Cyclotron physics theory design rules

a 4 T magnetic field for the maximum achievable energy of 8.5 MeV in the AMIT cyclotron corresponds to a beam-rest-frame electric field of E = 160 MV/m. This entails a marginal beam fraction loss per unit length of 1.42e-6 m-1

Calvo et al., Beam Stripping Interactions in Compact Cyclotrons — PRAB 24, 090101 (2021) — p. 7-8, 14

Editorial note, tabletop extrapolation: At 0.889 T and 500 keV the rest-frame field is ~9 MV/m, where the exponential makes the lifetime effectively infinite - Lorentz stripping can be ignored entirely for a next machine.

These factors limit the accuracy of determination of the most probable energy to a few hundred kilovolts. The high energy portion of the energy distribution can be determined with almost equivalent accuracy

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

Editorial note, tabletop extrapolation: The asymmetry - high-energy side of an absorber spectrum better determined than the low-energy tail - is the shape to remember, but the historical few-hundred-keV floor belongs to that apparatus: for a PIPS-plus-degrader setup, build the detector-and-degrader response matrix and quote separate uncertainties for mode, upper edge and tail rather than scaling ORNL's numbers.

In scattering from light nuclei, this introduces appreciable kinematic broadening of peaks as the entrance aperture is opened.

Alford, Bilaniuk & Hawrylak, Broad Range Spectrograph for Use with the Rochester 27-inch Cyclotron — NYO-9683 (1961) — p. 8

Editorial note, tabletop extrapolation: For a next machine's Rutherford-scattering station the same knob exists: closing the entrance aperture trades count rate for resolution. State the broadening honestly - dE ~ |dE/dtheta|*dtheta depends on beam energy, angle and kinematics in general; only the FRACTIONAL elastic broadening at fixed masses and angle drops out energy-independent - so compute it for the actual geometry rather than quoting a mass-ratio shortcut.

The first factor in (1) is exact. It does not depend on the exact shape of the response curve (pulse-height vs proton recoil energy) of the scintillator.

Fulbright et al., A Fast Neutron Time of Flight System for Use with Cyclotrons — NYO-9360 (1962) — p. 15

Editorial note, tabletop extrapolation: A lovely teaching derivation - a two-factor closed form students can test against a calibration reaction - and the transferable habit is identifying which factor rests on which assumption. Validate against calibration or Monte Carlo for the actual detector and energy range before leaning on it.

The maximum energy of the current cyclotron, using (6), is 150 keV for deuterons, which puts the cross section near the maximum. ... [the d(d,n)3He reaction is] exothermic with 2.227 MeV released for each deuteron pair.

Cressman, The Design and Construction of a Small Cyclotron — Houghton College thesis (2006) — p. 49

Editorial note, tabletop extrapolation: Working back from the excitation curve converts machine energy from a bragging number into a requirement - the transferable design move. D-D is the standout low-energy neutron reaction because its cross-section is already usable near 100 keV; take Q-values and cross-sections from live evaluated data (per site policy), not from the thesis's figures.

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.

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.

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.

Since Br is equal to zero on the median plane, the following approximation is valid for small axial displacements

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

Editorial note, tabletop extrapolation: This is why a median-plane-only field survey suffices for a first-cut axial-focusing model - the off-plane field follows from Maxwell to first order in z. The 1963 authors also flagged its limit - the approximation degrades for large axial amplitudes and unknown off-plane field shape.

It might be expected that the combined effect of the cyclotron's resonant acceleration and our embedded Wien Filter's velocity selection might separate the mass ambiguity. However, this is not the case, as the cyclotron itself acts a velocity filter at each of its radii. […] It also holds that at any given radius, both the deuterium and helium cover the same angular distance and thus must have the same angular velocity to keep in step with the oscillating RF voltage. Now it is easily seen that the velocity of either deuterium or helium will be the same at the entrance to the deflection channel. Note, while the two ions have the same velocity, they obviously do not have the same energy.

Koeth, Ponter, Hoffman, Schneider & Krutzler, Rutgers 12-Inch Cyclotron Electrostatic Deflector (2010, rev. 2011) — p. 2

Editorial note, tabletop extrapolation: Directly relevant to a hydrogen-fed tabletop machine, where the contaminant species of interest are H2+ and H3+ rather than deuterium and helium: an internal E x B channel will separate q/m = 1 from q/m = 1/2, but will not distinguish two species sharing a q/m. Species identification has to come from elsewhere (gas fill, source chemistry, or a downstream detector), a limit worth knowing before building the channel as a diagnostic.

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".)

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.

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).

Many nuclear reactions produce neutrons, but perhaps the simplest is d-d reaction, with a broadly peaked cross section at a mere 180 keV. With an incident energy beam, in the regime of 180 keV, the reaction d(d,n)He3 reaction produces 2.45 MeV neutrons quasi-isotropically.

Koeth, Neutron Production with a 12-Inch Cyclotron (2017) — p. 1

Editorial note, tabletop extrapolation: The species distinction is the load-bearing radiological fact: deuterons on a deuterated target make fast neutrons at only ~180 keV, while common stable targets have (p,n) thresholds above 1 MeV (7Li(p,n) at ~1.88 MeV is among the lowest) — so a sub-MeV proton machine's neutron picture hinges on verifying the beam really is protons (deuterium contamination opens the D–D channel), what the beam actually strikes, and the truthful maximum energy; no blanket neutron-free claim follows. The cross-section characterization and the quasi-isotropic 2.45 MeV figure are the source's own: at finite beam energy the neutron energy is angle-dependent, and quantitative cross-section shapes should be taken from evaluated data at design time, not from this description. The source states its laboratory move was what provided the radiological controls to permit fast-neutron generation (abstract, p.1).

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.

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.

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.

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.