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.

The limitation for H¯ acceleration is the energy of about 70 MeV, as further acceleration would result in magnetic dissipation of H¯ ions, and require reduction of magnetic field

Karamyshev, A New Design of PET Cyclotron — IPAC2022, paper THPOMS016, JACoW (2022) — p. 3

Editorial note, tabletop extrapolation: The 70 MeV figure is the author's limit for this magnet regime. Lorentz stripping depends on energy and field together through the rest-frame electric field E′ = γβcB, so a tabletop H- design compares a cited stripping lifetime at its own E′ with the transit time rather than assuming the loss negligible; residual-gas stripping is a separate loss mechanism.

In all cases, the estimated accuracies of the recommended data are believed to have been conservatively estimated, and are intended to represent upper limits.

Barnett et al., Atomic Data for Fusion, Volume 1: Collisions of H, H2, He and Li Atoms and Ions with Atoms and Molecules — ORNL-6086/V1, Oak Ridge National Laboratory (1990) — p. 15

Editorial note, tabletop extrapolation: A beam-survival estimate inherits the per-reaction accuracy line printed under each table (10 % to 45 % for the reactions relevant here, or "unknown"); carry that band through to the predicted survival fraction rather than quoting a single number. That the stated bands are upper limits is the compilation's own judgment.

Accuracy: E < 100 eV/amu - Unknown; E > 100 eV/amu - 10%

Barnett et al., Atomic Data for Fusion, Volume 1: Collisions of H, H2, He and Li Atoms and Ions with Atoms and Molecules — ORNL-6086/V1, Oak Ridge National Laboratory (1990) — p. 44

Editorial note, tabletop extrapolation: Because the cross section drops about thirtyfold between 10 and 100 keV, the proton loss in a 100 keV-1 MeV machine is concentrated in the early, low-energy turns near the centre; a survival integral must therefore weight each turn by sigma at that turn's energy rather than use one cross section for the whole spiral. Raising the energy gain per turn shortens exactly the part of the path where sigma is largest.

For convenience, least-squares Chebyshev polynomial fits to the recommended cross-sections and rate coefficients are presented for each reaction. These polynomial fits are indicated graphically, and a statement is made concerning the accuracy of each fit to the recommended data.

Barnett et al., Atomic Data for Fusion, Volume 1: Collisions of H, H2, He and Li Atoms and Ions with Atoms and Molecules — ORNL-6086/V1, Oak Ridge National Laboratory (1990) — p. 16

Editorial note, tabletop extrapolation: A loss calculator coded from the fitting coefficients is convenient, but the fit discrepancy is an additional approximation uncertainty on top of the data accuracy (the two are not simply additive); for the proton capture fit the largest deviations sit at eV energies irrelevant to the accelerated beam, so check the fit against a few table points over the machine's own 1–100 keV range before preferring either the fit or log-log interpolation.

Accuracy: H2+ + H2 - 30% E < 1x10^3 eV/amu; 20% E > 1x10^3 eV/amu

Barnett et al., Atomic Data for Fusion, Volume 1: Collisions of H, H2, He and Li Atoms and Ions with Atoms and Molecules — ORNL-6086/V1, Oak Ridge National Laboratory (1990) — p. 74

Editorial note, tabletop extrapolation: In a fixed-field cyclotron an H2+ reaches half the lab energy of a proton at the same radius, i.e. a quarter of the energy per amu, so over the whole spiral it sits much nearer the 1-10 keV/amu region where loss cross sections are largest. Reading an H2+ table at lab energy instead of per-amu energy misplaces the ion by a factor of two in velocity-equivalent energy.

This cross section is the sum of cross sections for all reactions that destroy the H2+ molecular ion in passage through H2.

Barnett et al., Atomic Data for Fusion, Volume 1: Collisions of H, H2, He and Li Atoms and Ions with Atoms and Molecules — ORNL-6086/V1, Oak Ridge National Laboratory (1990) — p. 629

Editorial note, tabletop extrapolation: An H2+ beam disappears by dissociation as well as by capture, and dissociation grows in relative importance with energy; a survival estimate built on the capture table alone can understate the H2+ loss several-fold in the tens-of-keV/amu range. The same page notes that large variations in ion-source operating conditions changed the measured cross sections by up to 10 %.

This cross section is the sum of cross sections for all reactions that destroy the H3+ molecular ion in passage through H2.

Barnett et al., Atomic Data for Fusion, Volume 1: Collisions of H, H2, He and Li Atoms and Ions with Atoms and Molecules — ORNL-6086/V1, Oak Ridge National Laboratory (1990) — p. 627

Editorial note, tabletop extrapolation: Species comparison for a survival calculation: at the same energy per amu the three hydrogen ions have loss cross sections of the same order (several 1e-16 cm2 at a few keV/amu). The species difference in survival therefore comes from each ion's energy-per-amu history along its spiral together with its path length and turn count — integrate n·Q_d(E/A) over each species' actual orbit rather than inferring survival from cross-section magnitude alone. The table covers only 1.6–35 keV/amu (single reference); outside that range the compilation gives no H3+ total.

It is well known that the H2+ electron capture cross sections are dependent on the vibrational levels of the incident H2+ ion. See the following set of data for the magnitude of this effect which increases as the energy decreases.

Barnett et al., Atomic Data for Fusion, Volume 1: Collisions of H, H2, He and Li Atoms and Ions with Atoms and Molecules — ORNL-6086/V1, Oak Ridge National Laboratory (1990) — p. 74

Editorial note, tabletop extrapolation: The vibrational-state data stop at 0.5 keV/amu, so for the accelerated beam the effect enters as an additional uncertainty on the H2+ loss rather than a correction; it matters most in the source and the first gap crossings, where H2+ energies per amu are lowest.

(6) Rate coefficient k at thermal energy for production of H3+ is: k = [2.0 ± .15]x10^-9 cm3/s

Barnett et al., Atomic Data for Fusion, Volume 1: Collisions of H, H2, He and Li Atoms and Ions with Atoms and Molecules — ORNL-6086/V1, Oak Ridge National Laboratory (1990) — p. 656

Editorial note, tabletop extrapolation: This reaction acts predominantly before acceleration — in the source plasma and extraction region, where H2 density is highest and H2+ is slow — shifting the extracted species mix toward H3+ as source pressure rises. The cross section falls steeply but the table ends at 7 eV/amu rather than at zero, so it is usually much less important after the first substantial acceleration; include it when estimating H2+ transmission through any dense, low-energy path.

Although some entries in the tables are given to several significant figures, all entries should be considered uncertain to at least ± 5%.

Phelps, Cross Sections and Swarm Coefficients for H+, H2+, H3+, H, H2, and H− in H2 for Energies from 0.1 eV to 10 keV — J. Phys. Chem. Ref. Data 19, 653 (1990) — p. 3

Editorial note, tabletop extrapolation: Treat +/-5 % as a floor, not an error bar. Where Phelps and the Red Book overlap (H+, H2+, H3+ in H2 up to 10 keV laboratory energy) the spread between the two compilations is a useful cross-check; Phelps states on the same page that it made extensive use of the Barnett compilations, so agreement between them is not fully independent confirmation.

Unless otherwise specified, all energies are laboratory energies EL rather than relative, center-of-mass, or "collision" energies.

Phelps, Cross Sections and Swarm Coefficients for H+, H2+, H3+, H, H2, and H− in H2 for Energies from 0.1 eV to 10 keV — J. Phys. Chem. Ref. Data 19, 653 (1990) — p. 3

Editorial note, tabletop extrapolation: With the target taken at rest, the cross section applies directly with the beam speed: a keV ion moves hundreds of times faster than room-temperature H2, so a gas-kinetic sqrt(2) relative-speed factor (appropriate when both partners are thermal) does not belong in the beam mean-free-path formula.

For energies from 2 to 10 keV, our recommended values approach the tabulated values for fast H production from Barnett et al.

Phelps, Cross Sections and Swarm Coefficients for H+, H2+, H3+, H, H2, and H− in H2 for Energies from 0.1 eV to 10 keV — J. Phys. Chem. Ref. Data 19, 653 (1990) — p. 6

Editorial note, tabletop extrapolation: Proton capture near the cross-section peak is a well-determined input to a survival estimate, with the two compilations agreeing inside the Red Book's 10 % band (and not fully independently, see dg-1916). The hydrogen density along the path — gauge gas correction, pressure gradients — is an additional uncertainty to be quantified for the machine alongside the cross-section band, not ranked above or below it without an uncertainty budget.

The cross section for the destruction of fast H2+ is slightly larger than that shown for charge transfer at energies from 1 to 20 keV.

Phelps, Cross Sections and Swarm Coefficients for H+, H2+, H3+, H, H2, and H− in H2 for Energies from 0.1 eV to 10 keV — J. Phys. Chem. Ref. Data 19, 653 (1990) — p. 10

Editorial note, tabletop extrapolation: For a survival estimate use a total destruction cross section rather than a single channel: the capture-only numbers differ by ~45 % between the two sources (7.0e-20 m2 vs 4.8e-16 cm2), outside the Red Book's 20 % band, while the totals (9.4 vs ~8.2e-16 cm2) differ by ~15 %, inside it. Phelps's recommended table ends at 10 keV laboratory energy (the destruction measurements it cites run to 20 keV); the Red Book tabulates its total through 50 keV/amu.

The dominant cross section for low-energy H2+ in H2 is that for the formation of H3+ + H.

Phelps, Cross Sections and Swarm Coefficients for H+, H2+, H3+, H, H2, and H− in H2 for Energies from 0.1 eV to 10 keV — J. Phys. Chem. Ref. Data 19, 653 (1990) — p. 10

Editorial note, tabletop extrapolation: H2+ ions below roughly 3–4 eV in the source or extraction gap convert mainly to H3+; above the crossover charge transfer dominates and H3+ formation falls to about 0.15 % of the charge-transfer value by 18 eV in the table (0.015e-20 vs 10.2e-20 m2). Its absolute probability after acceleration follows from that cross section times the gas column the ions traverse, and the loss physics of the preceding rules takes over.

for EL > 2.5 keV and is approximately equal to an appropriately weighted sum of the reaction cross sections shown.

Phelps, Cross Sections and Swarm Coefficients for H+, H2+, H3+, H, H2, and H− in H2 for Energies from 0.1 eV to 10 keV — J. Phys. Chem. Ref. Data 19, 653 (1990) — p. 14

Editorial note, tabletop extrapolation: A single dissociative event yields several fast fragments, so unweighted product cross sections overcount destruction; the fast-H column alone (7.8e-20 m2 at 10 keV) already exceeds the Red Book total (~4.7e-16 cm2 at the same velocity). For H3+ survival use the Red Book G-44 total.

Data showing a strong dependence of the cross section for collisional dissociation of H2+ on the degree of vibrational excitation have been obtained

Phelps, Cross Sections and Swarm Coefficients for H+, H2+, H3+, H, H2, and H− in H2 for Energies from 0.1 eV to 10 keV — J. Phys. Chem. Ref. Data 19, 653 (1990) — p. 12

Editorial note, tabletop extrapolation: A filament or discharge source produces H2+ in a spread of vibrational states, so the dissociation part of the H2+ loss in a working machine may differ from values measured with state-selected ions; carry this as an unquantified state-distribution uncertainty on H2+ survival — its direction depends on which states the source populates relative to the reference data. The Red Book's separate note that ion-source operating conditions shifted its measured H2+ totals by up to 10 % (G-46, p. 629, carried on dg-1912) is the only scale indication in these sources.

Furthermore, an intensity comparison in terms of mR/mAs is really not fair to the W anode, because its spectrum has fewer of the low energy photons which are more efficient at ionizing air.

Fewell & Shuping, Handbook of Mammographic X-Ray Spectra — HEW Publication (FDA) 79-8071, Bureau of Radiological Health (1978) — p. 13

Editorial note, tabletop extrapolation: Exposure in roentgens weights soft photons heavily — the point of the authors' remark. For shielding, the quantity that matters behind a wall is the hard tail of the actual spectrum: use a measured or conservatively modelled source spectrum and transport it through the real wall, including hard-tail, fluorescence and scatter contributions. Unfiltered tube output is not a conservative bound for a chamber, and filtered-beam data are not automatically realistic for it.

This field strength is produced, with beams of the ordinary intensity, if there are four positive ions for each electron in the stream, a condition which seems not unreasonable.

Johnson, A Low Voltage Cathode Ray Oscillograph — Bell System Technical Journal 1(2), 142–153 (1922) — p. 7

Editorial note, tabletop extrapolation: Johnson's four-ions-per-electron figure belongs to his assumed beam (1 degree divergence, 2 x 10^-5 A) and stays there. The transferable point is that positive-ion compensation can oppose electron space-charge spreading; how much compensation a given beamline needs is found experimentally or from a geometry- and species-specific ionization and transport model, since the ion yield per electron depends on energy, gas, pressure and loss mechanisms rather than falling monotonically with beam speed.

It has been found that the electron beam efficiently excites the test gas to produce visible fluorescence throughout the flow field with a brightness which is dependent upon the local gas density.

Hillard, Ocheltree & Storey, Spectroscopic Analysis of Electron-Beam-Induced Fluorescence in Hypersonic Helium Flow — NASA TN D-6005, Langley Research Center (1970) — p. 3

Editorial note, tabletop extrapolation: Beam-induced emission scales with the number of target atoms per unit path, so beam brightness in a fill gas is set by number density (pressure and temperature together), not by pressure alone. The tunnel's number densities correspond to 2.65–5.92 kPa at room temperature — three decades above a few-pascal visible-beam fill — so the Langley brightness regime is not that of a room-temperature bench fill, and the brightness–density relation reported is the source's for its own regime. The measurement is in helium; the source has no neon data.

the intensity of the molecular nitrogen radiation observed is too high to be attributed to direct excitation by the electron beam.

Hillard, Ocheltree & Storey, Spectroscopic Analysis of Electron-Beam-Induced Fluorescence in Hypersonic Helium Flow — NASA TN D-6005, Langley Research Center (1970) — p. 8

Editorial note, tabletop extrapolation: Under the Langley conditions (26 keV beam, cryogenic flowing helium), 40–400 ppm N2 produced strong N2+ emission and was the main radiator away from the beam, apparently through an indirect helium-mediated process the authors suggest but do not establish; atomic helium stayed the primary radiator within the beam. A spectrometer check of the emission is a sensitive impurity monitor whose sensitivity and concentration scale have to be calibrated for the actual fill and beam; the suggested transfer reactions are helium-specific, so any impurity effect in neon must be measured separately.

(3)Atomic helium was identified as the primary radiator in the electron beam, whereas molecular helium was the primary radiator near the beam.

Hillard, Ocheltree & Storey, Spectroscopic Analysis of Electron-Beam-Induced Fluorescence in Hypersonic Helium Flow — NASA TN D-6005, Langley Research Center (1970) — p. 11

Editorial note, tabletop extrapolation: The apparent width of a luminous beam can exceed the electron core: in the Langley helium flow, scattered electrons and trapped resonance radiation spread atomic-line emission over several core diameters, and afterglow species drifted with the gas flow. Size a visible beam from the bright core, not the halo; the halo extent in neon is not measured in this source.

It is the total effective absorbing area in square centimeters of all the molecules in a cubic centimeter of gas at a pressure of 1 mm of Hg

Brode, The Absorption Coefficient for Slow Electrons in Gases — Physical Review 25, 636 (1925) — p. 5

Editorial note, tabletop extrapolation: Converts Brode's curves into beam path lengths at the reference temperature of his constant: at 1 Pa (7.5 x 10^-3 mm Hg), lambda (cm) = 1/(a x 0.0075). Brode's 'absorption' counts any electron deflected out of the narrow beam past a finite collector, so it is an effective beam-removal (attenuation) length — not a total-scattering cross-section, since small-angle forward scattering within the collector's acceptance is not counted, and not an energy-loss length. For another gas temperature convert a to a cross-section per molecule and use lambda = 1/(n sigma).

In argon, the coefficient reaches a sharp maximum at about 12 volts, the value for 2 volts being only one-fifth of the maximum value.

Brode, The Absorption Coefficient for Slow Electrons in Gases — Physical Review 25, 636 (1925) — p. 2

Editorial note, tabletop extrapolation: Referee arithmetic at 1 Pa argon: lambda ~ 2.4 cm near 12 eV and ~ 19 cm at 360 eV. At fixed pressure, operating where Brode's removal coefficient is lower lengthens the unscattered path; whether the beam is visible over that path is a separate question of excitation and emission cross sections, beam current and viewing geometry.

In helium there is a maximum at 4 volts. Methane gave a curve similar to these, with a maximum at about 7.5 volts

Brode, The Absorption Coefficient for Slow Electrons in Gases — Physical Review 25, 636 (1925) — p. 2

Editorial note, tabletop extrapolation: Referee arithmetic at 1 Pa helium: lambda ~ 6 cm near 4 eV and roughly half a metre at a few hundred eV. Brode measured no neon; his introduction only cites Akesson (two maxima in Ne) and Ramsauer and Mayer (Ne coefficient nearly constant, decreasing slightly with velocity), so a neon mean free path must come from another source.

The curves for nitrogen and carbon monoxide are nearly identical, both having a maximum at about 18 volts and a minimum at about 9 volts. The curve for hydrogen shows no maximum, the coefficient increasing steadily down to 2 volts.

Brode, The Absorption Coefficient for Slow Electrons in Gases — Physical Review 25, 636 (1925) — p. 2

Editorial note, tabletop extrapolation: Residual air (N2) or hydrogen in a noble-gas fill adds beam loss below a few hundred volts in proportion to its partial pressure; in pure nitrogen at 1 Pa (a ~ 30) the removal length is about 4-5 cm at 10-40 eV (referee arithmetic). Keep air leaks and outgassing hydrogen low if the beam must survive tens of centimetres.

The ranges of Nelms are on the low side, ours on the high side, and the discrepancies give an indication of the error likely to result from lack of knowledge of low-energy stopping behavior.

Berger & Seltzer, Tables of Energy Losses and Ranges of Electrons and Positrons — NASA SP-3012 (1964) — p. 19

Editorial note, tabletop extrapolation: Use these tables for the keV end of a low-energy electron beamline (ranges, stopping, bremsstrahlung yield), not for the tens-to-hundreds-of-eV regime they do not cover. The 10 keV range discrepancy the authors quantify is for H2, Al, Ag and Pb (1–12 %); for neon, argon and air it is unquantified here — compare against modern ESTAR-class data before assigning a percentage.

It is difficult to estimate very closely the error of the data in Table I.

Berger & Seltzer, Tables of Energy Losses and Ranges of Electrons and Positrons — NASA SP-3012 (1964) — p. 25

Editorial note, tabletop extrapolation: The authors estimate roughly 1–2 % for their calculated collision stopping powers; that does not establish a few-percent range accuracy for neon, argon or air, because ranges also carry the explicitly uncertain low-energy closure near the 10 keV lower limit — check ranges there against modern data. The radiation-yield column at 10–20 keV inherits the ±20 % bremsstrahlung uncertainty and is an estimate.

0.010 1.791E 01 6.784E-03 1.792E 01 3.226E-04 2.190E-04

Berger & Seltzer, Tables of Energy Losses and Ranges of Electrons and Positrons — NASA SP-3012 (1964) — p. 66

Editorial note, tabletop extrapolation: Referee arithmetic for a neon fill at 1 Pa and 293 K (density ~ 8.3 x 10^-9 g/cm^3): a 10 keV electron loses only ~0.15 eV per cm and its path length to stop is ~390 m, so a 10 keV beam crosses a bench-scale vessel essentially without energy loss and its light is a thin-target signal; at 100 Pa the range is still ~4 m.

0.010 1.457E 01 1.112E-02 1.458E 01 4.050E-04 4.425E-04

Berger & Seltzer, Tables of Energy Losses and Ranges of Electrons and Positrons — NASA SP-3012 (1964) — p. 72

Editorial note, tabletop extrapolation: Referee arithmetic for argon at 1 Pa and 293 K (~1.64 x 10^-8 g/cm^3): 10 keV path length ~250 m, energy loss ~0.24 eV per cm. The higher radiation yield than neon matters only for X-ray estimates; about 4 x 10^-4 of a stopped 10 keV beam's energy goes to bremsstrahlung in argon itself.

0.010 1.970E 01 5.012E-03 1.971E 01 2.892E-04 1.463E-04

Berger & Seltzer, Tables of Energy Losses and Ranges of Electrons and Positrons — NASA SP-3012 (1964) — p. 130

Editorial note, tabletop extrapolation: Referee arithmetic at atmospheric density (1.205 x 10^-3 g/cm^3): 10 keV electrons travel about 2.4 mm of path in air and 20 keV about 8.1 mm (c.s.d.a. path, so projected penetration is less), so a beam vented or escaping through a thin window stops within millimetres of air. The 1.46 x 10^-4 radiation yield is for electrons stopped in air only; bremsstrahlung from the exit window, the target or any other higher-Z material intercepting the beam is assessed separately, and the 20 keV range is the table's own row (page image).

multipacting relies only on secondary electron emission from the electrodes. Thus, it is not dependent on the type of gas in which the breakdown takes place.

Woo, Final Report on RF Voltage Breakdown in Coaxial Transmission Lines — JPL Technical Report 32-1500, Jet Propulsion Laboratory, prepared under NASA contract NAS 7-100 (1970) — p. 10

Editorial note, tabletop extrapolation: Within Woo's collisionless, field-free two-surface model the gas species drops out and fd is the scaling parameter, so results from other machines scale by fd for a fixed impedance rather than by voltage alone. Whether a given gap actually multipacts still depends on the RF voltage band and phase, geometry, the electrodes' secondary-emission yield and surface condition, the pressure-dependent collision probability near the mean-free-path limit, and any magnetic or dc field — a gap inside a cyclotron magnet needs its own check (dg-1986, dg-1988).

The mean free path limit represents the condition that the electron mean free path is equal to d. The mean free path limit, therefore, separates the ionization and multipacting breakdown regions.

Woo, Final Report on RF Voltage Breakdown in Coaxial Transmission Lines — JPL Technical Report 32-1500, Jet Propulsion Laboratory, prepared under NASA contract NAS 7-100 (1970) — p. 17

Editorial note, tabletop extrapolation: Referee arithmetic from Fig. 7 (p.19): the air mean-free-path line follows p x d ~ 0.025 torr-cm, so at 1 Pa (7.5 x 10^-3 torr) a gap below ~3 cm is on the collisionless side and at 0.1 Pa almost any bench gap is — a collision-regime estimate only. fd below 70 MHz-cm excludes Woo's field-free two-surface mode (a 3 cm gap at 9 MHz is fd ~ 27); fd above it merely permits multipacting, which also needs a resonant voltage band and sufficient secondary yield, and a cyclotron's magnetic field changes the trajectories and allows other modes. A different fill gas shifts the line with its electron mean free path; the report places multipacting generally below 10^-2 torr (p.9).

This is equivalent to saying that each minimum occurs at a pressure given in microns (1 micron = 10^-3 torr) equal to the frequency given in MHz.

Woo, Final Report on RF Voltage Breakdown in Coaxial Transmission Lines — JPL Technical Report 32-1500, Jet Propulsion Laboratory, prepared under NASA contract NAS 7-100 (1970) — p. 20

Editorial note, tabletop extrapolation: Referee arithmetic: the diffusion-controlled minimum would fall near 9 microns (~1.2 Pa) at 9 MHz and near 27 microns (~3.6 Pa) at 27 MHz, so a 0.1-1 Pa fill sits at or below it, on the side where ignition gets harder as pressure falls. The rule holds for fd > 100 MHz-cm (a gap above ~11 cm at 9 MHz); bench gaps at 9 MHz are fd < 100, where Woo's oscillation-amplitude limit raises breakdown further. Air data in coaxial lines.

The present measurement gives a location of the Paschen minimum at (pL)* = 1.2 Torr cm with VBR,min = 290 V

Korolov & Donkó, Breakdown in hydrogen and deuterium gases in static and radio-frequency fields — Physics of Plasmas (2015), doi:10.1063/1.4929858; arXiv:1512.08726 — p. 4

Editorial note, tabletop extrapolation: Referee arithmetic: a 0.1–1 Pa hydrogen fill across a few-cm gap is pL ~ 0.002–0.02 Torr cm, two to three decades left of this minimum, where the source gives no breakdown data. The Paschen/Townsend curve describes gas breakdown only — field emission, surface flashover and particulates can break down an electrode assembly well below any extrapolated gas-breakdown voltage — so it is not a hold-off or electrical-safety guarantee; test the actual assembly conservatively. Electrode material and cleanliness move the minimum and especially the low-pressure branch.

we find a turning point (point "B" in figure 4(a)) beyond which we experience a sudden increase of the breakdown voltage

Korolov & Donkó, Breakdown in hydrogen and deuterium gases in static and radio-frequency fields — Physics of Plasmas (2015), doi:10.1063/1.4929858; arXiv:1512.08726 — p. 4

Editorial note, tabletop extrapolation: In this 13.56 MHz, 1 cm hydrogen experiment the low-pL branch is multi-valued: ignition at fixed pressure under rising voltage and at fixed voltage under rising pressure can occur at different points. In comparable systems characterise ignition with both sweeps rather than assuming a unique threshold.

at lower frequency the electron oscillation amplitude is higher at otherwise same conditions, therefore the bending point "B" moves to higher pL at lower f

Korolov & Donkó, Breakdown in hydrogen and deuterium gases in static and radio-frequency fields — Physics of Plasmas (2015), doi:10.1063/1.4929858; arXiv:1512.08726 — p. 5

Editorial note, tabletop extrapolation: Referee inference, limited: Korolov & Donkó tie the shape of the RF curve to the electron oscillation amplitude relative to the gap, which depends on the RF field, frequency, collision rate and geometry — and, in a cyclotron's 0.5–1 T field, on the strong magnetization of electrons at MHz frequencies, which the unmagnetized 1 cm bench case does not include. The surface-governed regime therefore cannot be assumed for a magnetized dee gap: calculate the electron motion for the actual field, frequency, gas and gap, or test ignition in the magnetized gap. The paper's data are for hydrogen at L = 1 cm and pL of roughly 1–30 Torr cm.

a factor of 1.8 is found (71 V vs. 125 V) between the minimum breakdown voltage values.

Korolov & Donkó, Breakdown in hydrogen and deuterium gases in static and radio-frequency fields — Physics of Plasmas (2015), doi:10.1063/1.4929858; arXiv:1512.08726 — p. 5

Editorial note, tabletop extrapolation: Two careful measurements of the same nominal system differed by a factor of about 1.8, which the authors attribute to systematic error or impurities in the other data set — the reason to characterise ignition in the actual cell, not a general uncertainty cap. The authors conditioned their stainless electrodes with ~8 mA dc discharges of both polarities for about 10 minutes (p. 2) — their preparation, with its own electrical and surface-modification hazards, not a universal prescription. The yield is quoted as 0.008 in the text and the Fig. 4(a) legend; the Fig. 4(b) legend prints 0.08.

50 us pulses used at these measurements. Use of longer pulses makes no sense because the fill time of the cavity is nearly 10 us.

Smirnov, Ivanov, Krasnov, Polikhov, Rezanov & Sharkov (Siemens Research Center, Moscow), 72 MHz Solid-State Amplifier Power Test — IPAC2014, paper WEPME008, JACoW (2014) — p. 2

Editorial note, tabletop extrapolation: The relation transfers and favours low frequency: at fixed loaded Q the fill time grows as 1/f, so a 6-30 MHz dee has a longer settling time than a 72 MHz cavity of the same Q. A soft-start or pulsed test on a tabletop dee should use pulses several fill times long, computed from that dee's measured loaded Q and frequency.

In class B, the amplifier only consumes power when it is driven by RF. This provides a much better efficiency, approaching 78.5% for an ideal transistor.

Jacob (ESRF), Radio Frequency Solid State Amplifiers — CAS–CERN Accelerator School: Power Converters, CERN-2015-003, pp. 197–216 (2015); arXiv:1607.01570 — p. 6

Editorial note, tabletop extrapolation: Transfers to a tabletop dee drive: push-pull class AB LDMOS is the standard compromise. A high-Q dee filters near its fundamental but cannot be assumed to suppress harmonics that land on other cavity or feed-line modes, so the output network and the modes deserve a survey. Class A wideband amplifiers (common as bench drivers) draw at least 2 W of DC per RF watt at full output (efficiency <= 50 %), against ~1.4 W for a 70 %-efficient class-AB stage.

the odd characteristic Ids(−Vgs) = −Ids(Vgs) minimizes the second harmonic H2 and higher even harmonics.

Jacob (ESRF), Radio Frequency Solid State Amplifiers — CAS–CERN Accelerator School: Power Converters, CERN-2015-003, pp. 197–216 (2015); arXiv:1607.01570 — p. 6

Editorial note, tabletop extrapolation: Explains why a push-pull LDMOS stage at 6-30 MHz can be expected to show a suppressed second harmonic relative to the odd ones; the third harmonic of a 9 MHz drive (27 MHz) may fall where a dee resonator or its feed line has other modes, so it deserves attention in output-filter design. The ESRF specification (same source, p16) nonetheless puts H3 lower than H2 for that system, so measured levels must come from the actual amplifier.

So, the power adds up correctly at the output of a combiner only by constructive interference, i.e. if and only if the signals at their inputs have the same amplitude and phase.

Jacob (ESRF), Radio Frequency Solid State Amplifiers — CAS–CERN Accelerator School: Power Converters, CERN-2015-003, pp. 197–216 (2015); arXiv:1607.01570 — p. 10

Editorial note, tabletop extrapolation: Relevant only when a tabletop drive sums two or more modules: phase-matched cables and equal module gain decide combining efficiency, and a non-directional combiner of the kind the source analyses has input-port coupling and no intrinsic isolation, so one failing module disturbs the others. Inspect the actual combiner's scattering matrix and isolation/load arrangement (hybrids obtain isolation through a terminated port) rather than judging by the presence of resistors. A single-module drive avoids combiner effects.

It is worth noting that the efficiency drops fast when operating the amplifier at reduced power, and is only about 47% at 100 kW, i.e. at ⅔ of the nominal power.

Jacob (ESRF), Radio Frequency Solid State Amplifiers — CAS–CERN Accelerator School: Power Converters, CERN-2015-003, pp. 197–216 (2015); arXiv:1607.01570 — p. 14

Editorial note, tabletop extrapolation: Transfers as a question to measure for a tabletop class-AB LDMOS stage: back-off efficiency depends on bias, matching, supply and device, so take the selected pallet's efficiency map — a dee needing ~150 W from a 1 kW-capable pallet (an assumed example) may run far below peak efficiency and dissipate correspondingly more heat per RF watt; sizing the device or its supply voltage to the actual requirement helps. The percentages are this 352 MHz system's.

the total transient reflection therefore peaks at more than 1.5 times the incoming pulse amplitude, corresponding to more than twice the input power;

Jacob (ESRF), Radio Frequency Solid State Amplifiers — CAS–CERN Accelerator School: Power Converters, CERN-2015-003, pp. 197–216 (2015); arXiv:1607.01570 — p. 17

Editorial note, tabletop extrapolation: Transfers to any tabletop dee driven by a soft-start, pulsed or interlock-tripped drive, at resonance and after the cavity has filled: switching the drive off is itself a reflected-power event — the emitted amplitude at switch-off is 2β/(1+β)·(1−e^(−t_p/T_f)), about 1 at β = 1 and above 1 for an over-coupled resonator once filled, smaller for a short pulse — so trips and foldback must not rely on reflected power falling when the drive is removed, and the device must tolerate the switch-off transient. Evaluate the actual coupling, detuning, pulse length and interlock response time; at β = 1 on resonance the steady reflection is zero, off resonance it is not.

The calculations below show that it is necessary for these ions to pass to the filament if arc currents of several amperes are to be obtained […] For V = 150 volts and d = 0.5 cm (usual calutron conditions), the emission current density is approximately 19 ma per cm² of the filament. Since the total area of a 140-mil filament facing the collimating surface slot is just over 0.5 cm², space-charge-limited current is of the order of 10 ma. When charge gas is present and the arc is struck and an ion sheath is formed, arc currents of 5 amps are easily obtainable. This is a current of 10 amp/cm² of effective filament surface. [Decimal points and the cm² superscripts are lost in the typescript scan; restored here.]

Yonts, Filament Life in Low Pressure Arc Type Ion Source — ORNL-1758, Oak Ridge National Laboratory (1954) — p. 11

Editorial note, tabletop extrapolation: The same physics governs a small filament source: at the calutron's 150 V across 0.5 cm, space-charge-limited emission into vacuum is of the order of milliamperes, and a different gap and voltage must be put through Child's law on their own; drawing much more current than that needs ion neutralization near the filament. The ion flux that permits the high current is the flux that sputters the filament, so emission current and filament erosion are coupled.

It has also been shown that operation of the full diode type reduces filament wear to a minimum but also reduces ion output to the point of uselessness

Yonts, Filament Life in Low Pressure Arc Type Ion Source — ORNL-1758, Oak Ridge National Laboratory (1954) — p. 42

Editorial note, tabletop extrapolation: The fitted coefficient and exponent belong to that calutron, charge material and operating range, and the fit is for the normalized loss J/I, so total erosion J has no simple sub-linear law in P unless arc current is held fixed. The transferable content is the method — plot weight loss per ampere-hour against arc power and fit — and the finding that the operating mode that minimizes wear is not the one that maximizes useful output, so the choice is a compromise set by measurement.

The filaments give extremely long life, provided the velocity of the positive ions does not exceed some critical value (22 volts in mercury vapor diodes) which depends on the type of gas in the tube.

Yonts, Filament Life Extension in High-Intensity Ion Sources — ORNL-1525, Oak Ridge National Laboratory (1953) — p. 5

Editorial note, tabletop extrapolation: The physics behind low-voltage operation as Yonts presents it: below some ion energy, cathode sputtering is small enough that low-work-function cathodes survive. The 22 V is his reported operating limit for mercury-vapour diodes, not a universal threshold: sputtering depends on projectile and cathode material, incidence angle, charge state and surface condition, so any hydrogen-source limit must be established for the particular ion species, cathode material and sheath conditions; low ion energy only tends to reduce sputtering, and an oxide cathode can also fail by poisoning or chemical attack.

However, atomic ions H+ are readily produced by secondary dissociation of molecular ions H2+ even when the maximum electron energy is just above 16 volts.

Smyth, Primary and Secondary Products of Ionization in Hydrogen — Physical Review 25, 452 (1925) — p. 1

Editorial note, tabletop extrapolation: Smyth's conclusion is a 1925 reading of one apparatus. Modern data separate the channels: H2+ formation opens at the H2 ionization energy, 15.426 eV, and direct dissociative ionization H2 + e → H+ + H + 2e opens at an appearance energy of 18.078 eV (NIST Chemistry WebBook, hydrogen ion-energetics page, retrieved 2026-10-25), so H+ is not only a secondary product; H3+ is chiefly made by the ion–molecule reaction H2+ + H2 → H3+ + H. For a filament source in hydrogen the species mix is therefore set by electron energy (which primary channels are open) together with gas density, the field and the ions' path and residence time (how much secondary chemistry occurs).

Curve B in Fig. 3 taken at p=.053 mm shows the H2+ peak almost suppressed but H3+ and H+ strong.

Smyth, Primary and Secondary Products of Ionization in Hydrogen — Physical Review 25, 452 (1925) — p. 6

Editorial note, tabletop extrapolation: The pressures are those of Smyth's ionization region with fixed fields, electron energy and path length: there, raising pressure suppressed H2+ and strengthened H3+ and H+. What transfers to another source is the collision-depth comparison (gas density times path length, or residence time), not the direction of the H+ change, which depends on the electron-energy distribution, ion collision energy and competing reactions and must be measured or modelled for the source in hand.

the distance an H2+ ion travels on an average before being dissociated by the proximity of another molecule is about 1/15 the ordinary molecular free path.

Smyth, Primary and Secondary Products of Ionization in Hydrogen — Physical Review 25, 452 (1925) — p. 10

Editorial note, tabletop extrapolation: The short destruction length means H2+ survives only where gas density times path length is small: on Smyth's fitted value, about 0.9 cm of path at 1e-3 mm Hg and about 9 cm at 1e-4 mm Hg (formula). Where a source's pressure and ion path put the product near or above one destruction length, the molecular-ion fraction will track pressure strongly; the fitted rate belongs to Smyth's fields and ion energies and must be re-measured for another geometry.

It was found that H+ was always relatively much stronger at higher values of V2

Smyth, Primary and Secondary Products of Ionization in Hydrogen — Physical Review 25, 452 (1925) — p. 12

Editorial note, tabletop extrapolation: The transfer is a testable hypothesis, not a prediction: in a cyclotron source the local extraction or bias field at the source exit — distinct from the dee voltage itself, which is RF, phase-dependent and may be screened inside the source — changes ion residence time and so may alter the species mix. Faster removal can also preserve H2+ rather than shift H3+ toward H+. This is a 1925 tube result; the direction and size of any shift in a particular source must be measured.

an apparent critical potential for H+ or H3+ may be observed anywhere between 16 and 21 volts if conditions are properly chosen and not varied.

Smyth, Primary and Secondary Products of Ionization in Hydrogen — Physical Review 25, 452 (1925) — p. 15

Editorial note, tabletop extrapolation: A species threshold measured on one source by sweeping electron energy says as much about that source's pressure and geometry as about hydrogen; the 16–21 V range is Smyth's observed apparent-onset range. Modern thresholds separate the channels: H2+ opens at 15.426 eV and direct dissociative production of H+ at 18.078 eV (NIST Chemistry WebBook, hydrogen ion-energetics page, retrieved 2026-10-25), so electron energy does change which primary channels are open, while pressure and residence time set the secondary chemistry. Just above threshold the H2 ionization cross section rises with electron energy; it peaks and then falls, so more energy is not simply more yield.

At low pressures only H2+ ions are formed; as the pressure is increased the percentage of H3+ increases in proportion to the pressure. In the apparatus used, the percentage of H+ increased with pressure but did not exceed 4 percent

Hogness & Lunn, The Ionization of Hydrogen by Electron Impact as Interpreted by Positive Ray Analysis — Physical Review 26, 44 (1925) — p. 1

Editorial note, tabletop extrapolation: This is the second independent 1925 measurement agreeing with Smyth that H2+ dominated the low-pressure spectrum and that H3+ was secondary. The low H+ fraction is specific to their tube, drawing-out field and sensitivity: H+ also has a direct dissociative-ionization channel from H2 (appearance energy 18.078 eV, NIST Chemistry WebBook, hydrogen ion-energetics page, retrieved 2026-10-25), open at their electron energies, and both papers show more H+ with stronger fields along the ion path.

Electrometer readings for H+, H2+ and H3+ peaks, at various pressures.

Hogness & Lunn, The Ionization of Hydrogen by Electron Impact as Interpreted by Positive Ray Analysis — Physical Review 26, 44 (1925) — p. 2

Editorial note, tabletop extrapolation: The table gives the measured shape of the species-versus-pressure curve for one positive-ray tube. The crossover sits at a lower pressure than the figure reading of Smyth's Fig. 4 recorded under dg-2085 (H3+ overtaking H2+ near 0.012–0.015 mm); different ion path lengths are one plausible apparatus-dependent cause, not an established explanation, and a source's own crossover depends on its geometry and other parameters and must be measured.

When the field V3 for drawing the positive ions from the ionization chamber was increased to such an extent that only those electrons in the upper portion of that chamber had ionizing energy, then the intensity of (H+ + H3+) increased with respect to that of H2+ as might have been expected since the H2+ ions formed had greater chance of collision having had a longer average path to traverse. Moreover the ratio of the intensity of H+ to that of H3+ increased considerably. This can be explained by assuming that under these conditions H2+ breaks up more often in accordance with reaction (4). This would happen if when the H2+ ion acquires sufficient energy before colliding, the H+ formed by disruption were not trapped by the H2 molecule to form H3+ but passed on into the resolving chamber.

Hogness & Lunn, The Ionization of Hydrogen by Electron Impact as Interpreted by Positive Ray Analysis — Physical Review 26, 44 (1925) — p. 5

Editorial note, tabletop extrapolation: Ion path length through gas and the energy ions gain along it both move the species mix; in a cyclotron source these are set by where the filament's electrons ionize relative to the exit and by the field at the exit. The mechanism is the authors' historical reading (the fast H3+ channel in modern terms is H2+ + H2 → H3+ + H, not capture of H+); the effect is described qualitatively, and its size in a given source must be measured.

It was found that the percentage of H+ was much greater than in pure hydrogen at any pressure employed (see the last line of Table 1). […] For H+, large potential difference and the addition of some inert gas such as helium; for H2+, low pressures; for H3+, higher pressure and small potential difference.

Hogness & Lunn, The Ionization of Hydrogen by Electron Impact as Interpreted by Positive Ray Analysis — Physical Review 26, 44 (1925) — p. 9

Editorial note, tabletop extrapolation: This gives the direction of each knob for a hydrogen filament source: pressure, and the applied potential in their tube (the draw-out field V3), which is not interchangeable with a generic source field without identifying which electrodes and ion energies are affected. The helium-admixture result shows that collision partners matter as well as density; using it on a cyclotron would change gas load and discharge behaviour, and the paper does not test that.

The increase in the percentage of H3+ as (V1+V2) was decreased may be explained as due to the characteristics of the discharge tube. The now greater retarding potential V3 decreased the velocity of those electrons that had penetrated into the lower part of the ionization chamber to such an extent that they could not ionize the gas. The H2+ ions were then formed at a greater average distance from the gauze J, and having had a longer average path to traverse had greater chance to collide.

Hogness & Lunn, The Ionization of Hydrogen by Electron Impact as Interpreted by Positive Ray Analysis — Physical Review 26, 44 (1925) — p. 6

Editorial note, tabletop extrapolation: In a filament source, changing electron energy can also change where ionization happens, so an apparent energy dependence of the species mix may be a geometry effect; whether an arc or bias scan moves the place of ion birth depends on the electrode geometry and must be checked rather than assumed.

current density in the coils scales inversely proportional to the linear dimensions L of the magnet. Since superconductors have an upper limit for the current density j that can be carried, and dissipative coils have an upper limit for j due to the need to remove the dissipated power, j needs to be reduced below that prescribed by simple scaling when L reaches a certain small value that depends, of course, on many details of the magnet design. […] PM, on the other hand, can be scaled to any size without any loss in field strength. From this follows that when it is necessary that a magnetically significant dimension of a magnet is very small, a permanent magnet will always produce higher fields than an electromagnet.

Halbach (Lawrence Berkeley Laboratory), Use of Permanent Magnets in Accelerator Technology: Present and Future — LBL-23556 (1987) — p. 3

Editorial note, tabletop extrapolation: For a small cyclotron the magnetically significant dimensions are the gap (centimetres) and the pole radius; the argument says where permanent magnets gain most. The critical size depends, per Halbach, on field, configuration and available material, so it must be evaluated for the specific design rather than assumed.

Cited in: Cyclotron Magnet Design

for this kind of solenoidal field magnet, the line integral of H along the whole vertical centerline is, because of Ampere's Law, always identically zero.

Halbach (Lawrence Berkeley Laboratory), Use of Permanent Magnets in Accelerator Technology: Present and Future — LBL-23556 (1987) — p. 10

Editorial note, tabletop extrapolation: A permanent-magnet cyclotron with an axial-injection bore through pole, magnet layer and yoke is this case: the gap field along the axis must be balanced by reversed axial field elsewhere on the axis, so the injection line passes through field reversals that a coil-driven magnet (whose axial H integral equals its ampere-turns) is not forced to impose. Injection optics through the bore should be computed with the full on-axis field.

Cited in: Cyclotron Magnet Design

Eqn. (5) shows the same hard limit on the achievable field strength that charaterizes all pure REC magnets: for given geometric parameters, the obtainable field strength is limited by the properties of the REC material.

Halbach (Lawrence Berkeley Laboratory), Physical and Optical Properties of Rare Earth Cobalt Magnets — LBL-11393 (1980) — p. 8

Editorial note, tabletop extrapolation: Halbach's hard limit is stated for iron-free REC magnets. A PM-energized cyclotron is a hybrid, in which high-permeability poles may permit flux concentration beyond the corresponding open-magnet field; whether they do, and what limits first — pole or return-yoke saturation, the magnet recoil line and demagnetization margin, leakage — has to come from a magnetic-circuit or FEM analysis early in the design.

Cited in: Cyclotron Magnet Design