Design Guide › Space charge
Cyclotron space charge design rules
9 of the guide’s 2170 rules carry the space-charge tag.
Rules for the beam's own field: the vertical-tune floor a milliampere-class machine sets against space-charge defocusing, and how far that requirement relaxes at the nanoampere-to-microampere currents of a tabletop machine.
Each rule keeps its formula where the source gives one, a verbatim quote, a page-level
citation, and a stable identifier (dg-NNNN) that resolves here and on the
all-in-one guide. Where an editorial note says
“the reference machine”, its parameters are on the
guide’s front page.
By applicability level: level 2 (2) · level 3 (5) · level 4 (1) · level 5 (1) — levels rank breadth, never license to skip (method). Related domains, by shared rules: Beam dynamics (8), Ion source (3), Physics theory (2), Vacuum (2), Controls & instrumentation (1). To combine tags or levels, open this domain in the filterable view.
Verify before use. Every rule here is a source extract in the vocabulary of the editorial methodology — faithful to its cited page, not an independently validated engineering requirement. Re-read any rule that drives a real design decision at the cited page before committing metal, money, or high voltage to it. The editorial note under each quote is this site’s extrapolation to a tabletop machine, not something the source said: an editor’s judgement, audited for overreach, never a citation.
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VECC set the design aim of vertical betatron tune above 0.5 at all radii to handle space-charge defocusing at the ~5 mA average current; high flutter provides the strong vertical focusing. Fig. 6 (p. 3) shows the computed vertical tune reaching ~0.6–0.9 outside r ≈ 13 cm but falling below 0.5 inside it (figure reading), so the aim is not met at the innermost radii.
Source quote & editorial note
provide the vertical betatron tune > 0.5 at all radii. This is necessary for handling the beam space charge defocusing force
Editorial note, tabletop extrapolation: The > 0.5 target was set for a milliampere-class average current. At the far lower currents of a tabletop machine the space-charge contribution is to be evaluated, not assumed small: estimate the tune depression from peak bunch current, injection energy and beam size before deciding how much vertical focusing to provide. A weak-focusing uniform-field machine takes its vertical tune from the field index rather than from this criterion.
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A low-voltage thermionic electron beam can be focused by a trace of gas instead of a magnetic lens: in Johnson's 1922 Western Electric oscillograph tube (300–400 V per the synopsis, p. 1; argon-filled, p. 8), gas at a few thousandths of a millimetre of mercury shrank a spot that would be 1 cm across in a high-vacuum tube to 1 mm diameter, with the focus controlled by beam current.
spot 1 cm (high vacuum) -> 1 mm (few 1e-3 mm Hg gas); beam 300-400 VSource quote & editorial note
The method of focusing that is used in the present tubes grew out of the suggestion by Dr. H. J. van der Bijl, that a small amount of gas be introduced into the tube. This gas, at a pressure of a few thousandths of a millimeter of mercury, serves to reduce to 1 mm. diameter a spot which would be 1 cm. across in a high vacuum tube. The sharpness of the spot depends also upon the current in the electron stream so that the focus may be controlled by the cathode temperature.
Johnson, A Low Voltage Cathode Ray Oscillograph — Bell System Technical Journal 1(2), 142–153 (1922) — p. 2
Editorial note, tabletop extrapolation: Gas focusing is a no-magnet way to keep a few-hundred-volt electron beam narrow in a fill gas. A visible-beam line run in argon at a similar pressure may show this ion-channel focusing whether or not it is wanted — the effect depends on beam current and energy, geometry, residence time and pressure, so its presence and useful range are measured for the actual geometry, with fill pressure one of the knobs. Johnson worked in argon; the source has no neon data.
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Gas focusing works by ion space charge: positive ions made by beam collisions drift out only at thermal speed and accumulate in the beam. Johnson's estimate for a 1 degree initial divergence at 2 x 10^-5 A is a radial field of about 1 V/cm, supplied if there are about four positive ions per beam electron; a crossed-beam test in his tube measured a deflection equivalent to about 3 V/cm.
E_r(required) ~ 1 V/cm for 1 deg divergence, I = 2e-5 A; ~4 ions per beam electron; measured ~3 V/cm (crossed beams)Source quote & editorial note
This field strength is produced, with beams of the ordinary intensity, if there are four positive ions for each electron in the stream, a condition which seems not unreasonable.
Johnson, A Low Voltage Cathode Ray Oscillograph — Bell System Technical Journal 1(2), 142–153 (1922) — p. 7
Editorial note, tabletop extrapolation: Johnson's four-ions-per-electron figure belongs to his assumed beam (1 degree divergence, 2 x 10^-5 A) and stays there. The transferable point is that positive-ion compensation can oppose electron space-charge spreading; how much compensation a given beamline needs is found experimentally or from a geometry- and species-specific ionization and transport model, since the ion yield per electron depends on energy, gas, pressure and loss mechanisms rather than falling monotonically with beam speed.
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In a gas-focused beam the focal length is set by the beam current: the number of ions per electron stays roughly constant, so more current means more total ionization, a stronger radial field and a focus at shorter distance. Johnson used current (via cathode temperature) as the focus control and saw the convergence point move as predicted.
Source quote & editorial note
When the current is increased, therefore, the total positive ionization of the beam increases, the field around the beam becomes stronger, and the electrons are brought to a focus in a shorter distance.
Johnson, A Low Voltage Cathode Ray Oscillograph — Bell System Technical Journal 1(2), 142–153 (1922) — p. 7
Editorial note, tabletop extrapolation: In Johnson's ion-focused tube raising the current shortened the focal distance, so intensity and focus may be coupled in similar gas-filled low-energy beamlines: an emission-current change can move the waist. If focus and intensity must be set separately, provide an independent focusing element (a magnetic or electrostatic lens, or another characterised focusing parameter) — an independent current control alone does not decouple them.
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Ion-channel focusing needs time to build: Johnson calculated about 10^-6 s for the beam to create its focusing ions, and a beam swept by the deflector plates stayed sharp at 10^5 Hz but visibly widened at 10^6 Hz, which he ascribed to imperfect focusing at that sweep speed.
ion build-up time ~1e-6 s (p.7-8); sharp at 1e5 Hz sweep, widened at 1e6 HzSource quote & editorial note
At a frequency of 10^5 cycles per second the line was still sharp, but at 10^6 cycles there was a noticeable widening of the line
Johnson, A Low Voltage Cathode Ray Oscillograph — Bell System Technical Journal 1(2), 142–153 (1922) — p. 8
Editorial note, tabletop extrapolation: In Johnson's sweep geometry the trace stayed sharp at 100 kHz and widened at 1 MHz: moving a beam quickly onto previously un-ionized paths can outrun its ion channel. Whether an RF-modulated or stationary beam keeps its focus depends on ion creation and decay times in the actual gas and geometry — a stationary path can reuse an ion channel that persists across many cycles — so measure rather than assume either way.
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Heavier fill gases focus a low-voltage electron beam far more readily than light ones: Johnson found focus much easier in nitrogen, argon or mercury vapour than in hydrogen or helium, consistent with the slow heavy ions staying in the beam longer.
Source quote & editorial note
a focus is much more readily obtained in the heavier gases having slow molecules, such as nitrogen, argon or mercury vapor, than in hydrogen and helium where the mean velocity of the molecules is greater.
Johnson, A Low Voltage Cathode Ray Oscillograph — Bell System Technical Journal 1(2), 142–153 (1922) — p. 8
Editorial note, tabletop extrapolation: Choice of fill gas sets focusing strength as well as emission colour: Johnson found helium and hydrogen focus poorly and argon well. He did not test neon, and atomic mass alone does not order the result (ionization cross sections, charge exchange, ion mobility and metastables all enter), so neon's behaviour is unknown from this source — scan gas species, pressure and current on the bench rather than expecting a predictable compensation.
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Johnson filled his tubes with argon, which he described in 1922 as 'the heaviest available permanent gas which does not attack the electrodes' (krypton and xenon are heavier inert gases), and for his tube length and beam current found the best pressure to be 5 to 10 microns of mercury (about 0.7–1.3 Pa), a range wide enough to leave latitude for focusing by current.
5-10 micron Hg = 0.67-1.33 Pa (conversion: 1 micron Hg = 0.133 Pa)Source quote & editorial note
The tubes are therefore filled with argon, the heaviest available permanent gas which does not attack the electrodes. The best pressure for the length of tube adopted and for the current which can be obtained in the beam is 5 to 10 microns, and this leaves considerable latitude for the adjustment of the electron current to get a sharp focus.
Johnson, A Low Voltage Cathode Ray Oscillograph — Bell System Technical Journal 1(2), 142–153 (1922) — p. 8
Editorial note, tabletop extrapolation: A historical anchor for the 0.1-1 Pa regime: a 300-400 V beam focused by argon at roughly 1 Pa over the length of a bench tube. The optimum is tied to his tube length and current, so treat it as a starting point for a pressure scan, not a universal value.
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Yonts's Child's-law estimate for the calutron: with no gas the space-charge-limited emission at 150 V across 0.5 cm is about 19 mA/cm^2 (as printed; the printed inputs give ~17), i.e. of the order of 10 mA from the ~0.5 cm^2 of a 140-mil filament facing the slot, whereas the struck arc easily draws 5 A (~10 A/cm^2); the ion sheath at the filament is what makes arc currents of amperes possible, so ions must reach the filament. The report then solves for the sheath thickness and prints x = 6.7 x 10^-2 cm, "67 mm" and "about 7 mm" (decimal points lost in the typescript; evidently 0.67 mm and 0.7 mm) [referee: its x^2 step is ×10 high; the same inputs give x ≈ 2.1 x 10^-2 cm ≈ 0.2 mm].
j = 2.334e-6·V^1.5/d^2 [j in A/cm^2, V in volts, d in cm] (printed "2 334 x 10-6"; decimals lost in the typescript). Units pass: V = 150 V → V^1.5 = 1837 V^1.5; d^2 = 0.25 cm^2 → j = 2.334e-6 × 1837 / 0.25 = 0.0172 A/cm^2 = 17 mA/cm^2 (with the report's own V^1.5 ≈ 1.9e3 it is 17.7 mA/cm^2; the printed 19 mA/cm^2 is not reproduced from the printed inputs). × 0.5 cm^2 → ~9 mA, "of the order of 10 ma" as stated. Sheath: x^2 = 2.334e-6 × 1.9e3 / 10 = 4.4e-4 cm^2 → x = 2.1e-2 cm = 0.21 mm. The report prints x^2 = "2.334 x 10^-7 x 1.9 x 10^3 = 44 x 10^-4" (the product is 4.4 x 10^-4), then x = 6.7 x 10^-2 cm (= sqrt of 44e-4), "67 mm" (p.11) and "about 7 mm" (p.12): one ×10 slip in x^2 plus dropped decimal points. Corrected sheath ≈ 0.2 mm.Source quote & editorial note
The calculations below show that it is necessary for these ions to pass to the filament if arc currents of several amperes are to be obtained […] For V = 150 volts and d = 0.5 cm (usual calutron conditions), the emission current density is approximately 19 ma per cm² of the filament. Since the total area of a 140-mil filament facing the collimating surface slot is just over 0.5 cm², space-charge-limited current is of the order of 10 ma. When charge gas is present and the arc is struck and an ion sheath is formed, arc currents of 5 amps are easily obtainable. This is a current of 10 amp/cm² of effective filament surface. [Decimal points and the cm² superscripts are lost in the typescript scan; restored here.]
Editorial note, tabletop extrapolation: The same physics governs a small filament source: at the calutron's 150 V across 0.5 cm, space-charge-limited emission into vacuum is of the order of milliamperes, and a different gap and voltage must be put through Child's law on their own; drawing much more current than that needs ion neutralization near the filament. The ion flux that permits the high current is the flux that sputters the filament, so emission current and filament erosion are coupled.
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At constant pressure and filament current, measured ion current at the receiver peaked and then fell as arc current rose, because rising ion density in the exit region produced strong defocusing; Yonts warns that the optimum arc condition is specific to the electrode configuration and the accelerating and focusing voltages.
Source quote & editorial note
Total ion output probably does continue to increase with increasing arc current, but as ion density in the ion exit region increases, a strong defocusing action arises to reduce the ion current entering the receiver. […] The exact values of arc current and voltage at which strong defocusing sets in depends on both the configuration of the accelerating electrode system and on the accelerating and focusing voltages applied. Consequently, the optimum arc conditions found for a particular source unit may not apply to a unit having a different electrode configuration, nor to the same unit operated at different accelerating or focusing voltages
Editorial note, tabletop extrapolation: The useful figure of merit is the beam that arrives where it is wanted, not arc current: pushing the arc harder can lower delivered beam. In a cyclotron the dee voltage affects first-gap extraction, acceleration and focusing in the way the calutron's accelerating and focusing voltages do, so a source optimum found at one dee voltage should be re-measured at another.