Cyclotron Modeling Design: 166 Sourced Rules

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

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

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

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

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

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

it was not possible to operate the model magnet at the extremely low (3.5 kilogauss) field levels at which we might like to operate the full scale machine.

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

Editorial note, tabletop extrapolation: Modern translation for the FEMM pipeline: a model (physical or FEM) validated at one excitation does not certify another — iron saturation state changes the profile shape, so re-run the field solution at every planned operating point, especially the lowest, and verify the real magnet across its full excitation range.

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

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

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

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

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

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

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

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

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

Dees were simulated by a capacitor connected from the end of each dee stem to ground... Other circuit components were selected to have approximately the same values as those in a full-scale operation.

Howard (ed.), Electronuclear Research Division Semiannual, period ending 20 September 1953 — ORNL-1663 (1954) — p. PDF p. 19 as cited (printed p. 10, section 'VARIABLE-ENERGY HEAVY-PARTICLE CYCLOTRON')

Editorial note, tabletop extrapolation: A bench-scale dee-stem mockup (pipe sections plus padding capacitors) lets the next machine's oscillator/coupling scheme be raced against alternatives for pocket change - the same measure-on-model philosophy as the deferred filament-coupling rule, one report earlier in hardware form.

The one-quarter-scale model magnet is of the closed-yoke type. ... Its total weight will be 14.4 tons; 12.7 tons will be iron and 1.7 tons will be copper. The magnet gap will be in a vertical plane to provide the greatest access for making field measurements. The pole tips are removable so that shims of any shape can be inserted readily.

Howard (ed.), Electronuclear Research Division Semiannual, period ending 20 September 1954 — ORNL-1795 (1954) — p. 19

Editorial note, tabletop extrapolation: For any next-machine shim-test rig (or a scaled FEMM-validation magnet), design for the measurement campaign: open sightlines for the Hall probe, pole tips that unbolt, gap oriented for jig access - the orientation serving the probe rather than mimicking the final machine is the editorial reading of ORNL's choice.

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

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

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

Performance of the model is considered sufficiently satisfactory to proceed with the full scale design and construction based on the model dimensions.

Anderson, Half-Scale Model Tests on the Three Quarter Wave R.F. System — UCRL-31 (1947) — p. 16

Editorial note, tabletop extrapolation: A next machine's dee/stem/tank is already benchtop-sized, so the transferable form is the mockup itself — a cheap RF-only copy (no vacuum) of the dee-liner geometry, swept with a VNA before the vacuum parts are machined. Same lineage as UCRL-64 and MDDC-1045 already in this collection.

Cited in: Choosing Your Machine

Only the radio frequency circuit was simulated in the model, the vacuum system and purely mechanical equipment was not included. Insulation required for the application of bias voltage to the dee and condenser rotor was not included as it had no radio frequency function.

Anderson, Half-Scale Model Tests on the Three Quarter Wave R.F. System — UCRL-31 (1947) — p. 7

Editorial note, tabletop extrapolation: License to mock up the next machine's RF cavity in bare copper/aluminum on a bench plate - no chamber, no pumps - provided every electromagnetic boundary that shapes the mode is reproduced: RF-current surfaces (liner included), coupling structures, and any dielectric near high fields. A bare-metal model validates resonance and field geometry; Q, loss and breakdown under vacuum still need the real thing.

For 30 kv on the full scale system the above power input figures become 146 kw at 9.25 mc, 132 kw at 12 mc, 146 kw at 15 mc, and 98 kw at 23 mc for continuous operation.

Anderson, Half-Scale Model Tests on the Three Quarter Wave R.F. System — UCRL-31 (1947) — p. 15

Editorial note, tabletop extrapolation: The V^2 term is the live part for power budgeting: measured drive power at a safe low dee voltage extrapolates as (V_target/V_test)^2 on the SAME matched, linear, unloaded cavity - so a 1500-V measurement anchors the 5-13 kV LDMOS requirement, with beam/plasma loading, thermal drift of losses and amplifier efficiency budgeted on top, and the extrapolation ending where multipactor or breakdown begins.

the inductance inherent in the 304-TL triodes is large compared with that in the 9C21 triodes to be used in the final oscillator. ... The power measurements at this stage in the experiments were found to be very inaccurate due to differences in plate capacity on the two 304-TL triodes and the consequent difference in distribution of r.f. currents.

Anderson, Half-Scale Model Tests on the Three Quarter Wave R.F. System — UCRL-31 (1947) — p. 12

Editorial note, tabletop extrapolation: When bench-testing a next machine's RF with a stand-in amplifier or without the real chamber wall, write the fidelity caveats into the test log - the model predicts the cavity, not the parts that were substituted. (The filament-line impedance discontinuity previously listed here is dropped pending re-read - the scan discusses filament-line length effects but not that specific claim.)

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

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

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

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

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

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

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

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

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

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

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

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

modifications to <B> to yield isochronism out to the 29th entry in the radial table and with the flutter field modified by a small amount to smooth out effects of measurement errors. In addition, Fourier components of argument greater than 99 have been dropped since these components are sufficiently small to have a negligible effect on the particle motion. The radial spacing of the table entrys is interpreted as increased by the factor 64/8.75 corresponding to the ratio of pole diameters ... In the Fourier analysis the measured field has been assumed to have perfect 120 [deg] symmetry.

Blosser & Gordon, Computational Study of a Resonant Extraction System for a 3-Sector Cyclotron — MSUCP-9 (1961) — p. 6

Editorial note, tabletop extrapolation: The historical analog of the CadQuery->FEMM->field-map pipeline, plus the habit worth copying exactly as MSU practiced it: document every cleanup applied to the field the tracker ate. For an as-built machine, keep the RAW map too - symmetrizing and smoothing erase the very error harmonics that drive resonances - and use the cleaned copy only for idealized nominal studies; check magnetic similarity (saturation behavior) before radially scaling any model field.

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

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

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

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

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

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

beams entering the extraction region approximately centered on the equilibrium orbit behave as well or better than beams entering with substantial displacement.

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

Editorial note, tabletop extrapolation: The project-level lesson for a next machine — run the cheap simulation before committing hardware to any orbit-dynamics intuition, including intuitions published by people as good as Blosser and Gordon.

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

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

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

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

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

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

The iteration process suggested by Murray and Ratner for calculation of X1 converges slowly for small values of a, and does not converge at all for large a. An iteration process, described below, suggested by M.M. Gordon was used and was found to work well for all cases. ... This iteration is Newtonian in character such that if a given X1 has an error of order e, then X1 given by equation 9 will have an error at order e squared. ... X1 ~ X/(1+a) for X1 small ... X1 ~ X - a for X1 large.

Beal, Computation of Electric Field and Potential of an Idealized Dee Geometry — MSUCP-12 (1961) — p. 7

Editorial note, tabletop extrapolation: Copy the iteration and its initial-guess branch into the tracker's field routine; iterate to a set tolerance - quadratic convergence is local, so the branch guess is what makes it robust - cheap enough to call per integration step, or use once to build a spline.

Table 1 gives values of electric field and potential for a wide range of dee gap arrangements. [tables span PDF pages 11-19]

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

Editorial note, tabletop extrapolation: Unit-test targets for the tracker's gap-field routine AND an independent check on FEMM electrostatic runs: model the same idealized geometry once and agree within a DOCUMENTED convergence tolerance - set by a mesh/domain convergence study against the table's demonstrated accuracy - before trusting FEMM on the real electrode shapes; the table's five printed digits are formatting, not a five-digit acceptance criterion — a five-figure match to the printed table would mean an implementation reproducing the paper's rounding errors (see the rule's dated erratum).

A fixed point computer program written by D.A. Johnson for use on MISTIC was used to calculate the above equations for Ex and Vx at the point (X,0) with alpha<4. [defs: alpha = 1/A ; A = a^2/(1-a^2) ; (pi/2)(k/h) = cosh^-1(1/a) + (1-a^2)^1/2 / a^2]

Beal, Computation of Electric Field and Potential of an Idealized Dee Geometry — MSUCP-12 (1961) — p. PDF 8 (printed p. 5), Sec. IV Results, for the alpha<4 statement; alpha's definition on PDF 7 (printed p. 4), and the k/h relation on PDF 5 (printed p. 2)

Editorial note, tabletop extrapolation: For use on a next machine, the real deviations to check against FEMM are finite dee thickness, the rounded tip, and the curved gap line near the source at small radius - run the comparison over the actual geometry rather than assuming the analytic solution's quality at any radius; near center the ion-source chimney dominates the field regardless.

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

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

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

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

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

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

The design has used a 1/2 scale model, in conjunction with detailed computer calculations, to establish all parameters which can be varied

Rainwater et al., The Columbia University Nevis Synchrocyclotron Major Modification — NEVIS-189 / R-774 / CU-295 (1971) — p. 9

Editorial note, tabletop extrapolation: At tabletop size the "scale model" is the full-size mockup on the bench — cold-test a next machine's dee/stem with a VNA before power exists; the mode-spectrum audit transfers verbatim.

The value of mu used was found experimentally by measuring the field from a precisely known configuration of conductors

Rainwater et al., The Columbia University Nevis Synchrocyclotron Major Modification — NEVIS-189 / R-774 / CU-295 (1971) — p. 13

Editorial note, tabletop extrapolation: A model-calibration pattern for the FEMM pipeline: one known-geometry measurement (a wire loop, a known coil) in the actual gap BENCHMARKS the model at that operating point - repeat at several magnet currents and locations before trusting the saturation model across the map; one point pins one point, not the whole BH curve.

The Q of the model will be 0.7 times the Q of the actual installation and so will require 1.4 times as much power for a given dee voltage.

MacKenzie, Preliminary Report on the “Three Quarter Wave” R.F. System for Frequency Modulated Cyclotrons — AECD-1850, University of California (1947) — p. 6

Editorial note, tabletop extrapolation: Bench-model a dee-stem or resonator geometry at reduced size before cutting full-size copper, applying the sqrt(scale) Q correction to power comparisons - and determine coupling separately, from impedance or measured external Q: the power ratio says nothing directly about what a tap or loop must pick up.

It was not excited satisfactorily due to the fact that at 100 megacycles the transit time effects were quite noticeable on the fundamental mode.

MacKenzie, Preliminary Report on the “Three Quarter Wave” R.F. System for Frequency Modulated Cyclotrons — AECD-1850, University of California (1947) — p. 6

Editorial note, tabletop extrapolation: Cold measurements scale approximately - eigenfrequencies and field patterns follow geometry, while Q, losses, contacts and probe loading need corrections or direct measurement. Any POWERED model test needs a driver checked for transit angle and loading at the model frequency; otherwise the model exhibits modes the real system never sees, and vice versa.

dimensions can be calculated fairly exactly whereas in the system shown in Figure 5 one must depend on model tests (which are safer anyway).

MacKenzie, Preliminary Report on the “Three Quarter Wave” R.F. System for Frequency Modulated Cyclotrons — AECD-1850, University of California (1947) — p. 6

Editorial note, tabletop extrapolation: Transmission-line formulas ignore end effects, bends, and support hardware; for any resonator whose geometry is not a textbook line, a cheap model measurement outranks the calculation it checks.

The 1/2 scale model uses about 400 watts input to the oscillator at 48 megacycles and 600 watts input at 18 megacycles to produce 1500 volts on the dee.

MacKenzie, Preliminary Report on the “Three Quarter Wave” R.F. System for Frequency Modulated Cyclotrons — AECD-1850, University of California (1947) — p. 16

Editorial note, tabletop extrapolation: Dee power scales as voltage squared: measure watts-per-volt-squared on the bench and the amplifier requirement for any target voltage falls out. Mind the quantity - the model figures are OSCILLATOR INPUT, so the scaled 28-42 kW carries the model oscillator's efficiency inside it: separate wall loss from drive-chain overhead when budgeting a modern amplifier (dg-313, dg-316), and budget for joint quality and surface material shifting Q.

It was desirable to know the magnetic field accurately to within 0.5 in. on the full-scale magnet. This corresponded to 1/32 in. on the models.

Wakerling & Guthrie (eds.), Magnets and Magnetic Measuring Techniques — TID-5215, Radiation Laboratory, University of California (1949) — p. 73

Editorial note, tabletop extrapolation: For Hall-mapping a shim edge: the sensor's active area averages across the gradient, so where the gradient scale approaches the sensor size, either model the convolution or verify with a smaller probe; jig position repeatability sits in the same budget as its own line. Neither substitutes for the other.

a linear scale model built from steel with the same magnetic properties as planned for the prototype magnet and operated at the same field strength will give results directly applicable to the prototype.

Wakerling & Guthrie (eds.), Magnets and Magnetic Measuring Techniques — TID-5215, Radiation Laboratory, University of California (1949) — p. 25

Editorial note, tabletop extrapolation: The physics that lets FEMM stand where models stood — and the terms of validity are the same for both: correct B-H data and correct geometry. Any cheap sub-scale mock-up of a planned magnet obeys it too, provided the steel matches and B is held, not NI.

since the X Beta model was built to 1/8 scale it seemed true that the results would be more accurate than those which could be obtained on a 1/16-scale model.

Wakerling & Guthrie (eds.), Magnets and Magnetic Measuring Techniques — TID-5215, Radiation Laboratory, University of California (1949) — p. 149

Editorial note, tabletop extrapolation: The mesh-refinement decision in physical form: coarse resolution for excitation, force and leakage questions; fine resolution only for the finest field-uniformity region - spending fine-model effort on questions the coarse model already answers is waste in either medium. Accuracy also rides on geometric similarity, material scaling and construction error, not scale alone.

The usual measurements made included a magnetization curve, a comparison of gap performance at different points in the magnet, uniformity contour maps, determination of magnetic forces, the density of flux through various parts of the magnet

Wakerling & Guthrie (eds.), Magnets and Magnetic Measuring Techniques — TID-5215, Radiation Laboratory, University of California (1949) — p. 26

Editorial note, tabletop extrapolation: A ready-made deliverables checklist for a FEMM campaign on a new magnet: produce the quoted five (B-H behavior, gap comparisons, uniformity contours, forces, member-by-member flux audit) and add the modern staples - efficiency and leakage accounting and a stray-field map - as the extended set; the point is a defined deliverables list agreed before the runs, not after.

With 3400 oersteds in the gaps the efficiency obtained was 95.4 +/- 2.0 per cent. Within experimental error they were the same at both field strengths. This indicates that the magnet design is rather conservative.

Wakerling & Guthrie (eds.), Magnets and Magnetic Measuring Techniques — TID-5215, Radiation Laboratory, University of California (1949) — p. 140

Editorial note, tabletop extrapolation: A two-point excitation scan (measured B vs I against the linear NI prediction) is the coarse global check on an H-frame - it flags that saturation is happening somewhere, not where: localizing the saturating member takes FEM or local flux measurements. Local saturation can also hide inside an unchanged global efficiency, so treat a clean two-point result as necessary, not sufficient.

By dividing the flux difference between any two loops by the area enclosed in the difference of the two loops, the average leakage flux density in that area can be calculated. Through the proper selection of pairs, either the vertical component of the leakage flux or the horizontal component may be found.

Wakerling & Guthrie (eds.), Magnets and Magnetic Measuring Techniques — TID-5215, Radiation Laboratory, University of California (1949) — p. PDF 142 (printed p. 132) for the quoted 10,000-15,000 gauss judgment (cited page 141 is off by one); method on PDF 140-143 (printed 130-133), Sec. 2.5 'Flux-density Measurements'

Editorial note, tabletop extrapolation: The 10-15 kG working band for structural mild steel is the same number modern small-magnet guidance gives (cf. Wouters; Zickler CAS) — and the loop-audit method is the measurement twin of integrating B over member cross-sections in a FEMM postprocessor: every member gets a number, every number gets a verdict.

The permeability of the material of the full-scale unit is higher than that of the model. This indicates that a slightly better performance could be expected from the full-scale unit than from the model.

Wakerling & Guthrie (eds.), Magnets and Magnetic Measuring Techniques — TID-5215, Radiation Laboratory, University of California (1949) — p. 146

Editorial note, tabletop extrapolation: The sign-audit habit transfers to simulation directly: list every model-vs-hardware difference (B-H table provenance, fillets, packing factor, joint gaps) with the direction it biases the prediction, so measured-vs-predicted discrepancies arrive pre-explained - direction-audited, which is weaker than bounded: a bound needs magnitudes for every term, not just signs.

The magnetic performance of the track is better than predicted from the model experiment.

Wakerling & Guthrie (eds.), Magnets and Magnetic Measuring Techniques — TID-5215, Radiation Laboratory, University of California (1949) — p. 191

Editorial note, tabletop extrapolation: The historical calibration point for a predict-then-verify magnet pipeline: one faithful same-steel scaled-model campaign landed close on global quantities and erred conservative because the prototype's iron out-performed the model's. One campaign is precedent for the METHOD - predict, then verify at full scale - not an accuracy guarantee for models or FEM in general: each pipeline earns its own error bars (dg-080's 3% benchmark, dg-817's first-beam case).

the excellent results obtained in testing track 1 showed that a dimensional check of the shim positions was entirely adequate.

Wakerling & Guthrie (eds.), Magnets and Magnetic Measuring Techniques — TID-5215, Radiation Laboratory, University of California (1949) — p. 191

Editorial note, tabletop extrapolation: The economic payoff of validation, with its boundary drawn correctly: dimensional-only acceptance covers exact replicas made under the same materials, tooling and process - a CHANGED shim geometry is a new magnetic configuration and gets its own field map. In one-off tabletop practice, that means the caliper substitutes for the gaussmeter only when re-making the same part, never when revising it; keep periodic magnetic audits regardless.

It would appear that the only feasible method of making corrections when the necessity arises is by trial and error.

Wakerling & Guthrie (eds.), Magnets and Magnetic Measuring Techniques — TID-5215, Radiation Laboratory, University of California (1949) — p. 181

Editorial note, tabletop extrapolation: A 1944 warning that survives every FEMM run: analysis predicts changes to an as-built field only if the change is modeled as executed - and even then B-H uncertainty, hysteresis, stress and omitted 3-D features can dominate. Plan shimming as measure-cut-measure iterations with FEM as the starting estimate, and keep shim stock adjustable.

the amount of testing required to arrive at a final shim design was much less than anticipated. This was due in part to the fact that the geometry of parts limited the possible variations more closely than was expected.

The University of Washington 60-Inch Cyclotron: Progress and Status Report of Design and Construction — AECU-1951, University of Washington (1951) — p. 17

Editorial note, tabletop extrapolation: With FEMM the model-magnet role is filled by simulation (ucrl-31 showed the scale-model method itself; MacKenzie AECD-1850 the model-test discipline), but the balanced verdict is the lesson — physical iteration budget should go where the computable model is least trustworthy (saturation, real steel, mechanical tolerances).

The steel for both the cyclotron magnet and the model was poured from the same heat and it can be assumed magnetic properties are identical.

The University of Washington 60-Inch Cyclotron: Progress and Status Report of Design and Construction — AECU-1951, University of Washington (1951) — p. 18

Editorial note, tabletop extrapolation: The transferable rule is representative material between test article and final article: a next machine's FEMM model should use a B-H curve measured on the actual purchased steel - on coupons matching the real stock's processing and orientation where possible - not a library curve for the nominal grade; same-heat stock reduces one variability source, it does not guarantee identity after different forging and machining.

showed that the model data could be used as a basis of prediction with confidence.

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

Editorial note, tabletop extrapolation: The FEMM-era version — survey the bare magnet, reconcile with the simulation, THEN machine shims from the reconciled model. Same model-then-verify discipline as MacKenzie's aecd-1850.

there are many uncertainties in the exact determination of the constants of the equivalent circuit ... the calculated values ... serve well as a guide

The University of Washington 60-Inch Cyclotron: Progress and Status Report of Design and Construction — AECU-1951, University of Washington (1951) — p. 82

Editorial note, tabletop extrapolation: A tabletop resonator IS the scale model — build the dee/stem mockup on the bench, measure f and Q before committing to vacuum hardware, and trust lumped calculations as guides not gospel. Berkeley used the same quarter-scale method on the 88-inch (ucrl-9435), which also confirms the Q-degradation-at-joints lesson (ornl-2648).

The circuit elements of the rf system were adjusted so that the first two higher modes would not be excited by an oscillator harmonic.

Smith, The RCA 6949 as a Self-Excited Cyclotron Oscillator — UCRL-9435, Lawrence Radiation Laboratory (1960) — p. 5

Editorial note, tabletop extrapolation: For a fixed-frequency machine: sweep the dee system on a VNA, list the modes WITH their widths and couplings, and check them against the drive's measured harmonic spectrum - detune offenders with a strap or stub before blaming the amplifier for instability. Same mode-vs-harmonic discipline as the msucp-9 and ornl-2648 lines.

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]

Prechtl & Wolf, Das Lehr-Zyklotron COLUMBUS — Mit einem Teilchenbeschleuniger Physik und Technik erleben, Springer (2020) — p. 32-33

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.

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]

Prechtl & Wolf, Das Lehr-Zyklotron COLUMBUS — Mit einem Teilchenbeschleuniger Physik und Technik erleben, Springer (2020) — p. 42-43

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.

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]

Prechtl & Wolf, Das Lehr-Zyklotron COLUMBUS — Mit einem Teilchenbeschleuniger Physik und Technik erleben, Springer (2020) — p. 54

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.

Nach einem Ansatz von Maxwell wird dies durch einen Faktor √2 im Nenner berücksichtigt [tr.: following Maxwell this is accounted for by a factor sqrt(2) in the denominator]

Prechtl & Wolf, Das Lehr-Zyklotron COLUMBUS — Mit einem Teilchenbeschleuniger Physik und Technik erleben, Springer (2020) — p. 80-81

Editorial note, tabletop extrapolation: For a beam ion crossing slow gas the stationary form is the better physical model (relative speed ~ ion speed) - but only with the right sigma: gas-kinetic hard-sphere values are not loss cross-sections, so a beam-loss budget needs sigma_loss(E) from evaluated data (dg-460), whatever the sqrt(2) bookkeeping.

Der differentielle Ionisierungswirkungsquerschnitt σ liegt in der Größenordnung von 1–3 1/(cm·mbar) [tr.: the differential ionisation coefficient is of order 1-3 per cm mbar]

Prechtl & Wolf, Das Lehr-Zyklotron COLUMBUS — Mit einem Teilchenbeschleuniger Physik und Technik erleben, Springer (2020) — p. 63-64

Editorial note, tabletop extrapolation: Compare with measurement honestly: at 2-5 mA emission the sigma=1 estimate gives ~0.3-0.7 uA against measured 1-3 uA - a factor of a few, not an order of magnitude, and the gap closes further once the CHIMNEY pressure (well above chamber pressure) and the sigma range are used. The formula omits extraction efficiency and losses in both directions - calibrate it per source rather than reading it as floor or ceiling.

Diese Gleichung gilt allerdings nur unter folgenden Voraussetzungen [tr.: this equation holds only under the following assumptions]

Prechtl & Wolf, Das Lehr-Zyklotron COLUMBUS — Mit einem Teilchenbeschleuniger Physik und Technik erleben, Springer (2020) — p. 84-85

Editorial note, tabletop extrapolation: Evaluate the sum numerically for your actual k (from injection energy, final energy and effective gain per crossing - higher voltage means FEWER crossings for the same energy); phase slip and gap curvature make the real path longer than the formula, so add margin before comparing with the mean free path.

It was assumed that the magnetic field strength at all points would scale linearly with the magnetic field at the center.

Cressman, The Design and Construction of a Small Cyclotron — Houghton College thesis (2006) — p. 36

Editorial note, tabletop extrapolation: One map plus one excitation curve replaces a full map at every operating point, a large time saving; the shortcut degrades as iron saturates, so maps taken near maximum excitation should be spot-checked rather than scaled.

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

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

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

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

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

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

The change of differential- and circuit-equations is completed as long as the calculated ions meet a virtual obstacle. Such an obstacle can be the ion source, a shield or the vacuum chamber itself.

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

Editorial note, tabletop extrapolation: Encoding the source housing and chamber wall as kill surfaces in an orbit code is a cheap way to predict which starting phases survive the first turns and where lost beam lands.

As shown in Fig. 1 the simulation consists of three sections, the input-layer, the simulation-layer and the presentation-layer. … The input-layer incorporates three groups of parameters which define the ions, the experiment and the geometry of the cyclotron. So the simulation can be easily adapted to different situations. … The simulation engine described above has three modules for the evaluation: XY-Plot, Intensity-Plot and Spectrometer-Plot. Every module - apart from the graphical output - allows also a data export to a file so that the measured values can be processed by external programs.

Wolf, Rueß & Prechtl, Simulation and Detection of the Helical Ion-Paths in a Small Cyclotron — MOB02, Proceedings of Cyclotrons2016 (2016) — p. 1, 2

Editorial note, tabletop extrapolation: Separating ion, experiment, and geometry parameter groups lets one code serve several machine configurations and several experiments; the data-export hook is what allows direct overlay of simulated and measured detector scans.

If B = 66 mT, the simulation predicts a set of closely spaced trajectories of H2+ - ions that have 1/5 of the maximum possible speed. This meets the corresponding measurement

Wolf, Rueß & Prechtl, Simulation and Detection of the Helical Ion-Paths in a Small Cyclotron — MOB02, Proceedings of Cyclotrons2016 (2016) — p. 3

Editorial note, tabletop extrapolation: This sharpens the false-beam trap for I(B) sweeps on hydrogen machines: a strong cup signal at an unexpected field may be partially accelerated H2+ on closely spaced slow orbits, and a first-turns simulation predicts where such impostor peaks should appear. Treat the simulation as one hypothesis, not a verdict - a Faraday cup is not species-resolving, so confirmation needs a q/m-dependent field-frequency scan or another species-sensitive check before a peak is accepted or discounted.

In the corresponding plot one can see how close the orbits are to each other. So it is obvious that at an Intensity-Plot I = I (x) in reality will be an almost continuous chart as shown in Fig. 6. This is qualitatively confirmed by the model, too.

Wolf, Rueß & Prechtl, Simulation and Detection of the Helical Ion-Paths in a Small Cyclotron — MOB02, Proceedings of Cyclotrons2016 (2016) — p. 3

Editorial note, tabletop extrapolation: Absence of clean turn separation in a radial probe scan is not by itself evidence of a fault: when turn spacing is small against beam width and probe resolution, a smeared continuous profile is the expected signature - though phase and energy spread, emittance and detector response smear it further.

the quantitative analysis of the results must be considered very carefully, because there are some assumptions in the simulation that are not met in reality, such as a constant ion beam or the conservation of particles, which is not satisfied, due to the lack of focusing. Finally, the measurement accuracy in the determination of some parameters was not considered in the simulation. Nevertheless, the present simulation offers qualitatively a good idea of the acceleration process

Wolf, Rueß & Prechtl, Simulation and Detection of the Helical Ion-Paths in a Small Cyclotron — MOB02, Proceedings of Cyclotrons2016 (2016) — p. 3

Editorial note, tabletop extrapolation: When a home-built tracker and a probe scan disagree in amplitude, the model's stated idealizations (constant current, no losses) are one candidate cause - so are field maps, geometry, RF phase and calibration; check both sides. Use the simulation for locations and trends, not absolute currents, and treat even peak locations as unvalidated until compared against measurement.

Initially, a program which analytically calculated the indirect and direct magnetic fluxes from various candidate configurations was used to define the dimensions of the magnets, poles, and yoke. The computer program POISSON was then used to verify and optimize this solution

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

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: the analytic pass explores the design space cheaply; the FEA pass is reserved for verifying one or two survivors. Free 2-D solvers fill the POISSON role today — noting that a 2-D axisymmetric model verifies the nominal design only, so discrete-block and assembly asymmetries need 3-D modeling or a measured field map.

Cited in: Cyclotron Magnet Design

they are injected axially using a spiral electrostatic inflector, Figure 3. The inflector geometry has been optimized with the computer codes CASINO, RELAX3D and Poisson so that the emittance of the ion beam coming out of the inflector matches the acceptance of the cyclotron

Clark, Halbach, Kunkel, Leung, Li & Young, A Compact Permanent Magnet Cyclotron for Accelerator Mass Spectrometry — Proceedings of Cyclotrons'95, Cape Town (1995) — p. 2

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: emittance matching at the inflector exit — the output phase-space distribution oriented so it lies within the cyclotron acceptance, not mere geometric survival — is the design criterion; the code roles (electrode field solve plus 3-D trajectory integration in the real fields) map onto modern open tools.

calculations of the magnetic field using the computer program POISSON indicate that the field should be uniform to approximately +/- 2 parts in 104 throughout the acceleration region, and +/- 1 part for the majority of the trajectory

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

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: axisymmetric 2-D FEA predicts the nominal field of an azimuthally symmetric PM magnet only — segmentation, assembly and material-variation errors are 3-D and need their own tolerance analysis or a measured map. The companion as-built paper (dg-1553) measured 7e-4 pre-trim, 3.5x this prediction, attributed to deliberately oversize barrel magnets awaiting cut-back — prediction and measurement reconcile only through that shim provision, not as direct agreement.

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

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

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

The magnet prototype is 150 mm in length. At room temperature (20 °C), the measured central magnetic field is 0.7545 T, and the integrated field is 0.13964 T·m. These measurements (Fig. 3) closely align with the simulation predictions, confirming the accuracy of both the magnetic and mechanical design ... [Figure 3 caption:] Z scan of magnetic field measurement

Hsu, Jan, Chu & Lin, Integrating Permanent Magnets and Electromagnets — A Hybrid Dipole Magnet Design — WEBD3, Proceedings of IPAC2025 (2025) — p. 2

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: measurement-versus-simulation agreement on a field scan is a core acceptance test that closes a PM magnet build — one test, complemented as applicable by alignment and repeatability checks, integrated-field or multipole mapping, and temperature characterization. Quoting the measurement temperature alongside the value is essential practice for PM systems because of their temperature coefficient.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The gradient, ∂B/∂r, of the magnetic field of the ISU cyclotron was measured with the field and gradient meter developed by Thoburn (5). The magnetic field, B, was obtained by numerical integration of this gradient.

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

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: orbit quantities (focusing, phase slip) depend on the gradient, and numerically differentiating a noisy point-by-point field survey amplifies error — measuring the gradient directly, or fitting before differentiating, is the robust order of operations for gradient-dependent quantities. The integration to B(r) needs an absolute anchor (a calibrated field value at some radius) and accumulates probe baseline and spacing errors, so check the integrated map against independent absolute-field measurements. A two-coil differential probe is buildable at hobby scale.

Actually, several important phenomena have been neglected in these calculations. For one, phase grouping, as described by Cohen (6), has not been considered ... Also, the assumption that all protons with negative initial phases would be lost from the beam and that no protons with positive initial phase would be lost from the beam because of defocusing is not entirely justified ... The intensity curves shown in Figure 4 are expected to give a correct qualitative description of the beam in the cyclotron and to provide a rough quantitative description

Mueller, Proton Orbit Calculations for the Iowa State University Cyclotron — Proceedings of the Iowa Academy of Science 68(1), 492–501 (1961) — p. 8, 9

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: declaring a model's omissions up front turns experiment-theory disagreement into information — each discrepancy gets a candidate attribution to a listed omission instead of eroding trust in the whole calculation. These are proposed explanations to test (add effects one at a time, compare residuals), not validated causes. This is the working pattern for pairing simple orbit models with commissioning data.

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

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

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

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

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

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

A magnetic field approximation developed by Dr. D. E. Hudson was used in this study. This relationship is shown graphically in Figure 2; the mathematical expression is: (2) B(r) = [17,000 + 0.25cm−2r2 + 0.232cm−3r3 − 0.0118cm4r4 + 6.21cm−11.6 10−10 r11.6] gauss.

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: Editorial note, tabletop extrapolation: an analytic field fit gives an orbit integrator smooth, differentiable input — critical because focusing depends on dB/dr — and the polynomial-plus-steep-power form captures the flat-center/sharp-edge shape typical of small unshimmed poles. The same functional form fits modern FEA field maps. Before using any transcribed fit, verify it reproduces the source's own quoted landmarks (here, n = 0.2 at r = 10.1 cm).

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.

the electric field is considered as concentrated in a region of zero width at the center of the dee gap. The dee-to-dee voltage is defined as V; therefore, a proton will receive a boost of energy, ∆E = eV, when it crosses the dee gap. It is assumed that the momentum is altered only in the direction perpendicular to the dee gap and parallel to the median plane ... During this instantaneous acceleration r, and z are not altered ... If an orbital radius were to exceed that of an initial deceleration, the proton energy would increase since velocity and radius are proportional; however, this contradicts the loss of energy in deceleration. Hence, it is assumed that once a proton suffers an electric deceleration it will never reach a greater energy.

Moses, Proton Orbits in a Small Cyclotron — Proceedings of the Iowa Academy of Science 70(1), 403–414 (1963) — p. 7

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: the impulse-at-gap approximation is the standard trick that keeps a homebrew tracker fast — integrate smooth magnetic motion between gaps and apply discrete energy kicks; this study checked its implementation against one-step analytic predictions. First-deceleration is the study's termination convention, backed by its radius-energy argument — for another machine verify that argument holds (or track on to exclude later recovery), and benchmark the zero-width gap against a finite-gap/transit-time estimate where the gap is not small compared to the orbit.

Since the solution of the actual cyclotron problem is not known, a problem was devised for which a solution could be found simply ... It was checked by entering a uniform magnetic field. The results should be circular motion with r0 = mv/eB. There should be no change in the coordinates of the apogees and perigees (no precession), and the average angular velocity should be that predicted by the cyclotron equation ... The program for axial motion was checked by entering a constant term for the gradient (dB/dr). The maximum of |z| should be constant and the frequency of oscillations should be that predicted (ωz = φ̇ √n(r0)). The program that simulates the electric accelerations was checked by entering it many times, as in a lengthy orbit study. The results of each acceleration were used as initial conditions for the next. Then the cyclotron equation and momentum transfer hypotheses were used to predict the final results in one step. The two sets of results were then compared for accuracy.

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

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: a minimum smoke-test suite — each physics module gets a limit case with a known closed-form answer before the modules are combined, and it still catches sign, unit and stepsize errors that field-map runs would mask, for an afternoon's cost. A modern tracker adds timestep/order convergence, invariant-drift checks, the relativistic rho = p/(qB) benchmark, and field-map interpolation tests on top of these.

All of the tests made on the program indicated that the errors resulting from computational methods would not be as significant as those due to the experimentally determined quantities such as field values. This does not include the errors resulting from the magnetic field approximation for points away from the median plane of the dee box. These errors can be studied when more is known about the magnetic field shape.

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

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: in this verified calculation, measured-field uncertainty was judged to dominate computational error — the transferable habit is stating which error source dominates and which remains unquantified (here, the off-median-plane field approximation was explicitly deferred). For a new tracker, test integration convergence, map interpolation and measurement uncertainty separately before deciding where effort goes; integrator error can still dominate with unsuitable algorithms or step sizes.

The magnitude of za is the amplitude of the axial oscillations. Note that Equation 5 is linear; therefore, the units of za are arbitrary. Hence, only the direction of za is significant in forming a uniform distribution ... values of σz′ were chosen to be σz′ = Nπ/8 where N = 0, 1, 2, ... 7. Eight calculations were performed while only σz′ was varied. Other integral values of N would give the same results with a possible reversal in sign. The true axial motion is not quite simple harmonic, but this gives a reasonable distribution approximation ... the computer outputs r and φ at apogees and perigees in radial positions. It gives z and φ at maxima of |z|. It also outputs r, φ and θ at selected dee gap crossings ... their relations to one another are more useful to study since these have more experimental significance ... only one graph of r versus φ is needed when variations in σz′ are considered

Moses, Proton Orbits in a Small Cyclotron — Proceedings of the Iowa Academy of Science 70(1), 403–414 (1963) — p. 8, 9

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: exploiting linearity and symmetry collapsed a two-parameter scan to eight runs on a 1962 vacuum-tube computer — the same economy matters when scanning initial conditions with a desktop tracker today, and event-based output (turn-by-turn at gaps and extrema) is what maps directly onto probe measurements. Eight phase samples suffice under the linearized model; a nonlinear tracker needs a denser scan.

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

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

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

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

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

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

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

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

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

The designed field has been modelled with CST Microwave Studio for 3D and POISSON code for 2 D related designs. The final measured value should match the designed value of 1.2 T. Relevant documents of design can be supplied, if the vendor requires during technical bid discussion.

IUAC, e-Tender 09/GOR/2024–25 — H-Dipole Water-Cooled DC Electromagnet for the Table-Top Cyclotron: Engineering Specification and Acceptance Tests (2024) — p. 43

Editorial note, tabletop extrapolation: Editorial note, tabletop extrapolation: confirms that a modern professional teaching-magnet design still rests on a 2D POISSON-class solve cross-checked in 3D — the same two-tier workflow available to amateurs through free 2D field solvers plus selective 3D checks. The acceptance criterion is written against the simulation, making the model the effective contract baseline — though 'match the designed value' is stated without a numeric tolerance, which a real acceptance procedure needs.

A 2-dimensional (X-Z plane) PSF magnetic field file describing the cyclotron's magnet field was imported into SIMION. SIMION then azimuthally rotated the 2D field about the z-axis creating the complete full 3-D volume. The chamber lids, dummy DEE, and ion source chimney were drawn as a single 3D solid in AutoCAD, again imported into SIMION. Finally, because of the simplicity of the DEE geometry, it was drawn in the SIMION graphics editor. […] The magnetic field, DEE voltage, and angular frequency were set to nominal 12-inch cyclotron settings. Ions, of unity mass and charge (i.e. protons) were launched with zero kinetic energy at the position of the aperture.

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

Editorial note, tabletop extrapolation: A workable modeling recipe at exactly this scale: a 2-D axisymmetric magnetostatic solve (Poisson-Superfish then, FEMM now) supplies the field, CAD supplies the electrode solids, and the tracking code rotates the field into 3-D. The zero-kinetic-energy launch from the aperture is a useful BASELINE — it isolates what the geometry alone does to the earliest ions — not a validated convention: before treating the model as predictive, run sensitivities over plausible initial energy, direction, position and RF phase, since real plasma ions carry all four spreads.

[3] Obviously the 2D model was severely limited due to complex 3D geometry, but the model was at least valid in the X-Z plane from the symmetry about that plane, and confirmed a vertical electric field of order estimated above. Taking this calculation further required a full 3D code. […] Want of a 3D E&M modeling code with the ability to fly and track ions prompted the purchase of SIMION - a full 3D E&M particle tracking code.[4]

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

Editorial note, tabletop extrapolation: A scoping pattern, conditionally: where the relevant symmetry plane is defensible (as it was for this dee-and-chimney silhouette), a free 2-D electrostatic solve can confirm the ORDER of a parasitic field — often all a go/no-go decision needs. Reach for 3-D tracking when out-of-plane geometry materially shapes either the field magnitude or the trajectory — which is exactly why this group bought SIMION to learn where their ions actually landed.

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

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

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

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

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

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

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

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

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

The new chimney's wall was thinned near the aperture to increase the amount of field that penetrates into the plasma column. To further take increase the local electric field, angled brass plates were mounted on the face of the DEE near the ion source aperture. The plates were named "pullers" for their obvious role in ion extraction. A simple PSF model showed that the field at the plasma sheath increased by a factor of 760. According to the Langmuir-Childs' (LC) law a 130 fold increase in the peak emitted ion source should result.[5] Already measurements show approximately two orders of magnitude increase, and significantly more is expected once better initial steering is accomplished (discussed later).

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

Editorial note, tabletop extrapolation: The highest-leverage source modification in this memo, entirely within tabletop means: angled brass pullers on the dee face plus a thinned chimney wall near the aperture. The prediction chain is the source's own — modeled sheath field ×760, a Langmuir-Child-based prediction of ×130 in peak emitted current, measured ≈×100 — and its internals are not fully spelled out: a naive I ∝ V^3/2 scaling of a ×760 equivalent-voltage gain would predict far more than ×130, so the source's figure evidently folds in the real extraction geometry. Carry the design move and the measured two-orders-of-magnitude result; re-derive any prediction for your own geometry with your own field model.

It is clear from these views (figures 10 and 14) that the pullers are vertically offset; measurement shows they are 0.32mm high. As a result, the ions are introduced closer to the bottom puller, where the non-zero, off-plane, vertical gradient strongly pulled the beam down. A PSF model in which the DEE and pullers are raised by 0.32 mm illustrates the experienced vertical field; see figure 15.

Koeth, Hanebuth, Hoffman & Schneider, Rutgers 12-Inch Cyclotron Ion Source Studies: Part II (2007) — p. 5

Editorial note, tabletop extrapolation: A sobering tolerance number for tabletop builders: a third of a millimetre of vertical misalignment between puller and aperture was enough to dominate the injection dynamics on a 12-inch machine. The authors' own conclusion is that adjustability, not tighter machining, is the answer — see their planned 4-axis bellows adjuster. Note the figure-based measurement technique (photograph the source through a port, subtract a background image, measure against a known dimension) is itself the cheap part.

This determines the necessary electric field; we must multiply the electric field by the HV electrode-septum gap spacing to determine the required applied potential. Using the values of ρo and ρ1 listed above, we find that for protons in a 1.0 Telsa magnetic field, a transverse electric field of 4.2 MV/m is required. The nominal electrode-septum spacing is 0.31 inches, thereby setting the electrode voltage at 33 kV.

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

Editorial note, tabletop extrapolation: The sizing equation a tabletop extraction design starts from, checked against the printed numbers: with rho_0 = 0.1016 m, rho_1 = 0.1778 m the magnitude comes to 4.17 MV/m, and times 0.31 inch gives 32.8 kV — agreeing with the printed 33 kV (computed). Scaling: with field scaled by b and ALL lengths by s, the required field goes as b²s and the voltage as b²s² — so half the field at two-thirds scale needs ~1/6 the field and ~1/9 the voltage, which is what makes a modest HV supply workable on a smaller machine. ("Telsa" is the source's typo for Tesla.)

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

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

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

Because of the round pole tips, it seemed natural to take the 2D slice of the magnet in the plane where the pole tips were the widest – at one half of the depth of the magnet. Again, the 2D approximation is only valid when the magnet is considered in the non-saturated regime. With a nominal operating field of 1 Tesla this approximation becomes suspect. It should be noted that the pole tip material is fully annealed, hot rolled 1006 steel, possessing a very large µ.

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

Editorial note, tabletop extrapolation: Transferable with the author's own hedges intact: for round poles he took the 2-D slice where the tips are widest (half the magnet depth) — a natural choice for that geometry — and warned the planar approximation 'becomes suspect' at the nominal 1 T because saturation breaks it. Modern practice softens the cliff: include real B-H data and validate against measurement or a 3-D solve near the knee (dg-1691). Fully annealed hot-rolled 1006, chosen here for its very large µ, is the pole-tip material of record.

made to be denser (thus higher resolution) in the region of interest, namely, between the poles, while setting a less dense mesh for regions of little interest. A greater number of horizontal mesh lines, as compared with vertical mesh lines, were required to resolve the slight inclination of the pole tips.

Koeth, Report on the 12-Inch Cyclotron Magnet Study: Measurements, Modeling, and Future Plans (c. 2005) — p. 5

Editorial note, tabletop extrapolation: Concrete meshing guidance for exactly this problem: the taper physics lives in a 0.008-inch gap change over 5 inches (the retrofit spec, dg-1680), so resolution along the gradient direction is what buys a correct modeled field index — in Poisson/Superfish that meant more horizontal than vertical mesh lines. The principle transfers to FEMM as LOCAL refinement in the gap and along the tapered pole boundary (its unstructured triangles have no line-count knob); in any code, finish with a mesh-convergence check on Bz and dBz/dr before trusting n(r).

The coil current density was empirically set. The construction of the actual 12-inch cyclotron coils is unknown, as the coils came from a surplus source. The current density was varied in PSF through several points, until the peak 1.22 Tesla was achieved in the center of the gap. This corresponded to a PSF setting of 30,000 Ampere-turns. ... A comparison of PSF’s output with the linear portion of the actual measured B(i) curve can yield insight into the construction of the coils, which was determined to be about 850 windings per coil.

Koeth, Report on the 12-Inch Cyclotron Magnet Study: Measurements, Modeling, and Future Plans (c. 2005) — p. 5

Editorial note, tabletop extrapolation: A recoverable-datasheet method for surplus coils: fit a magnetostatics model's excitation until it reproduces the measured field, then read effective turns from matched ampere-turns over the linear region — N = (fitted A-turns)/I, with the per-coil-versus-total convention stated explicitly, which this memo leaves ambiguous: 30,000 A-turns over 850 turns implies ~35 A on a per-coil reading, while the document's stated ~32 A nominal (dg-1687) with 850 turns gives 27,200 — a bookkeeping tension to resolve on your own magnet, not an error to copy. The 850 turns is the inferred construction of THESE coils, not sizing guidance.

Fig.6 PSF B(i) curve, note lack of saturation ... Fig.7 Actual measured B(i) curve

Koeth, Report on the 12-Inch Cyclotron Magnet Study: Measurements, Modeling, and Future Plans (c. 2005) — p. 5

Editorial note, tabletop extrapolation: A cautionary pair at the target scale: the same 2-D model that matched the measured radial field SHAPE missed the excitation curve's saturation knee entirely — as run, evidently without material nonlinearity doing its job. The correct lesson is narrower than 'knees cannot be modeled': a nonlinear 2-D solve with a real B-H curve can capture saturation (3-D leakage it cannot), so give the code proper steel data, then validate BOTH B(i) and the field shape against measurement through the knee. (Measured curve endpoints — roll-over above ~0.03 kA, ~1.17 T at 0.05 kA — read from the rendered Fig. 7, whose x-axis is printed in kA.)

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

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

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

As shown in Figure 13 the profiles of the measured field and the modeled field are precisely matched in the region utilized for acceleration. This is an encouraging result, as pointed out earlier; the measured field profile followed a single line directly facing the magnet, while the modeled profile followed a single line 90o azimuthally from the measured path. If there was to be a discrepancy between the measured and modeled data, it would have been expected to be at a maximum difference between these two paths. A discrepancy does become pronounced at a radius greater than 6-inches, the “lower” strength field is the measured field. This is just as one would expect, as the measured path does does not have a vertical yoke piece to coral in the field lines, and thus they leak out easier.

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

Editorial note, tabletop extrapolation: A validation result with a built-in lesson about where the comparison stops being fair: measured (open-side azimuth) and modeled (yoke-side) profiles matched precisely inside the acceleration region and split beyond 6 inches, the open side reading lower — no yoke there to corral the return flux. Practice for an H-frame: take scans at several azimuths, compare like against like where possible, quantify residuals, and EXPECT 2-D/3-D disagreement in the fringe — interior agreement on one cut is encouraging, not proof.

The first set was a simple, pure-radial sector design of periodicity four. Their geometry was the least expensive to machine and provided the maximum field variation achievable within practical constraints. Not expected to host beam, their purpose was to benchmark simulations, measurement techniques, and test analysis code.

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: A process rule worth more than most hardware numbers: build the cheap, geometrically simple article first and use it to shake down the toolchain — solver, field mapper, analysis scripts — before spending shop time on the expensive curved part. The radial set rehearses most of the pipeline; what it cannot validate is the spiral-specific machining and edge-field modeling, which the real article still tests (the sequence that produced AKG270, dg-1717).

The newly written MatLab analysis code plots Bz about a circle of any requested radius in 5° increments – the center of the circle is intuitively chosen. Although the data lies on a rectangular grid, a MatLab provided 2-D linear interpolation routine was used to determine the field at any requested location. ... In the weak focusing case, the magnet center was determined by sweeping the center of the circle first in the x and then the y directions. The standard deviation of the values about the measurement circle was calculated and stored. After a sweep in x or y that included the magnet center, the data was fit to a parabola, from which the minimum standard deviation, i.e. the center locations, could be inferred as seen is Figure 4.

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. 3

Editorial note, tabletop extrapolation: A reusable analysis for near-axisymmetric maps: you need not align the probe stage to the magnetic centre — find it afterwards in software by minimizing the azimuthal standard deviation of Bz (sweep the circle centre in x, then y, fit parabolas). The source applies it to the WEAK-FOCUSING case, where azimuthal uniformity is the expectation; on an AVF map the same minimization would chew on real sector harmonics, so centre those maps by fiducials or a symmetry-aware fit (the program's own N-harmonic method, dg-1792). On this magnet the correction moved the centre about half a grid step — (28.0, 27.0) to (28.5, 27.4), read from the rendered Figs. 3-5 annotations — and that half-step separated an apparent azimuthal error from a flat field (dg-1702).

SIMULATIONS Form start to finish, four software tools have been employed to simulate the beam dynamics of in these magnetic fields. SolidWorks was used to mechanically model the magnet, Maxwell 3D was uses to solve the field problem and generate the needed field report for SIMION to fly and track the ions in. Post processing was performed in MatLab. ... Maxwell 3D (M3D) was first benchmarked against our weak focusing PSF simulations. A 3-D magnet model, which included the weak focusing pole tips was designed in SolidWorks and then imported into Maxwell 3D. The problem was solved to have a nominal peak central field of 1 Tesla. To within measurement errors the models agreed.

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

Editorial note, tabletop extrapolation: A four-stage pipeline — CAD, 3-D field solver, tracker, analysis — with the transferable discipline being the BENCHMARK step: before trusting the 3-D solver on new geometry, reproduce the old validated result on the old geometry (here Maxwell 3D reproduced the Poisson/Superfish weak-focusing field within measurement errors — the FIELD model, not the tracking chain, is what that comparison validates). Free-tool substitutions: FEMM only where a planar/axisymmetric approximation is defensible — a radial-sector AVF field is intrinsically 3-D, so budget for Elmer or another 3-D solver there — and verify the field-transfer and tracking layers separately (dg-1712's trap).

The Maxwell 3D current was nearly set the same, differences between the two resulting average field reports were aligned by normalizing the simulated data central Bz value to exactly match the measured central value. ... Figure 14. Comparison of measured and simulated average field of the radial sector tips. Good agreement is noted over the range of the ions travel, at most 1% deviation is seen at r=2.5.

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

Editorial note, tabletop extrapolation: A quantified model-versus-measurement figure at the target scale, precisely bounded: after normalizing the simulated central Bz to the measured value, the AVERAGE-FIELD SHAPE agreed within 1% over the ion region (worst at r = 2.5 in). That is shape validation, not absolute-excitation validation — and not yet flutter, harmonic or tune validation, which need their own comparisons (dg-1721 does the 2-D map). The 45,000 A-turns for 1 T with these sector tips versus the Poisson model's 30,000 for 1.22 T with solid tips is suggestive of what valleys cost, but the two figures come from different codes and endpoints — measure the penalty on matched geometry before budgeting it. (The 45,000 A-t / 1 T / 0.25-inch-step statements are on p.5; the normalization sentence and Fig. 14 caption are on p.6.)

The resolution of the imported field has been set at 1 mm to conveniently match the 1mm:1 SIMION grid unit ratio. The M3D report file is a 6-column a comma separated variable file reporting x, y, z, Bx, By, and Bz at each grid point, with a spacing of 1-mm between grid points. To fully cover the ion accessible region in our cyclotron, the field region must span a volume with a radius up to 4.5 inches – the point of interception of deflector. Therefore the extent of the report spans ±115 mm (~ 4.55-inches) X ±115 mm X (~ 4.55-inches) X 53 mm (~ 1.04-inches) which contains 231 X 231 X 53 rows of data, an excess of 2.8 million points - causing the simple text data to become unwieldy, in excess of 500 MB.

Koeth, Hine, Hoffman, Krutzler, Ponter, Rosenberg, Ruisard & Schneider, Comparison of Azimuthally Varying with Constant Gradient Magnetic Fields with the Rutgers 12-Inch Cyclotron (2011) — p. 7

Editorial note, tabletop extrapolation: Concrete sizing for a tracker's field-map file: 1 mm resolution over the ion-accessible volume of a 12-inch machine is 231 × 231 × 53 points — 2.8 million rows, over 500 MB as text — so plan a binary or compressed intermediate format from the start. The printed axial figures do not reconcile (53 mm ≈ 2.09 in, yet the parenthetical prints "~1.04-inches", plausibly a half-extent; unresolved in the source — inspect your own file's z bounds rather than inferring). Size the map to cover the COMPLETE tracking domain out through every loss surface and relevant fringe region, not merely the aperture the beam is supposed to occupy.

The Maxwell 3D magnetic field’s median plane is the x-y plane while the median plane is y-z in SIMION. A careful vector rotation is required in translating the M3D report into the usable SIMION file. ... While the magnetic field is being loaded, SIMION populates the vector field by cycling through x the fastest, y the second, and finally z. This requires data sorting that cycles through x for every increment of y, and cycles through y once per increment of z.

Koeth, Hine, Hoffman, Krutzler, Ponter, Rosenberg, Ruisard & Schneider, Comparison of Azimuthally Varying with Constant Gradient Magnetic Fields with the Rutgers 12-Inch Cyclotron (2011) — p. 7

Editorial note, tabletop extrapolation: A high-cost trap for any solver-to-tracker bridge, in its general form: axis conventions between two codes are YOURS to reconcile, the mismatch is silent — the file loads, the ions fly, the answer is wrong in a way that looks like physics — and row-order-encoded coordinates mean a sorting error produces a plausible-looking scrambled field. Both argue for the same insurance: smoke-test every new bridge on a known analytic field (a uniform B, a simple dipole) before believing anything it produces.

Multiple ions can be launched together, however for these trace space simulations it was found best, especially while searching for the stability limits, to track single ions. The SIMION simulation run terminates once the ion is lost. An ion is declared lost if it exceeds the boundary of the DEE structure or falls outside of the magnetic field volume. If the ion’s orbit is stable, it will continue to circulate indefinitely and the simulation will need to be manually terminated.

Koeth, Hine, Hoffman, Krutzler, Ponter, Rosenberg, Ruisard & Schneider, Comparison of Azimuthally Varying with Constant Gradient Magnetic Fields with the Rutgers 12-Inch Cyclotron (2011) — p. 8

Editorial note, tabletop extrapolation: Practical tracker-design advice for a tabletop orbit code: define loss against real apertures (the dee, not an abstract radius), and build in a turn-count or wall-clock stop, because a stable orbit is an infinite loop. The single-ion preference when mapping stability boundaries is a workflow point — bunches obscure which initial condition failed.

As is seen in figure 18, the weak focusing field had the smallest trace space area, the radial AVF had the greatest, and the spiral AVF field was slightly less than the radial sector. It is also interesting to note the appearance of the grouping in several of the incomplete trace space contours, this indicates that the ratio of vertical fraction of an oscillation to the revolution frequency is near a rational fraction, however, given sufficient time they would completely fill in their contour. These trace space orbits are exhibiting near-resonant behavior of the order of the grouping number.

Koeth, Hine, Hoffman, Krutzler, Ponter, Rosenberg, Ruisard & Schneider, Comparison of Azimuthally Varying with Constant Gradient Magnetic Fields with the Rutgers 12-Inch Cyclotron (2011) — p. 9

Editorial note, tabletop extrapolation: A free screening diagnostic from plots a tracker user already makes: once-per-turn trace-space points clumping into n groups suggest a tune near a rational p/n — near-resonant behaviour of about that order. It is a clue, not a verdict: finite tracking, aliasing and plotting cadence can also group points, so confirm by extending the run and extracting the turn-by-turn phase advance (or a spectrum) before naming the resonance. (Attribution flag: the source cites 'figure 18' — captioned Radial Trace Space — inside its vertical-stability paragraph; Figure 19 is the vertical plot, so the printed figure number is almost certainly a misprint and the comparison is of vertical trace space.)

The AKG270 radial trace space was explored first to identify the equilibrium orbits in 50 keV increments. Even, at 50 keV, non-linear behavior is noted in the larger stable orbits, exhibiting four-sided contours, behavior which is consistent with pole tips that have a sector periodicity of four. The corners of the four-sided nonlinear orbits become more pronounced and bulbous with increasing energy. By 250 keV four closed contours formed in the protracted corners, displayed in Figure 23. Thus, in addition to the primary Equilibrium Orbit, there are four off-center stable orbits.

Koeth, Hine, Hoffman, Krutzler, Ponter, Rosenberg, Ruisard & Schneider, Comparison of Azimuthally Varying with Constant Gradient Magnetic Fields with the Rutgers 12-Inch Cyclotron (2011) — p. 10

Editorial note, tabletop extrapolation: A phenomenon to look for on any sectored machine, from this worked case: the four-sector AKG270's radial phase space showed four-sided nonlinear contours already at 50 keV, sharpening with energy until four closed islands formed by 250 keV — genuine off-centre stable orbits alongside the primary one. Whether YOUR sector count produces islands, and at what energy, depends on the field harmonics and tunes: survey trace space at energy steps fine enough to resolve your calculated resonances (50 keV was this study's choice), and follow with RF-on tracking to learn whether real accelerating beam gets captured by them (a beam parked on an island reads as mis-steered, dg-1793).

Both plots were normalized such that the peak central fields were 1 Tesla – this required at most a 5% adjustment to either data set. Figure 29 subtracts the measurement from the simulation. ... Figure 29. Subtracting the measurement from the simulation reveals 14% discrepancies at the outer pole tip edge. The two agree within 1% in the ion region.

Koeth, Hine, Hoffman, Krutzler, Ponter, Rosenberg, Ruisard & Schneider, Comparison of Azimuthally Varying with Constant Gradient Magnetic Fields with the Rutgers 12-Inch Cyclotron (2011) — p. 11

Editorial note, tabletop extrapolation: The most useful validation figure in this pair of documents, precisely bounded: after each map was normalized to a 1 T central peak (≤5% adjustment either way), the SHAPES agreed within 1% over the ion region and split by 14% at the outer pole-tip edge — cause not identified by the source, with fringe and edge effects the natural suspects but unproven. Budget trust accordingly: normalization means absolute solver accuracy is NOT bounded by the 1%, and the pole edge — exactly where an extraction deflector sits — earned measurement on this magnet and will on yours. (The measurement grid: 1/8-inch step at 30 A, from the rendered Figs. 27-28 plot titles.)

The project slated for Spring 2012 will develop a beam phase measurement probe. This experiment measures the beam arrival time with respect to the RF cycle. Measuring variations in the beam’s arrival time along a radial line is a method of characterizing the field’s isochronism. At the time of this writing, the authors are exploring an FFT based method to derive the radial and axial tune values from the SIMION simulations. Such a method would be quicker in the evaluation of the magnetic field configurations, reserving the tedium of trace space plot generation for only the most promising candidates.

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. 12

Editorial note, tabletop extrapolation: Both items are stated as the authors' intent at the time of writing, not results, and should be read as such. The beam-phase-probe method is nevertheless a described technique a tabletop builder can adopt: radial variation in arrival phase is a direct, measurable isochronism error. The FFT tune-extraction point is a workflow recommendation for anyone running an orbit tracker — screen candidate fields by tune, then spend trace-space effort only on survivors.

We recall from section II that the vertical betatron frequency follows the square root of the field index multiplied by the RF frequency: f_beta-vert = sqrt(n) f_0 We extend that relationship to their respective periods of oscillation: T_beta-vert = (1/sqrt(n)) T_0 Thus for a given n it take 1/sqrt(n) RF periods or ion revolutions to complete one vertical betatron oscillation.

Koeth, Hanebuth, Schneider & Hoffman, Observation of Betatron Motion in the Rutgers 12-Inch Cyclotron (2006) — p. 6

Editorial note, tabletop extrapolation: The hand calculation that tells a builder how many TURNS per vertical oscillation to expect: 1/√n turns (equal to RF periods only on fundamental-harmonic operation, h = 1, as here; at harmonic h it is h/√n RF cycles). With this machine's measured n ≈ 0.025-0.042 near 8.6-9.7 cm, that is about 4.9-6.3 turns per oscillation — a local estimate where n varies. The quote's equation glyphs are transcribed in plain-text form here. (The nu_x = sqrt(1-n) statement and the radial-stability remark are on p.2; the nu_z material is on p.6.)

The profiling of the radial dependence of the magnetic field between the pole pieces was executed with a Hall probe mounted on a computer controlled motorized platform. A LabView program wrote the Hall probes value and probe's position into a text file. ... The LANL Finite Element code Possion Superfish's (PSF) [6] output was compared to our measurement. Strong agreement between the McClain & Friedman's measurement with the PSF justified the use of the computer model for further analysis, see figure 3. [3]

Koeth, Hanebuth, Schneider & Hoffman, Observation of Betatron Motion in the Rutgers 12-Inch Cyclotron (2006) — p. 3

Editorial note, tabletop extrapolation: The measure-then-validate-then-model workflow to copy with FEMM or Superfish: the model earns trust for downstream analysis only after a mapped Hall-probe profile agrees with it (here the downstream use included the field-index work of the following sections). Note the source prints "Possion Superfish" (a typo for Poisson Superfish); the quote is transcribed as printed.

Operating at 300 Watts of RF power on resonance at 14.8640 MHz (Corresponding to a B-field of 0.977 Tesla), the DEE Vp-p that develops is 7,500 V, thus ∆E is 7,500eV. Taking q=1.6E-19, and m=1.67E-27, so we can expect: ... ∆r(r) = (8.2E-5) 1/r

Koeth, Hanebuth, Schneider & Hoffman, Observation of Betatron Motion in the Rutgers 12-Inch Cyclotron (2006) — p. 6

Editorial note, tabletop extrapolation: The single most useful sizing formula for probe and cup design: turn spacing at any radius from dee voltage and field — it sets how thin an intercepting tip must be and whether turns separate on a screen. Conditions: nonrelativistic ions, approximately uniform B, small per-turn gain, with ΔE the effective energy gain per turn (this machine's single-dee convention takes it as the 7,500 V peak-to-peak; multiple gaps or off-crest phase change it). Caution: the printed substitution line shows the charge as (1.6E-27) in the denominator, a source misprint for 1.6E-19 (the text above states q=1.6E-19); recomputing with 1.6E-19 reproduces the printed 8.2E-5 coefficient.

Using a fourth order polynomial fit and our equation for ∆r(r) we can create table 1. The first measured peak of the vertical betatron oscillation was at ro = 8.6cm, and we denote that as the start of the betatron period. We then allow one RF period, hence one ion revolution, to process, after which, using our equation for ∆r(r), we reevaluate the new radius and that radius' field index n. It can easily be shown that the fraction that the betatron period advances at a given n is just sqrt(n). … Integer values of fractional betatron periods indicate the full completion of a vertical betatron oscillation. … Noting the radii at which these occur the reader immediately sees the same values that were observed at r1 and r2 as reported in section V.

Koeth, Hanebuth, Schneider & Hoffman, Observation of Betatron Motion in the Rutgers 12-Inch Cyclotron (2006) — p. 6

Editorial note, tabletop extrapolation: A worked piecewise-tracking recipe implementable in a spreadsheet — no orbit code — and checked by its authors against the streak photo: integer betatron periods land at the observed r1 and r2. The tabulated n values over 8.6-9.7 cm run 0.025 to 0.042 (Table 1, read from the rendered page). Table 1 prints 9.2 cm in two consecutive rows (n = 0.031 and 0.033) — most likely rounding of nearby unrounded radii rather than a misprint.

The edge-focusing adds a term to νz2 depending on the "flutter" (mean square deviation of B(θ) ... where <B> is the θ-averaged axial magnetic field. ... The spiraling changes the edge crossing angles and the sector focusing contribution to the axial tune becomes ν2sector = F2 (1 + 2tan2 ε) where F is defined in Eq. 3 and ε is the spiral angle.

Hernalsteens, Ponter, Beaudoin, Koeth, Ruisard & Miller, Betatron Tune Characterization of the Rutgers 12-Inch Cyclotron for Different Magnetic Poles Configurations (2016) — p. 2

Editorial note, tabletop extrapolation: The minimum analysis needed to turn a measured or simulated AVF field map into a predicted axial tune for a tabletop machine. The quoted line reflects the PDF's text-layer rendering of typeset superscripts; the equation as set on the page is nu_sector^2 = F^2 (1 + 2 tan^2 epsilon).

These data were in turn fitted with an harmonic signal using a moving window technique. The fitting technique takes the measurement errors into account: lower weights were associated with the data points corresponding to large beam envelopes, as the determination of the centroid of these data points are not as precise as those were the beam is at a focus. Image-by-image examination of the fit quality showed that the best results are obtained using a 5-point window. That relatively low number of data points allowed to extract frequency information for many radial position as the typical signal length is around 15 turns long.

Hernalsteens, Ponter, Beaudoin, Koeth, Ruisard & Miller, Betatron Tune Characterization of the Rutgers 12-Inch Cyclotron for Different Magnetic Poles Configurations (2016) — p. 2

Editorial note, tabletop extrapolation: Calibration context from the Rutgers analysis: their typical signal was ~15 turns and image-by-image checks favoured a 5-point moving window. On another machine, pick the window by fit residuals and uncertainty (synthetic-signal tests are cheap); the transferable part is the method — error-weighted moving-window harmonic fits with envelope-dependent weights — not the two numbers.

Fourteen pole piece conceptions were modeled during the semester long project. … After examining field profiles and particle motion, the original design was adjusted or discarded, and a new design analyzed identically. Due to the short project duration (1 semester), each of the 3 students established competency in one program and worked as a team in interpreting results.

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 transferable part is the loop and its discipline: CAD → field solver → profile/flutter inspection → tracking → adjust or discard, then re-analyse identically — with the labour split so nobody had to master every tool. Fourteen concepts in one semester is what three students at a university managed with that structure; treat it as an existence proof of the loop's throughput, not an amateur productivity quota.

This configuration demonstrated a reasonably flat profile, with a variation of ~4% from 1.5 out to 4 inches

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. A reference AVF geometry at exactly tabletop scale — four sectors, Archimedean, 270 degrees of sweep, ~4% average-field flatness over the 1.5–4 inch working annulus on THIS 12-inch magnet. The same spiral cut for a different gap, excitation or yoke will not reproduce the flatness; re-run the 3-D model and map the result (the paper's own loop, dg-1788). SUSPECTED SOURCE MISPRINT: Fig. 4's x-axis is labelled "radius [mm]" but runs 0–6, which is inches on a 12-inch (6-inch-radius) pole; read it as inches, consistent with the text's own "1.5 out to 4 inches".

Difference analysis reveals a 14% variation between simulation and measurement. However, the discrepancy is <1% within the ion region.

Ruisard, Hine, Koeth & Rosenberg, The Rutgers Cyclotron: Placing Student's Careers on Target — WE1PB02, Proceedings of Cyclotrons2013 (2013) — p. 4

Editorial note, tabletop extrapolation: PDF p.4 = printed p.294. A rare model-versus-measurement pair at tabletop scale, and the lesson is the split: a 14% global mismatch coexists with sub-1% agreement where the beam lives. Score a FEMM/Elmer validation over the ion region so a usable model is not condemned by its periphery — but keep the full-aperture residual map and read it: where the big errors sit (and whether they are fringe, boundary or saturation artifacts) matters for extraction and for trusting the model's edges.

The geometric center was identified using an FFT-based analysis that maximizes the fourth harmonic and minimizes all others.

Ruisard, Hine, Koeth & Rosenberg, The Rutgers Cyclotron: Placing Student's Careers on Target — WE1PB02, Proceedings of Cyclotrons2013 (2013) — p. 4

Editorial note, tabletop extrapolation: PDF p.4 = printed p.294. A purely computational alignment method for anyone with a mapped field: for an N-sector pole, choose the origin that concentrates power in the N-fold symmetric harmonics (N and its multiples are legitimate structure; everything else is error or mis-centering). It removes the guesswork from registering a hand-built mapper's frame to the pole — then cross-check against mechanical registration and, once beam exists, closed-orbit behaviour, since a construction error can put the symmetry center away from the orbit center.

Additional SIMION studies identified 6 kV peak voltage and 15.534 MHz frequency as the optimal working point for proton transport.

Ruisard, Hine, Koeth & Rosenberg, The Rutgers Cyclotron: Placing Student's Careers on Target — WE1PB02, Proceedings of Cyclotrons2013 (2013) — p. 4

Editorial note, tabletop extrapolation: PDF p.4 = printed p.294. Shows a tabletop-scale RF operating point being chosen from tracking rather than by trial: a dee-voltage/frequency pair reported to the nearest kilohertz (15.534 MHz) with its 6 kV partner. The transferable practice: the tracker picks the working point before the machine is fired, and the optimum is jointly a voltage AND a frequency. The pair itself belongs to this field map — re-derive yours from your own model.

Our phase shift observations agree qualitatively with simulation, but absolute agreement is yet to be achieved.

Gonski, Burcher, Lazarov, Krutzler, Koeth & Beaudoin, A Novel Optical Method for Measuring Beam Phase and Width in the Rutgers 12-Inch Cyclotron — WE1PB04, Proceedings of Cyclotrons2013 (2013) — p. 3

Editorial note, tabletop extrapolation: PDF p.3 = printed p.301. Honest calibration expectation for a tabletop builder running a tracker against a real machine: trends matched, absolutes did not. The input the authors chose to go measure was the dee voltage — the same poorly-known quantity on most amateur machines — but field map, source initial conditions and RF phase are equally capable of owning an absolute discrepancy; calibrate the important inputs before assigning blame to one.

This was verified by simulation: 385, 405, and 425 keV ions were admitted to the deflector resulting in a comparable target image … The bands are compilations of revolutions. Ions with sufficient radial extent in the nth turn are captured by the channel and form the outer (right most) band in Fig. 8. Those not intercepted continue on for another revolution, nth+1, of acceleration, and thus have a greater rigidity and hence are deflected less forming the second band, and so it goes for the third band, or nth+2 turn. … Considering the finite width of the deflector entrance slit and channel, the resolution has been calculated to be 10% at 500 keV.

Koeth, Rosenberg, Krutzler, Ponter, Schneider & Hoffman, Rutgers 12-Inch Cyclotron: Dedicated to Training Through Research and Development — WEPPT024, Proceedings of Cyclotrons2013 (2013) — p. 3

Editorial note, tabletop extrapolation: PDF p.3 = printed p.368. A small machine can display individual turns as separate bands on one screen — the source's own account: outermost band the nth turn, successive bands nth+1 and nth+2 at greater rigidity, verified by admitting 385/405/425 keV ions in SIMION. Two metrics must not be conflated: adjacent-band SEPARATION (~20 keV here) is an image-structure statement, while the calculated 10% at 500 keV (~50 keV) is the absolute-energy resolution set by the entrance slit and channel width — so the screen resolves turn structure without being a 20 keV spectrometer. Band spacing tracks energy gain per turn; converting it to dee volts needs the gap-crossing count and phase, not just the image.

A full 3D SIMION model has been developed and extensively verified with every configuration of our cyclotron.

Koeth, Beam Physics Demonstrations with the Rutgers 12-Inch Cyclotron — WEPPT025, Proceedings of Cyclotrons2013 (2013) — p. 1

Editorial note, tabletop extrapolation: PDF p.1 = printed p.369. The Rutgers pattern worth copying: one maintained 3-D model kept in step with every hardware configuration, rather than a fresh single-purpose simulation per experiment. Read across this collection, the payoff shows up as predictions that preceded hardware — the radial-sector phase-slippage failure, the AVF operating point, the deflector turn bands (each carried on its own card with its own source). Note the companion paper we1pb04 reports qualitative, not absolute, agreement for phase, so 'verified' is trend-level where checked.

The beam has been accelerated in the 3D magnetic and 3D RF electric field maps with initial amplitudes of betatron oscillations up to 5. Acceleration takes about 120 turns beginning from energy 1 MeV to 15 MeV. Total number of turns with 20 kV accelerating voltage was equal 125 turns. There were no losses of particles in any radius.

Karamyshev, A New Design of PET Cyclotron — IPAC2022, paper THPOMS016, JACoW (2022) — p. 2

Editorial note, tabletop extrapolation: Method transfers: an unconventional field or RF layout warrants tracking through computed 3-D field maps before hardware commitment. A tabletop machine with tens to hundreds of turns at sub-kV to few-kV dee voltage is equally checkable this way.

The preliminary dimensions of the magnet and the properties of the equilibrium orbits were first obtained using hard edge approximation. The primary size of the magnet was estimated using two-dimensional POISSON code. Finally a 3D code was utilized for the field optimization.

Pandit, Sing Babu, Goswami, Sarma et al., Magnetic Field Calculations of a 10 MeV High Current Compact Cyclotron — APAC 2007, paper WEPMA005, JACoW (2007) — p. 1

Editorial note, tabletop extrapolation: The same staged sequence (analytic hard-edge → 2-D FEM → 3-D FEM) fits a tabletop magnet. An H-frame is not axisymmetric, so its 2-D stage is an approximation either way — an axisymmetric pole model for the gap field or a planar cross-section for the yoke — and the return yoke, coil ends and any sector or shim geometry need the 3-D stage with a convergence check against the 2-D result.

Since the vertical focusing frequency is quite sensitive to the fringe field at the hill-valley boundary, we have modified the classical hard edge formula to include the soft edge effect.

Pandit, Sing Babu, Goswami, Sarma et al., Magnetic Field Calculations of a 10 MeV High Current Compact Cyclotron — APAC 2007, paper WEPMA005, JACoW (2007) — p. 2

Editorial note, tabletop extrapolation: Applies to any sector-focused tabletop study: a hard-edge estimate can misestimate vertical focusing materially, in either direction depending on how edges and fringe integrals are defined, so vertical tune should be checked against a soft-edge or FEM field before committing sector geometry.

The analysis shows that the phase excursion in the entire region is limited within ± 10deg.

Pandit, Sing Babu, Goswami, Sarma et al., Magnetic Field Calculations of a 10 MeV High Current Compact Cyclotron — APAC 2007, paper WEPMA005, JACoW (2007) — p. 3

Editorial note, tabletop extrapolation: The paper calls ±10° a reasonable limit for its design; a tabletop machine can use the same integrated-phase metric to set the field-flatness tolerance it shims to, choosing its own budget from its turn number and acceptance.

cross section for "destruction" or loss of ions or fast neutrals in m2.

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. 4

Editorial note, tabletop extrapolation: The exponential form is the integrated version of the step-wise dN = sigma*n*N*v*dt estimate; it matters once losses exceed ~10 %, where a linear estimate overstates the loss. Example arithmetic: n = 3.3e17 m^-3 at 1e-5 Torr of H2 and 293 K, and Q_d = 9.6e-20 m^2 (a proton near 7 keV) give alpha ~ 0.03 m^-1, i.e. about 3 % loss per metre of path near the cross-section peak. The relation alpha = n*Q_d is the standard beam-in-static-gas definition; Phelps's own swarm coefficients for drifting ions are not this quantity.

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.

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 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.

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.

bining the VNA measurements with the Rshunt/Q figure obtained from the finite element 3D model. Table 1 summarizes these results. […] Table 1: Low Power Measurements Under Vacuum — Minimum frequency (plungers up) 60.150 MHz; Maximum frequency (plungers down) 60.375 MHz; QL 2,333; Q0 4,763; Coupling factor 1.042; Rshunt 303 kΩ; Expected forward power at 60 kV 5.93 kW; Expected pickup power at 60 kV 32.0 dBm

Gavela et al. (CIEMAT), Fabrication and Tests of a RF Cavity for a Novel Compact Superconducting Cyclotron for Radioisotope Production — IPAC2017, paper THPVA062, JACoW (2017) — p. 3

Editorial note, tabletop extrapolation: Applicable to a tabletop dee: R/Q depends on geometry and comes from a full-wave eigenmode model (or a validated lumped derivation) of the dee, Q from the VNA; their product gives Rshunt = (R/Q)·Q0 under the source's voltage conventions and so the voltage-to-wall-loss scale. The pickup scale needs the measured pickup coupling or a calibrated S21 as well — it does not follow from R/Q alone. Cross-check the calibration by an independent method such as an X-ray endpoint (dg-2052 ff.), noting that electrons need not sample the full gap voltage and that the measurement deliberately produces X-rays requiring radiation controls.

An accurately determining of the maximum energy of X-ray spectrum was difficult because of background radiation from the induced activity of the internal elements of the cyclotron (Fig. 5).

Issatov, Kabytayeva et al. (FLNR JINR), Calibration of Measurement of Absolute Dee Voltage for U-400M Isochronous Cyclotron at FLNR JINR — RuPAC 2018, paper TUPSA31, JACoW (2018) — p. 2

Editorial note, tabletop extrapolation: Fitting a modelled endpoint shape rather than reading the last channel by eye reduces the endpoint's sensitivity to background; on a machine without beam-induced activation the background may instead be dominated by room radioactivity, as Miller & Kashy (1975) found at MSU. The FLUKA step is the U-400M authors' method; a simpler thick-target bremsstrahlung shape could serve the same purpose at low energy, which the paper does not test.

Since the two-dimensional model does not include the effect of fringe-field loss in the third dimension, it over estimated the central field by about 40%. Therefore the results calculated from the two-dimensional model shown here were normalized to fit the data. […] The final design was modeled in three-dimensions using the TOSCA magnet design code

Barlow, Kraus & Meyer (Los Alamos National Laboratory), Variable-Field Permanent-Magnet Dipole — LA-UR-93-3237 (1993) — p. 5

Editorial note, tabletop extrapolation: For a permanent-magnet design, where the driving material cannot be turned up after assembly, an absolute-field estimate needs a 3-D model or a measured correction; 2-D modelling is useful for preliminary shape studies but must be validated against 3-D modelling or measurements for the actual geometry, since fringe effects can change relative trends as well as the normalization. The 40 % belongs to this short C-dipole with 10 cm poles; a round cyclotron pole has a different 2-D/3-D relationship that must itself be checked.

Cited in: Cyclotron Magnet Design

It is assumed throughout, that the material is never driven beyond the knee of that curve.

Halbach (Lawrence Berkeley Laboratory), Use of Permanent Magnets in Accelerator Technology: Present and Future — LBL-23556 (1987) — p. 4

Editorial note, tabletop extrapolation: The design procedure for a permanent-magnet cyclotron must check the working point of every block, including blocks in reverse-field regions near the gap and pole edges, against the supplier's demagnetization curves over the entire specified temperature range and under all adverse-field conditions (assembly, fault, nearby ferromagnetic tools); the maximum temperature is often but not universally the limiting case — ferrites can be more vulnerable when cold. The 1987 remanence range predates today's NdFeB grades; use the supplier's curves for the actual material.

Cited in: Cyclotron Magnet Design

magnet properties can be described, in addition to "frozen magnetization" (Fig. 2) and B(H) curve (Fig. 3), by an anisotropic permeability very close to 1, plus either an impressed current or charge distribution, which, for the usual case of homogeneously magnetized material, corresponds to current or charge sheets on the surface. […] I design such systems with a design code that consists of an appropriately linked set of analytical formulas. This is possible only because the material is "analytical."

Halbach (Lawrence Berkeley Laboratory), Use of Permanent Magnets in Accelerator Technology: Present and Future — LBL-23556 (1987) — p. 9

Editorial note, tabletop extrapolation: An analytic flux-budget model is a legitimate first design tool for a permanent-magnet cyclotron magnet, fast enough to explore block layouts before a finite-element check; its accuracy depends on the material staying in its linear region and on the iron being far from saturation.

Cited in: Cyclotron Magnet Design

In real life, this assumption is violated only very slightly, particularly because the most damaging deviations from vacuum fields are sensitive only to the difference in differential permeability in the directions parallel and perpendicular to the easy axis.

Halbach (Lawrence Berkeley Laboratory), Physical and Optical Properties of Rare Earth Cobalt Magnets — LBL-11393 (1980) — p. 2

Editorial note, tabletop extrapolation: Superposition is what allows block-by-block field contributions to be summed, sorted and swapped during assembly and shimming. It holds for the magnet material itself; once iron poles are added (as in a cyclotron) the iron is nonlinear and the full model needs a field solver or an unsaturated-iron assumption.

Cited in: Cyclotron Magnet Design

If one replaces all REC inside a closed scalar potential surface by high permeability steel, the performance of the magnet will not be affected as long as the permeability of the steel is large enough.

Halbach (Lawrence Berkeley Laboratory), Physical and Optical Properties of Rare Earth Cobalt Magnets — LBL-11393 (1980) — p. 5

Editorial note, tabletop extrapolation: A cyclotron gap is the extreme wide, flat dipole, so the steel pole of a PM-energized cyclotron can be read by analogy as this substitution — valid where a closed constant-scalar-potential surface of the corresponding all-magnet solution can be identified and the steel inside it keeps high permeability. Avoiding saturation alone is not sufficient; verify the proposed pole substitution with a magnetic-circuit calculation or FEM rather than assuming that only the material outside the pole need be magnet.

Cited in: Cyclotron Magnet Design

each gradient magnet was measured with a Morgan (rotating) coil. Harmonics up to the 12-pole were measured. A FORTRAN program was written to calculate the shape of an end shim for each magnet that would adjust the quadrupole, skew quadrupole, sextupole, and octupole [6]. The program generated machine code for a wire EDM (Electric Discharge Machine) that cut the exact shape required for each magnet. The shims were ready within 24 hours after the first measurement. In general one iteration was required for each magnet. In rare cases two iterations were needed.

Volk (Fermi National Accelerator Laboratory), Experiences with Permanent Magnets at the Fermilab Recycler Ring — FERMILAB-TM-2497 (2012); arXiv:1209.4324 — p. 7

Editorial note, tabletop extrapolation: The loop - measure, compute a shim from a field model, machine it, re-measure - transfers from these ring gradient magnets to removing a PM cyclotron's first harmonic or trimming its radial profile, where there is no coil current to adjust; the Recycler's end-shim geometry and harmonic set do not transfer. The number of iterations a cyclotron needs depends on how well its field model predicts shim effect.

Cited in: Cyclotron Magnet Design

When an iron ‘wire’, infinite in z with a small circular cross-section, is inserted into a locally uniform magnetic field B0, it becomes magnetised with M = 2B0, assuming perfect iron with µr = ∞.

Brooks, Mahler, Cintorino & Jain (Brookhaven National Laboratory), Production of Low Cost, High Field Quality Halbach Magnets — IPAC2017, paper THPIK007, JACoW (2017) — p. 2

Editorial note, tabletop extrapolation: The 2-D wire formula is specific to shims in the bore of BNL's Halbach multipole magnets; in a cyclotron gap, small iron pieces on or near the pole act the same way in principle, but their effect on the median-plane field has to come from a field solver or measurement.

Cited in: Cyclotron Magnet Design

Assembled magnets such as those in Figs. 3 and 4 were measured on BNL’s rotating coil and the observed errors added on to the 2D field model. Then between 32 and 64 iron wires were added to the simulation just inside the magnet bore and their radii optimised in order to reduce the harmonic errors to zero again, which was achieved almost exactly when wires up to 63mil (1.6mm) diameter were allowed.

Brooks, Mahler, Cintorino & Jain (Brookhaven National Laboratory), Production of Low Cost, High Field Quality Halbach Magnets — IPAC2017, paper THPIK007, JACoW (2017) — p. 2

Editorial note, tabletop extrapolation: The method avoids tight block tolerances by correcting the as-built magnet, which suits a PM cyclotron assembled from catalogue blocks: map the assembled field, add the measured error to the model, and solve for shim sizes. What transfers is the loop; the bore-wire geometry and rotating-coil measurement belong to the Halbach quadrupoles, and a cyclotron needs a gap field map and a model of shims on or near the pole.

Cited in: Cyclotron Magnet Design

The optimiser (for both the wedges in the BD magnet and the iron wire radii) is based on an SVD decomposition of the linear response matrix of the simulated harmonic errors to small changes in the magnet.

Brooks, Mahler, Cintorino & Jain (Brookhaven National Laboratory), Production of Low Cost, High Field Quality Halbach Magnets — IPAC2017, paper THPIK007, JACoW (2017) — p. 2

Editorial note, tabletop extrapolation: The same linear-response approach can in principle be applied to cyclotron shimming: tabulate how each candidate shim changes the first harmonic and the radial field profile, then invert with regularisation so small, noisy measured errors do not demand large shims. The source's goal fields are FFAG multipoles; using an isochronous radial profile as the goal is an extension it does not test.

Cited in: Cyclotron Magnet Design

A value Br1 < Br is used to compensate for the fact that µr ≃ 1.02–1.05 in reality, which reduces |M|.

Brooks, Mahler, Cintorino & Jain (Brookhaven National Laboratory), Production of Low Cost, High Field Quality Halbach Magnets — IPAC2017, paper THPIK007, JACoW (2017) — p. 2

Editorial note, tabletop extrapolation: A fast fixed-magnetisation model of a PM cyclotron's block array could be calibrated once against a full nonlinear solve and then used for shim and sorting optimisation, as Brooks et al. did for Halbach multipoles without iron; it does not replace the nonlinear solve where a cyclotron's iron poles saturate.

Cited in: Cyclotron Magnet Design

it is not necessary to specify the gas.

Sharipov & Seleznev, Data on Internal Rarefied Gas Flows — J. Phys. Chem. Ref. Data 27, 657 (1998) — p. 21-22 (printed 676-677)

Editorial note, tabletop extrapolation: With G_P read from the tables, the mass throughput of a long, uniform tube follows from its radius, the local pressure gradient and β for the gas. At equal G, pressure gradient, radius and temperature Ṁ scales as β ∝ m^(1/2), so a light gas such as hydrogen carries a smaller mass flow but a larger pV throughput Ṁ·k_B·T/m ∝ m^(−1/2) (computed, not in the source); G itself also differs between gases at the same pressure because δ contains μ and m. The result holds for a long tube (L = l/a ≫ 1); short stubs and apertures need the finite-length treatment in later sections.

Equations (3.9) have been obtained without any assumption on the pressure and temperature drops. Even with large drops, the representation (3.9) remains valid.

Sharipov & Seleznev, Data on Internal Rarefied Gas Flows — J. Phys. Chem. Ref. Data 27, 657 (1998) — p. 19-20 (printed 674-675)

Editorial note, tabletop extrapolation: A long, thin pumping or gas-feed line whose two ends sit at very different pressures can still be treated with the local-δ tables, provided its length is many radii, so that the pressure changes little over one radius everywhere along it (the review's |ξ_P| ≪ 1); the review notes the assumption may be violated near the ends and neglects that only for long capillaries; δ then changes along the line and the flow must be integrated, not evaluated at one pressure.

any isothermal gas flow can be successfully calculated with the help of the BGK model

Sharipov & Seleznev, Data on Internal Rarefied Gas Flows — J. Phys. Chem. Ref. Data 27, 657 (1998) — p. 13, 28 (printed 668, 683)

Editorial note, tabletop extrapolation: For pressure-driven conductance of an isothermal line, BGK-based tables or a BGK solver are what the review considers adequate; for estimates involving temperature gradients (thermal transpiration between a warm gauge and a cold line) the review recommends the S model for monatomic gases only. For a polyatomic gas such as hydrogen it calls for special model equations, so a BGK or S-model creep value there is not supported by the review.

The great shortcoming of the method is that one cannot estimate the error of the variational solution.

Sharipov & Seleznev, Data on Internal Rarefied Gas Flows — J. Phys. Chem. Ref. Data 27, 657 (1998) — p. 11-12, 16-18 (printed 666-667, 671-673)

Editorial note, tabletop extrapolation: A test-particle Monte Carlo of a port or baffle is the review's free-molecular method and assumes gas–gas collisions can be discounted. At higher pressure in the same geometry the review describes transition-regime methods instead, such as the discrete-velocity method (good results over the entire Kn range) or DSMC. The review's two DSMC conditions are a summary, not a complete recipe: a DSMC result still needs its own convergence checks in cell size, time step and particle number.

This approach is justified only in the free-molecular regime. In the transition regime both approaches give approximate results which can be used for an estimate.

Sharipov & Seleznev, Data on Internal Rarefied Gas Flows — J. Phys. Chem. Ref. Data 27, 657 (1998) — p. 19 (printed 674)

Editorial note, tabletop extrapolation: For a hydrogen feed with residual water and nitrogen, the review's free-molecular approach is to compute each species' flow separately, from that species' own partial pressures and the single-gas theory, and to sum the flows; the conductance coefficients themselves are not summed. The approach is justified only where the passage is free-molecular for its own geometry, which high vacuum alone does not establish (check δ for the passage, dg-2171); at transition pressures the result is an estimate only.

One can see that the disagreement between the model equations (BGK and S model) and the exact BE is within 2%.

Sharipov & Seleznev, Data on Internal Rarefied Gas Flows — J. Phys. Chem. Ref. Data 27, 657 (1998) — p. 24-25 (printed 679-680)

Editorial note, tabletop extrapolation: For a long narrow slot (height a, width ≫ a), the per-unit-width reduced flow rate at fixed pressure gradient passes through a shallow minimum near δ ≈ 1 (1.5379 at δ = 1.1), so the slot's conductance per unit gradient varies non-monotonically with pressure there.

model equation solutions and that obtained from the BE is within 2%, which can be considered reasonable.

Sharipov & Seleznev, Data on Internal Rarefied Gas Flows — J. Phys. Chem. Ref. Data 27, 657 (1998) — p. 26-27 (printed 681-682)

Editorial note, tabletop extrapolation: For a long round tube the reduced flow rate dips about 8.5 % below its free-molecular value near δ ≈ 0.3 and then rises. A constant free-molecular G = 1.5045 stays within about 10 % of the Table 7 BGK values up to δ ≈ 2 and is about 20 % low at δ = 3 (computed, not in the source: 1.5045/1.6608 = 0.906, 1.5045/1.8850 = 0.798).

since they do not coincide with all the rest of the data, even with the data obtained later by himself with collaborators

Sharipov & Seleznev, Data on Internal Rarefied Gas Flows — J. Phys. Chem. Ref. Data 27, 657 (1998) — p. 27-28 (printed 682-683)

Editorial note, tabletop extrapolation: If a gas–wall pair has α well below 1, a diffuse-wall conductance for a long line near the molecular end can be low by tens of percent (about 20 % at α = 0.90 and δ = 0.01, Table 9 against Table 7, computed: 1.791/1.4768 = 1.21). The size depends on α, which the review notes is usually measured indirectly through capillary flow rates.

obtained by the two quite different methods are in good agreement between them

Sharipov & Seleznev, Data on Internal Rarefied Gas Flows — J. Phys. Chem. Ref. Data 27, 657 (1998) — p. 25 (printed 680)

Editorial note, tabletop extrapolation: The same accommodation sensitivity applies to slot-shaped passages; the effect remains visible even at δ = 10, so it is not confined to the molecular end of the pressure range.

Comparing (3.88) and (3.89) one can see that the coefficients of the asymptotic formulas essentially depend on the kinetic model equation.

Sharipov & Seleznev, Data on Internal Rarefied Gas Flows — J. Phys. Chem. Ref. Data 27, 657 (1998) — p. 33–34, 36 (printed 688–689, 691)

Editorial note, tabletop extrapolation: For a long circular tube in laminar, small-gradient flow at the high-pressure end of its range, the local reduced coefficient G*P approaches δ/4 plus a slip constant of order one; a real roughing line with a large pressure drop needs that coefficient integrated along it (dg-2174), and the result does not cover bends, valves or turbulent flow; the 1/δ terms are where model choice shows. The thermal creep term σT/δ falls off with pressure, so creep effects in a long line weaken as δ grows.

can be successfully used to calculate the mass flow rate G_ΔP^tb at any pressure difference if and only if T_I = T_II.

Sharipov & Seleznev, Data on Internal Rarefied Gas Flows — J. Phys. Chem. Ref. Data 27, 657 (1998) — p. 36, 38 (printed 691, 693)

Editorial note, tabletop extrapolation: For a long pumping line (length many radii) held at one temperature between a chamber and a pump inlet at different pressures, the review's shortcut estimates the throughput with one small-drop coefficient taken at the mean of the two end δ values. The review states it without an error bound; where the pressure ratio is large and the answer matters, integrate the local coefficient along the line (Eq. 3.108).

The data given in Tables 11–20, 22, 24, and 25 can be applied to monatomic gases only.

Sharipov & Seleznev, Data on Internal Rarefied Gas Flows — J. Phys. Chem. Ref. Data 27, 657 (1998) — p. 39–40, 48 (printed 694–695, 703)

Editorial note, tabletop extrapolation: Isothermal pressure-driven conductance data from the review carry over to a hydrogen gas load; thermal creep, thermal slip and TPD values at intermediate δ do not carry over on the review's own reading. A hydrogen thermal-transpiration correction in the transition regime needs a polyatomic-specific source or a measurement.

The formulas cover the entire range of the rarefaction parameter

Sharipov & Seleznev, Data on Internal Rarefied Gas Flows — J. Phys. Chem. Ref. Data 27, 657 (1998) — p. 41, 43 (printed 696, 698)

Editorial note, tabletop extrapolation: For a slit in a plate whose thickness is negligible beside the slit width, at small pressure difference, these formulas give conductance relative to its molecular value across the whole range; at δI = 1 it is already about 1.4 times the molecular value in the BGK table. A gap with real depth is a channel, not a slit (dg-2180, dg-2198).

The results by Clausing are overstated for large L.

Sharipov & Seleznev, Data on Internal Rarefied Gas Flows — J. Phys. Chem. Ref. Data 27, 657 (1998) — p. 44–45, 48 (printed 699–700, 703)

Editorial note, tabletop extrapolation: A tube or nipple in molecular flow conducts W times the conductance of an orifice of the same radius; with L defined on the radius (not the diameter), a tube as long as its diameter (L = 2) passes about half the orifice value (0.5142). The review prefers Eq. 5.9 to Clausing's values, which are overstated at large L.

Since the capillary ends create an additional resistance for the flow, the idea was to substitute the real length by some effective length.

Sharipov & Seleznev, Data on Internal Rarefied Gas Flows — J. Phys. Chem. Ref. Data 27, 657 (1998) — p. 46–47 (printed 701–702)

Editorial note, tabletop extrapolation: For the review's ideal case, a straight circular tube between large reservoirs in viscous flow at small pressure drop, a short tube behaves like a long one lengthened by about 3π/8 ≈ 1.18 radii (computed, not in the source: 3π/8 = 1.178). In the transition regime the review supports the effective-length shortcut only for the channel and only when δI·L ≥ 20. The added length is not a correction for other entrance geometries.

There are not any rigorous theoretical data for the gas flow through a tube of finite length.

Sharipov & Seleznev, Data on Internal Rarefied Gas Flows — J. Phys. Chem. Ref. Data 27, 657 (1998) — p. 47-48 (printed 702-703)

Editorial note, tabletop extrapolation: A straight cylindrical tube of finite length in the transition regime had only empirical formulas behind it on the 1998 review's account, so results from these formulas are estimates. They describe a straight tube characterised by L = l/a with diffuse walls and say nothing about elbows, reducers, valves or centring-ring restrictions. The review's §5.6 states the G_P finite-length data and Eq. 5.21 apply to polyatomic gases.