Trichel pulses: a study primer

Project folder: Researchers/Trichel. Primer author: Step 4. Verdicts are the arbiter's (arbiter.md); this primer adds none. Revised after QE (qe.md, verdict MINOR) to match the arbiter's "QE revision" section. Tags: [read] from the folder's document; [web] from the Internet, with URL (most were found by the Investigator, see sites.md and research/); [memory] recalled, with confidence; [calc] arithmetic reproduced in example.py.

Source note, repeated from the earlier steps. The folder holds one document: P. Sattari, G. S. P. Castle and K. Adamiak (called SCA below), "Numerical Simulation of Trichel Pulses in a Negative Corona Discharge in Air", ESA Annual Meeting on Electrostatics 2010, paper K4, University of Western Ontario (UWO). It is fully readable. Nothing written by G. W. Trichel himself is in the folder. What is said here about his 1938 work rests on SCA's description, on his paper's abstract [web] https://journals.aps.org/pr/abstract/10.1103/PhysRev.54.1078, on Loeb's 1939 book [web] archive.org (link in Further reading), or on [memory]. Much of the deeper checking used Sattari's 2011 PhD thesis, which the Investigator saved (research/files/Sattari-2011-PhD-thesis-UWO.pdf) [web] https://ir.lib.uwo.ca/etd/215.

This is not an "alternative science" project in the usual sense. Trichel pulses are real and mainstream. The SCA paper is an ordinary engineering model. What makes it worth studying is that it shows, in small, how a computer model gets compared with an experiment, and how that comparison can quietly go wrong.

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1. The idea in one paragraph

Put a sharp needle a centimetre above a metal plate in ordinary air and make the needle a few thousand volts negative. A faint glow appears at the point, and a small current flows. If you look at that current with a fast oscilloscope, it is not steady. It comes as a train of sharp, evenly spaced spikes, each lasting tens of nanoseconds, repeating every few to a few tens of microseconds. G. W. Trichel photographed these spikes at Berkeley in 1938, and they carry his name. His explanation is a self-switching cycle. A burst of ionization at the tip makes clouds of charged particles. The clouds shield the tip and the burst dies. The charge then drifts slowly away, the shielding fades, and the next burst fires. In Trichel's own account, as SCA (the authors of the folder's paper, see below) report it, the positive-ion cloud next to the tip does the switching off. The textbook version that later became standard puts the weight on the slow negative-ion cloud, an emphasis that goes back at least to Loeb's group in 1941 (§3.2). Seventy years later the SCA group wrote a two-dimensional computer model of this cycle. It does produce a regular pulse train. The paper says the model's spacing between pulses is about half the measured one. The review found that the "measured" numbers it compared against are not traceable to measurements, so the size, and even the sign, of that error is not settled.

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2. What you need to know first

Electrostatics with conductors (Laplace and Poisson equations). In a region without charge, the electric potential V obeys Laplace's equation, ∇²V = 0. With a charge density ρ (coulombs per cubic metre, C/m³) it obeys Poisson's equation, ∇²V = −ρ/ε₀, where ε₀ = 8.854×10⁻¹² F/m is the vacuum permittivity. The electric field is E = −∇V. Sharp conductors concentrate field: the field at a tip of radius R is roughly V/R times a slowly varying logarithm. The method of images lets you find the field of a charge near a conducting sphere. Read: D. J. Griffiths, Introduction to Electrodynamics, chapters 2 and 3 (electrostatics; potentials, Laplace's equation and images) [memory, high].

Drift and mobility. An ion in a gas does not accelerate freely; it collides constantly and reaches a steady drift velocity proportional to the field, v = μE. The constant μ is the mobility, in m²/(V·s). Ions in air at 1 atm have μ ≈ 2×10⁻⁴ m²/(V·s), so in 10⁶ V/m they move at about 200 m/s. Electrons are about 250 times more mobile (SCA use 0.05 m²/(V·s)). Diffusion (random spreading) is linked to mobility by the Einstein relation D = μk_BT/e, where k_B is Boltzmann's constant, T the temperature and e the electron charge. Read: Y. P. Raizer, Gas Discharge Physics (Springer, 1991), the early chapters on drift, diffusion and ionization [memory, high for the book; chapter numbers not given].

Ionization, attachment and the Townsend avalanche. A free electron in a strong field gains enough energy between collisions to knock another electron off a molecule. The two electrons do it again, and so on: an avalanche. The rate is written as a frequency k_i (per second) or as the Townsend coefficient α = k_i/(μ_e E), the number of new electrons per metre of travel. Oxygen molecules can also capture an electron to form a slow negative ion O₂⁻; this is attachment, rate k_a (per second). Air is "electronegative" because of this capture. Rates depend mainly on the reduced field E/N, the field divided by the gas number density N (molecules per m³). E/N is measured in townsend (Td): 1 Td = 10⁻²¹ V·m². An avalanche that sustains itself needs a feedback: positive ions hitting the cathode knock out new electrons with probability γ (the secondary-emission coefficient). Townsend's condition for self-sustained discharge is γ(e^K − 1) = 1, where K = ∫(α − η)ds is the number of e-folds of multiplication across the ionization zone and η = k_a/(μ_e E) is the attachment coefficient. Read: Raizer as above; or Lieberman and Lichtenberg, Principles of Plasma Discharges and Materials Processing [memory, high for the book].

Corona discharge. A corona is a partial discharge: ionization happens only in a thin layer where the field is high (near the sharp electrode); the rest of the gap carries current only by ion drift. "Negative corona" means the sharp electrode is the cathode. Onset voltage is the voltage at which the corona first lights. Peek's empirical formula (1910s) estimates onset for wires and spheres. Read: L. B. Loeb, Fundamental Processes of Electrical Discharge in Gases (Wiley, 1939), the corona chapters, which reproduce Trichel's oscillograms (free on archive.org) [web].

Numerical modelling of drift-diffusion (very brief). Each species (electrons, positive ions, negative ions) obeys a continuity equation: ∂n/∂t + ∇·(n v − D∇n) = sources − sinks. Coupled to Poisson's equation, this is a "drift-diffusion-Poisson" model. Steep, moving density fronts make ordinary schemes oscillate. Flux-corrected transport (FCT) is a standard fix that blends a smooth low-order scheme with a sharp high-order one while preventing new peaks. FEM means the finite element method, which handles curved geometry such as a needle. You do not need to be able to code this to follow the primer. Read: D. Kuzmin and M. Möller, "Algebraic flux correction I" (in Flux-Corrected Transport, Springer), which SCA cite [read: p.15].

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3. The textbook view

3.1 The phenomenon

Trichel's 1938 abstract states that the negative point-to-plane current consists of discrete pulses whose size and frequency depend on current, point size and pressure; that the frequency is "independent of gap length" and "not affected by the electrical constants of the external circuit"; and that the periodicity comes from "space-charge formation and clearing" [web] https://journals.aps.org/pr/abstract/10.1103/PhysRev.54.1078 (abstract, via the Investigator). Loeb's 1939 book reproduces his oscillograms (Fig. 243) and a sketch of the glow (Fig. 247) [web] https://archive.org/stream/in.ernet.dli.2015.74804/2015.74804.Fundamental-Processes-Of-Electrical-Discharge-In-Gases_djvu.txt. SCA summarize later measurements: "very short rise times (as short as 1.3 ns [Zentner 1970]) and very short durations (tens of ns [Soria 2007]) separated by longer inter-pulse periods (tens of us)" [read: p.1].

SCA also attribute three empirical laws to Trichel [read: p.1]: the pulse frequency "varies linearly with the corona current"; at the same current it varies inversely with the radius of the point; and it "is not a function of gap length". These are stated at fixed current. The Lama-Gallo law below is stated at fixed voltage, so the two need not conflict (arbiter QE revision R6) [memory, low; not checked].

Which gases pulse. SCA write: "The Trichel pulses only occur in electronegative gases, i.e. those which have affinity for electrons. In electropositive (non-attaching) gases such as nitrogen, the corona current in the external circuit would be continuous without pulses" [read: p.1; no reference given, and note the conditional "would be"]. They add: "Loeb et al [6] confirmed that these regular pulses exist only in electronegative gases" [read: p.2], citing Loeb, Kip and Hudson, Phys. Rev. 60, 714 (1941), without saying which gases Loeb's group used. They also note that pulses in SF₆, a strongly attaching gas, are "more irregular" [read: p.1, citing Van Brunt and Leep 1981]. The arbiter found a direct two-sided test: Weissler, Phys. Rev. 63, 96 (1943), reports that Trichel pulses "are not observed in pure H₂ and N₂ but are immediately produced upon the addition of traces of O₂" [web] https://journals.aps.org/pr/abstract/10.1103/PhysRev.63.96 (abstract as relayed by a search summary; full text unread). Against these, Zhang et al. (2017) report pulses "in various gases rather than only in electronegative gases" (see §6, row 1b).

Earlier explanations, as SCA describe them [read: p.2, deep read C]. This is SCA's account of prior work, written while introducing their own model; nobody in this review has read the original papers. - Loeb's group (1941): successive avalanches, each giving rise to about three successors. SCA say it "could not explain the fast rise time of 1.5ns observed in air" (p.1 quotes 1.3 ns from Zentner; the two figures are not reconciled). - Aleksandrov (1963): several avalanches developing in parallel; predicts the fast rise and a slowly rising "foot" before the main pulse. - Ogasawara (1966): an analytic solution for the early rise; shows only exponential growth, no pulse train, and no role for negative ions. - Kekez, Savic and Lougheed (1982): an equivalent electric circuit that reproduces the succession of pulses but not the mechanism. - Morrow (1985): a one-dimensional finite-difference plus FCT model; got the first pulse only, then the current just decayed; SCA say he ignored ion secondary emission (yet they take their γ = 0.01 from him). - MacAlpine and Yim (1995) and others: simplified models of pulse chains. - Napartovich et al. (1997): a "1.5D" model (one dimension with an assumed radial spread) that produced pulse sequences in dry air for gaps under 1 cm. - Soria et al. (2007): a particle-in-cell simulation of Trichel pulses in pure oxygen (cited for pulse durations, not discussed). This history is the background to SCA's claim to the "first successful" two-dimensional three-species simulation at several voltages; Napartovich 1997, MacAlpine and Yim 1995 and Soria 2007 are the nearest prior art against which that claim could be checked (arbiter QE revision R5).

The best-known quantitative study is Lama and Gallo, J. Appl. Phys. 45, 103-113 (1974), cited by SCA as ref. [7] [read: p.14]. Nobody in this review read it. Sattari's thesis quotes their empirical law as [web] thesis Ch. 8.2:

f = K₁ V (V − V₀) / (r d²)

where f is the pulse frequency, V the magnitude of the needle voltage, V₀ the onset voltage (2.3 kV in their data, as quoted), r the tip radius, d the gap, and K₁ ≈ 27 kHz·mm³/kV². The period is T = 1/f.

3.2 The standard mechanism, step by step [memory, high]

1. Field at the tip. For a hyperboloid needle of tip radius R facing a plane at distance d, the Laplace (no space charge) field at the tip is close to

E_tip ≈ 2V / [R ln(4d/R)].

Near the tip, along the axis, it falls off as E(s) ≈ E_tip · R/(R + 2s), where s is the distance in front of the tip. 2. Avalanche. Where E/N exceeds the threshold at which ionization beats attachment (about 87 Td in SCA's corrected coefficients, see §5, step 2), a stray electron multiplies. 3. Positive ions removed. The positive ions are created within tens of micrometres of the tip and fall into it within nanoseconds. That ends the burst once ionization stops. 4. Negative ions block. The electrons run outward into weaker field and are captured by O₂, forming a slow negative-ion cloud. Its charge reduces the field at the tip below threshold. 5. Clearing. The cloud drifts away at v = μE. When it is far enough, the tip field recovers and the cycle repeats. The period is roughly the cloud's clearing time.

Whose mechanism is this? Steps 4 and 5, with the negative-ion cloud setting the period, are the later textbook emphasis. Trichel's own account, as SCA quote it, gives the central role to the positive cloud: "A negative cloud of space charge is formed far from the point and a positive cloud is formed near the point. The presence of positive ions near the point increases the field in this area and decreases the electric field between the positive ion cloud and the negative ion cloud. So, no ionization can take place beyond the positive ion cloud. The positive ions move towards the point under the effect of the electric field and narrow the region of enhanced electric field and finally the ionization process stops" [read: p.2]. On p.1 SCA also credit Trichel with "the shielding effect produced by the positive ion cloud near the cathode" [read: p.1]; the two passages are worded differently and SCA do not reconcile them. The 1941 Loeb, Kip and Hudson abstract says Trichel's theory "is applicable except as modified by the influence of negative ions" [web] https://link.aps.org/doi/10.1103/PhysRev.60.714 (search summary). So Zhang et al.'s 2017 positive-ion account (§6) is closer to Trichel's own wording than to the later negative-ion emphasis (arbiter QE revision R2). One caution: SCA's reference for this passage is printed as Trichel's positive-corona paper (Phys. Rev. 55, 382), so it is not certain which Trichel paper the quoted mechanism comes from (§6, "Gaps in the record").

3.3 The model equations (SCA, as they should read)

With n_e, n_p, n_n the electron, positive-ion and negative-ion number densities (per m³) and e = 1.602×10⁻¹⁹ C:

• Poisson: ∇²V = −e(n_p − n_e − n_n)/ε₀, E = −∇V.
• Electrons: ∂n_e/∂t + ∇·(n_e W_e − D_e∇n_e) = (k_i − k_a) n_e + k_d n_n − β n_e n_p
• Positive ions: ∂n_p/∂t + ∇·(n_p W_p) = k_i n_e − β n_e n_p
• Negative ions: ∂n_n/∂t + ∇·(n_n W_n) = k_a n_e − k_d n_n

Here W = μE are drift velocities (μ_e = 0.05, μ_p = μ_n = 2×10⁻⁴ m²/(V·s)); D_e is electron diffusion; k_i, k_a, k_d are ionization, attachment and detachment frequencies (k_a = 6×10⁶ /s, k_d = 0); β = 5×10⁻¹³ m³/s is electron-ion recombination. Boundary conditions: V = V_c − I·R_ext on the needle (V_c applied voltage, I circuit current, R_ext an external resistor), V = 0 on the plate; electron density at the needle set by secondary emission with γ = 0.01 [read: p.3-5, page images p.4-5, via deep read].

Source of the constants: SCA cite most rates to a 1997 report by Akishev and colleagues at TRINITI (a Russian research institute) under contract to ABB, an engineering company [read: p.4, p.15]. It is not published and not online. ABB is a commercial party with a stake in corona modelling; the report cannot be checked by this review.

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4. Where the authors depart, side by side

"Inventor" here means two parties: Trichel (the phenomenon and its explanation) and SCA (the model). Trichel's starting point is the textbook. SCA's departures are of two kinds: printing slips in the equations, and the way they compared with experiment.

What is solid in SCA's starting point: the drift-diffusion-Poisson framework is standard physics; the mechanism is Trichel's; the timescales come out right (§5). What is fragile: the printed equations, the unstated circuit, and above all the reference data.

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5. Worked example: the Trichel period at −7 kV, by hand

Goal: estimate the time between pulses for SCA's geometry from first principles, then compare it with the model and with the two different "experimental" values the same authors used. Every number below is printed by example.py:

.venv/bin/python review/Trichel/example.py

Inputs: tip radius R = 100 μm = 1×10⁻⁴ m; gap d = 1 cm = 1×10⁻² m; |V| = 7000 V; N = 2.46×10²⁵ /m³; μ_e = 0.05 and μ_ion = 2×10⁻⁴ m²/(V·s); k_a = 6×10⁶ /s; γ = 0.01 [read: p.3-7].

Step 1. How strong is the field at the tip?

E_tip ≈ 2V / [R ln(4d/R)]. Here 4d/R = 4×10⁻² / 10⁻⁴ = 400, and ln 400 = 5.99. So

E_tip ≈ 2 × 7000 / (10⁻⁴ × 5.99) = 2.34×10⁷ V/m.

The exact hyperboloid formula (in example.py) gives 2.35×10⁷ V/m. The average field across the gap, V/d, is only 7×10⁵ V/m, so the tip multiplies the field about 34 times. SCA's Fig. 10 shows a peak tip field of 2.2 to 2.45×10⁷ V/m [read: page image p.11, via deep read]. The hand estimate matches the model.

Step 2. Where does ionization beat attachment?

Set k_i = k_a with the corrected law: 11.93 − 449.3/EN = log₁₀(6×10⁶) = 6.778, so 449.3/EN = 5.152 and EN = 87.2 Td. Convert: E_th = 87.2 × 10⁻²¹ × 2.46×10²⁵ = 2.14×10⁶ V/m.

With the misprinted law, k_i ≈ 6.6×10¹¹ /s even at 41 Td (10⁶ V/m), 100,000 times k_a. There would be no threshold, no onset voltage and no pulsing. That is how we know the printed version is a slip: it contradicts the paper's own results.

A sanity check on the corrected law: at 10⁷ V/m, k_i = 6.7×10¹⁰ /s, so α = k_i/(μ_e E) = 1.3×10⁵ per metre, a plausible value for air at that field [memory, medium].

Step 3. How big is the ionization zone, and is 7 kV far above onset?

Ionization beats attachment where E(s) > E_th. With E(s) ≈ E_tip R/(R + 2s):

s_zone = (R/2)(E_tip/E_th − 1) = 50 μm × (10.9 − 1) ≈ 0.50 mm (exact: 510 μm).

That is in the Laplace field, before any space charge; the space charge of a pulse shrinks it. The multiplication across the zone, K = ∫(α − η)ds, is 21.9 e-folds at 7 kV (example.py integrates it numerically). Townsend's self-sustaining condition needs only K = ln(1 + 1/γ) = ln 101 = 4.6. So 7 kV is far above onset.

A side result, offered as an exercise, not a verdict: solving K(V) = 4.6 with the model's own coefficients gives an onset of about 3.25 kV [calc]. That lies between Gallo's measured 2.3 kV and SCA's stated 4.7 kV. It supports the arbiter's remark that the 4.7 kV was not computed from the model's coefficients. The criterion is rough (Laplace field, no photoionization, constant γ and μ_e).

Step 4. How fast do the positive ions leave?

Positive ions are made 25 to 50 μm from the tip, where E ≈ 10⁷ V/m. Their speed is μE = 2×10⁻⁴ × 10⁷ = 2000 m/s. Time to fall back: 25×10⁻⁶ / 2000 = 13 ns, or 25 ns from 50 μm. This matches SCA's "a few nanoseconds" [read: p.9] and the measured "tens of ns" pulse widths. The pulse ends when these ions are gone.

Step 5. How long does the negative-ion cloud take to clear?

(a) Drift time. A negative ion starting at the tip moves at ds/dt = μE(s) = μE_tip R/(R + 2s). Separate and integrate:

(R + 2s) ds = μ E_tip R dt ⇒ R s + s² = μ E_tip R t ⇒ t = (R s + s²) / (μ E_tip R).

The denominator is 2×10⁻⁴ × 2.34×10⁷ × 10⁻⁴ = 0.468 m²/s. Then:

(b) How far must it go? Model the tip as a sphere of radius R held at fixed potential and the cloud as a point charge q at distance D = R + s from the sphere's centre. Using the method of images, the charge and its image add a field at the nearest point of the sphere of

ΔE = q (D + R) / [4πε₀ R (D − R)²].

It points against the needle's own field, so it lowers the tip field. Take q = 115 pC, the charge per pulse the thesis reports for r = 100 μm [web, thesis, via reviewers]. At s = 1 mm: ΔE = 8.99×10⁹ × 1.15×10⁻¹⁰ × 1.2×10⁻³ / (10⁻⁴ × 10⁻⁶) = 1.2×10⁷ V/m. At s = 2 mm: 5.7×10⁶ V/m. The tip field itself is 2.3×10⁷ V/m. So a cloud that is still a millimetre or two away is still holding the tip field down by a large fraction. The cloud must travel millimetres, not micrometres, before the next pulse. SCA's own Fig. 10 shows the tip field only recovering to about 1.1×10⁷ V/m before each new pulse [read: page image p.11, via deep read]. That fits this picture: the remaining clouds from earlier pulses never fully clear.

Hand estimate: a clearing distance of 1 to 2 mm gives a period of about 2 to 9 μs. This is crude. The cloud is spread out, not a point; the needle is not a sphere; and charge from earlier pulses piles up. It gives the right order of magnitude and nothing more. For contrast, crossing the whole gap in a uniform field would take d²/(μV) = 71 μs.

Step 6. Compare with the model and the two reference sets

Lama-Gallo law as quoted in the thesis, with r = 0.1 mm and d = 10 mm: K₁/(r d²) = 27/(0.1 × 100) = 2.7 kHz/kV². At 7 kV with V₀ = 2.3 kV: f = 2.7 × 7 × 4.7 = 88.8 kHz, T = 11.26 μs. With V₀ = 4.7 kV instead: f = 2.7 × 7 × 2.3 = 43.5 kHz, T = 23.0 μs.

(All times in μs. Model and "experimental" from SCA Table 1 [read: p.13]; the rest [calc].)

Fit 1/(TV) against V for the SCA "experimental" column: the line has slope 2.848 kHz/kV² (K₁ = 28.5) and crosses zero at V₀ = 4.705 kV, with residuals below 0.008 kHz/kV. Four points sit on a two-parameter curve almost exactly. The same fit to the model's own periods gives V₀ = 4.64 kV but K₁ = 62 and residuals up to 5.7 kHz/kV (the 9 kV point is off). Model divided by the law with Gallo's V₀ = 2.3 kV: 1.01, 0.78, 0.55, 0.64.

What this shows: - The physics checks out in order of magnitude. A hand estimate of 2 to 9 μs, a model of 11.3 μs, and references of 11.3 or 21.9 μs are all in the same decade at 7 kV. Trichel's clearing picture gives the right timescale. - The "factor of two" depends entirely on which reference you choose. The SCA column behaves like the Lama-Gallo formula evaluated with the model's own 4.7 kV onset. Against Gallo's own 2.3 kV, the model is within 1% at 7 kV and too short by 22 to 45% at higher voltages. - Both columns moved between the authors' two documents, not only the reference. The thesis (Table 7.1, R_ext = 100 kΩ) reports simulated periods of 13.5, 9.9, 6.6 and 6.2 μs at 7, 8, 9 and 10 kV, with standard deviations of 2.09, 0.81, 0.61 and 0.80 μs, against "Gallo's T" of 11.3, 8.1, 6.1 and 4.8 μs [web, thesis p.111, checked by the arbiter]. So in the thesis the model is 8 to 29% too long, the opposite sign to the paper. And the ESA model periods (11.33, 6.31, 3.39 μs) are only 0.84, 0.64 and 0.51 of the thesis model's at 7, 8 and 9 kV [calc, arbiter]. The thesis model used different ion mobilities, a circuit equation with R_ext = 100 kΩ and C_d = 0.055 pF, and the 449.3 ionization law; what the 2010 code used is not known. The thesis's scatter of 0.6 to 2.1 μs is as large as some of the differences being discussed; the ESA paper gives no scatter. The 9 kV ESA value (3.39 μs, from three intervals) is the point that moved most. - Nobody knows yet how SCA produced their column. Gallo, whom SCA thank [read: p.14], may have supplied fitted values for a different geometry. That is why the arbiter would not call it "refuted": a calculation cannot refute a number whose origin is unknown. The fix is to read Lama and Gallo's 1974 tables.

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6. Evidence table

Verdicts from arbiter.md. "Tested" means a controlled comparison with experiment.

Related evidence kept in the record (no verdict of its own): - Gallo's narrow channel. C. F. Gallo's previously unpublished microscope observations, reported in one source, Sattari's thesis Ch. 8.3 (relayed by his collaborators; no independent publication found yet): an "extremely narrow bright ionization channel" 0.12 to 2 μm wide at a negative tip, absent in positive corona; the channel wobbles on very sharp tips, which the source offers as the cause of the "mode shifts" Loeb and Trichel described; onset at −2.3 kV roughly independent of tip radius. The thesis authors write that this "is not described by present theory or simulations and may possibly be confined magnetically by the high current density" [web, thesis]. With an effective tip radius of 10 μm fitted in place of 100 μm, the thesis model matched Gallo's period to 15% and current to 23% (a tuned parameter, so consistency rather than proof). The arbiter's rough check: a 1.8 mA current in a 1 μm channel makes B ≈ 0.36 T at the edge, giving an electron cyclotron frequency (~6×10¹⁰ rad/s) below the collision frequency (~10¹² /s [memory, medium]), so magnetic confinement looks doubtful; it has not been tested by experiment. If the channel is real, a model with a 5 μm mesh round a 100 μm tip does not resolve the physics that matters. - Circuit independence. Trichel's abstract says the pulses are "not affected by the electrical constants of the external circuit"; the thesis model's period depends on R_ext. Yet R_ext C_d = 100 kΩ × 0.055 pF = 5.5 ns, far shorter than the period [calc]. Unresolved. - Gap dependence. Trichel: frequency "independent of gap length"; Lama and Gallo: f ∝ 1/d². Probably different regimes [memory, medium]; unresolved. - Two onsets in the experimental record. Gallo −2.3 kV; Dordizadeh, Adamiak and Castle 2017: −4 to −6 kV for 19 to 55 μm tips [web] https://www.sciencedirect.com/science/article/abs/pii/S0304388616301735 (abstract). Different tips, so a difference rather than a proven contradiction. The same 2017 paper reports measured pulse shape independent of radius, voltage and gap, and "some discrepancies" with simulation.

Gaps in the record (absence-of-evidence analysis from the arbiter, kept): - Trichel's research stops after 1939. Gervais William Trichel (1898-1990) was a serving U.S. Army officer: Berkeley Ph.D. 1938; "Captain, Ordnance Corps" in Loeb's 1939 preface; Frankford Arsenal fire control 1938-43; chief of Army Ordnance rocket development 1943-45; later Chrysler 1947-63 [web] Redmond and Smith, Project Whirlwind (archive.org); Loeb 1939 preface (archive.org); https://prabook.com/web/gervais_william.trichel/90642 (a compiled biography). Explanations weighed: return to military service (best supported, three sources); wartime secrecy (plausible for his later ordnance work, but it does not explain why the corona papers stopped; no document found); suppression or dismissal (no evidence: Loeb reproduced his oscillograms and the work has been cited for 85 years); commercial secrecy (no evidence). The 1938 dissertation itself was not found (WorldCat blocked). The Berkeley catalogue or ProQuest would settle the timeline. - The TRINITI/ABB report behind the rate constants is not online. A copy, or the later Napartovich/Akishev papers, would decide whether the misprint is SCA's or inherited. - Gallo's observations appear in the 2011 thesis but not in the 2010 paper, which thanks him. Possible reasons: not yet written up; in conflict with the smooth-tip model; informal. Evidence for each is thin. Gallo's lab records or a later paper would decide. - Trichel's citation. SCA's ref. [2], "Phys. Rev. 55, 382", is Trichel's positive-corona paper (1939); the negative-corona paper is Phys. Rev. 54, 1078 (1938) [web] https://link.aps.org/doi/10.1103/PhysRev.55.382. The thesis repeats the error.

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7. What a learner should take away

Solid. Trichel pulses are real, regular and reproduced for 85 years. Trichel's 1938 explanation, ion space charge switching the tip field off and on, gives the right timescales from nothing more than Griffiths-level electrostatics plus v = μE: nanoseconds for the positive ions, microseconds for the negative cloud (§5). A drift-diffusion-Poisson model with three species reproduces a pulse train. The tip field in the model matches a one-line hand formula.

Wrong, and why it is instructive. - Printed equations with the wrong density in the source terms and an exponent off by a factor of 100. The lesson: check every printed formula against its consequences. A rate law with no threshold cannot produce a corona onset, so the printed version cannot be the one that ran. - A comparison with "experiment" whose numbers are indistinguishable from a formula evaluated with the model's own onset voltage (where they came from is still unknown). The lesson: when four data points fit a two-parameter curve to three significant figures, they are probably not raw measurements. Always ask where the reference column came from, and whether the geometry and circuit match. - Comparing one number (the period) when the pulse has several (rise time, width, peak current, charge, onset). A model can get one right for the wrong reason. The thesis shows simulated widths about ten times the measured ones (in its −9 kV pressure study). - An "explanation" of a discrepancy ("approximations in the numerical technique") that names no approximation and tests nothing.

Genuinely open. - What are Lama and Gallo's actual periods for a 100 μm tip at 1 cm, and their V₀ for it? - Do negative ions matter at all (Zhang 2017 says positive ions dominate and negative ions are "not necessary")? Which gases did Zhang test, and how pure were they? How does that square with Weissler's 1943 report of no pulses in pure N₂ and H₂? - Is SCA's "first successful" 2D simulation really first? Napartovich 1997, MacAlpine and Yim 1995 and Soria 2007 would decide. - Is Gallo's micrometre channel real, and does it set the period? - Is the big first pulse a real start-up spike or a numerical artefact of the seed charge? - Why do Trichel (no circuit effect) and the thesis model (period depends on R_ext) disagree? - Why are measured pulses tens of ns wide and simulated ones hundreds?

Each of these has a concrete test, listed in arbiter.md "What would settle it". The cheapest for an amateur: build a needle-plane gap with a 1 MΩ ballast resistor and a 50 Ω current shunt, look at the pulses on a scope of at least 100 MHz bandwidth, and measure period against voltage. Then add a small fan blowing across the tip. If negative-ion clearing sets the period, sweeping the ions away should shorten it.

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8. Check yourself

1. A needle has tip radius 50 μm and sits 2 cm from a plate at −6 kV. Estimate the Laplace field at the tip. Is it above the 2.1×10⁶ V/m ionization threshold?
2. Why does the printed SCA ionization law, k_i = 10^(11.93 − 4.493/EN), rule itself out as the law the code used?
3. Using t = (Rs + s²)/(μE_tip R), how long does a negative ion take to move 1.5 mm from the 7 kV, 100 μm tip of §5? Is that in the right range for the period?
4. If four "measured" periods fit f = C·V(V − V₀) with residuals 1000 times smaller than the spread of the data, what does that suggest, and what should you do?
5. Name two pieces of evidence, kept in this record, that compete with the "negative-ion clearing sets the period" picture.

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9. Further reading

Mainstream sources [memory, high unless tagged]: - D. J. Griffiths, Introduction to Electrodynamics: Laplace's equation, images. - Y. P. Raizer, Gas Discharge Physics, Springer 1991: ionization, attachment, Townsend breakdown, corona. - M. A. Lieberman and A. J. Lichtenberg, Principles of Plasma Discharges and Materials Processing, Wiley. - L. B. Loeb, Fundamental Processes of Electrical Discharge in Gases, Wiley 1939, with Trichel's oscillograms (Figs. 243, 247, 248) [web] https://archive.org/stream/in.ernet.dli.2015.74804/2015.74804.Fundamental-Processes-Of-Electrical-Discharge-In-Gases_djvu.txt - R. Morrow, "Theory of negative corona in oxygen", Phys. Rev. A 32, 1799 (1985) [read: ref. list p.14]. - A. P. Napartovich et al., "A numerical simulation of Trichel-pulse formation in a negative corona", J. Phys. D 30, 2726 (1997) [read: ref. list p.15]. - W. L. Lama and C. F. Gallo, "Systematic study of the electrical characteristics of the Trichel current pulses from negative needle-to-plane coronas", J. Appl. Phys. 45, 103-113 (1974), DOI 10.1063/1.1662943 [read: ref. list p.14; web, bibliographic only]. The paper to read first if you want to settle the open questions. - D. Kuzmin and M. Möller, "Algebraic flux correction I", in Flux-Corrected Transport, Springer [read: ref. list p.15].

The inventor's and authors' own key works: - G. W. Trichel, "The mechanism of the negative point to plane corona", Phys. Rev. 54, 1078 (1938) [web] https://link.aps.org/doi/10.1103/PhysRev.54.1078 (title [memory, medium]; abstract via Investigator). - G. W. Trichel, positive point-to-plane corona, Phys. Rev. 55, 382 (1939) [web] https://link.aps.org/doi/10.1103/PhysRev.55.382. - P. Sattari, G. S. P. Castle, K. Adamiak, ESA 2010 paper K4 (this folder). - P. Sattari, PhD thesis, University of Western Ontario, 2011 [web] https://ir.lib.uwo.ca/etd/215 (copy in research/files/). - Dordizadeh, Adamiak, Castle, J. Electrostatics 88 (2017) 49-54, measured pulse shapes [web] https://www.sciencedirect.com/science/article/abs/pii/S0304388616301735 - Zhang, Xia, Jiang, Ouyang, Sci. Rep. 7:10135 (2017), the competing positive-ion account [web] https://pmc.ncbi.nlm.nih.gov/articles/PMC5579005/ - Shaygani and Adamiak, J. Phys. D 56 (2023) 385202, later model comparing rate-coefficient sets [web] https://iopscience.iop.org/article/10.1088/1361-6463/acdd0f/meta

Experimental papers on the gas question (found by the arbiter; full texts unread): - L. B. Loeb, A. F. Kip, G. G. Hudson, "Pulses in negative point-to-plane corona", Phys. Rev. 60, 714 (1941) [read: ref. list p.14] [web] https://link.aps.org/doi/10.1103/PhysRev.60.714 - Weissler, "Positive and Negative Point-to-Plane Corona in Pure and Impure Hydrogen, Nitrogen, and Argon", Phys. Rev. 63, 96 (1943) [web] https://journals.aps.org/pr/abstract/10.1103/PhysRev.63.96 - "Pulsed mode of a negative corona in nitrogen: I. Experiment", Technical Physics (2001) [web] https://link.springer.com/article/10.1134/1.1378130 (title only)

Related work found by the Investigator, not read by this review (listed so it is not lost; sites.md, research/2026-10-04-google.md): - Kokovin et al. 2021: multi-fluid model of the transition from Trichel pulses to a stationary corona. - Friebe et al. 2021 (MDPI), "Trichel Pulse Analysis: Physical Calculation and Validation". - Ferreira et al. 2025: stability of low-current needle corona. - Eindhoven University of Technology thesis, 1987, "Negative corona discharges: a fundamental study" (pure.tue.nl). - Amin, J. Appl. Phys. (1954): on pre-onset streamers, "sought in vain by G. W. Trichel in 1937" (Google snippet only), a trace of Trichel's own research programme.

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Answers

1. 4d/R = 4 × 0.02 / 5×10⁻⁵ = 1600; ln 1600 = 7.38. E_tip ≈ 2 × 6000 / (5×10⁻⁵ × 7.38) = 3.3×10⁷ V/m. Yes, about 15 times the threshold.
2. As printed, k_i is about 7×10¹¹ /s even at 41 Td, 100,000 times the attachment rate. Ionization would beat attachment everywhere in the gap at any voltage, so there would be no onset voltage and no pulsing. The paper reports an onset and pulses, so the code must have used something else; the thesis prints 449.3.
3. Rs + s² = 10⁻⁴ × 1.5×10⁻³ + (1.5×10⁻³)² = 1.5×10⁻⁷ + 2.25×10⁻⁶ = 2.4×10⁻⁶ m². Divide by 0.468 m²/s: t ≈ 5.1 μs. Yes: the same order as the model's 11.3 μs and the reference values of 11.3 or 21.9 μs.
4. They are probably values computed from a fitted formula, not raw measurements (raw data scatter). Find out where they came from, which V₀ and geometry were used, and compare with the original paper's tables before trusting any ratio.
5. Any two of: Zhang et al. 2017 (pulses in various gases, positive ions predominant, negative ions "not necessary"); Trichel's own account as SCA quote it, which gives the positive-ion cloud near the tip the switching role (§3.2); Gallo's micrometre ionization channel and radius-independent −2.3 kV onset (one source, the thesis); Trichel's own finding that the circuit does not affect the pulses, against a model whose period depends on R_ext. Note the other side too: Weissler 1943 (no pulses in pure N₂ and H₂, pulses once traces of O₂ are added) and Loeb's group 1941 (as cited by SCA) support the idea that an electron-attaching gas, and so negative ions, is needed for pulsing at all. That supports the weak form, not the claim that negative-ion clearing alone sets the period.