Design Guide › level 1
Cyclotron design rules, level 1: fundamental
42 of the guide’s 1878 rules sit at level 1: fundamental — governs the feasibility of any cyclotron.
The level ranks how early and how universally a rule binds a cyclotron design — breadth,
never weight. It is not permission to skip a rule whose trigger a machine has, and safety
rules are never skippable on level alone; how the levels were assigned and audited is on
the methodology page. Each rule keeps its
formula where the source gives one, a verbatim quote, a page-level citation, and a stable
identifier (dg-NNNN) that resolves here, on its
subsystem page, and on the
all-in-one guide. Where an editorial note says “the
reference machine”, its parameters are on the
guide’s front page.
This level’s rules by subsystem — each link opens just that subset, in the all-in-one guide’s filters: Beam dynamics (23) · Magnet (17) · RF (11) · Cyclotron general (9) · Project management (4) · Materials (3) · Modeling (3) · Pedagogy (3) · Physics theory (3) · Dee (2) · Fabrication (2) · Ion source (2) · Safety (2) · Vacuum (2) · Vacuum chamber (1) · Extraction (1) · Shielding (1). A rule carrying several tags is counted under each; a subsystem’s complete rule set, across all levels, is on its own page in the subsystem directory. To add a search term or a second subsystem, open this level in the all-in-one guide with filters, which carries every rule and filters in the browser.
Verify before use. Every rule here is a source extract in the vocabulary of the editorial methodology — faithful to its cited page, not an independently validated engineering requirement. Re-read any rule that drives a real design decision at the cited page before committing metal, money, or high voltage to it. The editorial note under each quote is this site’s extrapolation to a tabletop machine, not something the source said: an editor’s judgement, audited for overreach, never a citation.
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Compute achievable proton energy as T(MeV) = 3.12e-4 x B^2(kilogauss) x R^2(inches), where R is the radius of usable UNIFORM field, not the physical pole radius.
T(MeV) = 3.12e-4 * B^2(kG) * R^2(in) for protons; 1.56e-4 for deuteronsSource quote & editorial note
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.
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Keep the field index n = -(r/B)(dB/dr) between 0 and 1 everywhere ions circulate; both axial and radial oscillations are stable only in this band.
B = B0*(r0/r)^n (constant-n form); stability requires 0 < n < 1; f_axial = sqrt(n)*f0, f_radial = sqrt(1-n)*f0Source quote & editorial note
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
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Keep the magnetic circuit out of saturation: the source's 1060 steel saturates around 1.7 T, above which they treat further excitation as wasted, so their design keeps peak fields in the iron under about 1.6 T.
B_local(iron) < B_sat; B_sat(1060 steel) ~ 1.7 T (alloy- and treatment-dependent)Source quote & editorial note
Our magnet is constructed out of 1060 steel, which saturates at around 1.7 T; above this magnetic flux density the yoke is unaffected by further excitation.
Heuer & Baumgartner, Design of a 2 MeV Cyclotron (2009) — p. 6, 29
Editorial note, tabletop extrapolation: The binding quantity is local flux density in the iron (narrowest yoke section, pole roots), not the gap field: at 0.59 T in the gap the reference machine is far from saturation everywhere, but a next machine pushing the gap past ~1.5 T must check each cross-section of the return path against its own steel's saturation curve - saturation onset is gradual and alloy-dependent, not a hard wall at 1.7 T.
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Magnetic pressure is B^2/(2*mu0) - attractive along field lines, repulsive normal to them - and at 0.5 T it is already ~99.5 kPa = 14.4 psi, about one atmosphere pulling the poles together.
P = B^2/(2*mu0); 0.5 T -> 99,472 N/m^2 ~ 1 atmSource quote & editorial note
pressure @ 0.5T 99,472 Newton/m2... ~ 1 atmosphere
Editorial note, tabletop extrapolation: At the reference machine's 0.59 T the poles attract with ~1.4 atm over the 8-inch pole face — about 4,500 N (~1,000 lbf); clamps and any pole-retraction scheme must carry that load (chamber lids carry the separate atmospheric load — see the lid-deflection calculator). [Corrected 2026-08-20: previously printed as "~4500 lbf", the newton value mislabeled.]
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Start every cyclotron magnet from the rigidity relation B*rho = sqrt(T^2 + 2*T*E0)/(300*Z) (B in tesla, rho in m, T and rest energy E0 in MeV) to fix the field-radius product before any geometry is drawn.
B*rho = sqrt(T^2 + 2*T*E0)/(300*Z)Source quote & editorial note
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).
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Ferromagnetic materials lose their advantage above their saturation field (typically ~2 T): incremental permeability falls toward 1, so added excitation buys little more than it would in an air-core coil - the reason iron-dominated designs stay below saturation.
mu_r -> 1 as B approaches saturation (typically ~2 T)Source quote & editorial note
ferromagnetic materials lose their advantages above their saturation field (typically 2 T).
Humphries, Principles of Charged Particle Acceleration (1986) — p. 104, 108
Editorial note, tabletop extrapolation: Sets the practical scale of the iron-magnet approach for a next machine: above the saturation region, further field comes almost entirely from added ampere-turns at air-core rates - which is why higher-field machines move to superconducting coils. Below about 1.5 T the iron does most of the work.
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Shape the magnet for field index 0 < n < 1 through the beam region - the weak-focusing band the source's bending magnets were shaped to: n > 0 gives vertical focusing, n < 1 keeps radial focusing.
0 < n(r) < 1; nu_r = sqrt(1-n), nu_z = sqrt(n) (azimuthally symmetric weak-focusing model)Source quote & editorial note
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.
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Non-relativistic cyclotron energy is Tmax[MeV] = 48*(Z*R[m]*B[T])^2/A - energy scales as the square of both field and radius.
Tmax[MeV] = 48*(Z*R*B)^2/ASource quote & editorial note
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.
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For axial stability the field must decrease with radius (n > 0, i.e. dB/dr < 0) - achievable with a flat-pole H-magnet's natural falloff - and oscillation solutions are real only for 0 < n < 1, with tunes nu_r = sqrt(1-n), nu_z = sqrt(n).
nu_r = sqrt(1-n), nu_z = sqrt(n); require 0 < n < 1Source quote & editorial note
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.
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Pick pole size by mission: 6-9 inch poles are the economical educational range; go to 12-15 inches if you want enough energy for neutron-yielding light-element reactions.
educational: 6-9 in poles; light-element/neutron reactions: 12-15 inSource quote & editorial note
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).
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The gap drives the field: for a gap-dominated, unsaturated magnet B ~ mu0*NI/g, so keep the pole gap as small as the vacuum chamber, dee clearance and beam aperture allow, even at the cost of a harder chamber design - the source calls its tight spacing 'essential' despite the chamber difficulty it caused. [Corrected 2026-08-23: earlier text also asserted the magnet is 'the single most expensive subsystem', which the quote does not say.]
B ~ mu0*NI/g for a gap-dominated, unsaturated circuit; real magnets add fringe, yoke reluctance and saturationSource quote & editorial note
it is advantageous to keep the gap between the magnet poles small. This tight spacing made the design of the vacuum chamber more difficult, but it was essential.
Baumgartner, The Cyclotron Kids' 2 MeV Proton Cyclotron — Cyclotrons 2013, WE1PB05 (2013) — p. 1-2
Editorial note, tabletop extrapolation: The central trade for a next machine: each millimetre of gap saved is field (at fixed ampere-turns), and energy scales as B^2 (at fixed radius and species) - but only within the unsaturated, gap-dominated regime, and only after dee-voltage clearance, pumping and field quality have had their say. Verify the saturation and fringe terms in FEMM before banking the gain. [Note revised 2026-08-23: an earlier text called the gain 'for free'; the chamber redesign it costs is the quote's own point.]
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The proton RF frequency is 15.2 MHz per tesla; use a table of f = 15.23*B MHz to co-design magnet field and RF tuning range (Cyclotron Kids' table: 1.0-1.7 T maps to 15.2-25.9 MHz, with matching capacitance 166 pF down to 57 pF for their fixed tank inductance).
f(MHz) = 15.23 * B(T) for protonsSource quote & editorial note
B (Tesla) 1 ... 1.6 ... f (MHz) 15.23 ... 24.36
Baumgartner & Heuer, The Cyclotron Kids 14-Inch Accelerator (2010) — p. 12
Editorial note, tabletop extrapolation: The reference machine's 0.59 T resonates at ~9.0 MHz; a next machine's field choice fixes the synchronous frequency via this 15.23 MHz/T constant (fundamental-harmonic protons). The tank tuning range then follows from the chosen inductance - the source's capacitance column is specific to theirs.
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Once iron poles saturate - about 2 T in the source's accounting - added excitation buys little further field and maximum energy grows mainly with radius; iron-pole designs therefore plan around fields below saturation.
pole saturation ~2 T (source's figure; onset is alloy- and geometry-dependent and gradual)Source quote & editorial note
once the iron magnet poles become saturated (at about 2 T) the maximum energy is determined by R
Loucks, Initial Results from the Houghton College Cyclotron — Houghton College thesis (2007) — p. 18
Editorial note, tabletop extrapolation: Frames the next machine's tradeoff space: pushing the reference machine's 0.59 T toward 1.2-1.5 T is cheap energy gain (E ~ B^2 at fixed radius), while near pole saturation the iron stops helping and pole diameter becomes the effective lever.
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Keep the classical-cyclotron field index n = -(r/B)(dB/dr) between 0 and 1 everywhere inside the acceleration region - n<0 loses axial focusing, n>1 loses radial stability - and empirically n should rise roughly linearly from 0 toward 1 with radius, shaped by shimming.
n = -(r/B)dB/dr; 0 < n < 1, rising ~linearly with r; f_z = sqrt(n)*f0, f_r = sqrt(1-n)*f0Source quote & editorial note
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
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If using an ion-source chimney, verify the first half-turn clears the chimney body: with a 0.5-in dee gap and Rs = 0.8 ohm, calculated first ions clear at ~200 W RF (50 W is far too low, 500 W comfortable).
First-turn radius from x,y solutions with E = Vpeak/gap; thresholds: 50 W too low, ~200 W first ions clear, 500 W sufficientSource quote & editorial note
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.
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Set RF frequency from the cyclotron resonance relation: for protons f(MHz) = 1.52 x B(kilogauss); tune B (not f) during operation to find resonance.
f = eB/(2*pi*m); protons f(Mc) = 1.52*B(kG); deuterons and alphas (4He2+) f = 0.76*B(kG)Source quote & editorial note
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.
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Cyclotron resonance frequency is f0 = 15.2 * B[T] * Z / A MHz - about 10 MHz per tesla region for protons (15.2 MHz at 1 T).
f0[MHz] = 15.2 * B[T] * Z/ASource quote & editorial note
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.
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Relativistic phase slip caps a fixed-frequency cyclotron at Tmax = sqrt(16*q*V0*mi*c^2/pi) with optimal detuned injection - so the maximum energy grows only as the square root of dee voltage (100 kV -> ~31 MeV for deuterons; the practical cure is more volts per turn).
Tmax = sqrt(16*q*V0*mi*c^2/pi); f_rf/f_g0 = 1/(1+Tmax/2mi c^2)Source quote & editorial note
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.
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Low-energy protons orbit at 15.23 MHz per tesla (f = qB/2*pi*m); scale RF frequency linearly with the orbit-averaged field for a classical proton cyclotron on the fundamental harmonic.
f(MHz) = 15.23 * B(T) for protonsSource quote & editorial note
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.
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The classical fixed-frequency cyclotron is limited to under ~25 MeV protons because phase slip accumulates at ~360*(gamma-1) degrees per turn; at 21 MeV that is ~8 deg/turn, losing a peak-phase ion in 11 revolutions unless energy gain per turn is enormous (360 kV for the LBL 60-inch).
dphi/dn = 360*(gamma-1) deg/turn; classical limit E < ~25 MeVSource quote & editorial note
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.
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Higher dee voltage raises the fixed-frequency energy ceiling by reducing the number of turns (and thus accumulated relativistic phase slip); the particle survives while phase slip < pi/2, giving a maximum around 15 MeV for protons at 50 kV peak-to-peak.
accept while phase shift < pi/2; ~15 MeV max for protons at 50 kVppSource quote & editorial note
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.
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Set the vacuum requirement so the beam's mean free path is at least an order of magnitude longer than the total spiral flight distance; compute the flight distance as the sum of the spiral's per-turn circumferences (for r proportional to sqrt(E), about two-thirds of turn count times the final circumference). [Corrected 2026-08-23: an earlier version repeated the source's conclusion that ~2e-3 torr is adequate for a fast machine. That figure follows from the thesis reading a 5 km mean free path off its own plot at 2e-3 torr, which implies a cross-section near 3e-20 cm2 - four orders of magnitude below the measured proton electron-capture cross-section in hydrogen (8.7e-16 cm2 at 10 keV, ORNL-6086 p. A-28). With the measured value the capture mean free path at 2e-3 torr is about 0.2 m, shorter than one turn. The criterion stands; the number does not, and the thesis machine reported no beam. Compute the mean free path with the capture cross-section, never the gas-kinetic one; see /learn/vacuum/.]
MFP >= 10 * flight path; lambda = kT/(P*sigma) with sigma the charge-exchange cross-section [the source's '2e-3 torr -> ~5 km' uses a sigma four orders too small; see correction]Source quote & editorial note
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.
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
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Relativistic detuning budget: a 10 MeV proton is only ~1% heavier, but that 1% frequency shift accumulated over hundreds of turns is what caps fixed-frequency cyclotrons near 20 MeV - a small per-turn effect below ~1 MeV, though it still accumulates with turn count.
dm/m ~ T/(938 MeV); cyclotron limit ~20 MeVSource quote & editorial note
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.
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In a classical (azimuthally symmetric) cyclotron, keep the field-decay index n between 0 and 1 at all working radii; only then are both radial and axial motion stable, with tunes Qr = sqrt(1-n) and Qz = sqrt(n).
0 < n < 1; n = -(dB/dr)(r/B); Qr = sqrt(1-n), Qz = sqrt(n)Source quote & editorial note
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
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A classical cyclotron's final proton energy is limited to 10-15 MeV with one or two dees at practically realizable dee voltages; set the RF generator frequency below the central-field revolution frequency so the phase slides negative and turns around near -90 degrees, maximizing radius before phase loss.
E_max(protons, classical) ~ 10-15 MeV; choose f_rf < f(0) so phase turnaround occurs near -90 degSource quote & editorial note
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.
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Set beam energy as an explicit compromise among cost, the physics value of higher energy, and the fraction of beam you can extract - the source's three axes; current they set separately, from what the research program needed.
Source quote & editorial note
The beam energy is really a three way compromise between cost, the advantages of higher energy, and the ability to extract a large fraction of the beam.
Oak Ridge National Laboratory, A Proposal for the Mc² Isochronous Cyclotron — ORNL-3540 (1963) — p. 20
Editorial note, tabletop extrapolation: Directly transferable process rule (their answer, 810 MeV / 100 uA, is not): write down the compromise axes for a next machine's energy point instead of inheriting a number.
Cited in: Choosing Your Machine
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Adapting existing equipment mortgages the machine: ORNL's own five-point verdict on the 63-inch - built fast from adapted parts, it ended up unshieldable, with marginal field (median plane drifts, hard to keep shimmed), dee-to-ground capped at ~40 kV by its bushing insulators, a 6-in. gap half of what was needed, and a single-species rf system — and none of the five "can readily be corrected".
Source quote & editorial note
a large amount of existing equipment was adapted for use in the accelerator, and many design compromises were accepted. Consequently, this machine lacks the versatility and reliability which are essential ... which cannot readily be corrected: 1. The cyclotron is not and cannot be shielded ... [the dee stems] must enter the vacuum through bushing insulators. This limits the maximum dee-to-ground potential to about 40 kv ... 4. The magnetic field gap is only 6 in., less than half of what it should be to obtain the desired output.
Editorial note, tabletop extrapolation: The counterweight to thrift - surplus-equipment compromises in shielding provisions, magnet gap, and insulator ratings are the ones a finished machine cannot shed. When designing a next machine around salvaged parts, check each against this five-item list; anything on it deserves new hardware.
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Size shielding around the SECONDARY radiation: the beam's interaction with the target, the accelerator structure, or the shielding itself 'most often' determines the type and magnitude of shielding required - and primary-beam containment is still assessed wherever extraction, a thin window, or an abnormal loss could make ions accessible.
shield for secondaries (X-rays, neutrons) produced where the beam is lost, not for the primary ionsSource quote & editorial note
Secondary radiations produced as a result of the interaction of the primary beam with a target, portion of the accelerator, or the shielding most often determine the type and magnitude of the shielding.
Editorial note, tabletop extrapolation: For the reference machine and a next machine the primary protons stay inside the chamber in normal operation, so the external radiation field is dominated by secondaries - dee-gap electron bremsstrahlung today; reaction products (the 11B(p,alpha) alphas) and any (p,n)-capable contaminants joining the inventory at a next machine's energies.
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Begin the radiation-protection program at the CONCEPTION of the facility — safeguards incorporated during funding/design/construction cost significantly less than safeguards superimposed on an existing facility.
RP designed-in at concept << RP retrofitted (cost)Source quote & editorial note
if proper safeguards are incorporated into the construction of the accelerator facility the cost of safety will be significantly lower then if such safeguards are superimposed upon already existing facilities.
Editorial note, tabletop extrapolation: Why a hazard analysis belongs at desk phase rather than after first beam: the enclosure, interlocks and monitoring get designed into the machine rather than left to whoever installs it.
Cited in: Choosing Your Machine
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Copy a proven machine when one exists at your scale: the UW 60-inch worked from a complete set of Berkeley Crocker plans, followed 'closely on the magnet design', drew sustained advice from the originating lab, and reached assembled-ready-for-test in three years - a schedule the report credits to exactly that inheritance; original design effort went to the subsystems where the precedent was silent.
Source quote & editorial note
We have had available for our use a complete set of the Berkeley plans which was kindly placed at our disposal by Professor E. O. Lawrence.
Editorial note, tabletop extrapolation: The strategy transfers directly: for any new machine, start from the closest documented working design - this corpus and the builds census exist to make that possible - and spend novelty only where the precedent is silent.
Cited in: Choosing Your Machine
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Know the fixed-frequency niche boundary: a 60-inch pole at ~15 kG is "about the optimum dimensions in which deuterons may be accelerated profitably without resorting to frequency modulation" — beyond this scale relativistic phase slip forces FM/synchro operation. Below it, constant-frequency operation buys large beam currents.
Source quote & editorial note
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.
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.
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Treat magnetic rigidity zeta = B*rho (T m) as the magnet system's design variable: an ion that reaches radius rho carries p = q*B*rho and, nonrelativistically, E = q^2*(B*rho)^2/(2m) - the field-and-geometry CEILING on energy; dee voltage sets turn count and whether the ceiling is reachable, not the ceiling itself.
p_max = q*B*rho ; E_max = q^2*(B*rho)^2/(2*m) ; v_max = (q/m)*B*rhoSource quote & editorial note
Sie bestimmt die maximal erreichbare Energie der Ionen und diese ist somit nur vom Magnetfeld und dem Radius der Austrittsbahn abhängig [tr.: energy depends only on field and exit radius]
Editorial note, tabletop extrapolation: For 0.5 T and 10 cm usable radius, zeta = 0.05 T m gives ~120 keV protons; doubling either B or rho quadruples the ceiling. Low dee voltage doesn't lower it - but capture, phase acceptance and losses can keep the beam from ever reaching rho, which is the caveat behind 'regardless of dee voltage'.
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Below a rigidity of about 0.3 T m (protons: 0.31 T m, 4.7 MeV, v = 0.1c) the source treats the machine as non-relativistic; the relativistic regime would demand fields of 3-6 T on 50-100 mm poles and is out of reach for small magnets. The boundary is a tolerance statement, not a switch: at beta = 0.1 the cyclotron frequency is already ~0.5% low.
zeta_rel = m*(0.1c)/q = 0.31 T m (H+), 0.63 T m (H2+)Source quote & editorial note
Für ζ ≤ 0,3 Tm ist man demnach im nichtrelativistischen Bereich [tr.: for zeta <= 0.3 T m one is in the non-relativistic regime]
Editorial note, tabletop extrapolation: A tabletop proton machine (zeta ~ 0.03-0.12 T m) sits comfortably below the bound - by a factor of 2.5 at the top of that range, not an order of magnitude. Constant-mass orbit codes are fine for geometry, but check the RF phase budget: even the ~1e-3-class frequency shift at 0.12 T m accumulates over hundreds of turns, so run the accumulated-phase check alongside the field-shape one rather than crediting all slip to field errors.
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Dee voltage does not set the final energy (ideal on-crest model): the magnet and usable radius fix the ladder height, the voltage is the rung spacing - k = E_max/(q*U0) crossings, first-orbit radius r1 = sqrt(2*(q/m)*U0)/omega_cyc.
k = E_max/(q*U0) ; v1 = sqrt(2*(q/m)*U0) ; r1 = v1/omega_cycSource quote & editorial note
Die Endenergie der Ionen ist so etwas wie die Höhe einer Leiter und die Beschleunigungsspannung ist dann der Abstand der einzelnen Sprossen [tr.: final energy is the ladder height, voltage the rung spacing]
Editorial note, tabletop extrapolation: Recomputed for 1000 V protons at 185 mT: r1 = 24.7 mm, matching the book. A 150 keV machine at 1 kV needs 150 ideal crossings; at 5 kV only 30 - relaxing vacuum and field-error tolerance roughly in proportion. The idealization to keep visible: real voltage also moves capture, turn separation and whether the top rung is reachable at all (dg-1376's ceiling-vs-attainability).
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The minimum dee amplitude is the one whose first orbit clears the ion source: on COLUMBUS, protons clear from U0 >= 200 V and H2+ from U0 >= 400 V on the first turn (r1 ~ 11 mm at their respective fields, against the 20 mm chimney region).
r1 = sqrt(2*m*U0/q)/B; clearance threshold U0_min ~ (B*r_clear)^2*(q/m)/2 (ideal full-qU0 first kick)Source quote & editorial note
Protonen ab U0 ≥ 200 V und H2+-Ionen ab U0 ≥ 400 V – bereits beim ersten Umlauf – hinreichend weit von der Ionenquelle entfernt [tr.: clear of the source from 200 V / 400 V on the first turn]
Editorial note, tabletop extrapolation: The scaling is the useful transfer: at 0.6 T and a 15 mm clearance radius the same ideal estimate gives ~3.9 kV for protons - so a sub-kV dee on a higher-field machine would NOT clear a 15-mm-class source housing under these assumptions; trace the actual source and gap geometry (launch phase, initial position, 3-D fields) before trusting the ideal number either way.
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The book's vacuum criterion is a path-length condition: the mean free path of the accelerated ion must be at least the total spiral path length to final radius, s_ges = r1*pi*sum_{i=1..k} sqrt(i) + k*gap - so lower dee voltage (more turns) demands lower pressure.
l_bar >= s_ges = r1*pi*sum(sqrt(i), i=1..k) + k*gap (the source's criterion; note lambda = s means ~37% survival, not arrival)Source quote & editorial note
muss die mittlere freie Weglänge für die betreffenden Ionen größer oder gleich der gesamten Bahnlänge sein [tr.: the mean free path must be >= the total path length]
Editorial note, tabletop extrapolation: The qualitative lever is real - raising dee voltage shortens the spiral and relaxes the pump requirement - but pressure thresholds don't follow from radius and voltage alone: compute survival as exp(-integral n*sigma_loss(E) ds) with the charge-exchange cross-section for the actual species, gas and energies (dg-460's lesson), and pick pressure from an explicit acceptable loss fraction.
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A teaching machine need not extract the beam - the book's point exactly: 'the particle beam does not even need to be extracted' - an internal probe, species identification by specific charge, and a visible running accelerator met the project's pedagogical objectives.
Source quote & editorial note
Der Teilchenstrahl braucht dabei nicht einmal ausgelenkt zu werden [tr.: the particle beam does not even need to be extracted]
Editorial note, tabletop extrapolation: A radial probe drive with a Faraday cup is the first detector to build on any small machine, and extraction is properly a separate later project - which, when undertaken, teaches its own lessons (septum design, transport, external diagnostics; the COLUMBUS Wien-filter plan, dg-1512, is that next chapter). Wait until internal beam is reproducible across days.
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Resonant acceleration requires omega_RF = k*omega_cyc with k odd (1, 3, 5, ...); the fixed-frequency property (period independent of radius and velocity) holds while the accumulated phase slip from gamma - 1 stays acceptable for the chosen turn count and phase window - at 0.1c the frequency is already ~0.5% low, which may or may not matter depending on turns.
T = 2*pi*m/(q*B) ; omega_RF = k*(q/m)*B, k = 1, 3, 5, ...Source quote & editorial note
ωHF = k · ωZyk = k · v/r = k · (q/m) B mit k = 1; 3; 5; ... [tr.: RF frequency equals an odd multiple of the cyclotron frequency]
Editorial note, tabletop extrapolation: Third-harmonic operation (k = 3) lets a 0.2 T magnet accelerate protons with a 9 MHz resonator, at the cost of a narrower phase window per crossing; it is the formal basis of the sub-harmonic peaks in I(B) spectra (dg-1424).
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Practical relativistic ceiling as the thesis cites it: beyond raising electrode voltage, relativity can be countered by shaping the magnetic field, and 'the actual relativistic limit for magnetic resonators accelerating protons is closer to 25 MeV' - a historical (Rose-era) estimate for that machine class, not a universal constant.
Source quote & editorial note
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.
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State a teaching cyclotron's requirements as two conditions before any dimensioning, per the 2013 design account: every operating parameter (vacuum, magnetic field, frequency) kept low enough that standard commercial components suffice, and the final energy kept small enough that no harmful radiation can arise, so that students can experiment at the running machine; the whole parameter table is then presented as the consequence of these two conditions.
Source quote & editorial note
In order to build such a small cyclotron one has to meet two conditions: Vacuum, magnetic field, frequency etc. must be so low that one can use standard components as far as possible, otherwise the costs will go to infinity; The final energy of the cyclotron must be small enough so that no harmful radiation can arise, so that the students can do experiments with the cyclotron. Table 1 shows the technical data of COLUMBUS. One can easily recognize that COLUMBUS meets all the conditions mentioned above.
Editorial note, tabletop extrapolation: A hobby-scale build benefits from the same requirements discipline; writing the cost condition and the radiation condition down first turns every later component choice into a check against them. The radiation condition itself needs its own verification, not just an energy number: whether 'no harmful radiation can arise' at a given operating point is the paper's claim for its machine, and X-rays begin when high voltage or RF is energized, before any beam.
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Complete transverse stability in a constant-gradient (weak-focusing) cyclotron requires 0 < n < 1, where the field index n = -(r/B)(dB/dr); the axial tune is nu_z = sqrt(n) and the radial tune is nu_x = sqrt(1-n), both from the Kerst-Serber equation.
n = -(r/B)(dB/dr); d2z/dt2 + n w^2 z = 0; d2x/dt2 + w^2 (1-n) x = 0; nu_z = sqrt(n); nu_x = sqrt(1-n)Source quote & editorial note
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
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.
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A weak-focusing cyclotron only meets the cyclotron condition at one point in the ion's flight from source to target; the accumulated error is tolerable as long as the overall integrated phase slippage stays under 90 degrees, and raising the accelerating dee voltage reduces the number of turns and hence the accumulated slippage. Alternatively, starting the ions in a field that is too high lets the slippage run one way, meet the condition midway, then reverse to net zero.
Source quote & editorial note
This error is acceptable, as long as the overall integrated phase slippage is less than 90°.
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.