference machine to see where the beam sits vertically, and expect the early turns to deserve the tuning effort - on that machine, beam surviving to half radius mostly escaped further DEE loss; extraction, phase and radial channels are separate ledgers.

an increase in bias potential on the dees increases the beam accelerated to maximum radius by a factor of as much as 30% when the cyclotron is operated without an accelerating slit (rf) mounted on the dee.

Howard (ed.), Electromagnetic Research Division Quarterly, period ending 31 March 1952 — ORNL-1339 (1952) — p. 16

Editorial note, tabletop extrapolation: Worth a cheap experiment on the reference machine - with a proper RF-rated bias-injection network (choke/filter, insulation, supply protection), never a bare DC supply on a live dee. ORNL's stated reading is that bias pulls badly-phased ions deeper into the gap field; with a slit installed they saw no bias effect. The source does not rank the two approaches - test both on the actual machine.

It was found that distortion of the magnetic field when the tank is evacuated is negligible. Latest measurements of the magnetic field reveal a first harmonic inhomogeneity of approximately 0.03%.

Howard (ed.), Electromagnetic Research Division Quarterly, period ending 31 March 1952 — ORNL-1339 (1952) — p. 15

Editorial note, tabletop extrapolation: Two transfers: verify a next machine's field map with the chamber assembled and pumped (pole deflection under vacuum load is a real worry that proved negligible for them - measure once to confirm); and read 0.03% as what a carefully shimmed classical machine ACHIEVED - the new machine's allowable first harmonic comes from its own orbit-centering budget, and note 0.03% of a 0.5-1 T tabletop field is 1.5-3 G, so gauss-level targets and fractional targets must be kept straight.

Current measurements beyond 18.5" were unreliable; the current at the maximum radius, 25.5", is estimated to be of the order of one microampere.

Howard (ed.), Electromagnetic Research Division Quarterly, period ending 30 June 1952 — ORNL-1345 (1952) — p. 10

Editorial note, tabletop extrapolation: Calibrates expectations qualitatively, not in absolute scale: an untuned machine can lose orders of magnitude between small radius and full radius, so log the whole I(r) curve - its shape (where the loss happens) is the tuning roadmap. This is one machine's commissioning history, not a norm to be satisfied with.

the ion source position is limited to a maximum radius of 2.5 inches. This forces operation at comparatively low dee voltages (20-30 kv) in order to center the orbits.

Jungerman, Kibbe & Peek, Central Region Studies for Incorporating an Axial Ion Source in the Davis 76-in. Cyclotron — UCD-CNL-49 (1966) — p. 5

Editorial note, tabletop extrapolation: The design logic transfers directly to a next machine's central-region layout: pick dee voltage and source-puller radius TOGETHER from the first-orbit geometry. It also cuts the other way for the reference machine's 5-13 kV upgrade: raising dee voltage moves the optimum source position outward — re-scan source position after the RF upgrade.

The isochronous fields are obtained with trim coil settings computed by a linear program, and their greatest deviation from isochronism is less than 15 gauss in all cases.

Jungerman, Kibbe & Peek, Central Region Studies for Incorporating an Axial Ion Source in the Davis 76-in. Cyclotron — UCD-CNL-49 (1966) — p. 6

Editorial note, tabletop extrapolation: Calibration, not criterion: what any machine tolerates is the accumulated RF phase slip - the signed integral of the frequency error over ITS acceleration history - so run the phase-slip integral in the tracker for the actual field map, turn count and dee voltage, and let that set the gauss tolerance; a few-tens-of-turns classical machine and a hundreds-of-turns AVF machine land in different places by exactly that arithmetic.

The starting conditions for all cases were estimated by starting the ions on an equilibrium orbit of 12 inch radius and de-accelerating them to the center.

Jungerman, Kibbe & Peek, Central Region Studies for Incorporating an Axial Ion Source in the Davis 76-in. Cyclotron — UCD-CNL-49 (1966) — p. 6

Editorial note, tabletop extrapolation: Directly implementable in the Python orbit tracker: find the equilibrium orbit at modest radius (well-conditioned), integrate backwards keeping the RF phase time-consistent, and read off CANDIDATE source-slit and puller coordinates - then validate with a full central-region field model and forward tracking; the backward pass suggests the geometry, it doesn't determine it.

the beam measured at extraction is augmented by possibly 10% by using 22 amps in trim coil number 1 rather than -145 amps. The former produces the central radial bump.

Jungerman, Kibbe & Peek, Central Region Studies for Incorporating an Axial Ion Source in the Davis 76-in. Cyclotron — UCD-CNL-49 (1966) — p. 6

Editorial note, tabletop extrapolation: Consistent with the classical-cyclotron instinct - a small central bump (field falling with radius from turn one) focuses the early turns where the ORNL 22-inch z-studies located most dee loss - as a HYPOTHESIS the correlation supports, not a demonstrated mechanism. Empirically checkable on the reference machine with shim washers at the pole center: calculate the phase-slip cost first, map the shimmed field, and measure both transmission and where the losses move.

The particles pass through the 3/3 resonance at a radius of 6-7 inches. The computer calculations show that the radial oscillation build-up at resonance is not excessive and soon damps to 0.3 inch.

Jungerman, Kibbe & Peek, Central Region Studies for Incorporating an Axial Ion Source in the Davis 76-in. Cyclotron — UCD-CNL-49 (1966) — p. 6

Editorial note, tabletop extrapolation: Method for the CYCLOPS-lite tracker: don't just plot nu_r(r) and forbid resonance lines - integrate through them with realistic errors and acceleration rate, and report maximum excursion in millimeters against the aperture. A fast-crossed resonance can be acceptable if the complete envelope keeps clearance; 'damps' in the historical usage reflects detuning and adiabatic effects, not dissipation.

the validity of the calculations and magnetic field data is supported by the fact that we obtained an internal beam of 21 MeV H2+ ions using the computed field on the first attempt.

Jungerman, Kibbe & Peek, Central Region Studies for Incorporating an Axial Ion Source in the Davis 76-in. Cyclotron — UCD-CNL-49 (1966) — p. 7

Editorial note, tabletop extrapolation: The 1966 encouragement for a next machine's compute-first pipeline (field map -> tracker -> build): careful measurement plus an honest tracker produced first beam on the computed field, first attempt, on that machine. One result is precedent, not promise - keep shim stock on hand, and let the pipeline earn trust machine by machine.

This region, called the "gap," should have a diameter about nine times as great as its axial dimension. ... n = - (R/H)(dH/dR) << 1/5 ... The desired field is one which decreases linearly with increasing radius to the outside "edge" of the gap.

Foss et al., Cyclotron Component Design Technical Reports — TID-454 (1952) — p. PDF p.6 = printed p.6 (TID-454, Technical Report No. 1, Sec. 1.1 Pole Tips, 'Introduction')

Editorial note, tabletop extrapolation: CORROBORATING, not new - the same premises underlie Livingston-Blewett and Wouters (corpus already carries 0<n<1 stability). TID-454's working condition is the stricter n<<1/5; note its own 130-in/14-in example is 9.3x. The reference machine's 8-in poles over a wide gap fall far short of 9x, which is exactly why usable radius is scarce; a next machine's gap choice should respect this proportion.

The operation of the oscillator determines the variation of capacity with time which will keep the ion phase constant. This variation is not very critical.

Foss et al., Cyclotron Component Design Technical Reports — TID-454 (1952) — p. 152

Editorial note, tabletop extrapolation: For a small synchrotron's RF ramp the pattern maps with its own tolerances: f(t) and the allowable phase/frequency error come from the magnetic ramp, synchronous orbit and RF-bucket acceptance - the cited looseness belongs to that FM oscillator, not to synchrotron ramps in general; the duty-factor lesson (reset time is pure tax, minimize it within hardware limits) transfers as stated.

Measurements of the internal beam of the ORNL 86-inch cyclotron very early indicated that the energy of the proton beam might vary as much as +/-10% from H-rho calculations. ... The energy of maximum intensity was found to vary by as much as several hundred kilovolts with minor adjustments of the ion source position, dee voltage, magnetic field tuning, and oscillator frequency.

Cohen, Measurement of Beam Energy and Energy Distribution on an Internal Cyclotron Target — ORNL-1347 (1952) — p. 5

Editorial note, tabletop extrapolation: The direct historical support for this collection's energy-convention discipline: the reference machine's '150 keV-class computed' is a convention, not a measurement, and its own discrepancy must be measured, not assigned ORNL's +/-10%. For a next machine's B11(p,alpha) work, where yield vs energy is steep, measure energy AT the target (absorber stack in front of the PIPS, or foil methods) every time tuning changes.

From (5) the measurement of dr is essentially a determination of the phase.

Cohen, Spatial Distribution of Current on an Internal Cyclotron Target — ORNL-1348 (1952) — p. 9

Editorial note, tabletop extrapolation: Energy-independent physics: a differential probe (shadowed double tip) or the sectioned-target map gives dr, and with the dee voltage - stated in ONE convention, peak dee-to-dee or dee-to-ground, never mixed - that is a direct measurement of ion RF phase, the quantity a next machine's field-tolerance budget protects. A rare experimental handle on phase for machines with no beam-position monitors.

leaving the ion source ungrounded has a very substantial effect, since it then floats nearer the potential of the accelerating slit which is attached to the dees. This reduces the effect of the latter and the radial width approaches the theoretical predictions for a cyclotron without an accelerating slit.

Cohen, Spatial Distribution of Current on an Internal Cyclotron Target — ORNL-1348 (1952) — p. 9

Editorial note, tabletop extrapolation: Direct lesson for the reference machine's central-region debugging: the source body's electrical state - DC connection AND RF return impedance, since a floating body near driven dees picks up RF capacitively - is a real optics knob (or a real gremlin). Verify and log the filament/chimney ground path; an intermittent source ground would masquerade as day-to-day beam irreproducibility of exactly the kind ORNL-1347 warns about.

the ions which were the chief contributors to the current at resonance are lost by defocusing and ions of more positive phases are now the chief contributors.

Cohen, Spatial Distribution of Current on an Internal Cyclotron Target — ORNL-1348 (1952) — p. 9

Editorial note, tabletop extrapolation: Explains an observation class on the reference machine: probe current can look tolerant of field/frequency error while the surviving phase distribution, turn spacing, transmission and attained radius shift underneath - reinforcing ORNL-1347's rule that current on target is not evidence the energy is what B-rho says (at a FIXED radius the momentum is still ~qBr; what moves is which ions get there and how). Whether self-selection broadens your tuning curves is testable with phase- or energy-sensitive measurements - treat the width cautiously either way.

Six- and eight-sector machines are free from strong essential resonances ... also, the symmetry easily permits four accelerating gaps per revolution, a situation well suited to rf cavities.

Oak Ridge National Laboratory, A Proposal for the Mc² Isochronous Cyclotron — ORNL-3540 (1963) — p. 266-267

Editorial note, tabletop extrapolation: 810-MeV specifics (vr climbing to 2, spiral sectors) do not scale down; the method — list resonances crossed by your vr/vz trajectory before fixing N, then let RF layout break ties — applies to any AVF design.

electric fields are relatively ineffective at high particle velocities, but the force on an ion due to a magnetic field is proportional to velocity.

Oak Ridge National Laboratory, A Proposal for the Mc² Isochronous Cyclotron — ORNL-3540 (1963) — p. 151

Editorial note, tabletop extrapolation: At 150 keV protons (v ~ 5.4e6 m/s) the comparison runs strongly toward electrostatic: modest septum fields equal coil fields that are awkward to engineer at that scale, which is why documented small machines extract electrostatically. Run the B = E/v arithmetic before copying any big-machine magnetic-channel scheme - the crossover is a computation, not a law.

When the horizontal component of the geomagnetic field was canceled, this attenuation was eliminated.

Oak Ridge National Laboratory, A Proposal for the Mc² Isochronous Cyclotron — ORNL-3540 (1963) — p. 276

Editorial note, tabletop extrapolation: Earth's field (~0.5 G total; the horizontal component is what mattered there) is normally a negligible perturbation inside the reference machine's 0.59-T gap, but real for any low-field electron-analogue experiment, long injection path, or fringe-field beamline - measure the local vector rather than assuming.

The data show individual orbital positions from the first orbit up to the twelfth, the upper limit being determined by the potential on the dees.

Howard (ed.), Electromagnetic Research Division Quarterly, period ending 31 December 1951 — ORNL-1269 (1952) — p. 40

Editorial note, tabletop extrapolation: A candidate measurement for the reference machine - measured turn spacing plus the field map gives effective energy gain per turn, which would anchor its uncalibrated ~1.3 kV dee voltage (via gap count, synchronous phase and transit-time factors, not directly). First check feasibility: at ~1.3 kV the inner-turn spacing may be smaller than the existing probe wire - compute dr against probe width before promising resolution. Fig. 12 (PDF p.41) shows the 22-inch doing this at 9.2-12 kV dee-to-dee.

a choice of radius, 11 in. or 20 in., is thus obtained by shifting the position of the dees and target. In either case the ion source position remains unchanged and the beam orbits remain centered

Howard (ed.), Electronuclear Research Division Semiannual, period ending 20 September 1953 — ORNL-1663 (1954) — p. 18

Editorial note, tabletop extrapolation: Variable energy WITHOUT retuning B or rf - E ~ r^2 at fixed field and frequency - by repositioning the dee assembly AND target together as ORNL did; a target-only intercept at reduced radius is a simpler tabletop variant (an extrapolation, not ORNL's method), and either way the delivered energy is verified from the mapped field and measured target radius, not assumed calibrated.

Plans are being made for an electron-model accelerator to be used in assessing the importance of imperfection resonances and the feasibility of their penetration.

Howard (ed.), Electronuclear Research Division Semiannual, period ending 20 March 1955 — ORNL-1884 (1955) — p. 35

Editorial note, tabletop extrapolation: A historical instance of risk-ordered development: when the open question is orbit dynamics, a cheap electron model is one way to attack it before proton iron is bought - scaled properly (dg-892's rigidity caveat). The tabletop program's equivalent instruments are the tracker and measured field maps.

in order to produce a suitable object for the analyzing magnet, we introduce a second magnet whose sole function is to focus as much of the beam as possible on a precision slit.

Bromley & Bruner, The Design of a Focusing and Analyzing System for the 27-inch Cyclotron Beam — NYO-3823 (1954) — p. 4

Editorial note, tabletop extrapolation: DIRECT for any ANALYZED external line on a next machine - the canonical two-stage architecture: a condenser focuses as much beam as possible onto a precision slit, and that illuminated slit becomes the analyzer's cleanly defined object. Lines that only transport or irradiate skip the apparatus; the slit's shielding is the companion rule (dg-966).

A considerable amount of undesirable radiation will be produced by that part of the beam intercepted by the slit system. ... it is also desirable that the analyzer image distance be large, in order to reduce the angular spread of the beam at the image point. ... [placing the condenser farther from] the cyclotron port ... required larger condenser pole pieces in order to accommodate the horizontally spreading beam.

Bromley & Bruner, The Design of a Focusing and Analyzing System for the 27-inch Cyclotron Beam — NYO-3823 (1954) — p. 5

Editorial note, tabletop extrapolation: DIRECT and cheap to honor at layout time, nearly impossible later: treat every defining aperture as a place where beam power - and therefore radiation - concentrates. At 150-170 keV the intercepted beam makes mostly heat plus thick-target bremsstrahlung whose X-ray yield climbs steeply with voltage, so the slit belongs with the shielded, surveyed components, wherever the survey ranks it that day.

Cited in: Shielding a Small Cyclotron

the weight and power requirements would each be less than the corresponding sector-magnet requirements by at least a factor of ten.

Bromley & Bruner, The Design of a Focusing and Analyzing System for the 27-inch Cyclotron Beam — NYO-3823 (1954) — p. 6

Editorial note, tabletop extrapolation: DIRECT - the as-built comparison (p.32) was 355 lb for the doublet pair vs an estimated 3 tons for a sector condenser. At a next machine's rigidity (~7x lower than Rochester's 4e5 G-cm) a doublet becomes a benchtop object; the no-deflection tunability argument is the one to remember when laying out the line.

the effective 'point' source in the vertical plane does not coincide with that in the horizontal plane and is such that the vertical-plane object distance V is greater. ... The second is simply that the common image be real and beyond the magnets themselves. ... Since it is known that the angular spread is greater in the horizontal than in the vertical plane, we make the first lens convergent in the horizontal plane.

Bromley & Bruner, The Design of a Focusing and Analyzing System for the 27-inch Cyclotron Beam — NYO-3823 (1954) — p. 14

Editorial note, tabletop extrapolation: DIRECT design input for next-machine transport modeling: fit separate horizontal/vertical source points and divergences from measured beam profiles (or a quadrupole scan) rather than assuming a stigmatic waist at the extraction channel - then let the measured two-plane phase space, not the historical ordering, choose the first quad's polarity.

With this as a guide we apply an extra factor of safety and take a = 3, c = 1.5, hence the magnet poles are given by xy = +/- 2.25 cm^2

Bromley & Bruner, The Design of a Focusing and Analyzing System for the 27-inch Cyclotron Beam — NYO-3823 (1954) — p. 29

Editorial note, tabletop extrapolation: DIRECT method, not numbers: measure the real beam first, then stack explicit margins on the way to the pole constant. A next machine's envelope comes from its own extraction simulation or measurement; margin-then-round-up is what prevents discovering an undersized bore after the coils are wound.

Measurements have shown that the effective length of the magnets is as much as 20% greater than the physical length.

Bromley & Bruner, The Design of a Focusing and Analyzing System for the 27-inch Cyclotron Beam — NYO-3823 (1954) — p. 29

Editorial note, tabletop extrapolation: DIRECT: for short quads the fringe extension is a first-order effect, not a correction - and it scales with aperture, which is why short, fat quads see the largest effect. Get l_eff per magnet from the FEMM/tracker pipeline; ignoring it produces significant focal errors.

For fields above about 8 kilogauss, saturation effects begin to set in, and the field becomes less uniform near the pole boundaries.

Bromley & Bruner, The Design of a Focusing and Analyzing System for the 27-inch Cyclotron Beam — NYO-3823 (1954) — p. 43

Editorial note, tabletop extrapolation: DIRECT sizing rule with scale caveat: the 8 kG threshold is geometry- and steel-specific, but the logic (uniformity budget, not raw B_sat, sets the working field; bend radius follows) applies to any analyzer dipole on a next machine. At ~170 keV protons rho is a few cm even at modest fields - the analyzer becomes a bench magnet.

The effect of the fringe field is to shift the effective pole boundary outward, and this is taken into account empirically by adding to the right-hand side of III-20(b) a term 0.4 G

Bromley & Bruner, The Design of a Focusing and Analyzing System for the 27-inch Cyclotron Beam — NYO-3823 (1954) — p. 43

Editorial note, tabletop extrapolation: For a next machine's analyzer designed in FEMM, the sanity check is the concept, not the constant: compute the effective field boundary from the longitudinal field integral of the simulated fringe and compare against the steel edge - an offset of very roughly half a gap is the expected order. Do not equate the 0.4G term with a tracking code's FINT parameter (FINT conventions carry fringe focusing integrals, a different quantity).

the methods for correcting for the fringe fields effects ... are not sufficiently precise to warrant basing the magnet design on the second-order calculations.

Bromley & Bruner, The Design of a Focusing and Analyzing System for the 27-inch Cyclotron Beam — NYO-3823 (1954) — p. 50

Editorial note, tabletop extrapolation: DIRECT design-philosophy rule for the whole next-machine campaign - match model order to input-data quality. Also note the companion check they DID run (pp.49-50), that the bent beam clears the back of the magnet with ~5 cm margin for the full 6 cm beam - a 30-second calculation that catches a catastrophic layout error.

The operation of the wedge analyzer has been checked using the standard current carrying wire technique.

Bromley & Bruner, The Design of a Focusing and Analyzing System for the 27-inch Cyclotron Beam — NYO-3823 (1954) — p. 52

Editorial note, tabletop extrapolation: The wire method is a superb zero-beam measurement of magnet optics for a teaching lab or a first analyzer - state tension, sag and field-orientation assumptions when using B-rho = T/I. Budget alignment and tuning provisions into any multi-element line as a design habit; the cited line's transmission and energy-spread figures await the re-read before serving as benchmarks.

the problem of obtaining a pulsed beam usually does not arise, because the beam of a conventional cyclotron is already naturally bunched.

Fulbright et al., A Fast Neutron Time of Flight System for Use with Cyclotrons — NYO-9360 (1962) — p. 6

Editorial note, tabletop extrapolation: DIRECT and foundational for the experiment catalog: the reference machine at 9 MHz delivers phase-bunched beam at the RF period - a measurable, teachable property and the enabling fact for gated counting. 'Usually' is operative: time-of-flight at fine resolution, or experiments needing low repetition rate, can still require pulse selection or extra bunching - check bunch width and period against the experiment's timing demands.

In our case reducing the frequency of beam pulses to about 3 mc can virtually eliminate the problem.

Fulbright et al., A Fast Neutron Time of Flight System for Use with Cyclotrons — NYO-9360 (1962) — p. 16

Editorial note, tabletop extrapolation: Check overlap arithmetically for the actual spectrum and flight path - t[ns] ~ 72.3*L[m]/sqrt(E[MeV]), so even an all-sub-MeV spectrum overlaps at a 10 MHz rate over a 1 m path, and D-D work adds multi-MeV neutrons on a sub-MeV machine. The subharmonic plate pair is a genuinely cheap cyclotron chopper for periodic bunch rejection and duty-cycle control - single-bunch selection needs a gating scheme beyond a sinusoidal drive - and the odd/even zero-crossing subtlety is real circuit physics worth teaching.

the acceleration can be considered to good approximation as being a simple impulsive change in the energy of the particle at the azimuth of the accelerating gap

Blosser & Gordon, Computational Study of a Resonant Extraction System for a 3-Sector Cyclotron — MSUCP-9 (1961) — p. 4

Editorial note, tabletop extrapolation: The core architecture for CYCLOPS-lite - thin-gap kicks alternating with magnetic coasting maps - with phase carried as a dynamical variable from the first line of code; MSUCP-12's analytic gap field upgrades the kick to a distributed one when transit time matters, and a comparison against distributed-gap tracking on the actual geometry is the validation step, not the 1961 result alone.

The Fixed Point Code locates closed orbits by means of a highly effective linear transfer matrix procedure, the General Orbit Code tracks arbitrary orbits as desired either with or without acceleration effects. For both routines the magnetic field is described by tables of Fourier coefficients as functions of radius; each uses equations of motion which are exact in the median plane.

Blosser & Gordon, Computational Study of a Resonant Extraction System for a 3-Sector Cyclotron — MSUCP-9 (1961) — p. 5

Editorial note, tabletop extrapolation: This is the CYCLOPS architecture in embryo (the lineage the planned "CYCLOPS-lite" copies); a next machine's tracker should likewise separate the equilibrium-orbit /tune solver from the general tracker, sharing one Fourier-vs-radius field representation fed by FEMM.

the general orbit code is employed to trace forward and backward in time orbits with initial conditions displaced slightly from the unstable fixed points.

Blosser & Gordon, Computational Study of a Resonant Extraction System for a 3-Sector Cyclotron — MSUCP-9 (1961) — p. 5

Editorial note, tabletop extrapolation: Directly reusable in a Python tracker (a symplectic or invertible integrator makes the backward branch trustworthy); the efficient way to draw the r-pr stability picture near resonances - cross-check with a scatter of ordinary orbits where the manifolds tangle, since in a nonintegrable map they can intersect and form stochastic layers rather than clean boundaries.

the small axial beam space in a cyclotron constrains the particles to move in a region where the z dependence of the field is quite linear.

Blosser & Gordon, Computational Study of a Resonant Extraction System for a 3-Sector Cyclotron — MSUCP-9 (1961) — p. 3

Editorial note, tabletop extrapolation: Build the next machine's first tracker around (r, pr, E, phase) PLUS linearized (z, pz) from the outset - the aperture does not hold particles near the median plane, it deletes the ones that leave, so vertical tune, resonance crossings and the physical aperture decide transmission. Full-3D spot checks then benchmark the linear model over representative launches; a handful of them is the check on the linearization, not a license to omit z.

The powerful effect of a cos 0 field component in a cyclotron ... is clearly evidenced by the large changes in the phase plot which result when the small 1% bump is added.

Blosser & Gordon, Computational Study of a Resonant Extraction System for a 3-Sector Cyclotron — MSUCP-9 (1961) — p. 22

Editorial note, tabletop extrapolation: Cuts both ways near nu_r = 1: the demonstration is why first-harmonic errors get 'great respect' - so Fourier-analyze the candidate field map and track the measured B1(r) through the local tune to learn what YOUR machine's shim asymmetries cost; and a deliberate bump coil is a powerful orbit-steering experiment once its ampere-turns are sized from that same analysis, not assumed few-turn-cheap.

introduction of the field bump has caused a discontinuous jump in the location of the central stable orbit in the phase diagram

Blosser & Gordon, Computational Study of a Resonant Extraction System for a 3-Sector Cyclotron — MSUCP-9 (1961) — p. 24

Editorial note, tabletop extrapolation: The conceptual mechanism to have in hand before a regenerative extraction attempt on a next machine: it needs nu_r to pass unity with a controlled first harmonic, both of which a FEMM-fed tracker can compute for a candidate pole design. It is one extraction method - electrostatic deflection, stripping, or simply large natural turn separation do not require crossing nu_r = 1. [Note revised 2026-08-23: earlier wording read as if this were prerequisite to any extraction.]

stretching and rotation are fine but not twisting, filamentation, etc.

Blosser & Gordon, Computational Study of a Resonant Extraction System for a 3-Sector Cyclotron — MSUCP-9 (1961) — p. 24

Editorial note, tabletop extrapolation: The right figure of merit for any next machine's beamline or extraction simulation - track a grid of particles and judge the deformed shape, not just the centroid. Five to two dozen particles sufficed in 1961 for the smooth cases; check convergence by refining the grid where the map is nonlinear, since a sparse grid can miss filamentation entirely.

The essential design requirement of such a system is a turn program which avoids regions of large flow rate gradient in the static phase space.

Blosser & Gordon, Computational Study of a Resonant Extraction System for a 3-Sector Cyclotron — MSUCP-9 (1961) — p. 45

Editorial note, tabletop extrapolation: The doctrine transfers to schemes where a static phase-plot analysis applies: superimpose the accelerated beam path on static phase plots (cheap tracker post-processing), identify the large-flow-gradient regions - especially near unstable fixed points - and compare candidate turn programs by accelerated tracking. Crossing faster reduces exposure to a bad region but can excite other resonances non-adiabatically, so test, don't assume.

volts per turn substantially lower than the designed 280 kev/turn would result in sharp reduction of both extraction efficiency and optical quality.

Blosser & Gordon, Computational Study of a Resonant Extraction System for a 3-Sector Cyclotron — MSUCP-9 (1961) — p. 45

Editorial note, tabletop extrapolation: The quantitative ancestor of 'dee volts buy extraction': the reference machine's uncalibrated ~1.3 kV dee is one reason it is internal-beam-only, and a next machine's 5-13 kV target is what would make an extraction scheme thinkable - thinkable, not feasible, until the turn separation (delta_r ~ r*delta_E/2E), phase width, septum clearance, tune and bump design are actually computed.

The sinusoidal voltage, it is seen, shifts the final position of the beam spot but has almost no effect on the distortion.

Blosser & Gordon, Computational Study of a Resonant Extraction System for a 3-Sector Cyclotron — MSUCP-9 (1961) — p. 40

Editorial note, tabletop extrapolation: A permission slip for CYCLOPS-lite staging — start with constant energy gain per gap to get the radial dynamics right, then add cos(phi) gain and phase slip as a second-stage refinement, checking that conclusions survive.

the result is seen to fairly accurately predict the actual amplitude and 0 of this point of the grid

Blosser & Gordon, Computational Study of a Resonant Extraction System for a 3-Sector Cyclotron — MSUCP-9 (1961) — p. 44

Editorial note, tabletop extrapolation: Two lessons - build point analytic cross-checks into a next machine's tracker test suite (transfer-matrix estimates against tracked orbits), and note that in the idealised 180-degree-symmetric two-dee case this particular excitation cancels to first order; check the real machine with its measured field, RF balance and gap geometry rather than assuming it. [Note revised 2026-08-23: earlier note called the physics 'benign' for a symmetric tabletop field.]

This paper presents in summary formulas for the computation of electric fields and potentials of an idealized cyclotron dee geometry.

Beal, Computation of Electric Field and Potential of an Idealized Dee Geometry — MSUCP-12 (1961) — p. 6

Editorial note, tabletop extrapolation: TRACKER SEED (flagged): this is directly implementable as the gap-field model in the tiny and the next machine's Python trackers — roughly ten lines plus a Newton solve — replacing or validating FEMM electrostatic maps. Identify 2h with the dee aperture, 2k with the dee-to-dummy-dee gap, 2V0 with the full dee-to-dummy-dee voltage (a grounded dummy dee is the same solution shifted by a constant, V0 = V_dee/2).

Therefore, equations 2 and 4 coupled with equations 3 and 5 can be used to determine the electric field and potential at a point X, Y of the dee region.

Beal, Computation of Electric Field and Potential of an Idealized Dee Geometry — MSUCP-12 (1961) — p. 5

Editorial note, tabletop extrapolation: E_y(x,y) is what a tracker needs for the Rose/Wilson electric gap-focusing term - available analytically at any point, no field map required. Whether that term dominates first-turn axial stability on a sub-kV machine is for the axial-stability calculation to say; implement, verify against finite differences, and let the tracking decide.

Table 1. k/h = 0.1: at x/h = 0, E/(V0/h) = 0.99388 [values verified against page image]

Beal, Computation of Electric Field and Potential of an Idealized Dee Geometry — MSUCP-12 (1961) — p. 11

Editorial note, tabletop extrapolation: Sets the ceiling on CENTERLINE gap field for a dee redesign: with a 1-inch aperture (h = 0.5 in) and 2.5 kV dee-to-dummy, the median-plane peak cannot exceed ~2 kV/cm however tight the gap. Two cautions: local surface fields at electrode edges run above the centerline value - breakdown cares about those (dg-353) - and widening the aperture trades centerline field for beam height by the table's factors, not one-for-one at every k/h.

Table 1, k/h = 0.1: V/V0 = 0.73760 at x/h = 1.0, 0.94468 at 2.0 [verified against page image]

Beal, Computation of Electric Field and Potential of an Idealized Dee Geometry — MSUCP-12 (1961) — p. 11

Editorial note, tabletop extrapolation: Transit-time factors and gap-crossing phase errors must be computed on this extended profile - and whether a delta-kick model is adequate is the transit parameter's call: evaluate omega*L_eff/v for the actual first-turn velocities and the ~2h-long field region, and let that number, not a blanket assumption, decide when the distributed kick is needed.

The process is a trial and error search, with general guidelines and test criteria for success.

Rainwater et al., The Columbia University Nevis Synchrocyclotron Major Modification — NEVIS-189 / R-774 / CU-295 (1971) — p. 5

Editorial note, tabletop extrapolation: Directly transferable design-process pattern for any pole or shim work on a next machine: let FEMM play the role of the Nevis model magnets, with analytic tune formulae as the accept/reject criteria - within FEMM's 2-D limits (azimuthal structure needs a 3-D model or the measured map; the playbook's tracker closes that loop). The final accept/reject is the measured field, exactly as it was at Nevis.

first evaluated using the Smith-Garren formula, checked at critical places, especially at larger r, by exact orbit motion computer solutions.

Rainwater et al., The Columbia University Nevis Synchrocyclotron Major Modification — NEVIS-189 / R-774 / CU-295 (1971) — p. 5

Editorial note, tabletop extrapolation: Exactly the field-solver-plus-orbit-tracker pipeline an amateur design can run. The Nevis precedent: spend the expensive tracking where the cheap formulae are least trustworthy - large radius and the extraction region on their machine - and anywhere else the smooth approximation visibly strains.

alerted by the ORNL studies of observed beam loss in the ORIC cyclotron to try to avoid

Rainwater et al., The Columbia University Nevis Synchrocyclotron Major Modification — NEVIS-189 / R-774 / CU-295 (1971) — p. 4

Editorial note, tabletop extrapolation: Method transfers directly — a tune plot should carry resonance lines sourced from operating-experience literature, not just textbook theory; a weak-focusing tabletop crosses fewer lines but the audit habit is the point.

a slow linear reduction (turn off) of the RF amplitude there will result in a near adiabatic spreading out of the phase angle

Rainwater et al., The Columbia University Nevis Synchrocyclotron Major Modification — NEVIS-189 / R-774 / CU-295 (1971) — p. 5

Editorial note, tabletop extrapolation: Swept-frequency machinery, not fixed-frequency CW territory - the design space it illustrates (adiabatic capture, slow parameter ramps judged against the phase-oscillation period, not a fixed microsecond count) belongs to any future synchro- or synchrotron-class RF program.

It has always bothered me that a film should ever break at its thickest port [sic]; (Ramsay, "Alternatives to Thin Film Carbon Foils")

Fifth Annual Conference of the International Nuclear Target Development Society — LA-6850-C, Los Alamos Scientific Laboratory (1977) — p. 81

Editorial note, tabletop extrapolation: When a thin internal target or probe foil dies, read the wreckage before assigning the cause: the radial-crease/thickened-spot signature points to Ramsay's stress mechanism, while melting, sputtering, charging marks or a failed frame each tell a different story - the mechanisms coexist and beam conditions pick the winner.

the foil breakage time is dependent on the total number of bombarding ions (Livingston, Berry & Thomas, "Thin Carbon Foil Breakage Times Under Ion Beam Bombardment"; reprint of NIM 148 (1978) 125)

Proceedings of the Sixth Annual Conference of the International Nuclear Target Development Society — LBL-7950, Lawrence Berkeley Laboratory (1978) — p. 152

Editorial note, tabletop extrapolation: Plan foil replacement by integrated charge where the fluence law holds - and verify it holds: dose-rate heating, spot profile and mounting can break the current-independence outside the tested window. Thickness buys nothing in beam life WITHIN the cited range, so choose it from mechanics, handling and dispersion - not as a lifetime lever.

If the beam is focused to an area of about 0.2 cm2, the resulting specific energy depositions amounts to 1 kW/cm2.

International Nuclear Target Development Society Workshop — ANL/PHY-84-2, Argonne National Laboratory (1983) — p. 29

Editorial note, tabletop extrapolation: The reference machine at ~3 nA / ~150 keV deposits ~0.5 mW - no realistic solid target is troubled by half a milliwatt. Rerun the two-line arithmetic at every upgrade, using the energy LOST IN the target rather than incident beam power where targets are thin: a 10 uA / 1 MeV machine puts up to 10 W into a mm-scale spot, which is rotating-target or water-cooled-backing territory.

target heating can be a problem and efforts are made to diffuse the beam on the target.

International Nuclear Target Development Society Workshop — ANL/PHY-84-2, Argonne National Laboratory (1983) — p. 58

Editorial note, tabletop extrapolation: Spot size is a powerful cooling knob - average flux is P/(pi*r^2), so doubling the radius quarters it - but use it inside a checked budget: compute beam power and allowable target temperature first, confirm the full swept or defocused beam still lands on target (not the holder), and treat active cooling, temperature monitoring and beam-trip protection as their own requirements rather than things defocus postpones.

thermal runaway would occur and the stored beam would be lost.

International Nuclear Target Development Society Workshop — ANL/PHY-84-2, Argonne National Laboratory (1983) — p. 218

Editorial note, tabletop extrapolation: Directly relevant to the synchrotron campaign, not the cyclotrons: compare target heating with the circulating current and intercepted fraction - a nanoamp of circulating current is a nanoamp at the target - and apply the traversal multiplier where it belongs, to how fast the stored inventory or injected charge is consumed.

The properties of a magnetic field in space make it impractical to obtain a properly shimmed field if the gap length is much greater than one-half the radius p.

Wakerling & Guthrie (eds.), Magnets and Magnetic Measuring Techniques — TID-5215, Radiation Laboratory, University of California (1949) — p. 12

Editorial note, tabletop extrapolation: An 8-12 in. pole with a 1-2 in. gap sits far inside this limit, which is why small cyclotron shims work at all; the rule bites for any short-radius bending/analysis magnet where a generous gap is tempting for access.

is the maximum allowable space rate of change of the magnetic field in a direction toward the edge of a gap.

Wakerling & Guthrie (eds.), Magnets and Magnetic Measuring Techniques — TID-5215, Radiation Laboratory, University of California (1949) — p. 15

Editorial note, tabletop extrapolation: The transferable move: translate a beam-geometry tolerance into a measurable dH/dx budget via the curl-free midplane relation - a quick LOCAL gradient check to run on a field map when flux-line bowing is the relevant tolerance. It is a calutron criterion, not a cyclotron field-quality spec: orbit, focusing, flutter and resonance checks still decide.

it was necessary for the values of the quantity integral h_z dx, experimental and theoretical, to differ by less than 3 cm, in terms of galvanometer deflection.

Wakerling & Guthrie (eds.), Magnets and Magnetic Measuring Techniques — TID-5215, Radiation Laboratory, University of California (1949) — p. 87

Editorial note, tabletop extrapolation: The discipline transfers: derive numeric field-map acceptance bands from the orbit tolerance (phase-slip or centering budget) BEFORE surveying, so the survey ends in pass/fail per region. Use integral criteria where the beam observable demonstrably depends on the integral, and keep pointwise limits where local gradients, resonances or extraction physics bite - both kinds of band, each where it belongs.

A magnet of this size when used with detuerons, is about the optimum dimensions in which deuterons may be accelerated profitably without resorting to frequency modulation.

The University of Washington 60-Inch Cyclotron: Progress and Status Report of Design and Construction — AECU-1951, University of Washington (1951) — p. 5

Editorial note, tabletop extrapolation: Any tabletop proton/deuteron machine sits far inside the fixed-frequency regime: phase slip there is dominated by field shaping and dee voltage, not relativity. Fix small-machine beam loss with shimming and volts-per-turn first (dg-273's summed-slip check), and reserve frequency modulation for the relativistic regime this rule bounds.

(1) in the median plane the variation of the intensity with radial distance must conform closely to a fairly well defined relation, and (2) inside the exit radius ... the field must be accurately symmetrical

The University of Washington 60-Inch Cyclotron: Progress and Status Report of Design and Construction — AECU-1951, University of Washington (1951) — p. 14

Editorial note, tabletop extrapolation: DIRECT — the same decomposition (radial law, azimuthal symmetry, median-plane flatness) is how a tabletop field survey should be organized, each with its own instrument and its own fix.

The function of the spikes is to produce a sharp increase in the induction at the center of gap without producing a minimum anywhere in the radial dependence.

The University of Washington 60-Inch Cyclotron: Progress and Status Report of Design and Construction — AECU-1951, University of Washington (1951) — p. 33

Editorial note, tabletop extrapolation: A central field bump gives axial focusing in the first turns, where small machines lose most of their beam - and a machined center button is one of the cheapest beam-current experiments available. The no-minimum constraint is the careful part: model and map B(r), check the field index, isochronism cost, and RF/vacuum clearance before installing.

Peak A was a result of the quarter cycle effect, B was the third harmonic of resonant ions, C was caused by secondary collisions

Cressman, The Design and Construction of a Small Cyclotron — Houghton College thesis (2006) — p. 12

Editorial note, tabletop extrapolation: When sweeping for first beam, a peak is not proof of fundamental resonance: check its position against prediction, look for companions (one-third field would support a harmonic assignment - absence doesn't refute the fundamental, since harmonic visibility depends on capture and geometry), and use a retarding potential or energy-sensitive check where possible (dg-502's discipline).

the electrode was 1 cm thick, and the ion beam produced was less than one mm wide

Cressman, The Design and Construction of a Small Cyclotron — Houghton College thesis (2006) — p. 26

Editorial note, tabletop extrapolation: Passive fringe-field focusing compressed a working machine's beam to millimetre scale - encouraging, but don't divide the two numbers: the 1 cm is the electrode's THICKNESS, not necessarily the clear aperture, and the survey doesn't give the beam-width direction. Whether a centimetre-class dee aperture bottlenecks a new machine is an envelope/acceptance calculation, not this datum.

For a deuteron in an accelerating potential of 1,000 volts, the energy limit is 1.94 MeV, for an accelerating potential of 10,000 volts, the limit is 6.13 MeV, and for an accelerating potential of 20,000 volts, the limit is 8.67 MeV. ... the effects of relativity can be countered by more than just increasing the electrode voltage, it can also be mitigated by adjusting the shape and strength of the magnetic field. The actual relativistic limit for magnetic resonators accelerating protons is closer to 25 MeV.

Cressman, The Design and Construction of a Small Cyclotron — Houghton College thesis (2006) — p. 28

Editorial note, tabletop extrapolation: For sub-MeV machines relativity is far from limiting even at 1 kV dees on any convention - both the thesis's numbers and the halved re-derivation agree on that. If energies ever approach the MeV scale, dee voltage and field shaping are BOTH levers, per the thesis's own remark.

for smaller machines a larger increase index is more appropriate, to provide stronger focusing

Cressman, The Design and Construction of a Small Cyclotron — Houghton College thesis (2006) — p. 24

Editorial note, tabletop extrapolation: Budget the resonance-versus-focusing trade against the planned turn count quantitatively: integrate phase slip through the proposed B(r) for your turn count rather than assuming percent-level falloff stays cheap, and keep the index inside the weak-focusing stability window (0 < n < 1) everywhere the beam runs.

It ranges from zero in the center of the magnet to almost 3.5 in the fringe field.

Cressman, The Design and Construction of a Small Cyclotron — Houghton College thesis (2006) — p. 38

Editorial note, tabletop extrapolation: This magnet's measured profile - n near zero over most of the radius, rising steeply in the fringe - is what motivates shimming or pole shaping on flat-pole stock generally: no vertical magnetic focusing where n=0, local radial defocusing where n>1. Whether a given profile actually loses the beam is an orbit/tune calculation, not a glance at the n curve - run it before cutting shims.

The actual relativistic limit for magnetic resonators accelerating protons is closer to 25 MeV.

Cressman, The Design and Construction of a Small Cyclotron — Houghton College thesis (2006) — p. 28

Editorial note, tabletop extrapolation: Contextualizes the phase-slip table (dg-1468): field shaping buys real headroom beyond the uniform-field estimate - how much depends on the field design, so don't carry a fixed multiplier. For any machine, calculate cumulative phase slip from the actual energy gain per turn and B(r) rather than trusting an energy-class exemption; slip can bite below 1 MeV when the gain per turn is small.

the two-day workshop can also analyse other peaks e.g., the peaks that correspond with the third or fifth of the nominal velocity.

Wolf & Prechtl, COLUMBUS — A Small Cyclotron for School and Teaching Purposes — THPO001, Proceedings of Cyclotrons2022 (2022) — p. 2

Editorial note, tabletop extrapolation: When a field-sweep spectrum shows unexplained minor peaks, check near ONE-THIRD and ONE-FIFTH of the main peak's field (f_RF = h*f_c, so B_h = B_1/h for the same species - lower field, not higher) as the odd-harmonic hypothesis, alongside contaminant-species and instrument checks; a matching position is a candidate assignment, not proof (dg-502's discipline).

Table 1: Technical Data — Diameter of the Dees 140 mm (5.5 in); Flux-density of the magnetic field 0.38 T; Vacuum in the chamber 10-5 mbar; dto with H2 10-4 mbar; Cyclotron frequency 5.63 MHz; Number of revolutions 6-8; Voltage between the dees 2.0 -3.0 kV; Final energy 24 - 48 keV … The expected final energies of the protons are 24 - 48 keV after 6 - 8 revolutions. These energies don't produce any radiation outside the chamber.

Wolf, Frank & Held, COLUMBUS — A Small Cyclotron for School and Teaching Purposes — WE1PB03, Proceedings of Cyclotrons2013 (2013) — p. 1

Editorial note, tabletop extrapolation: A conference paper's date fixes the claim, not the beam; when reusing published small-cyclotron parameters, check whether the paper predates first beam and label such values as design predictions rather than demonstrated performance.

For the setup of the ion source, it is considered that the ion source remains adjustable in direction of the gap, so that the ideal position can be found by experiments. Normally the ion source is centred in the cyclotron. However, in our case – with our small cyclotron and such a small amount of revolutions (≤ 10) - it is better to optimize the starting position of the first path. Due to this fact the ion source will not be fixed mounted but it will be stuck under the dummy-dee instead

Frank, Wolf & Held, COLUMBUS — A Simple Ion Source — WEPPT021, Proceedings of Cyclotrons2013 (2013) — p. 1

Editorial note, tabletop extrapolation: With only a handful of turns there is no adiabatic settling; an adjustable source mount converts a machining guess about the first half-turn into a tunable parameter, and the optimum need not be the centred position.

the angle of emission shall be adjustable for a better acceleration and to prevent that the protons remain in the gap between the dees

Frank, Wolf & Held, COLUMBUS — A Simple Ion Source — WEPPT021, Proceedings of Cyclotrons2013 (2013) — p. 1

Editorial note, tabletop extrapolation: A rotatable head is a cheap second degree of freedom on top of source position; both exist because low-energy ions do not forgive alignment errors in the first gap.

In contrast to the simpler common school model that approximates the tracks in the acceleration gap by straight tracks, the presented simulation considers the deflection of the ions by the magnetic field in the acceleration gap. So a more realistic picture of the paths can be drawn, which will help to adjust the cyclotron and explore the initial orbits of the ions in detail.

Wolf, Rueß & Prechtl, Simulation and Detection of the Helical Ion-Paths in a Small Cyclotron — MOB02, Proceedings of Cyclotrons2016 (2016) — p. 1

Editorial note, tabletop extrapolation: Whether in-gap deflection matters scales with gap width against local gyroradius; on a small machine whose gap is a large fraction of the first-turn radius it shapes the first turns, which is exactly where this machine tunes. A home-built orbit code should integrate the coupled equations in the gap rather than assume straight crossings.

In the dee itself, or, in and behind the dummy dee there is no accelerating electric field so that the ion trajectories can be described by equations of a circle

Wolf, Rueß & Prechtl, Simulation and Detection of the Helical Ion-Paths in a Small Cyclotron — MOB02, Proceedings of Cyclotrons2016 (2016) — p. 2

Editorial note, tabletop extrapolation: The hybrid analytic-arc-plus-numerical-gap scheme is far cheaper than brute-force stepping the whole orbit and keeps the accelerating-field-free segments (uniform B, no E) exact; it suits a laptop-class tracker for a small machine.

A flat field without flutter was selected since it is a simpler configuration for this very low energy and the high harmonic provides adequate electrostatic axial focussing

Clark, Halbach, Kunkel, Leung, Li & Young, A Compact Permanent Magnet Cyclotron for Accelerator Mass Spectrometry — Proceedings of Cyclotrons'95, Cape Town (1995) — p. 3

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: removing hills and valleys is viable at low energy only where the RF harmonic and dee geometry demonstrably supply the axial focusing the flutter no longer provides — the LBNL machine ran at harmonic 15 with electrostatic focusing doing that job. Verify axial stability by analysis or tracking before deleting flutter from a design; low energy alone does not guarantee it.

In this design H is 15 and the minimum number of orbits is 40, giving the required R = 1800 ... The magnetic field in the midplane is 1 T. For high mass resolution, the orbits need to be isochronous; a flat magnetic field uniform to about 2 parts in 104 must therefore be maintained

Clark, Halbach, Kunkel, Leung, Li & Young, A Compact Permanent Magnet Cyclotron for Accelerator Mass Spectrometry — Proceedings of Cyclotrons'95, Cape Town (1995) — p. 1, 3

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: sets the scale of what field quality buys — the source pairs 2e-4 flatness with 40 turns at harmonic 15 to reach R = 1800. A machine running few turns on the fundamental tolerates far looser fields; derive the flatness budget from turn count, harmonic, and the allowed cumulative RF phase slip, not by copying this figure.

The field is slightly outside this range at a 4-5 cm radius, where the first 5 turns occur, but this contributes only a negligible amount of phase shift and axial defocusing

Clark, Halbach, Kunkel, Leung, Li & Young, A Compact Permanent Magnet Cyclotron for Accelerator Mass Spectrometry — Proceedings of Cyclotrons'95, Cape Town (1995) — p. 3

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: weight the flatness budget by turns spent at each radius — phase error integrates per turn, so a small out-of-spec zone crossed in a few turns can be tolerable while the many-turn outer region must meet spec. Confirm by computing cumulative phase slip and axial focusing through the zone; few-turn regions are not automatically free (coherent errors and resonance proximity can still matter).

a mass resolution of about 1800 is needed to separate 14 C from 13 CH. The resolution of a CMS is approximately: R ≈ 3 x n x H, where n is the number of turns that in-phase particles make in a synchronous field before extraction and H is the rf harmonic number

Clark, Halbach, Kunkel, Leung, Li & Young, A Compact Permanent Magnet Cyclotron for Accelerator Mass Spectrometry — Proceedings of Cyclotrons'95, Cape Town (1995) — p. 1

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: an estimating relation, not a law — the coefficient depends on the phase-slip criterion that defines an in-phase turn. Useful for order-of-magnitude estimates of how sharply a small machine discriminates species or off-resonance drive; derive the real number from a phase-history calculation for the actual field and RF program.

The overall size of the machine is dictated by the mass resolution needed, the turn separation needed to clear the center region, and the injection energy. Better center region clearance requires a larger magnet, which is more expensive, or it requires a smaller injection radius, making the transit time worse for high harmonics. So a compromise has to be made giving good transmission in the center region and a large enough injection radius to give a good transit time factor

Clark, Halbach, Kunkel, Leung, Li & Young, A Compact Permanent Magnet Cyclotron for Accelerator Mass Spectrometry — Proceedings of Cyclotrons'95, Cape Town (1995) — p. 1

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: on high-harmonic or low-voltage designs the center region can drive machine size alongside the final-orbit rigidity — check the transit-time factor at the first gap crossing before shrinking the injection radius to save magnet steel, and check that the extraction-radius rigidity still fits the pole.

Axial injection, in general, is very efficient in delivering the ions into the cyclotron midplane. We have designed a spiral inflector, an electrostatic channel which twists or "tilts" as it guides the ions down the axis of the machine and into the midplane ... This was accomplished using an ion trajectory program which takes into consideration the spatial variation of the magnetic fields in the cyclotron for the inflector design and a second trajectory program which calculates the cyclotron midplane trajectories, including electrostatic focusing effects

Young, Bertsche, Clark, Halbach, Kunkel, Leung et al., Development of a Compact Permanent Magnet Cyclotron for Accelerator Mass Spectrometry — Proceedings of PAC1993 (1993) — p. 2

Editorial note, tabletop extrapolation: Ignoring the real field map in the inflector region, or the electrostatic focusing in the first turns, breaks the emittance match even when the idealized design closes; both effects belong in the design loop from the start.

With modest energy gain per turn, 1500 V, it is possible to achieve this figure with an extraction radius of ≤ 9 cm. We have conservatively designed the instrument for an extraction radius of 12 cm, corresponding to an energy of 50 keV ... [Table 1:] Pole face radius 15 cm

Young, Bertsche, Clark, Halbach, Kunkel, Leung et al., Development of a Compact Permanent Magnet Cyclotron for Accelerator Mass Spectrometry — Proceedings of PAC1993 (1993) — p. 1, 3

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: radius margin keeps the working orbit away from the field-edge rolloff and leaves headroom in turn count and final energy above the bare requirement. It does not bend the rigidity relation — at fixed field the orbit radius for a given energy is fixed, so lower-than-planned dee voltage costs turns, not radius. Committing the magnet to the bare-minimum radius leaves no recovery path once it is built.

The variable frequency oscillator was tuned to the resonant frequency (f1) of the dee-box, and the r.f. supply adjusted to produce a peak voltage of 10 Kv from dee-to-dee ... The target radius (r2) was fixed, and the beam tuned in by varying the center magnetic field strength (B0). Beam intensities were determined for different values of B0 within the tuning range of the beam. This procedure was repeated for various values of r2 between six and eleven centimeters.

Burns, Experimental Program with the Iowa State University Undergraduate 1.5 Mev Cyclotron — Proceedings of the Iowa Academy of Science 68(1), 483–491 (1961) — p. 3

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: sweeping B rather than f leaves the RF system at its tuned point, and the field sweep costs nothing but magnet-supply adjustment — on machines whose magnet is adjustable at all (a fixed-PM machine has no such knob). The intensity-versus-field curve at each probe radius is the fundamental commissioning dataset for a fixed-frequency, adjustable-field machine; families of curves at several radii help localize losses when combined with source-output normalization and independent diagnostics — they do not by themselves separate central-region loss from phase slip, source drift or vertical loss.

The magnetic field (B1) at which the ions are in exact cyclotron resonance at the r.f. supply frequency (f1) is given by the cyclotron resonance equation, B1 = 2πmf1/e (MKS units) ... The difference between the actual center field value (B0) and the field B1 at some larger radius r1 is defined as the tune-down (δB): δB = B0 − B1 ... Fig. 2 shows that with r2 less than 8 cm, δBm is observed to be zero. As r2 is increased, the peak of the resonance curve (Bm) is seen to shift to the right and δBm increases. This shift is in agreement with theory and is due to the drop-off of the magnetic field strength with increasing radius ... [Figure 2 axis label:] B1=16,830 gauss

Burns, Experimental Program with the Iowa State University Undergraduate 1.5 Mev Cyclotron — Proceedings of the Iowa Academy of Science 68(1), 483–491 (1961) — p. 3-5

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: tune-down versus probe radius is a beam-based check on the integrated field profile — how much extra center field the ions need to stay near resonance out to a given radius. Compared against a curve computed from the field map with an orbit-and-phase model (RF-frequency error, injection phase and centering included), it is an end-to-end consistency check of field survey plus orbit model, not a standalone field measurement. The source itself rates δBm as less well established than the curve widths, with uncertainties over ten percent possible from reading Bm off the graphs.

Fig. 1 shows the theoretical and experimental shapes of the resonance curve at a target radius (r2) of 9 cm ... The two experimental curves are practically identical. They are about 150 gauss wide at half-maximum intensity. The theoretical curve (1), shown in broken lines, is quite a bit narrower-115 gauss at half-maximum intensity.

Burns, Experimental Program with the Iowa State University Undergraduate 1.5 Mev Cyclotron — Proceedings of the Iowa Academy of Science 68(1), 483–491 (1961) — p. 4

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: a measured tuning curve broader than the phase-window model predicts is not necessarily a field or metrology error — simplified phase-acceptance assumptions bias the prediction narrow. Treat a width discrepancy as a prompt to examine the phase-acceptance assumptions and other broadening mechanisms, not as evidence that a machine's field tolerances are looser than computed.

The beam intensity (I) drops off very rapidly at large values of r2 ... The value of I at the r2 of 6 cm is arbitrarily assigned the value of one. The beam falls off much more rapidly than predicted ... [Table 1, experimental vs theoretical maximum I:] 6.0: 1.0, 1.0; 7.0: 0.92, 1.0; 8.0: 0.86, 1.0; 9.0: 0.63, 1.0; 10.0: 0.14, 1.0; 10.5: —, 0.68; 11.0: 0.034, 0.11

Burns, Experimental Program with the Iowa State University Undergraduate 1.5 Mev Cyclotron — Proceedings of the Iowa Academy of Science 68(1), 483–491 (1961) — p. 6

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: budget for gradual transmission loss with radius even where idealized phase calculations predict none — measure transmission versus radius during commissioning and investigate centering, focusing, apertures, gas scattering and phase slip rather than presuming one mechanism. On this machine the largest radii kept only a few percent of the 6 cm intensity; treat usable pole-edge beam as something to demonstrate, not assume.

The resonance curves have ions of greatest positive phase contributing to the extreme right of the curve, while ions of least positive phase contribute to the left of the curve ... Ions of least positive phase should be lost as r2 is increased. This is shown experimentally by the gradual shift to the right of the left-hand side of the curves with increasing r2.

Burns, Experimental Program with the Iowa State University Undergraduate 1.5 Mev Cyclotron — Proceedings of the Iowa Academy of Science 68(1), 483–491 (1961) — p. 6

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: the shape and edge motion of intensity-versus-field curves at successive radii encode which phase groups survive — information obtainable with nothing but a probe and a field sweep, read through the orbit model's phase convention. It is a model-mediated diagnostic: check source stability and rule out aperture, centering and transport changes before reading edge motion as phase acceptance.

It was found that changes in the magnetic field strength of only a few gauss can result in a large reduction of the beam strength ... it can be noted in Figure 4 that the interval of δB values for which the relative intensity, I, is equal to 1 decreases with increasing target radius

Mueller, Proton Orbit Calculations for the Iowa State University Cyclotron — Proceedings of the Iowa Academy of Science 68(1), 492–501 (1961) — p. 2, 9

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: period support for gauss-level (parts-in-1e4) field-tolerance thinking in small-cyclotron design — computed for this machine's field profile, voltage and phase model, and consistent with its companion measured tuning curves. For another machine, derive the allowable field error from its own field map, RF voltage, turn count and phase-slip model, or measure it with an intensity-versus-field sweep. The transferable lesson is that the tolerance comes out in gauss rather than percent — the budget itself must be computed, not copied.

du/dr = πerB∆B(1−n)/(2mV0). This equation gives the rate of change of the sine of the phase lag, θ, as a function of r and the magnetic field, B. Integration gives u = (πe/2mV0) ∫ rB∆B(1−n) dr + u0. (1) From this result the phase of the proton can be obtained at any radius if the initial phase lag and the magnetic field are known

Mueller, Proton Orbit Calculations for the Iowa State University Cyclotron — Proceedings of the Iowa Academy of Science 68(1), 492–501 (1961) — p. 4

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: this single quadrature over the measured field map predicts phase history without tracking orbits, and it runs in a spreadsheet — under the model's assumptions (nonrelativistic centered orbits, continuous acceleration, initial phase restricted to the accelerating branch). The right first tool for choosing frequency and trim before any trajectory code is written; it bounds phase admissibility only — vertical loss, radial loss and scattering are separate budgets.

determining which initial phases will allow a proton to reach a given target radius ... it is assumed that all protons with negative initial phases are lost from the beam because of electric defocusing (4). It is also assumed that for all positive initial phases no protons are lost from the beam because of defocusing, and that the protons are distributed randomly with respect to initial phase θ0 ... Just those protons with initial phases such that sin−1(−1−um) < θ0 < sin−1(1−uM) will reach the target. Thus for the δB shown in Figure 3 the relative intensity, I, of the proton beam at the target radius is given by I = [sin−1(1−uM) − sin−1(−1−um)]/(π/2), (2) ... However, in general δB may be such that Equation 2 has to be modified ... it was necessary to develop empirical relations for um and uM as functions of δB for various target radii

Mueller, Proton Orbit Calculations for the Iowa State University Cyclotron — Proceedings of the Iowa Academy of Science 68(1), 492–501 (1961) — p. 5-8

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: this converts the phase integral into a predicted tuning curve directly comparable to a measured intensity-versus-field sweep — the cheapest model-versus-machine comparison a small cyclotron can make. Implement it with the piecewise clipping (initial-phase window bounded below by zero, intensity floored at zero); the bare central formula over-counts when phase excursions are small. Its documented biases (too narrow, too flat-topped) are known and explainable.

When n = −(r/B)(∂B/∂r) = 0.2 a resonance condition occurs between the vertical and radial oscillations of the proton. This resonance which occurs at r = 10.1 cm in the ISU cyclotron is possibly responsible for a major loss in beam intensity at large radii ... The effect of the resonant condition, n=0.2, is difficult to determine. The resonant condition does not occur until r=10.1 cm. At this point the beam intensity is quite low already; a detailed experimental study is to be carried out later

Mueller, Proton Orbit Calculations for the Iowa State University Cyclotron — Proceedings of the Iowa Academy of Science 68(1), 492–501 (1961) — p. 8

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: compute the radius where the measured field profile crosses n = 0.2 and treat it as a resonance-warning radius — whether appreciable coupling loss actually occurs there depends on coupling strength, crossing rate, field errors and orbit centering, so confirm with tracking or a transmission measurement before writing the region off. On steep-edged small poles this radius can arrive well inside the pole edge.

z ~ A = [πeV0 sin θ/E − π²er(∂Bz/∂r)/Bz]^(−1/4) (3) The envelope of these oscillations is given by Z = kA/Amax (4) where k is one-half the dee height and Amax is the maximum value of A ... It can be noted that the beam height does not change considerably with a change in the tune-down. In Equation 3 the second term is dominant for radii greater than 6 cm. Hence, the beam height is dependent almost completely on the ratio of the gradient of the magnetic field to the magnetic field. In the ISU cyclotron, which has a relatively large magnetic field gradient, the beam height vs. radius curve is approximately linear for radii greater than 6 cm

Mueller, Proton Orbit Calculations for the Iowa State University Cyclotron — Proceedings of the Iowa Academy of Science 68(1), 492–501 (1961) — p. 10, 11

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: the field map gives the relative axial-envelope shape — in this model beam height at radius follows (dB/dr)/B and barely responds to tuning — but an absolute vertical target size also needs the injected vertical phase space, apertures and RF-gap focusing propagated through. Use the map for the envelope shape and the compression trend; a pronounced field droop buys strong axial compression toward the target radius, at the cost of phase slip.

The conclusion is that resonant couplings between axial and radial oscillations should be studied by the calculation of proton trajectories. It is unnecessary to study electric decelerations with this method since circular orbit approximations appear to be sufficient.

Moses, Proton Orbits in a Small Cyclotron — Proceedings of the Iowa Academy of Science 70(1), 403–414 (1963) — p. 2

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: a tiered modeling strategy validated by direct comparison, not just convenience — run the cheap semicircle/phase-integral model for phase-limit questions and spend trajectory integration on the resonance region and anywhere else its assumptions break (wide fringe regions, strongly displaced starts, extraction). Benchmark the cheap model against a few full trajectories before trusting the division of labor. This sizes the orbit-code effort a small-machine project actually needs.

Protons may begin orbits over a centimeter from the center of the field. Since the field decreases rapidly near the maximum radius of 11.25cm, circular approximations of these orbits may introduce significant errors that would not appear for protons starting closer to the center or moving in a larger machine with a more uniform field ... Since the magnetic field is approximately uniform up to 5cm radius, orbits in this region will be considered as circular and as having constant displacement (δr)

Moses, Proton Orbits in a Small Cyclotron — Proceedings of the Iowa Academy of Science 70(1), 403–414 (1963) — p. 2, 4

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: small machines can be the worst case for textbook circular-orbit formulas — source offsets can be a large fraction of pole radius and the fringe region proportionally wide, as here (over 1 cm offset on an 11.25 cm machine). Map the field first, then let its flatness — together with orbit-centering and gap-kick estimates — decide out to what radius the simple formulas are trusted.

Note the amplitude expansion of axial oscillations in the region of 10.1cm. This is the predicted resonance at n = 0.2. Also at r = 10.3cm where n = 0.25 (ωz ≈ 2φ̇) there is a coupling due to the shifting of the orbit as a result of the electric accelerations. Since the two resonances are only about 0.2cm apart, the amplitude expansion cannot be considered as a result of only one of the two factors. In one plot of z the coupling had the opposite effect by reducing the amplitude of axial oscillation ... At the region of resonance, some of the oscillations have encountered phase localization; that is, all but the reduced oscillations are very nearly in phase.

Moses, Proton Orbits in a Small Cyclotron — Proceedings of the Iowa Academy of Science 70(1), 403–414 (1963) — p. 10

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: in the ISU geometry these two resonances crowded within 0.2 cm and their effects could not be separated — a warning that on a steep-edged small pole distinct resonances can overlap, making single-resonance analysis of beam loss underdetermined there. For another machine, locate each resonance from its own field map and track them separately and together before deciding whether they form one overlapping loss region.

To obtain Figure 6, a series of calculations was performed with r0i = 9cm, θ0 = 0, σz′ = Nπ/8 and δri varying from 0.1cm to 1.0cm. For each value of δri the maximum value of |z| was obtained. For example, if a proton entered the orbit with δri = 0.5cm, its axial amplitude should be less than 1/6.1 times the height of the dees if there is to be 100% certainty that the proton will not strike the dees.

Moses, Proton Orbits in a Small Cyclotron — Proceedings of the Iowa Academy of Science 70(1), 403–414 (1963) — p. 11

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: dee aperture is consumed multiplicatively by orbit-centering error — in this modeled geometry a half-centimeter centering error left about a sixth of the aperture usable through the resonance. The '100% certainty' is the model's own, within its tracked initial conditions and field approximation. Center the source and first turns well, or budget aperture for resonance-driven axial growth.

Note that the maximum energy decreased by about 3% as δri increased from 0.1cm to 1cm ... On comparing the two graphs in Figure 6 it was concluded that resonances have far more effect on premature termination of off-center orbits than do electric decelerations ... The study beginning with r0i = 5cm also indicated that δr increased to approximately 0.7cm at maximum radius; therefore, off-center orbits do not become circular as their radii increase.

Moses, Proton Orbits in a Small Cyclotron — Proceedings of the Iowa Academy of Science 70(1), 403–414 (1963) — p. 10, 12

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: in these simulations, centering errors cost transmission (vertical resonance loss) far more than final energy (phase slip), and the off-centering persisted to full radius rather than self-correcting. Read as a diagnostic prior, not a law — for another field and RF geometry check vertical aperture, radial interception, RF phase histories and energy gain together; an off-center orbit can also lose by phase slip or direct interception.

In the case of a centered orbit, the proton will strike the target when r0 exceeds rt, the target radius. Target energies would be approximately single-valued for centered orbits for which E = mv2/2 = [r0 e B(r0)]2/2m. However, off-center protons may strike the target whenever r0 + δr exceeds rt. It is then possible to have heterogeneous target energies since r0, and consequently E, may vary. Therefore, any simplified method of off-center orbit study must include a means for considering heterogeneous target energies.

Moses, Proton Orbits in a Small Cyclotron — Proceedings of the Iowa Academy of Science 70(1), 403–414 (1963) — p. 12

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: quoted beam energy from radius alone assumes centered orbits — a distribution of orbit offsets and betatron phases both shifts and broadens the energy arriving at a probe. Propagate the measured or assumed offset distribution through the local energy-radius relation, or read impacts from tracked trajectories; threshold-reaction measurements near the target radius smear accordingly.

[Magnet data table:] Field homogeneity at the median plane — better than 18 x10-3 up to radius of 120 mm (expected as per simulation) ... Field mapping in the median plane of the magnet should be carried out. Homogeneity of the magnetic field at different radial and angular positions w.r.t. the central field (B/B) shall be measured and compared with the results obtained using simulations ... Homogeneity of B/B ~18x10-3 over a radius of 120 mm of the pole is required as per design

IUAC, e-Tender 09/GOR/2024–25 — H-Dipole Water-Cooled DC Electromagnet for the Table-Top Cyclotron: Engineering Specification and Acceptance Tests (2024) — p. 13, 18

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: a design/spec number from a modern professional team for a small teaching cyclotron — 18e-3 out to ~79% of pole radius, expected from simulation and verified by mapping at acceptance. Context for what an unshimmed as-designed pole can look like, not a target to copy: derive the field-quality requirement from the machine's own phase-slip and orbit tolerances (the ISU worked example, dg-1547/dg-1588, ran at 2e-4), and give the mapping instrument resolution substantially finer than whatever criterion it must verify.

Evidence to back up the accusation presented itself when, after a particularly long beam run, a spiraled discoloration appeared on the copper chimney. The discoloration began at the aperture and wrapped in the direction of the beam rotation and with downward pitch as shown in figure 1. The discoloration is taken to be tracks of ions launched during the early portion of the RF phase that were not energetic enough to clear the chimney. It was suspected that the slight vertical asymmetrical geometry of the ion source chimney was the cause of the vertical electric field. In obtaining the order of magnitude of the vertical field a slope of 4.3 degrees was calculated from the spiral track.

Koeth, Hanebuth, Hoffman & Schneider, Rutgers 12-Inch Cyclotron Ion Source Studies: Part II (2007) — p. 1

Editorial note, tabletop extrapolation: Transferable to a small machine with an internal filament/chimney source: deposits and discoloration on the chimney are a free, retrospective record of where lost early-phase ions went. Photographing the chimney after a long run and measuring the spiral's pitch costs nothing and — as here, where the 4.3-degree track slope fed the field estimate of dg-1638 — can yield an order-of-magnitude number for the parasitic vertical field, PROVIDED the deposit's origin and timing can be argued. It is a track-pitch diagnostic, not a direct launch-angle measurement.

If one calculates that in one half of an RF cycle, the ion vertically declines 0.032 inches in height the effective integrated electric field is simply calculated from: [displayed equations F_z = ma_z = qE_z ; a_z = qE_z/m ; z = (qE_z/2m)t^2 ; E_y = 2dm/(qt^2) ; E_y = (2)(0.00081m)(1.6x10^-27 kg)/((1.6x10^-19)(40nS)^2) = 100 V/cm] […] When the ions have struck the chimney at this point they have a vertical energy of about 10eV.

Koeth, Hanebuth, Hoffman & Schneider, Rutgers 12-Inch Cyclotron Ion Source Studies: Part II (2007) — p. 1

Editorial note, tabletop extrapolation: The method, not the number, transfers: a half-RF-period vertical drop measured off a chimney track or first-turn photo converts into a field estimate via z = ½at². Two calibrations on the source's own arithmetic: the printed equation checks out for its inputs (2·0.00081·1.67e-27/(1.6e-19·(40e-9)²) ≈ 1.0e4 V/m = 100 V/cm — computed here, not stated), but 40 ns is not half a cycle at the memo's stated 14.90 MHz (33.6 ns is); rerunning the same constant-field model with 33.6 ns gives ≈144 V/cm. Treat 100 V/cm as the source's result under its own stated assumption. A machine at ~9-10 MHz has a longer half-period, so a given drop implies proportionally less field.

A chimney with three identical apertures was machined, one aperture was in the median plane as is typical of the normal ion source, and an aperture placed 2.5 mm above and below the median plane aperture. In addition to experiencing the vertical electric field the off-plane apertures gave the ions initial betatron amplitudes. The cyclotron was brought up to typical operating values – this time three stacked purple glows appeared fanning into the face of the DEE (figure 3). The intensity of the three apertures declined as they moved away from the filament, as plotted in figure 4. The off-plane sources intensity varied only +/- 7% from the median plane aperture.

Koeth, Hanebuth, Hoffman & Schneider, Rutgers 12-Inch Cyclotron Ion Source Studies: Part II (2007) — p. 2

Editorial note, tabletop extrapolation: A cheap, highly copyable experiment: one extra chimney with three apertures (median plane, ±2.5 mm) turns the source into a deliberate initial-condition generator, and the glow-intensity profile (Fig. 4) is the control — the off-plane sources matched the median one within ±7%, so survival differences are attributable mainly to optics, at that level of control. Pick your own offsets from your machine's modeled or measured vertical acceptance and the betatron amplitude you want to launch, not by scaling 2.5 mm to your gap.

A typical 15 second digital exposure of the fluorescent screen was made. Two (not three) sinusoidal patterns, slightly shifted in phase appeared. The third beam was lost to the DEE lid owning to the additive effects of large launch angle and betatron amplitude.

Koeth, Hanebuth, Hoffman & Schneider, Rutgers 12-Inch Cyclotron Ion Source Studies: Part II (2007) — p. 2

Editorial note, tabletop extrapolation: Relevant wherever vertical acceptance is a few millimetres: launch angle (from a parasitic field) and initial betatron amplitude (from an offset) superpose WITH SIGN AND PHASE — they can add or partially cancel, and the third beam here was the additive case, lost to the dee lid. For alignment tolerancing, budget the worst-case additive combination; for diagnosis, remember a surviving beam does not prove both errors are small. ("owning to" is the source's spelling of "owing to".)

For instance, the placement of the radially adjustable florescent screen at its present azimuthal location was dictated by practical limitations. By happenstance this position was very close to the azimuthal location of the maximum radial betatron amplitude (turn-to-turn spacing is at its greatest), thus providing the ability to discern the axial betatron motion.

Koeth, Hanebuth, Hoffman & Schneider, Rutgers 12-Inch Cyclotron Ion Source Studies: Part II (2007) — p. 2

Editorial note, tabletop extrapolation: A design rule worth applying deliberately rather than by luck, as it happened here: put the viewport/screen azimuth where turn-to-turn separation is greatest — that is where individual turns and the vertical oscillation can actually be told apart in a photograph. On a machine with only a handful of usable ports, model or measure the turn-spacing azimuth first and let that decide which port earns the diagnostic.

quickly losses the ability to distinguish the two different betatron paths. For a region of about 1.5 betatron periods (between 185 keV and 325 keV) the fluorescent screen intensity and turn to turn spacing were sufficient to capture an image useful for analysis. Calibration of the horizontal pixels indicates that the horizontal spacing of the two prominent betatron waveforms corresponds to a phase difference of 47 degrees.

Koeth, Hanebuth, Hoffman & Schneider, Rutgers 12-Inch Cyclotron Ion Source Studies: Part II (2007) — p. 3

Editorial note, tabletop extrapolation: Realistic expectations for photographic beam diagnostics: this image was analyzable for about 1.5 betatron periods (185-325 keV) — below that the glow was too faint, above it the turns crowded together — and within the window a calibrated pixel measurement resolved a 47-degree phase difference between the two launched waveforms. Pixel calibration against a known internal dimension (here the 0.25-inch chimney) is the enabling trick; find your own machine's usable window empirically, since it belongs to the screen, exposure and beam intensity, not the class.

Operation was at 600 watts at 14.90 MHz with a magnetic field of 0.977 Tesla. Plugging this data into the SIMION model we were able to reproduce the following plot (figure 6). […] Peak vertical displacement from the median plane was approximately 9 mm in both measurement and simulation.

Koeth, Hanebuth, Hoffman & Schneider, Rutgers 12-Inch Cyclotron Ion Source Studies: Part II (2007) — p. 3

Editorial note, tabletop extrapolation: The benchmark pattern to reproduce before trusting a tracker: one measurable, model-independent quantity — peak vertical excursion — agreeing between photograph and simulation at the ~9 mm precision reported (the memo states no uncertainty, so no stronger agreement claim is available). Note 14.90 MHz and 0.977 T are the proton fundamental (h = 1), a clean operating point; a machine at ~0.6 T sits near 9 MHz for the same harmonic (computed here).

From figure 6 we see that ions starting above the median plane (Z > 25.5 mm) were more likely to succeed to target. Ions that started at the lower aperture were very quickly lost. This analysis was pushed further to locate the optimal height from which to launch the ions from in this given geometry. From figure 9 it is seen that a height of 34 mm is the optimal location for a point source to launch from. This is 8 mm above the median plane.

Koeth, Hanebuth, Hoffman & Schneider, Rutgers 12-Inch Cyclotron Ion Source Studies: Part II (2007) — p. 3

Editorial note, tabletop extrapolation: Important cautionary result for tabletop source placement: the "obvious" choice of putting the aperture exactly in the median plane was not optimal in this machine, because the parasitic downward field means a deliberate upward offset recovers capture. The offset is specific to this geometry's field asymmetry, so a builder should scan launch height in their own model rather than copy 8 mm. Note the source is internally inconsistent about where the median plane sits — the text and Fig. 6 title use 25.5 mm while the Fig. 9 axis label reads "26=median plane", which is why 34 mm is described as 8 mm above it.

Fig. 9 Differing ion launch heights simulated in SIMION, green dots are location of measured peaks and valleys (dots heights are not representative of data height). Note height of DEE lid is at 36 mm. Also note beam blow up at r = 110 mm, this is where n = 0.2 resonance resides, see reference [2].

Koeth, Hanebuth, Hoffman & Schneider, Rutgers 12-Inch Cyclotron Ion Source Studies: Part II (2007) — p. 4

Editorial note, tabletop extrapolation: Both numbers are read from the figure and its caption. The transferable point is the method it illustrates: the useful radius of a weak-focusing machine is bounded not by the pole edge but by where the field index reaches a resonant value — on THIS machine, n = 0.2 at r = 110 mm of a 152 mm pole radius, and the simulation blows up there. Map your own n(r), find your own resonance radii, and place target and deflector inside the demonstrated usable radius; where n = 0.2 lands is your taper's choice (dg-1729).

A thin, curved, grounded sheet formed the septum and separates the main accelerating volume and the deflection channel. A slightly greater curved high voltage (HV) electrode was concentrically arranged to complete the deflection channel with and average 0.31 inch gap spacing. The deflector tangentially intercepted the spiraling cyclotron beam at a radius of 4.0 inches and transported the beam to a radius of 4.5 inches in 43° of azimuth. The deflection channel had a nominal radius of curvature of 7 inches.

Koeth, Ponter, Hoffman, Schneider & Krutzler, Rutgers 12-Inch Cyclotron Electrostatic Deflector (2010, rev. 2011) — p. 1

Editorial note, tabletop extrapolation: The most fully dimensioned extraction geometry in the amateur literature at this scale — a 12-inch machine intercepting at 4.0 inches. As orientation: the channel's radius of curvature is 1.75× the orbit radius, the gap ~7.5% of the orbit radius, and the channel spans 43° to gain 0.5 inch (ratios computed here). Applying those ratios to another machine is geometric illustration only — rigidity, turn separation, septum thickness and fringe fields all enter — so recompute the field and voltage (dg-1660) and verify by tracking before cutting metal.

This result is not understood, as only integer odd harmonic numbers support magnetic resonance acceleration. At an even harmonic, when acceleration occurs at a gap crossing, deceleration must occur at the subsequent crossing, yielding zero net accelerator per revolution. In the region between an even and odd harmonic, there is a balance of acceleration and phase slippage which the ions encounter. Operating a cyclotron near, but not on, an odd harmonic, can still lead to a successful resonantly accelerated beam, provided that the integrated phase slippage over all revolutions is less than 180 before hitting the target or extraction point. The greater the DEE voltage, the fewer the number of ion revolutions are needed to achieve the desired energy, thus the tolerance of phase slippage per turn increases with DEE voltage.

Koeth, Ponter, Hoffman, Schneider & Krutzler, Rutgers 12-Inch Cyclotron Electrostatic Deflector (2010, rev. 2011) — p. 6

Editorial note, tabletop extrapolation: Directly relevant to low-dee-voltage machines, in mirror image: many hundreds of turns means very little tolerable slip per turn, which is an operational argument for dee voltage beyond simple turn-count. State the physics as the source's gap phasing gives it: odd harmonics are the resonant condition for this conventional geometry, and near-harmonic operation can survive if the bunch stays inside the accelerating phase window — the 180-degree integrated-slip figure is an approximate span, conditional on where in phase the ions start and which way they slip. The reported 2.22 and 4.25 harmonic numbers are stated by the authors as "not understood" — an open anomaly, not a result. ("accelerator per revolution" is the source's typo for "acceleration".)

Initially, to satisfy the cyclotron resonance condition, the pole tips of the 12-Inch Cyclotron magnet were Blanchard ground to provide parallelism to better than 1 part in 10,000. As will be seen, this pure vertical field does not provide any beam focusing effects, all but a very few of the generated ions are lost on either the top of bottom plate of the DEE. Indeed, during commissioning of the cyclotron, only a trickle of beam current, less than a nano-ampere, made it to the periphery. Desiring beam currents of at least 10µA in intensity, a program to study and modify the cyclotron to achieve this goal is under way.

Koeth, Report on the 12-Inch Cyclotron Magnet Study: Measurements, Modeling, and Future Plans (c. 2005) — p. 1

Editorial note, tabletop extrapolation: The canonical educational-machine failure mode, and a machining-quality trap in reverse: extreme pole parallelism is exactly what leaves the beam without an axial restoring force (radial stability, with tune near 1, survives — it is the vertical plane that empties into the lids). The sub-nA periphery current is this machine's measured commissioning figure, a realistic 'before' anecdote rather than a class-wide baseline; the 10 µA goal is the authors' aspiration, not an achieved value anywhere in this document.

Complete transverse stability. It has thus been shown for axial stability, n must be greater than 0, and for radial stability n must be less than 1. Total transverse stability exists in the region of: 0 < n <1

Koeth, Report on the 12-Inch Cyclotron Magnet Study: Measurements, Modeling, and Future Plans (c. 2005) — p. 3

Editorial note, tabletop extrapolation: The design inequality for a weak-focusing machine, derived in this report from scratch: away from the central region, 0 < n < 1 buys simultaneous linear axial and radial stability (at r = 0 itself n = 0, as the source's own next passage states — the center is handled by other means, dg-1729/dg-1835). It is a LOCAL linear-stability window: resonances (dg-1682), acceleration and field errors still get their say. Sign convention: this document's leading minus makes n > 0 a falling field; the companion AVF paper uses k = d ln⟨B⟩/d ln R with opposite sign, so reconcile n = −k before mixing formulas.

For details beyond the scope of this document, coupled resonances between the transverse motions further limit the value of n. n values of 0.2, 0.25, 0.33, 0.5 (and others higher) need to be avoided. Since, the ions to be accelerated begin at r = 0, n = 0 and will only climb as the radius increases. If n = 0.2 needs to be avoided, then the rate at which Bz decreases must be moderated such that only near the maximum ion radius does n approach 0.2.

Koeth, Report on the 12-Inch Cyclotron Magnet Study: Measurements, Modeling, and Future Plans (c. 2005) — p. 3

Editorial note, tabletop extrapolation: Actionable sizing constraint for a taper design: it converts 'make the field droop' into 'droop slowly enough that the low-order resonances arrive only at the very end of the spiral.' The printed n list is physically standard — at n = 0.2 the tunes satisfy νr = 2νz (the Walkinshaw difference coupling), at 0.25 νz = 1/2, at 0.33 νr = √2·νz, at 0.5 νr = νz (all computed from νz = √n, νr = √(1−n)). Note the same document's p.7 attaches 0.2 and 0.5 to νz instead — the source is loose with its labels across pages, so identify resonances from BOTH tunes computed off your own n(r), never from a symbol's name.

We normalized the measured field profile for the three different operating currents: 20, 30, and 40 Amperes. Each field profile, as one would expect, had a peak field at r = 0. The data was linearly scaled to bring this peak field to unity. The simultaneous plotting of these normalized profiles, as shown in Figure 2, confirms that the field index’s (n’s) profile does not vary with field strength, even into the beginning of the saturated régime. This generously allows for just one analysis of the field profile.

Koeth, Report on the 12-Inch Cyclotron Magnet Study: Measurements, Modeling, and Future Plans (c. 2005) — p. 4

Editorial note, tabletop extrapolation: A genuine labour saver, within its validated window: on this magnet the normalized profiles overlaid across 20-40 A (into the onset of saturation), licensing one field-index analysis for the operating points inside that range. On another magnet, earn the shortcut the same way — normalized scans at several currents spanning YOUR operating point — and re-check before trusting it deeper into saturation than the comparison went (here ~1.16 T, the test endpoint, not a threshold).

The above analysis shows an ever increasing νz, with three clear regions of growth, see Figure 12. Initally, νz starts off at zero, climbs quickly up to a radius of 2 cm, then the increase takes on a slower rate of increase up to a radius of 9 cm. After 9 cm the rate if νz increase is exponential. Keep in mind that the maximum ion radius is 12.7 cm where νz reaches a value of 0.7 – well beyond the difference instability located at νz = 0.2, which comes at a radius of about 10 cm.

Koeth, Report on the 12-Inch Cyclotron Magnet Study: Measurements, Modeling, and Future Plans (c. 2005) — p. 6

Editorial note, tabletop extrapolation: The MODELED tune footprint of this machine's weak-focusing field: νz from zero, climbing fast to ~2 cm, a long gentle rise to 9 cm, then steeply beyond — 0.7 at the 12.7 cm maximum radius. Read it as the shape to expect from a tapered pole and recompute from your own B(r), not as a measured or transferable curve. Notation flag, computed: the passage puts 'the difference instability at νz = 0.2' at r ≈ 10 cm — where this machine's n ≈ 0.04 gives νz = √n ≈ 0.2, so the label is self-consistent as a TUNE — while the canonical Walkinshaw difference resonance sits at n = 0.2 (νz ≈ 0.45); the same document's p.3 uses n = 0.2 (dg-1682). The source mixes the two notations across pages; derive your resonance radii from computed νr and νz.

In the case that we do find an azimuthal field distortion, it will most likely have a periodicity of 2, which is inherently an unstable Azimuthal Varying Field (AVF) condition. A minimum periodicity of 3 is required for a stable operating point. ... The first option is to “shim” out the AVF. By use of the 2-D field mapper, we can identify lulls in the field and manually install thin iron shims to shorten the gap and bring up the field to the desired value.

Koeth, Report on the 12-Inch Cyclotron Magnet Study: Measurements, Modeling, and Future Plans (c. 2005) — p. 7

Editorial note, tabletop extrapolation: Both halves transfer with one correction. Diagnostic: determine the azimuthal harmonic CONTENT by Fourier analysis of a 2-D map rather than inferring it from the defect — an off-center pole shows up first as m = 1, a two-lobe (m = 2) component is the case this source singles out as inherently unstable, and its minimum-periodicity-3 statement is the author's claim, presented without derivation. Remedy: entirely amateur-accessible — thin iron shim stock laid in the mapped low spots to shorten the gap locally — followed by re-mapping, since the shims move the average field and the harmonics together.

The pink trace (lowest) in Figure 15 displays the sole effect of Thomas focusing, - AVF focusing without a spiral edge. It is interesting to note that in our case, weak focusing is in fact stronger than the colloquially termed AVF “strong focusing.” This peculiararity arises from the fact that our small (1.2MeV) cyclotron does not noticeably suffer from relativistic effects. If it did, the magnetic field would need to increase with radius, as opposed to our decreasing field, in order to keep the more “massive” ions in step with the RF.

Koeth, Report on the 12-Inch Cyclotron Magnet Study: Measurements, Modeling, and Future Plans (c. 2005) — p. 7

Editorial note, tabletop extrapolation: The qualitative result matters for a 100 keV-1 MeV machine and cuts against the modern instinct: with no relativistic detuning to fight, a non-relativistic machine may use a FALLING field, and this study found its tapered weak focusing stronger than its unspiralled Thomas alternative. No numeric ratio should be carried: Fig. 15's ordinate is printed 'field index - n' while text and caption call it νz, and its weak-focusing trace disagrees with the p.6 νz ≈ 0.7 value — if the plotted quantity were νz² the tunes would be its square roots — an internal inconsistency of the source, flagged. AVF earns its complexity when a rising (isochronous) field is needed, and can still be chosen at low energy for acceptance or tune control; this machine's own later spiral tips (dg-1745) are that choice made deliberately.

The light green trace (left and uppermost trace) in Figure 15 corresponds to the pole tip design shown in Figure 14. It is clear that νz grows very rapidly with even a slight spiral. Because of the cataclysmic beam instability at νz = 1, and other serious instabilities at νz = 0.2, 0.5 and so on, use of a spiral edge does not does not seem favorable.

Koeth, Report on the 12-Inch Cyclotron Magnet Study: Measurements, Modeling, and Future Plans (c. 2005) — p. 7

Editorial note, tabletop extrapolation: A caution, not a verdict, on spiral sectors at small radius: THIS slight-Archimedean design's modeled tune ramped so fast (νz = 1 by ~7 cm, read from the rendered Fig. 15's varchimedes trace) that the author judged spiral edges unfavourable for the machine, citing the νz = 1 instability and lines at 0.2 and 0.5. Whether a small pole has room to spread the ramp depends on sector count, flutter and spiral angle: plot the full tune trajectory against the resonance lines for YOUR field map and track through any crossing — the same group's 2011 study did exactly that and built a working 270° spiral (dg-1745), so treat this page as one design iteration's lesson.

AVF focusing can be used to supplement weak focusing. In this context, the weak focusing comes from the average radial gradient’s field index, denoted as k, where: k = d ln〈B〉/ d ln R . The tune is proportional to the relative focusing strength. Following the treatment of J.J. Livingood,[6] one can write the axial tune in terms of the average field index, flutter, and the instantaneous edge angle: νz² = -k + F(1+tan²ξ) The radial tune is written as νr² = 1+k … The rms variation of the vertical field is called flutter and is denoted as F. The azimuthal magnetic field component, Bθ, is also proportional to the flutter.

Koeth, Hine, Hoffman, Krutzler, Ponter, Rosenberg, Ruisard & Schneider, Comparison of Azimuthally Varying with Constant Gradient Magnetic Fields with the Rutgers 12-Inch Cyclotron (2011) — p. 2

Editorial note, tabletop extrapolation: The design equation for combining a weak-focusing taper with AVF sectors, showing the two contributions add. Two convention traps, both resolved here: (1) this paper calls F 'the rms variation' — for the linear-in-F tune formula to be the standard Livingood form, F must be the MEAN-SQUARE fractional variation ⟨((B−⟨B⟩)/⟨B⟩)²⟩, i.e. the square of the rms fraction, exactly as the same program's later paper defines it (F² there = ⟨…²⟩, tune quadratic in its F; dg-1746) — reconcile against Livingood before numeric use; (2) k = d ln⟨B⟩/d ln R is NEGATIVE for a falling field, opposite in sign to the magnet study's n, so n = −k. The tan²ξ factor is why edge angle is a powerful and dangerous knob — it grows without bound (dg-1697).

The first set of AVF pole tips measured were of the simplest design, and are shown installed with the cyclotron chamber removed in figure 10. With a periodicity of four, the hills and valley are each 45 degrees wide. They maintain a constant thickness out to the pole edge, except for a ¼ -inch chamfer to break the sharp corners. The data from the first scan is plotted in Figure 11. The four hills are prominent, however a small central bump is observed from the slug that ties the four vanes together. This weak focusing is required to promote a centrally localized focusing field since the flutter will be too small to be effective at the central convergence.

Koeth, Hine, Hoffman, Krutzler, Ponter, Rosenberg, Ruisard & Schneider, Comparison of Azimuthally Varying with Constant Gradient Magnetic Fields with the Rutgers 12-Inch Cyclotron (2011) — p. 5

Editorial note, tabletop extrapolation: The geometry as stated (four sectors, 45-degree hills and valleys, constant thickness, 1/4-inch chamfer, central slug) plus the central-region insight that matters most at small scale: flutter vanishes at r = 0, so a pure-AVF machine has no SECTOR focusing where ions are born — this design's central slug supplies a deliberate weak-focusing bump there, and the source states that requirement for its own field. Evaluate your own central region's full focusing budget (magnetic index plus RF-gap electric focusing and phase) rather than assuming the bump; most small AVF designs end up wanting one (the AKG270 kept it, dg-1717).

Figure 12 plots Bz(θ) over one quadrant displaying the relative evolution of the flutter with radius by individually plotting Bz(θ) for sixteen radii. The plot shows the emerging sinusoid flutter, despite the square stepped transitions between hills and valleys. Only for radii of 4-inches and greater does a ‘flat-top’ become apparent.

Koeth, Hine, Hoffman, Krutzler, Ponter, Rosenberg, Ruisard & Schneider, Comparison of Azimuthally Varying with Constant Gradient Magnetic Fields with the Rutgers 12-Inch Cyclotron (2011) — p. 5

Editorial note, tabletop extrapolation: An instructive measured fact about gap smoothing: square-cut sector iron produced a nearly sinusoidal Bz(θ) on this pole, with a flat top emerging only beyond 4 inches radius. The general lesson is that the gap filters sector geometry hard — machining need not chase a shaped profile blindly — but how much smoothing, where the designed flutter amplitude arrives, and what harmonics survive are set by gap-to-sector-width and radius ratios: solve or map YOUR geometry and take the flutter spectrum from that, rather than scaling this 4-inch mark by pole size.

SIMULATIONS Form start to finish, four software tools have been employed to simulate the beam dynamics of in these magnetic fields. SolidWorks was used to mechanically model the magnet, Maxwell 3D was uses to solve the field problem and generate the needed field report for SIMION to fly and track the ions in. Post processing was performed in MatLab. ... Maxwell 3D (M3D) was first benchmarked against our weak focusing PSF simulations. A 3-D magnet model, which included the weak focusing pole tips was designed in SolidWorks and then imported into Maxwell 3D. The problem was solved to have a nominal peak central field of 1 Tesla. To within measurement errors the models agreed.

Koeth, Hine, Hoffman, Krutzler, Ponter, Rosenberg, Ruisard & Schneider, Comparison of Azimuthally Varying with Constant Gradient Magnetic Fields with the Rutgers 12-Inch Cyclotron (2011) — p. 5

Editorial note, tabletop extrapolation: A four-stage pipeline — CAD, 3-D field solver, tracker, analysis — with the transferable discipline being the BENCHMARK step: before trusting the 3-D solver on new geometry, reproduce the old validated result on the old geometry (here Maxwell 3D reproduced the Poisson/Superfish weak-focusing field within measurement errors — the FIELD model, not the tracking chain, is what that comparison validates). Free-tool substitutions: FEMM only where a planar/axisymmetric approximation is defensible — a radial-sector AVF field is intrinsically 3-D, so budget for Elmer or another 3-D solver there — and verify the field-transfer and tracking layers separately (dg-1712's trap).

The resolution of the imported field has been set at 1 mm to conveniently match the 1mm:1 SIMION grid unit ratio. The M3D report file is a 6-column a comma separated variable file reporting x, y, z, Bx, By, and Bz at each grid point, with a spacing of 1-mm between grid points. To fully cover the ion accessible region in our cyclotron, the field region must span a volume with a radius up to 4.5 inches – the point of interception of deflector. Therefore the extent of the report spans ±115 mm (~ 4.55-inches) X ±115 mm X (~ 4.55-inches) X 53 mm (~ 1.04-inches) which contains 231 X 231 X 53 rows of data, an excess of 2.8 million points - causing the simple text data to become unwieldy, in excess of 500 MB.

Koeth, Hine, Hoffman, Krutzler, Ponter, Rosenberg, Ruisard & Schneider, Comparison of Azimuthally Varying with Constant Gradient Magnetic Fields with the Rutgers 12-Inch Cyclotron (2011) — p. 7

Editorial note, tabletop extrapolation: Concrete sizing for a tracker's field-map file: 1 mm resolution over the ion-accessible volume of a 12-inch machine is 231 × 231 × 53 points — 2.8 million rows, over 500 MB as text — so plan a binary or compressed intermediate format from the start. The printed axial figures do not reconcile (53 mm ≈ 2.09 in, yet the parenthetical prints "~1.04-inches", plausibly a half-extent; unresolved in the source — inspect your own file's z bounds rather than inferring). Size the map to cover the COMPLETE tracking domain out through every loss surface and relevant fringe region, not merely the aperture the beam is supposed to occupy.

The Maxwell 3D magnetic field’s median plane is the x-y plane while the median plane is y-z in SIMION. A careful vector rotation is required in translating the M3D report into the usable SIMION file. ... While the magnetic field is being loaded, SIMION populates the vector field by cycling through x the fastest, y the second, and finally z. This requires data sorting that cycles through x for every increment of y, and cycles through y once per increment of z.

Koeth, Hine, Hoffman, Krutzler, Ponter, Rosenberg, Ruisard & Schneider, Comparison of Azimuthally Varying with Constant Gradient Magnetic Fields with the Rutgers 12-Inch Cyclotron (2011) — p. 7

Editorial note, tabletop extrapolation: A high-cost trap for any solver-to-tracker bridge, in its general form: axis conventions between two codes are YOURS to reconcile, the mismatch is silent — the file loads, the ions fly, the answer is wrong in a way that looks like physics — and row-order-encoded coordinates mean a sorting error produces a plausible-looking scrambled field. Both argue for the same insurance: smoke-test every new bridge on a known analytic field (a uniform B, a simple dipole) before believing anything it produces.

Multiple ions can be launched together, however for these trace space simulations it was found best, especially while searching for the stability limits, to track single ions. The SIMION simulation run terminates once the ion is lost. An ion is declared lost if it exceeds the boundary of the DEE structure or falls outside of the magnetic field volume. If the ion’s orbit is stable, it will continue to circulate indefinitely and the simulation will need to be manually terminated.

Koeth, Hine, Hoffman, Krutzler, Ponter, Rosenberg, Ruisard & Schneider, Comparison of Azimuthally Varying with Constant Gradient Magnetic Fields with the Rutgers 12-Inch Cyclotron (2011) — p. 8

Editorial note, tabletop extrapolation: Practical tracker-design advice for a tabletop orbit code: define loss against real apertures (the dee, not an abstract radius), and build in a turn-count or wall-clock stop, because a stable orbit is an infinite loop. The single-ion preference when mapping stability boundaries is a workflow point — bunches obscure which initial condition failed.

As is seen in figure 18, the weak focusing field had the smallest trace space area, the radial AVF had the greatest, and the spiral AVF field was slightly less than the radial sector. It is also interesting to note the appearance of the grouping in several of the incomplete trace space contours, this indicates that the ratio of vertical fraction of an oscillation to the revolution frequency is near a rational fraction, however, given sufficient time they would completely fill in their contour. These trace space orbits are exhibiting near-resonant behavior of the order of the grouping number.

Koeth, Hine, Hoffman, Krutzler, Ponter, Rosenberg, Ruisard & Schneider, Comparison of Azimuthally Varying with Constant Gradient Magnetic Fields with the Rutgers 12-Inch Cyclotron (2011) — p. 9

Editorial note, tabletop extrapolation: A free screening diagnostic from plots a tracker user already makes: once-per-turn trace-space points clumping into n groups suggest a tune near a rational p/n — near-resonant behaviour of about that order. It is a clue, not a verdict: finite tracking, aliasing and plotting cadence can also group points, so confirm by extending the run and extracting the turn-by-turn phase advance (or a spectrum) before naming the resonance. (Attribution flag: the source cites 'figure 18' — captioned Radial Trace Space — inside its vertical-stability paragraph; Figure 19 is the vertical plot, so the printed figure number is almost certainly a misprint and the comparison is of vertical trace space.)

As is seen in Figure 20, the maximum vertical excursion of the proton in the weak field was nearly four times that of the proton in the radial sector AVF field. This has two immediate implications. First, to accommodate a given ion source, the magnet gap of AVF field can be drastically reduced, implying a smaller and less expensive magnet. Alternatively, the magnet gap can be maintained, and a greater vertical angular distribution can be accepted, implying greater beam intensity at the periphery.

Koeth, Hine, Hoffman, Krutzler, Ponter, Rosenberg, Ruisard & Schneider, Comparison of Azimuthally Varying with Constant Gradient Magnetic Fields with the Rutgers 12-Inch Cyclotron (2011) — p. 9

Editorial note, tabletop extrapolation: The clearest quantitative case for AVF at this scale, kept to what the simulation shows: one proton, identical launch, ±9 mm excursion in the weak-focusing field versus ±2.5 mm in the radial-sector field (read from the rendered Fig. 20; 'nearly four times' is the authors'). The source's two implications — a drastically reducible gap, or more accepted vertical angle — are design directions whose actual payoff needs full acceptance tracking and a self-consistent magnet redesign, since gap changes move excitation and field structure together. Note the apparent tension with the same program's finding that its weak-focusing νz exceeds its Thomas-field νz (dg-1696): tune and single-trajectory excursion are different measures, and the Fig. 15 labeling problem (same card) leaves the tune comparison unresolved.

While the constructed radial sector pole tips were not intended to support acceleration, with sufficient DEE voltage protons were successfully accelerated. The incurred phase slippage at normal operating conditions - namely a DEE voltage of 8 kV-peak - was indeed too severe to successfully bring ions to the full radius. However, a DEE voltage of 20 kV peak accepted ions over 20° of the RF cycle and accelerated … them to the periphery. This suggested that our first attempt is not too far from a practical design.

Koeth, Hine, Hoffman, Krutzler, Ponter, Rosenberg, Ruisard & Schneider, Comparison of Azimuthally Varying with Constant Gradient Magnetic Fields with the Rutgers 12-Inch Cyclotron (2011) — p. 9

Editorial note, tabletop extrapolation: Quantifies what a non-isochronous field costs in dee voltage, on this field and RF model: at the machine's normal 8 kV-peak the slippage was fatal; 20 kV-peak accepted a 20° RF window and carried protons to the periphery — a factor of 2.5, for this map. Recalculate the acceptance-versus-voltage curve for your own field, harmonic and RF waveform; the transferable shape is that voltage buys phase margin against a mismatched field (dg-1786 is the measured version of the same lesson). Simulation results, not measured beam.

SPIRAL AVF DESIGN Finally, we present the optimized design for a set of spiral pole tips that are intended to guide beam. The result was a four sector Archimedean spiral sweeping 270 degrees, and will herein be referred to as AKG270. With the exception of the weak focusing central region, these pole tips aimed to satisfy the isochronous condition, in order to minimize the phase slippage, and reduce the minimum DEE voltage while preserving axial stability throughout the accelerating region.

Koeth, Hine, Hoffman, Krutzler, Ponter, Rosenberg, Ruisard & Schneider, Comparison of Azimuthally Varying with Constant Gradient Magnetic Fields with the Rutgers 12-Inch Cyclotron (2011) — p. 10

Editorial note, tabletop extrapolation: The design pattern worth copying is the HYBRID: weak focusing kept in the centre where flutter cannot help, spiral-AVF outboard where isochronism pays — that is what minimized phase slippage and dee voltage while preserving axial stability here. The 270-degree four-sector Archimedean sweep is this magnet's optimized answer (the authors credit their machine shop for cutting it); another machine re-runs the optimization on its own field map and takes whatever sweep its tunes demand.

The AKG270 radial trace space was explored first to identify the equilibrium orbits in 50 keV increments. Even, at 50 keV, non-linear behavior is noted in the larger stable orbits, exhibiting four-sided contours, behavior which is consistent with pole tips that have a sector periodicity of four. The corners of the four-sided nonlinear orbits become more pronounced and bulbous with increasing energy. By 250 keV four closed contours formed in the protracted corners, displayed in Figure 23. Thus, in addition to the primary Equilibrium Orbit, there are four off-center stable orbits.

Koeth, Hine, Hoffman, Krutzler, Ponter, Rosenberg, Ruisard & Schneider, Comparison of Azimuthally Varying with Constant Gradient Magnetic Fields with the Rutgers 12-Inch Cyclotron (2011) — p. 10

Editorial note, tabletop extrapolation: A phenomenon to look for on any sectored machine, from this worked case: the four-sector AKG270's radial phase space showed four-sided nonlinear contours already at 50 keV, sharpening with energy until four closed islands formed by 250 keV — genuine off-centre stable orbits alongside the primary one. Whether YOUR sector count produces islands, and at what energy, depends on the field harmonics and tunes: survey trace space at energy steps fine enough to resolve your calculated resonances (50 keV was this study's choice), and follow with RF-on tracking to learn whether real accelerating beam gets captured by them (a beam parked on an island reads as mis-steered, dg-1793).

The off-center equilibrium orbits were experimentally verified using the wire-loop orbit technique.[7] A 30 AWG wire loop, with a circumference of 71 mm, was energized with a current of 2.5 amps and placed in the magnet gap. Myriad other stable orbits made it difficult to perform this experiment in the median plane; instead a clear acrylic sheet was placed on the bottom pole tip to provide a flat surface on

Koeth, Hine, Hoffman, Krutzler, Ponter, Rosenberg, Ruisard & Schneider, Comparison of Azimuthally Varying with Constant Gradient Magnetic Fields with the Rutgers 12-Inch Cyclotron (2011) — p. 10

Editorial note, tabletop extrapolation: An outstanding no-vacuum, no-beam diagnostic: a current-carrying flexible loop settles onto stable orbit shapes of a real measured field for the price of magnet wire and a bench supply — a physical check on the tracker before the chamber ever pumps down. Physics to hold onto: the loop obeys T/ρ = I·B, so its effective rigidity is set by tension over current — circumference constrains which closed shapes are available but does not by itself select a particle energy (the source says as much; its extra orbits are the point of dg-1720). Practicalities: the acrylic sheet keeps the loop on a plane (not the median plane — a known offset), and 2.5 A in 30 AWG dissipates real heat, so current-limit, keep the duty short, and mind magnet forces. Setup as run: 30 AWG, 71 mm circumference, 2.5 A.

The energized wire loop simply needed to be tossed towards the gap and it would reproducibly snap to the nearest stable orbit, one such off-center orbit is show in figure 24. An overlay of five loop images demonstrating five stable orbits is shown in figure 25. This technique found another four orbits (for a total of nine) located even further away from the center. Since the wire-loop technique does not discriminate based circumference (only the loop’s tension will vary), the further outlaying orbits are most likely lower energy equilibrium orbits.

Koeth, Hine, Hoffman, Krutzler, Ponter, Rosenberg, Ruisard & Schneider, Comparison of Azimuthally Varying with Constant Gradient Magnetic Fields with the Rutgers 12-Inch Cyclotron (2011) — p. 11

Editorial note, tabletop extrapolation: Interpretation guidance for the wire-loop method: the loop finds the orbit FAMILY, not one energy, so a bench survey should turn up more orbits than any single-energy simulation predicts — here nine against five, with the source judging the outliers 'most likely' lower-energy equilibria since the technique discriminates on tension, not circumference. Treat extra positions as candidates: compare against multi-energy tracking, and check loop mechanics (tension, friction, off-median-plane field) before either assigning an energy or reading a model discrepancy. (The figure references in this passage are off by one against the printed captions — the photographs are Figs. 25 and 26, not 24 and 25.)

Protons were flown with RF in SIMION with the AKG270 magnetic field. The trajectory of a single proton is shown in Figure 21. The driving frequency and amplitude were iteratively tuned to locate the minimum peak DEE voltage necessary to successfully accelerate the proton to the target. This lowest practical voltage found in the simulation was 6 kVpeak at a frequency of 15.534 MHz.

Koeth, Hine, Hoffman, Krutzler, Ponter, Rosenberg, Ruisard & Schneider, Comparison of Azimuthally Varying with Constant Gradient Magnetic Fields with the Rutgers 12-Inch Cyclotron (2011) — p. 11

Editorial note, tabletop extrapolation: Quantifies the payoff of designing for isochronism, within one simulation campaign: 6 kV-peak at 15.534 MHz sufficed in the AKG270 spiral field, versus the 20 kV-peak the non-isochronous radial-sector field needed and the machine's normal 8 kV (both from the same study's radial-sector section, dg-1716). If shunt impedance and loading were unchanged, cavity loss ∝ V² would differ by ~11× between 6 and 20 kV — a conditional estimate, computed here. The frequency checks: 15.534 MHz ↔ ~1.02 T for protons at the fundamental (computed). Field shaping as a lever on the RF budget is the transferable idea; single-particle simulation, not measured beam.

After locating the equilibrium orbits, a complete comparison of the focusing between the AKG270 poles and the weak focusing pole tips was performed using the simulated fields. The DEEs were removed from both cases to observe, if any, non-linear effects at large excursions. The radial and axial results are respectively shown in Appendix II-a and -b. In most of the radial cases the AKG270 radial trace space is bounded by the weak focusing pole tips. At the lower energies of 50 and 100 keV, the vertical trace space of the AKG270 poletips is larger than that of the weak focusing poles, indicating a greater angular acceptance from the ion source. This is due to the enhanced central focusing from the weak focusing bump.

Koeth, Hine, Hoffman, Krutzler, Ponter, Rosenberg, Ruisard & Schneider, Comparison of Azimuthally Varying with Constant Gradient Magnetic Fields with the Rutgers 12-Inch Cyclotron (2011) — p. 11

Editorial note, tabletop extrapolation: Where this hybrid field's acceptance advantage showed up: at the LOW-energy end — 50 and 100 keV vertical trace spaces larger than the weak-focusing poles' — and the source credits the AKG270's retained central weak-focusing bump, not the spirals. That is the hybrid logic confirmed at exactly the energies where source acceptance is decided. Methodological detail worth copying: the dees were removed from both simulations so the comparison probes field nonlinearity, not mechanical clipping. Radially, the weak-focusing field bounded AKG270 in most cases; simulated statics, not measured beam.

The average radial field profile, plotted in figure 6, is generated from the assembled average fields along the radius – this is needed to calculate the average field index, k. In the case of the axisymmetric weak focusing field, the average field index is the same as the instantaneous field index, n.

Koeth, Hine, Hoffman, Krutzler, Ponter, Rosenberg, Ruisard & Schneider, Comparison of Azimuthally Varying with Constant Gradient Magnetic Fields with the Rutgers 12-Inch Cyclotron (2011) — p. 4

Editorial note, tabletop extrapolation: Explains the tune shape the companion magnet study reported — near-zero νz to mid-radius, then a fast rise — and warns a designer who sizes a taper analytically: this machine's DESIGNED taper was a 2% droop (dg-1680), while the delivered profile falls ~12.5% by r = 5 inches because the pole-edge roll-off dominates the last stretch. Field index is the LOCAL derivative, not the accumulated drop — differentiate the measured profile to get n(r), and expect the edge, not the taper, to own the outer-radius focusing.

Figure 30. Average Bz as a function of radius for the AKG270 pole tips in the median plane.

Koeth, Hine, Hoffman, Krutzler, Ponter, Rosenberg, Ruisard & Schneider, Comparison of Azimuthally Varying with Constant Gradient Magnetic Fields with the Rutgers 12-Inch Cyclotron (2011) — p. 11

Editorial note, tabletop extrapolation: What a hybrid weak-focusing-plus-near-isochronous profile looks like in practice on a 12-inch pole, directly comparable with the same paper's weak-focusing profile (dg-1724): much flatter across the middle of the ion region. The design intent and its payoff — minimized slippage, the 6 kV-peak minimum dee voltage — are carried on their own cards (dg-1717, dg-1722). A flat average field approximates isochronism only in the nonrelativistic limit; a higher-energy design shapes ⟨B⟩ to track γ instead. Digitized values approximate.

For details beyond the scope of this document, coupled resonances between the transverse motions further limit the value of n. n values of 0.2, 0.25, 0.33, 0.5 (and others higher) need to be avoided. Since, the ions to be accelerated begin at r = 0, n = 0 and will only climb as the radius increases. If n = 0.2 needs to be avoided, then the rate at which Bz decreases must be moderated such that only near the maximum ion radius does n approach 0.2.

Koeth, Hanebuth, Schneider & Hoffman, Observation of Betatron Motion in the Rutgers 12-Inch Cyclotron (2006) — p. 2

Editorial note, tabletop extrapolation: The direct pole-tip taper criterion for a small weak-focusing machine: shape the taper so n approaches 0.2 only near maximum ion radius — under the author's stated premise of a profile whose n starts at 0 and only climbs. The resonance list (0.2, 0.25, 0.33, 0.5 and higher) is the author's claim, referred to Livingood for derivation, not a measurement from this machine; the field-index definition and 0 < n < 1 stability window are standard weak-focusing results stated here for context.

We recall from section II that the vertical betatron frequency follows the square root of the field index multiplied by the RF frequency: f_beta-vert = sqrt(n) f_0 We extend that relationship to their respective periods of oscillation: T_beta-vert = (1/sqrt(n)) T_0 Thus for a given n it take 1/sqrt(n) RF periods or ion revolutions to complete one vertical betatron oscillation.

Koeth, Hanebuth, Schneider & Hoffman, Observation of Betatron Motion in the Rutgers 12-Inch Cyclotron (2006) — p. 6

Editorial note, tabletop extrapolation: The hand calculation that tells a builder how many TURNS per vertical oscillation to expect: 1/√n turns (equal to RF periods only on fundamental-harmonic operation, h = 1, as here; at harmonic h it is h/√n RF cycles). With this machine's measured n ≈ 0.025-0.042 near 8.6-9.7 cm, that is about 4.9-6.3 turns per oscillation — a local estimate where n varies. The quote's equation glyphs are transcribed in plain-text form here. (The nu_x = sqrt(1-n) statement and the radial-stability remark are on p.2; the nu_z material is on p.6.)

The following measurements and modeling indeed confirm, at least in the assumption of azimuthal symmetry that our 12-inch magnet's field index runs from 0 to about 0.2 throughout the useful region for ion acceleration.

Koeth, Hanebuth, Schneider & Hoffman, Observation of Betatron Motion in the Rutgers 12-Inch Cyclotron (2006) — p. 3

Editorial note, tabletop extrapolation: Reference-machine geometry, not a target: on this 12-inch, maximum ion radius 5 inches sits an inch inside the 6-inch pole-tip edge, and the measured-and-modeled n runs 0 to about 0.2 across the acceleration region. Choose your own pole margin from magnetic modeling of your taper (the fringe rolls off inside the physical edge), and read 'about 0.2' as where THIS profile tops out — the design doctrine of keeping the 0.2 crossing near final radius is carried by dg-1729/dg-1835. The r = 5/6/8 inch markers are read from the Fig. 2 caption on the same page.

We then set the camera to a 15 second exposure and scanned the florescent screen slowly. The resulting image, Fig 10, clearly showed periodic behavior about the median plane with increasing frequency and decreasing amplitude as r increased. This was immediately identified as betatron motion.

Koeth, Hanebuth, Schneider & Hoffman, Observation of Betatron Motion in the Rutgers 12-Inch Cyclotron (2006) — p. 5

Editorial note, tabletop extrapolation: An almost free beam-dynamics diagnostic: a phosphor screen on a manual radial feedthrough plus a long-exposure camera through a viewport records vertical betatron structure across the scanned interval in one frame, no electronics. It is a QUALITATIVE record as taken; a tune number additionally needs calibrated radial coordinates and peak-spacing analysis (dg-1741 is this memo's own worked version).

The radial position of several peaks from the observed vertical betatron motion were recorded: ro = 8.6 cm (338keV) r1 = 9.2 cm (387keV) r2 = 9.6 cm (421keV) Relevant operating conditions: fo=14.8640 MHz (B = 0.977 Tesla) RF power = 300 Watts Magnet Current = 28.28 Amps … Precise “tuning” of the magnetic field yielded a peak beam current reading of 20nAmps.

Koeth, Hanebuth, Schneider & Hoffman, Observation of Betatron Motion in the Rutgers 12-Inch Cyclotron (2006) — p. 5

Editorial note, tabletop extrapolation: A calibrated benchmark set for the 100 keV-1 MeV band: field, frequency, radius, energy and beam current quoted together. The radius-energy pairs are internally consistent with E = q²B²r²/2m at B = 0.977 T (computed check: 8.6 cm gives 338 keV, 9.2 cm gives 387 keV, 9.6 cm gives 421 keV), so they can sanity-check another machine's energy bookkeeping.

Operating at 300 Watts of RF power on resonance at 14.8640 MHz (Corresponding to a B-field of 0.977 Tesla), the DEE Vp-p that develops is 7,500 V, thus ∆E is 7,500eV. Taking q=1.6E-19, and m=1.67E-27, so we can expect: ... ∆r(r) = (8.2E-5) 1/r

Koeth, Hanebuth, Schneider & Hoffman, Observation of Betatron Motion in the Rutgers 12-Inch Cyclotron (2006) — p. 6

Editorial note, tabletop extrapolation: The single most useful sizing formula for probe and cup design: turn spacing at any radius from dee voltage and field — it sets how thin an intercepting tip must be and whether turns separate on a screen. Conditions: nonrelativistic ions, approximately uniform B, small per-turn gain, with ΔE the effective energy gain per turn (this machine's single-dee convention takes it as the 7,500 V peak-to-peak; multiple gaps or off-crest phase change it). Caution: the printed substitution line shows the charge as (1.6E-27) in the denominator, a source misprint for 1.6E-19 (the text above states q=1.6E-19); recomputing with 1.6E-19 reproduces the printed 8.2E-5 coefficient.

Using a fourth order polynomial fit and our equation for ∆r(r) we can create table 1. The first measured peak of the vertical betatron oscillation was at ro = 8.6cm, and we denote that as the start of the betatron period. We then allow one RF period, hence one ion revolution, to process, after which, using our equation for ∆r(r), we reevaluate the new radius and that radius' field index n. It can easily be shown that the fraction that the betatron period advances at a given n is just sqrt(n). … Integer values of fractional betatron periods indicate the full completion of a vertical betatron oscillation. … Noting the radii at which these occur the reader immediately sees the same values that were observed at r1 and r2 as reported in section V.

Koeth, Hanebuth, Schneider & Hoffman, Observation of Betatron Motion in the Rutgers 12-Inch Cyclotron (2006) — p. 6

Editorial note, tabletop extrapolation: A worked piecewise-tracking recipe implementable in a spreadsheet — no orbit code — and checked by its authors against the streak photo: integer betatron periods land at the observed r1 and r2. The tabulated n values over 8.6-9.7 cm run 0.025 to 0.042 (Table 1, read from the rendered page). Table 1 prints 9.2 cm in two consecutive rows (n = 0.031 and 0.033) — most likely rounding of nearby unrounded radii rather than a misprint.

Radial betatron oscillations were not noticed as their period in the regime of low field index n is comparable to that of the ion revolution frequency.

Koeth, Hanebuth, Schneider & Hoffman, Observation of Betatron Motion in the Rutgers 12-Inch Cyclotron (2006) — p. 7

Editorial note, tabletop extrapolation: Expectation-setting for a weak-focusing machine: at small n, νx = √(1−n) is near 1, so radial betatron structure barely advances per turn and hides in a fixed-azimuth screen view — this memo saw none. It is a visibility statement, not an absence: turn-resolved diagnostics or the slow 1−νx beat can still expose radial motion (the program's later precession work, dg-1840, is exactly that physics put to use).

The natural question that should be asked: if the ion source aperture is in the median plane, why then the large vertical displacement? This can be attributed to the early ions sensitivity to any vertical component of the electrical field. Inspection of Fig 5 shows that the electric field from the ion source into the DEE diverges quickly. Thus a slight offset of the DEE with respect to the median plane will provide a significant vertical component. This can be mitigated by the installation of "pullers" on the DEE's aperture in the region of the ion source – a possible student project.

Koeth, Hanebuth, Schneider & Hoffman, Observation of Betatron Motion in the Rutgers 12-Inch Cyclotron (2006) — p. 7

Editorial note, tabletop extrapolation: Why a median-plane source aperture still launches vertically displaced beam: in the central source-to-dee region the extraction field diverges strongly, so any dee offset from the median plane hands the earliest ions a vertical kick. No tolerance number is given — the source's stated remedy is pullers on the dee aperture near the source (offered as a possible student project, not a demonstrated fix); tightening dee-to-median-plane alignment is the natural corollary a builder draws, not the source's measured mitigation.

Radial sectors pole-tips providing the so-called Thomas focusing [2] have been built but the beam could not be accelerated up to the deflector radius due to poor isochronicity. ... To successfully accelerate the beam up to the chamber's radius a new set of pole-tips was designed [5], at the same time providing additional focusing using a spiral sector design. ... An iterative design phase using a field solver and ion tracking lead to the machining of 270°spiral pole-tips

Hernalsteens, Ponter, Beaudoin, Koeth, Ruisard & Miller, Betatron Tune Characterization of the Rutgers 12-Inch Cyclotron for Different Magnetic Poles Configurations (2016) — p. 2

Editorial note, tabletop extrapolation: A documented negative result at exactly this scale: plain radial sectors on a small cyclotron can cost enough isochronism to prevent reaching full radius. If a tabletop builder wants AVF focusing, this collection's experience points to spiral sectors designed with a field solver plus tracking, not radial sectors alone. (The Archimedean-spiral / four-fold-symmetry statement is on p.1.)

The edge-focusing adds a term to νz2 depending on the "flutter" (mean square deviation of B(θ) ... where <B> is the θ-averaged axial magnetic field. ... The spiraling changes the edge crossing angles and the sector focusing contribution to the axial tune becomes ν2sector = F2 (1 + 2tan2 ε) where F is defined in Eq. 3 and ε is the spiral angle.

Hernalsteens, Ponter, Beaudoin, Koeth, Ruisard & Miller, Betatron Tune Characterization of the Rutgers 12-Inch Cyclotron for Different Magnetic Poles Configurations (2016) — p. 2

Editorial note, tabletop extrapolation: The minimum analysis needed to turn a measured or simulated AVF field map into a predicted axial tune for a tabletop machine. The quoted line reflects the PDF's text-layer rendering of typeset superscripts; the equation as set on the page is nu_sector^2 = F^2 (1 + 2 tan^2 epsilon).

The design of the source, a cold cathode Penning Ion Gauge (PIC) source, is reported in Ref. [4]. The aperture is circular with a 0.8mm radius and it can sustain a current of 5mA. A modified source chimney featuring an aperture offset along the vertical axis was built in order to provide a beam with an initial axial offset.

Hernalsteens, Ponter, Beaudoin, Koeth, Ruisard & Miller, Betatron Tune Characterization of the Rutgers 12-Inch Cyclotron for Different Magnetic Poles Configurations (2016) — p. 1

Editorial note, tabletop extrapolation: A spare chimney with a deliberately off-median aperture is a simple, purpose-built way to launch coherent axial oscillations for tune studies — the launch half of the measurement. Extracting a tune still needs adequate transmission and a diagnostic that resolves the oscillation turn by turn (here, the phosphor radial probe). The source prints the acronym "(PIC)" where "PIG" is standard; quote transcribed as printed.

The best linear fit in the measurement range reads νa = (0.086 ± 0.001) + (0.0012 ± 0.001) · (r − 45), where r is the radius expressed in millimeters in the range 45 to 80mm. The 90 % confidence interval is also shown revealing that the measurement resolution is sufficient to confirm the observed linear trend. ... The equation of the fit of the magnetic results (in the beam based measurement range) reads νa = (0.088 ± 0.004) + (0.0015 ± 0.0002) · (r − 45).

Hernalsteens, Ponter, Beaudoin, Koeth, Ruisard & Miller, Betatron Tune Characterization of the Rutgers 12-Inch Cyclotron for Different Magnetic Poles Configurations (2016) — p. 3

Editorial note, tabletop extrapolation: A validated model-versus-measurement pair for a weak-focusing machine in the target class: this machine's field map predicted its beam's axial tune within the measurement errors. For THIS field the fits put νz ≈ 0.09–0.13 over 45–80 mm (n ≈ 0.008–0.02) — comfortably below the n = 0.2 Walkinshaw coupling band that this collection's weak-focusing rules treat as the ceiling (dg-003, dg-138); another machine's margin comes from its own n(r), not these numbers. The printed slope uncertainty (±0.001 on a slope of 0.0012) is nearly as large as the value and looks like a source misprint given the stated 90% confidence in the trend.

It is interesting to note that the measurement technique that we use readily provides a turn-by-turn envelope beating information. This is contrasting the usual case of synchrotrons where that kind of data is not as easily accessible. This provides a second and independent mean of measuring the betatron tune. Indeed it is well known that the envelope beating signal has a frequency which is two times the betatron tune. ... Within the measurement errors the envelope-based result matches very well the centroid-based tune values.

Hernalsteens, Ponter, Beaudoin, Koeth, Ruisard & Miller, Betatron Tune Characterization of the Rutgers 12-Inch Cyclotron for Different Magnetic Poles Configurations (2016) — p. 3

Editorial note, tabletop extrapolation: A free cross-check for a machine already taking streak images: the envelope beats at 2ν, so fitting its modulation gives a second, independent tune number to compare with the centroid fit. One caution the source's comparison sidesteps: with once-per-turn sampling the 2ν component can alias, so fit it modulo the turn frequency and use the centroid tune (or an expected range) to unwrap before halving.

This solution only delivered a few nanoamps of current at the outer edge of the chamber.

Ruisard, Hine, Koeth & Rosenberg, The Rutgers Cyclotron: Placing Student's Careers on Target — WE1PB02, Proceedings of Cyclotrons2013 (2013) — p. 2

Editorial note, tabletop extrapolation: PDF p.2 = printed p.292 (the "dramatically increase deliverable beam current" phrase is on p.1). The easiest thing to machine — flat, parallel, precision-ground poles — is a documented failure mode at this scale: with a purely vertical field there is no axial restoring force, and this machine delivered only a few nanoamps to the chamber edge until a slight radial taper was cut. What another machine gets from flat poles depends on its own alignment, apertures and source; the transferable instruction is to evaluate axial tune and transmission from your own field map, expecting roughly this fate without a gradient.

Coupling resonances further restrict 0 < n < 0.2. In the existing tips, n = 0.2 occurs beyond the deflector radius.

Ruisard, Hine, Koeth & Rosenberg, The Rutgers Cyclotron: Placing Student's Careers on Target — WE1PB02, Proceedings of Cyclotrons2013 (2013) — p. 2

Editorial note, tabletop extrapolation: PDF p.2 = printed p.292. The pole-tip acceptance criterion under the ideal azimuthally-symmetric weak-focusing model (νr = √(1−n), νz = √n, so νr = 2νz at n = 0.2): do not just satisfy 0 < n < 1 — shape the taper so n stays under 0.2 out to the last useful radius, as this machine's tips do (n = 0.2 beyond the deflector radius). Confirm on the actual field map with a tune or tracking analysis; azimuthal variation, fringes and errors move the real resonance picture.

As predicted via simulation, phase slippage at standard DEE voltage (8 kV) was so severe that ions were not delivered to the deflector.

Ruisard, Hine, Koeth & Rosenberg, The Rutgers Cyclotron: Placing Student's Careers on Target — WE1PB02, Proceedings of Cyclotrons2013 (2013) — p. 2

Editorial note, tabletop extrapolation: PDF p.2 = printed p.292. A cautionary data point for anyone tempted by straight radial-sector AVF tips: on this machine the phase slippage was fatal at 8 kV on the dee — and, holding the same field-frequency mismatch and final radius, a machine with LESS energy gain per turn takes more turns and accumulates more slip, so a low-voltage build should expect this failure mode to bite harder, not softer. Check isochronism in the tracker before cutting sectored steel (the spiral redesign that followed is dg-1745's story).

This form is convenient for sectors defined by an Archimedean spiral, r = aθ^(1/n), for which tan ξ = dθ/dr.

Ruisard, Hine, Koeth & Rosenberg, The Rutgers Cyclotron: Placing Student's Careers on Target — WE1PB02, Proceedings of Cyclotrons2013 (2013) — p. 3

Editorial note, tabletop extrapolation: PDF p.3 = printed p.293 (the exponent 1/n is printed as a superscript; the quote transcribes it inline). The design equation for trading spiral tightness against vertical tune before cutting steel — with one correction applied: as printed, tan ξ = dθ/dr is not dimensionless; the standard edge-angle relation is tan ξ = r·dθ/dr, which for the stated Archimedean spiral evaluates to nθ. The flutter and approximation conventions are the paper's; verify against Livingood (its own ref [10]) before using the tune expression numerically.

The ideal average field profile decreases with radial distance from the center before flattening out at larger radii … This is necessary to provide weak focusing at the central region, where flutter is negligible. High flutter values were also desirable, to increase the vertical tune.

Ruisard, Hine, Koeth & Rosenberg, The Rutgers Cyclotron: Placing Student's Careers on Target — WE1PB02, Proceedings of Cyclotrons2013 (2013) — p. 3

Editorial note, tabletop extrapolation: PDF p.3 = printed p.293. The most useful shaping rule in this paper for a small AVF attempt: flutter is essentially zero on axis, so the central region must still weak-focus — the falling inner profile is not optional — and the flat outer region approximates isochronism only in the low-energy nonrelativistic sense (exact fixed-frequency isochronism wants the orbit-averaged field rising as γ; immaterial at this machine's energies, material by 20 MeV). Fig. 4 shows the resulting bump-plus-flat profile for the chosen 270-degree spiral, whose caption marks the isochronous region.

Multiple off-center stable orbits were found at particle energy 250 keV (nominal r=2.75”). … In Fig. 7, four stable fixed points can be seen surrounding the central fixed point. … The off-center islands quickly disappear at higher energies, and seem to have little effect on particle motion/stability during acceleration.

Ruisard, Hine, Koeth & Rosenberg, The Rutgers Cyclotron: Placing Student's Careers on Target — WE1PB02, Proceedings of Cyclotrons2013 (2013) — p. 4

Editorial note, tabletop extrapolation: PDF p.4 = printed p.294. Warns an AVF builder that a low-sector-count spiral geometry can grow parasitic off-center equilibrium orbits at intermediate energy (four of them here at 250 keV, matching the four-fold symmetry). The source's own hedged report: the islands "quickly disappear at higher energies, and seem to have little effect" during acceleration. Practical consequence: a beam that looks mis-steered at mid-radius may be sitting on an island — check with turn-by-turn tracking rather than assuming detrapping is clean.

Additional SIMION studies identified 6 kV peak voltage and 15.534 MHz frequency as the optimal working point for proton transport.

Ruisard, Hine, Koeth & Rosenberg, The Rutgers Cyclotron: Placing Student's Careers on Target — WE1PB02, Proceedings of Cyclotrons2013 (2013) — p. 4

Editorial note, tabletop extrapolation: PDF p.4 = printed p.294. Shows a tabletop-scale RF operating point being chosen from tracking rather than by trial: a dee-voltage/frequency pair reported to the nearest kilohertz (15.534 MHz) with its 6 kV partner. The transferable practice: the tracker picks the working point before the machine is fired, and the optimum is jointly a voltage AND a frequency. The pair itself belongs to this field map — re-derive yours from your own model.

This serendipitously provided a direct measure of the turn-to-turn (nth to nth+1) phase shift at a given radius.

Gonski, Burcher, Lazarov, Krutzler, Koeth & Beaudoin, A Novel Optical Method for Measuring Beam Phase and Width in the Rutgers 12-Inch Cyclotron — WE1PB04, Proceedings of Cyclotrons2013 (2013) — p. 2

Editorial note, tabletop extrapolation: PDF p.2 = printed p.300. Tabletop machines have small turn-to-turn spacing, so a partially intercepting probe is common — a double-peaked signal should raise the adjacent-turns hypothesis early, tested cheaply by changing probe insertion depth or plate size and watching whether separation and relative amplitude respond as turns would (species content, radial oscillations and bunch structure can also double a peak). Rutgers deconvoluted the two peaks with a dual-Gaussian fit and took the more intense (n+1) peak as the turn of interest.

we observe a tendency for bunch length to decrease with a rising magnetic field.

Gonski, Burcher, Lazarov, Krutzler, Koeth & Beaudoin, A Novel Optical Method for Measuring Beam Phase and Width in the Rutgers 12-Inch Cyclotron — WE1PB04, Proceedings of Cyclotrons2013 (2013) — p. 3

Editorial note, tabletop extrapolation: PDF p.3 = printed p.301 (Table 1; the 91 mm probe radius and ~100 keV termination energy are on PDF p.2). Rare published bunch-length numbers for a tabletop machine at almost exactly the 100 keV end of the target class: a beam tens of RF degrees long, with the measured means falling as field rises — the source states this as a tendency. The tabulated ±0.03 T is comparable to the spacing between the three field values; if that uncertainty were independent per row the settings would barely be distinguishable, so either it is largely common-mode (calibration) or ±0.003 T was intended — an unverified hypothesis, reported here as such with the printed value preserved.

We find that operating below the nominal magnetic field increased the turn-to-turn phase slippage.

Gonski, Burcher, Lazarov, Krutzler, Koeth & Beaudoin, A Novel Optical Method for Measuring Beam Phase and Width in the Rutgers 12-Inch Cyclotron — WE1PB04, Proceedings of Cyclotrons2013 (2013) — p. 2

Editorial note, tabletop extrapolation: PDF p.2 = printed p.300 (the linear fit is Fig. 5, PDF p.3 / printed p.301). Practical tuning guidance: on this machine, field trim and RF phase budget were one knob, with an approximately linear response over the measured 0.498–0.566 T and a definite sign — below nominal costs phase. On another machine, run the same local field scan (or a trajectory model) to get the slope and sign; the linearity is an observation over this range, not a law.

This error is acceptable, as long as the overall integrated phase slippage is less than 90°.

Gonski, Burcher, Lazarov, Krutzler, Koeth & Beaudoin, A Novel Optical Method for Measuring Beam Phase and Width in the Rutgers 12-Inch Cyclotron — WE1PB04, Proceedings of Cyclotrons2013 (2013) — p. 1

Editorial note, tabletop extrapolation: PDF p.1 = printed p.299. The governing constraint for any non-isochronous tabletop machine, under the source's convention: keep the integrated phase slippage inside the source's 90-degree budget, remembering the whole phase TRAJECTORY matters — a net-zero final slip does not save a beam that left the accelerating window mid-flight. Low dee voltage hurts twice (more turns against the same budget), and the deliberate start-above-nominal-field trick is best read as centering the phase excursion, not as a free correction.

The magnet’s upper and lower coils are independently energized for intentional field imbalance so as to shift the median plane. … Figure 6 shows three standard radial-draw beam images: the left frame top/bottom coil at 17/12 amps, the middle frame at 14.5/14.5 amps, and the right frame at 12/17 amps.

Koeth, Beam Physics Demonstrations with the Rutgers 12-Inch Cyclotron — WEPPT025, Proceedings of Cyclotrons2013 (2013) — p. 3

Editorial note, tabletop extrapolation: PDF p.3 = printed p.371 (design intent on PDF p.1 / printed p.369). A genuinely cheap axial-steering mechanism for a small machine: energize the two coils independently and trim the median plane. On this machine a 5 A top-to-bottom imbalance about the 14.5/14.5 A balance point still brought beam to the periphery — a demonstration that the knob has useful range, with transmission, centering and beam quality at each setting still to be measured on any machine that copies it.

The cyclotron chamber’s position can be moved horizontally with respect to the magnet’s center. … Initial ion radial-position errors can be introduced by a horizontal offset of the chamber, and hence ion source, with respect to the magnet center.

Koeth, Beam Physics Demonstrations with the Rutgers 12-Inch Cyclotron — WEPPT025, Proceedings of Cyclotrons2013 (2013) — p. 1

Editorial note, tabletop extrapolation: PDF p.1 = printed p.369 (purpose stated on PDF p.2-3 / printed p.370-371). An unusual design freedom: the chamber (and hence source) translates horizontally with respect to the magnet center, and the same adjustment that centers the source doubles as the deliberate-error knob for radial betatron studies. Copy it as a constrained, lockable, measurable translation — an unlocated chamber is not the feature; a controlled offset is — and remember one move shifts source, dees and probes together.

Values of n=0.2, 0.36, 0.5 (and others yet higher) need to be avoided.

Koeth, Beam Physics Demonstrations with the Rutgers 12-Inch Cyclotron — WEPPT025, Proceedings of Cyclotrons2013 (2013) — p. 2

Editorial note, tabletop extrapolation: PDF p.2 = printed p.370. The explicit forbidden-n list a weak-focusing pole-tip designer rarely sees written down: inside 0 < n < 1, the taper must also avoid 0.2 (Qx = 2Qz), 0.36 and 0.5. A well-chosen profile keeps n below 0.2 for nearly the whole acceleration (the source's own following prescription); the design questions are where n(r) crosses what, and how fast — compute or map n(r) rather than assuming which resonances are in play.

Since the ions begin their spiral journey at r=0 necessarily n also starts at 0, and will only climb as the radius increases; if n=0.2 is to be avoided (Qx=2Qz), then the rate at which Bz decreases must be moderated such that n=0.2 only near the final ion radius.

Koeth, Beam Physics Demonstrations with the Rutgers 12-Inch Cyclotron — WEPPT025, Proceedings of Cyclotrons2013 (2013) — p. 2

Editorial note, tabletop extrapolation: PDF p.2 = printed p.370. The actionable pole-taper prescription for a small weak-focusing machine, on the source's own premise that n starts at 0 and climbs with radius: moderate the fall-off so n = 0.2 arrives only near the final radius. A taper aggressive enough to buy strong axial focusing early reaches the coupling resonance early, and time spent near it with any driving asymmetry risks resonant amplitude growth — Rutgers built a deliberately bad pole set to demonstrate exactly that (dg-1841).

Small axial motion is initiated by a vertical electric field that imparts an upward kick to the ions immediately upon their exit of the chimney.

Koeth, Beam Physics Demonstrations with the Rutgers 12-Inch Cyclotron — WEPPT025, Proceedings of Cyclotrons2013 (2013) — p. 2

Editorial note, tabletop extrapolation: PDF p.2 = printed p.370. A controlled way to excite vertical motion for diagnosis rather than waiting for it to appear as a fault: an intentional electric kick at the chimney exit launches the oscillation, which a turn-resolving diagnostic (here, the radial-draw phosphor image) then converts into a tune number. The caution reads in reverse too: a stray vertical field near the source will do the same thing uninvited.

Thus for a given n it takes 1/√n ion revolutions to complete one vertical betatron oscillation

Koeth, Beam Physics Demonstrations with the Rutgers 12-Inch Cyclotron — WEPPT025, Proceedings of Cyclotrons2013 (2013) — p. 2

Editorial note, tabletop extrapolation: PDF p.2 = printed p.370. What converts a photograph into a number: count N revolutions between same-phase vertical maxima and νz ≈ 1/N — and, under the smooth azimuthally-symmetric weak-focusing approximation, n ≈ 1/N². It is an average over the interval, not a point measurement. On this machine's gently tapered poles νz ≈ 0.09 mid-radius (Fig. 3a), i.e. about 11 turns per oscillation, comfortably resolvable on its radial-draw images; treat that as Rutgers calibration context, not a class-typical value.

Figure 3a is a SIMION simulation of ions crossing a radial reference plane in our “good” poletips’ WF field, showing the beam coming to a focus near the DEE edge, where n=0.2. … The increased frequency of the axial oscillation with radius is a display of the growing field index, n. Both a) and b) exquisitely demonstrate adiabatic damping … The reader can estimate from Fig. 3a that Qz≈0.09 at r=65 mm.

Koeth, Beam Physics Demonstrations with the Rutgers 12-Inch Cyclotron — WEPPT025, Proceedings of Cyclotrons2013 (2013) — p. 2

Editorial note, tabletop extrapolation: PDF p.2 = printed p.370 (Fig. 3a axes: axial height ±6 mm versus radius 0-110 mm). A concrete worked example, not a class expectation: this machine's good weak-focusing set runs νz ≈ 0.09 mid-radius with millimetre-scale axial excursion, the oscillation frequency rising with radius as n grows, and both simulation and photograph showing adiabatic damping. Damping means the beam tightens vertically as it gains energy — which ARGUES the vertical acceptance question is decided early, near the source; verify it by tracking or measuring the envelope over the full radius, since apertures, field errors and resonances can still bite downstream.

Since Qx(r=0) begins at 1, any radial offset simply displaces the equilibrium orbit by the same. As the ions gain energy and spiral towards larger radii, Qx(r) begins to drop, causing the location of maximum radial displacement to azimuthally process. This continues until a tight inter-turn bunching occurs on one side of the machine while large turn to turn spacing develops on the other, as shown in Fig. 4. Historically this has been exploited to increase extraction efficiency by placing the septum between turns.

Koeth, Beam Physics Demonstrations with the Rutgers 12-Inch Cyclotron — WEPPT025, Proceedings of Cyclotrons2013 (2013) — p. 2

Editorial note, tabletop extrapolation: PDF p.2-3 = printed p.370-371 (the turn separation is photographed in Fig. 4). Directly useful to a small-machine builder attempting extraction: a deliberate radial offset makes the azimuth of maximum displacement precess as Qx falls, concentrating turns on one side and opening turn-to-turn gaps on the other — historically where the septum goes. Choose the azimuth by orbit tracking and low-current measurement; and note that on this machine the offset comes from translating the whole chamber, which moves dees and probes with it — offsetting the source alone is the finer instrument.

The n=0.2 point occurs at r=3.5 inches, well within the 5 inch DEE radius, so as to allow the ion displacement to grow.

Koeth, Beam Physics Demonstrations with the Rutgers 12-Inch Cyclotron — WEPPT025, Proceedings of Cyclotrons2013 (2013) — p. 3

Editorial note, tabletop extrapolation: PDF p.3 = printed p.371. The inverse of a design rule and the most instructive demonstration here: a taper whose n = 0.2 point lands at 3.5 inches instead of near the 5-inch dee edge converted a working configuration into one that grows axial displacement — and in the reported simulation and experiment the growth fed on a 3 mm chamber-to-magnet offset (the difference resonance bounds axial amplitude by the initial radial offset). What transfers is the mechanism and the method — locate n = 0.2 on the measured map, track orbits through it — not a fabrication tolerance or a universal seed threshold.

Uniformity of the magnetic field is extremely precise. The surfaces of the two poles are parallel with 0.001 inches. As will be seen later, the poles and yoke are slightly deflected due to the extreme pull of the electromagnetic force at high fields. No shimming of the magnetic field to increase the beam current has been attempted yet, however there are future plans to do so.

Koeth, The Construction and Operation of a Nine Inch Cyclotron (undated scan; the machine ran 1995–1999) — p. 2

Editorial note, tabletop extrapolation: What the reference machine did: pole faces parallel within 0.001 inch, no shimming attempted (a stated future plan), and beam achieved — an existence proof that this machine's field, as machined, sufficed for its ~184 keV operation. It is one machine's outcome, not a tolerance spec: map the assembled field under excitation (including the deflection under magnetic load the author himself flags, dg-1862) and let beam-dynamics requirements decide whether machining or shims are owed.

It is worth noting that even under maximum ion current conditions of 50 nanoamps no beam loading was noticed.

Koeth, The Construction and Operation of a Nine Inch Cyclotron (undated scan; the machine ran 1995–1999) — p. 5

Editorial note, tabletop extrapolation: A useful separation-of-concerns datum: at 50 nA this machine saw no detectable beam loading, so RF tuning and beam tuning decoupled cleanly — expect the same at nanoamp-class currents, as a practical matter rather than a law (loading scales with current and energy gain, and a sensitive enough RF measurement might resolve it). The corollary stands: at these currents beam loading is useless as a diagnostic; a collector is the instrument.

Further more, because of the very strong magnetic field parallel to the desired electron path, strong focusing occurs. Any electron that attempts to stray off of a vertical ascent or decent is immediately steered back towards the central axis of motion. Due to this corrective focusing, the electrons tend to oscillate back and forth in both X and Y while traveling downward in Z. Instead of following a linear path, the traversal then becomes a helical path with a very tight radius. … The electric fields of the ion source and accelerating RF sweep away the freed hydrogen electrons, leaving the massive protons behind.

Koeth, The Construction and Operation of a Nine Inch Cyclotron (undated scan; the machine ran 1995–1999) — p. 5

Editorial note, tabletop extrapolation: Why a crude filament-across-the-gap source works at all inside a cyclotron: the ~0.9 T field pins the ionizing electrons to tight helices about their field lines (the source describes this as steering back toward the central axis — strictly, gyration confines each electron about its own line rather than restoring it to a common axis), forming a thin ionizing sheet through the median plane right where ions must be born, with — the source's own statement — the ion-source and RF electric fields sweeping the freed electrons away. Corollary, scoped: emission, heating and vacuum behaviour bench-test fine outside the magnet; the magnetized TRANSPORT that makes the geometry work does not, so beam-relevant performance is a property of source plus field together.

Fig.9 shows the exponential emission of electrons in a test of the W-Th-Ir material. Hence slight changes in the filament current can produce great changes in thermionic emission. Ultimately changing the number of thermionic electrons available to ionize the hydrogen. In this way the cyclotron proton beam current can be controlled. As of yet the limiting factor in the maximum achievable beam current at the periphery is due to filament heating limitations.

Koeth, The Construction and Operation of a Nine Inch Cyclotron (undated scan; the machine ran 1995–1999) — p. 6

Editorial note, tabletop extrapolation: The practical control law: filament current is the beam-current knob, and the response is STEEP — Fig. 9's test filament went from essentially nothing below 4.5 A to ~3 mA emission at 5.0 A (read from the rendered figure, bias −300 VDC; the operating optimum on p.5 is −320 V). The physics under it is Richardson-Dushman: emission exponential in inverse temperature, temperature a nonlinear function of current — hence fine adjustment and a stable, monitored supply (current regulation is the natural choice; what matters is stable emission, however achieved). The author names filament heating as the beam-current limiter of record.

It is mounted such that the collector can be inserted radialy with a two inch travel, effectively determining the maximum ion radius. The minimum measurable ion radius, maximum insertion of the collector is 2.50 inches while the maximum ion radius, minimum collector insertion is 4.50 inches. A plot of beam current against radius, Fig.10, shows that the beam current linearly drops off as the radius grows.

Koeth, The Construction and Operation of a Nine Inch Cyclotron (undated scan; the machine ran 1995–1999) — p. 7

Editorial note, tabletop extrapolation: The cheapest radial beam-profile monitor a small cyclotron can have: the movable collector doubles as the radius-defining aperture, so one linear feedthrough yields current-versus-radius — an INVASIVE measurement, with energy then inferred from radius and the calibrated field rather than selected. On this run, collected current fell about eightfold from ~16 nA near 2.6 in to ~2 nA at 4.5 in (read from the rendered Fig. 10) — this machine's outer-turn attrition under its own source and pressure conditions, a shape to expect, not a universal loss factor. ("radialy" as printed.)

In run 91699C the resonant frequency was tuned to 13.590 MHz. Other parameters for run 91699C are listed below: fr 13.590 MHz / Forward RF Power 16 Watts / Theoretical B-field 0.889 Tesla / H2 Pressure in tank 5.1E-5 Torr / Filament Current 5.75 Amps / Filament Voltage 5.0 Volts / Filament Bias -320 Volts / Filament Emission 21.0 microAmps / Max. Ion Radius 7.0 cm / Max. Ion Energy 184 keV

Koeth, The Construction and Operation of a Nine Inch Cyclotron (undated scan; the machine ran 1995–1999) — p. 8

Editorial note, tabletop extrapolation: The single most valuable calibration point in the wave for a 100 keV-1 MeV tabletop design - a complete achieved operating point, not a design target. The energy is internally consistent: with B = 0.885 T (the measured peak) and r = 0.070 m, E = (qBr)^2/(2m) computes to 184 keV, matching the printed value. Note the whole machine ran on 16 W of RF and 21 microamps of filament emission. The forward slashes in the quote separate table rows; the microamp symbol is printed as a Greek mu.

The measured ion peak at 0.885 Tesla tightly corresponded with the theoretical value to 0.6%. ... However an unexpected peak at 0.449 Tesla developed. ... After an investigation into the matter, it was determined that indeed singly charged protons were being accelerated. ... the RF frequencies required for acceleration of the ions at the low magnetic fields, developed from excitation of higher frequency modes of oscillation in the tank circuit. ... If the applied frequency were double that of the fundamental, on it's second crossing of the gap the ion would receive a de-acceleration, thus gaining zero net acceleration. However, if the RF frequency were triple that of the fundamental it is seen that the electric field direction is again in sync with ion's travel. This effect holds true for any odd multiple of the fundamental cyclotron frequency.

Koeth, The Construction and Operation of a Nine Inch Cyclotron (undated scan; the machine ran 1995–1999) — p. 8

Editorial note, tabletop extrapolation: The most instructive diagnostic story in the document, with one open number. A builder ramping the magnet while watching a collector WILL see spurious low-field peaks and will suspect contaminant species; this source traced its extra peak to the RF tank ringing on harmonic modes, protons confirmed. Unresolved, computed here: 0.449 T is almost exactly half of 0.885 T — at fixed drive frequency that is an even multiple of the ion's fundamental, which the source's own two-crossing argument says gives zero net acceleration; a third-harmonic peak would sit near 0.295 T. So the qualitative lesson (check the RF spectrum and recompute candidate resonances via B = 2πmf/(qh) before blaming ion species) stands, while this particular peak's mechanism needs a nonideal ingredient the source does not supply. The author's reported consequences — harmonic operation sharpens the field peak; suppressing tank harmonics improves efficiency — were stated intent, not achieved results. Ellipses mark omitted intervening text.