Trichel: Mathematician review (Step 2)

Reviewer role: PhD mathematician. I check derivations, notation, units and hidden assumptions. Tags: [read] from the documents (I re-read page 4 of the ESA paper as an image and the Sattari thesis text), [web] from a URL, [memory] with confidence, [calc] my own arithmetic (scripts run in the project venv; numpy only, no scipy).

Source note: the project folder holds one document, the 2010 ESA paper by Sattari, Castle and Adamiak (SCA). Nothing by G. W. Trichel is in the folder. Statements about Trichel's 1938 work rest on that paper, on the Investigator's [web] leads, or on [memory]. Web check budget: one attempt (AIP page for Lama and Gallo 1974) returned HTTP 403, so the Lama and Gallo source itself was not read by me; its formulas are taken from the thesis's quotation of it.

Reader-level glossary (used below) - Trichel pulses: regular short bursts of current from a negatively charged needle in air, repeating every few to tens of microseconds (us). - Townsend (Td): unit of E/N, field E divided by gas molecule density N; 1 Td = 1e-21 V m^2. - Continuity equation: "rate of change of density plus divergence of flux equals sources minus sinks".

Corrections to Step 1

1. Misprint in the ionization law is confirmed, not just suspected. The deep read flagged ki = 10^(11.93 - 4.493/EN) as probably meaning 449.3/EN. I re-read the equation image on p.4 [read: page image p.4]: it prints 4.493. The authors' PhD thesis prints 449.3/EN (thesis Ch. 6, text line "449.3") [web: https://uwo.scholaris.ca/server/api/core/bitstreams/0c5eda5f-5b90-43e3-8010-ff0188ca2b1a/content, via research/files/Sattari-2011-PhD-thesis-UWO.pdf; I confirmed the 449.3 text in the PDF]. My recomputation [calc]: with 4.493, ki is about 6.6e11 /s already at 41 Td and saturates near 8.4e11 /s, far above ka = 6e6 /s at every field; with 449.3, ki = ka at E/N = 87 Td (E = 2.14e6 V/m = 21 kV/cm), which is a physically sensible air/oxygen ionization threshold, and ki/(mu_e E) (the Townsend coefficient alpha) is 1.3e5 /m at 1e7 V/m. So the printed ESA formula is a typo (internally inconsistent as printed); whether the code used the right one is not shown, but the thesis suggests it did.
2. The deep read says Table 1 shows the shortfall "48% to 63%". Correct per my ratios (0.519, 0.474, 0.373, 0.485 [calc]), accepted. Its note that the first 7 kV interval (11.5 us) is not the longest is also correct.
3. The deep read took the ESA "experimental" column at face value as Lama and Gallo data. It is not consistent with the thesis's own "Gallo's T" values (see Claim 5). This is a new finding, not an error of Step 1 (the Investigator surfaced the discrepancy; I located its likely cause).
4. Step 1 said the paper's Eq. (6)/(7) sources look inconsistent. Confirmed from the image: Eq. (6) reads "ki Np" and Eq. (7) reads "ka Nn - kd Nn" [read: page image p.4]. Physically they should be ki Ne and ka Ne - kd Nn. With kd = 0 and the printed Eq. (7), negative ions would grow as exp(ka t) from nothing-in-nothing-out, and no electron would be converted. I classify it as a typo (internal inconsistency in print), not in the method.
5. Unit point on N: Step 1 correctly notes 2.46e25 /m^3 is the total air density at 298 K and 1 atm ([calc] 2.461e25), called "oxygen molecules" in the paper. Oxygen alone would be about 5.2e24 /m^3 (21% of air) [memory, high]. The thesis repeats the label [web, thesis]. If the rate law was fitted to oxygen E/N, using total air density shifts E/N by about 4.7x. Effect on results unknown.
6. Step 1 caveat "geometry and R_ext not stated": partly resolved. The thesis states R_ext = 100 kOhm and gap capacitance Cd = 0.055 pF [web, thesis Ch. 6]. The ESA paper still does not.

Cross-check of the thesis's comparison set (my main new finding)

Lama and Gallo's empirical law as quoted in the thesis: f = 1/T = K1 V (V - V0)/(r d^2), K1 ~ 27 kHz mm^3 kV^-2, V0 = -2.3 kV; I = K2 V(V-V0)/d^2, K2 ~ 52 uA mm^2 kV^-2 [web: thesis Ch. 8.2, text lines 7837-7882]. Plug in r = 0.1 mm, d = 10 mm (the simulated geometry): - Law with V0 = 2.3 kV gives T = 11.26, 8.12, 6.14, 4.81 us at 7, 8, 9, 10 kV [calc]. The thesis column "Gallo's T" is 11.3, 8.1, 6.1, 4.8 [read: thesis Table 7.1]. Exact match, so the thesis column is just the law evaluated. Same for current: 52 x 32.9/100 = 17.1 uA, matching "Gallo's I" 17.1 uA at 7 kV [calc]. - The ESA column (21.85, 13.32, 9.09, 5.07 us at 7, 8, 9, 11 kV) fits the same functional form T = K/(V(V - V0)) with V0 = 4.71 kV and K = 350 us kV^2, reproducing all four values to within 0.01 us [calc, linear least squares of 1/T on V^2 and V]. 4.71 kV is the computed onset (4.7 kV) quoted in the ESA paper [read: p.7]. Fitting a 2-parameter model to 4 points leaves only 2 degrees of freedom, so an exact fit is suggestive, not proof. A real measured set would not normally be that clean; this looks like a curve evaluated from a formula in which the onset voltage was replaced by the calculated one (or an equivalent re-fit). K = 350 versus 370 from K1 = 27 (5% apart) shows it is not literally the quoted law either. - The simulated ESA periods themselves (11.33, 6.31, 3.39, 2.46) fit the same form less well (V0 about 4.3 kV; predicted 9.1, 5.8, 4.1, 2.35 [calc]); the model's 9 kV point (3.39) is low relative to its neighbours (log-log slope -3.4 numerical against -3.2 "experimental" [calc]). - Consequence: against the law with Lama and Gallo's own V0 = 2.3 kV, the ESA simulation at 7 kV (11.33 us) is within 1% of the law (11.26 us). The "underestimate by 50%" claim in the ESA paper depends on which onset voltage is used for the experimental curve. The thesis then reports the opposite sign (overestimate: 13.5 vs 11.3 us), with its own Chapter 8.2 saying both over- and under-estimate [web, thesis]. Hence "agreement within a factor of two" is not a reliable quantity from these documents. Caveat: I did not read Lama and Gallo 1974 (AIP, HTTP 403); I rely on the thesis's quotation. Neither document gives the original data. - Note also the Lama and Gallo law, as quoted, has T independent of the quantity the simulation most sensitively depends on: the thesis finds period depends on R_ext, whereas Trichel 1938 reported the pulses as unaffected by external circuit constants [web: https://link.aps.org/doi/10.1103/PhysRev.54.1078, abstract as relayed by the Investigator]. Not a math error, but a mismatch between model and early data.

Further checks on the ESA model (verified by me)

• Units of Eq. (1)/(4): Poisson's equation needs charge density (C/m^3). If N means number density (1/m^3) a factor e is missing; the figures use C/m^3 [read: p.9-10], so N appears to be charge density. Then the recombination term beta_ep Ne Np (beta = 5e-13 m^3/s) is dimensionally wrong by a factor 1/e: in charge-density units it should be beta Ne Np / e. Eqs (1), (4) and (5)-(6) cannot all be correct as printed. Worked check: Ne = Np = 1 C/m^3 in charge units is 6.2e18 /m^3; beta Ne Np/e term would be 5e-13 x 6.2e18 = 3.1e6 C/m^3/s, while the printed form gives 5e-13 C/m^3/s, a factor 6e18 smaller [calc]. Which one the code used is unknown, and it matters because recombination matters at the densities of the pulse peak (about 1e0 to 1e1 C/m^3).
• Diffusion, Eq. (11): D = kB T k / e0 contains an undefined "k"; Einstein's relation D = mu kB T/e requires k = mu_e. At 298 K, D = 0.05 x 0.02569 = 1.28e-3 m^2/s [calc]. Using the gas temperature rather than an electron temperature understates diffusion at these fields [memory, medium].
• Electron drift: mu_e = 0.05 m^2/Vs gives 5e5 m/s at 1e7 V/m, above typical air drift speeds of about 2e5 m/s [memory, medium]. Ion mobilities of 2e-4 m^2/Vs are standard in size; the thesis uses 2.24e-4 and 2.16e-4 [web, thesis].
• Timescale sanity check [calc]: at E = 1e6 V/m an ion with mu = 2e-4 m^2/Vs moves 200 m/s; it takes 5 us to cross 1 mm; at 3e5 V/m, 17 us. The tip-adjacent ion cloud at 0.025 mm [read: p.10] would clear its own region in under a microsecond at tip fields, but the relevant region for field recovery extends over a millimetre or more, where the field is 1e5 to 1e6 V/m. So period of 2 to 20 us is the right order of magnitude for ion drift, in line with the paper's mechanism. This is an order-of-magnitude check, not a proof.
• Tip field estimate: for a hyperbolic tip the tip field is of order 2V/(R ln(4d/R)) [memory, medium]; for R = 100 um, d = 1 cm, V = 7 kV this is 2.3e7 V/m [calc]. The paper's Fig. 10 shows peak tip field 2.2 to 2.45e7 V/m at -7 kV [read: page image p.11]. At V = 4.7 kV the same estimate gives 1.6e7 V/m, corresponding to ~200 Td at the tip surface, well above the 87 Td threshold. So a computed onset of 4.7 kV is not obviously consistent with an ionization threshold of 87 Td at the tip surface, unless the onset is defined by Peek's formula (the thesis says it was "evaluated from experimental Peek's formula" [web, thesis p.96]) or by a path integral of alpha-eta. The model's own onset was apparently not computed from the model's coefficients. Hidden assumption.
• Convergence tolerance 0.01 in text against 0.05 in Fig. 2; no mesh or time-step convergence study in the paper; mesh 5 um near a 100 um tip, with the main avalanche zone about 25 to 50 um thick [read: p.6, p.10]: only 5 to 10 elements. No error estimate is given for the periods. The thesis does a parametric study but I did not find a mesh refinement study in it.

Claim-by-claim (a to d)

Claim 1. Negative corona in air emits regular Trichel pulses, only in electronegative gases and a limited voltage range

a. Mathematics: not a mathematical claim; the model does not test it (it only simulates air/oxygen coefficients, and no pulse-free electropositive-gas run is shown). b. Experiments: pulses are well documented from Trichel 1938 onward, Loeb 1939 reproduced oscillograms [web: archive.org, Loeb 1939 Figs. 243, 247, 248, via Investigator]. "Only in electronegative gases" is contradicted by Zhang, Xia, Jiang and Ouyang, who report pulses in various gases and that negative ions are not necessary [web: https://pmc.ncbi.nlm.nih.gov/articles/PMC5579005/; I have not read it myself]. That is these authors' finding, from one group; no replication found yet. c. Tested: yes, many times (Trichel, Loeb et al. 1941, Lama and Gallo 1974, Zentner 1970, Dordizadeh 2017). The "only electronegative" part: tested by one group, disputed. d. If the Zhang finding stands, the textbook explanation in terms of negative ions would have to be reduced to a modifier of the period rather than a necessity. Verdict: split. Existence of regular pulses in air: Established. "Only in electronegative gases": Never properly tested as a general rule, and under direct challenge by one 2017 study (not replicated as far as found).

Claim 2. Mechanism: avalanche, positive ions absorbed in nanoseconds, slow negative-ion drift sets the period

a. Qualitative logic is sound and the order of magnitude of the clearing time is right [calc, above]. The equations as printed contain the typos listed above (Eqs. 6, 7, ki, units of recombination); with the corrections the structure (drift, attachment, Poisson feedback) is a standard drift-diffusion-reaction system and self-consistent. b. Consistent with Trichel's claim that periodicity comes from space-charge formation and clearing [web: Phys. Rev. 54, 1078 abstract via Investigator]. Not consistent in detail with Trichel's reported independence from external circuit constants, since the thesis simulations find R_ext matters (lower R_ext gives longer period) [web, thesis Ch. 7 conclusions]. Also Gallo's observed micron-scale channel at the tip (below) is not in the model. c. Reproduced numerically by SCA, and by earlier simulators (Morrow 1985, Napartovich 1997, Soria 2007 per refs [read: p.14-15]); I did not read these. SCA's two documents disagree with each other on the experimental benchmark. d. Nothing would need to change; it is the textbook picture. The positive-ion-dominated alternative (Zhang et al. 2017) would adjust it. Verdict: Plausible, untested in the strict sense (the mechanism is simulated, not isolated experimentally, e.g. by ion removal); the simulations are internally consistent once typos are fixed. I do not call it Established for this paper alone.

Claim 3. A 2D FEM-FCT three-species model reproduces the pulse train, post-pulse, large first pulse; first successful 2D FEM-FCT simulation at several voltages

a. Mathematics: the printed system has the defects listed (Eq. 6/7 sources, ki exponent, recombination units, undefined k, N_O2 labelling, tolerance 0.01 vs 0.05). Corrected, it is a coherent scheme. The authors state the post-pulse can "be a source of instability" requiring careful time step [read: p.8]; a post-pulse that the authors themselves associate with numerical instability is not evidence of a physical effect, and the paper does not claim to observe it experimentally. The very large first pulse (25 to 37 mA against 1.8 to 2 mA, a factor 12 to 20) arises from a charge-free start; since the initial condition is "small initial space charge density" with no value given [read: p.5], this is a start-up transient that depends on an unstated input. b. Experiment: only the period vs voltage is compared; no amplitude, charge per pulse, rise time or width comparison. The paper's simulated pulses are visibly hundreds of ns wide at base [read: page image p.9, deep read estimate]; the thesis gives 0.2 to 0.3 us pulse duration [web, thesis Table 8.3], against the "tens of ns" and rise times of about 1.3 ns quoted in the paper's own introduction [read: p.1]. Thus width and rise time disagree with the literature values the paper cites by an order of magnitude or more; this is not remarked on. The charge per pulse in the thesis is about 115 pC at r = 100 um [web, thesis Table 8.4]; my check [calc]: 115 pC / 11.3 us = 10.2 uA, matching the thesis's average current of 13.6 uA only roughly (ratio 0.75), so the period-current-charge triple is not exactly self-consistent in what I read; I did not trace why. c. Tested by the authors against Lama and Gallo (period only). Later group work: Dordizadeh, Adamiak and Castle 2017 report good agreement of their measurements with Lama and Gallo but "some discrepancies" with simulation [web: https://www.sciencedirect.com/science/article/abs/pii/S0304388616301735, abstract via Investigator; not read in full]. Independent replication by a different group of this specific FEM-FCT code: none found. Novelty claim "first" cannot be checked by me. d. Nothing in mainstream physics changes. Verdict: Plausible, untested as a quantitative predictor. Pulse existence and ordering of timescales: reproduced. Quantitative match: Never properly tested (benchmark data unstated and, as shown, inconsistent between documents).

Claim 4. Pulse frequency rises with voltage; period 11.33 us at 7 kV to 2.46 us at 11 kV, following Lama and Gallo's trend

a. Internally: the simulated values are monotone; averages in Table 1 are means excluding the first interval, as I verified (mean(13.38, 10.67, 10.85, 10.4) = 11.33; mean(3.53, 3.30, 3.35) = 3.39 [calc]). The 9 kV point rests on three intervals only. No uncertainty is given. The 8 and 11 kV rows have no waveform shown. b. Consistent in trend with the law f ~ V(V - V0) [web, thesis]. The magnitude depends on the benchmark chosen (above). c. Compared once by the authors; Lama and Gallo's data and Dordizadeh 2017 give the experimental side. Dordizadeh's reported onset of -4 to -6 kV for 19 to 55 um tips [web, ScienceDirect abstract via Investigator] differs from Lama and Gallo's V0 = -2.3 kV; the two experimental records are not reconciled in what I read. This matters directly for the benchmark discrepancy I found: a V0 near 4.7 kV would match the ESA column, a V0 of 2.3 kV matches the thesis column. d. Nothing. Verdict: Trend (frequency increases with voltage): Established. Specific numbers: Never properly tested as a quantitative benchmark; the reference values are not reproducible from the stated source.

Claim 5. Simulated periods are about half of measured; attributed to numerical approximations

a. The ratios are 0.52, 0.47, 0.37, 0.49 [calc]; "approximately 50%" is loose at 9 kV. The attribution to "approximations in the numerical technique" is untested [read: p.14]. Candidate causes with the strongest sensitivity: ion mobility (period scales as ~1/mu), R_ext, and the benchmark curve itself. The deep read ranked these; I add the benchmark (V0 = 4.7 vs 2.3 kV) as potentially accounting for the whole ESA gap at 7 kV, and the thesis shows the sign flips with R_ext = 100 kOhm and mobilities 2.2e-4 [web, thesis]. A factor-two discrepancy that changes sign between two papers by the same group and depends on a parameter (R_ext) which one paper omits cannot be assigned to numerics. b. Gallo's microscope observation of a 0.12 to 2 um bright channel at the negative tip, not seen in positive corona, with onset at -2.3 kV and a wobble that Gallo associates with the "mode shifts"; relayed in the thesis as previously unpublished observations by C. F. Gallo, one source (the thesis authors, collaborators of Gallo) [web, thesis Ch. 8.3]. Refitting with an effective tip radius of 10 um brings the period within 15% and the current within 23% (2.3 vs 2.0 us; 12.1 vs 9.8 uA) [web, thesis Ch. 8.4]. The fit uses a free parameter chosen after the fact (effective radius), so agreement is by adjustment. If real, it puts the physics at the tip outside the continuum fluid model. That is a documented observation worth weighing; I cannot evaluate the optics. The suggestion of magnetic confinement is the thesis authors' speculation. Checking it: a current of 1.8 mA through a 1 um radius channel gives a field of mu0 I/(2 pi r) = 2e-7 x 1.8e-3/1e-6 = 0.36 T at the surface [calc], large but not impossible; whether it confines an electron swarm at atmospheric pressure (collision frequency ~1e12 /s vs cyclotron 6e10 /s at 0.36 T) is doubtful because electrons are collisional: cyclotron frequency below collision frequency [calc, memory medium on collision frequency]. So magnetic confinement looks implausible by this estimate; that is only a rough bound. c. Not independently tested; "reasons being investigated" in 2010; the 2011 thesis offers parametric studies and the 2017 and 2023 papers (Shaygani and Adamiak, which compares three rate-coefficient formulations [web: https://iopscience.iop.org/article/10.1088/1361-6463/acdd0f/meta, via Investigator]) continue. d. Nothing in mainstream physics changes. A genuine micrometre channel would require a sub-continuum treatment. Verdict: Not testable as stated (cause unspecified), with the quantitative gap itself labeled Internally inconsistent across the authors' two documents (experimental reference values differ by almost a factor 2 at 7 kV: 21.85 us vs 11.3 us).

Summary of Investigator leads weighed

• Resolves the 449.3 misprint (kept; Corrections 1).
• Different experimental reference sets: kept and expanded above.
• Trichel citation error: the ESA paper cites Phys. Rev. 55, 382 (1938) with a "positive" title; the Investigator finds the negative-corona paper is Phys. Rev. 54, 1078 (1938) and 55, 382 is the 1939 positive-corona paper [web: https://link.aps.org/doi/10.1103/PhysRev.55.382]. Citation defect, not mathematical.
• Gallo's channel and R_ext dependence vs Trichel's independence from circuit constants: kept.
• Zhang et al. 2017 (pulses in non-electronegative gases): kept, reported as one group's claim.
• Historical record: Trichel (1898-1990), Army officer, record does not go dark; no mathematical content, nothing to add.
• Not read by me: Dordizadeh 2016/2017 full texts, Kokovin 2021, Ferreira 2025, Lama and Gallo 1974.

Verdict table