Mathematician review: Langmuir-Irving (Tonks and Langmuir, "Oscillations in Ionized Gases", Phys. Rev. 33, 195-210, Feb. 1929)

Reviewer role: PhD mathematician (Step 2). I check derivations, notation, units and hidden assumptions, and answer questions a-d for each central claim. I read the Step 1 deep read, then re-read for myself the experimental pages 13-17 (journal pp.207-210) from page images, which Step 1 had covered by digest only [read: page image pp.13-17]. I re-did the arithmetic with numpy. Source tags: [read], [web], [memory] with confidence.

Source gap, repeated from claims.md: the only primary document is one 17-page scan whose text layer is OCR. The theory was verified from page images in Step 1 and the experiments here. I did not read the 1928 PNAS paper, the June 1929 Note (Phys. Rev. 33, 990), or Tonks's 1931 papers (paywalled or not downloaded by the Investigator). Where a conclusion about later work rests on the Investigator's abstracts, I say so.

Reader aid, symbols used throughout: n = electron (and ion) number density, cm^-3; e = electron charge (esu, Gaussian units); m = electron mass; m_p = positive-ion mass; k = Boltzmann constant; T_e = electron temperature; nu = frequency in cycles/s (Hz); omega = 2 pi nu = angular frequency (rad/s); lambda = wavelength; c = speed of light; Gaussian units are used, so Poisson's equation reads div E = 4 pi rho (where SI would write rho/epsilon_0).

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Corrections to Step 1

Step 1's deep read of the theory (pp.1-9) is accurate. I rechecked its key steps and found no error in its algebra. Items to correct or add:

1. Digest p.17 and the transit-time claim is reversed. The digest says the plasma electrons' transit time through a wavelength (2.8e-8 s) is "much longer" than the oscillation period (5.5e-8 s). 2.8e-8 is shorter than 5.5e-8, by about a factor 2. The page says only that, "taken in conjunction with the enormous preponderance of plasma electrons", the beam cannot be thought to oscillate unaffected by them [read: page image p.15]. I re-derived the numbers: 14 cm / 5.1e8 cm/s = 2.7e-8 s; 1/18 MHz = 5.6e-8 s; lambda = v_e/(nu_d - nu_e) = 5.1e8/(37e6) = 13.8 cm. The printed numbers are right; only the digest's comparison word is wrong.
2. claims.md item 4, "two stable scattering states with frequencies in a ratio near 1:2". The paper's words are "almost twice", and the numbers are 1080 MHz (low scattering, 26 V) and 660 MHz (high scattering, 28 V) [read: page image p.14]. The ratio is 1.64, not 2. Since the formula gives nu proportional to n^(1/2), a ratio 1.64 means a density ratio of 2.7, not 4. The two states were also measured at different anode voltages (26 V and 28 V), so the frequency change is confounded with a voltage change. The paper does not interpret the ratio as a harmonic.
3. claims.md item 4, "wavelengths scale as relative ionization to the power -1/2, as the plasma-electron formula predicts" overstates the support. The paper says: "there was no way of finding the absolute value of the ion density in the present tube because of large edge effects"; only "relative values" from "an elaborate series of measurements" were computed; "Partial failure of some internal checks in these calculations show that too much reliance is not to be placed on them"; 13 of 15 points lie within the accuracy of the ionization determination, two do not [read: page image p.15]. See the evaluation of claim 4 below.
4. claims.md item 5, "oscillations explain the scattering". The authors did not claim this firmly. On p.13 (journal p.207) they write that the runs meant to correlate oscillations with scattering "are inadequate" for three reasons: the range covered was too narrow compared with the roughness of the results; the "temperature" of the scattered electrons was indeterminate; and there were no high-frequency checks on the crystal detector. Their only correlation: with 10 mA from a 0.0125 cm filament, oscillations "disappear in the same range of anode potential that the scattering does, namely between 60 and 75 v" [read: page image p.13]. Step 1's wording ("they suspect") is fair; the strength of the authors' own caveat should be kept.
5. Digest numeric/OCR items already caught by Step 1, confirmed: no epsilon in the formulas; dispersion term c^2/lambda^2; sound speed (kT_e/m_p)^(1/2); ion spacing 4.6e-4 cm (not e-5); the first occurrence of "2(2 pi)^(1/2) lambda_D" on p.7 is a misprint for 2 sqrt(2) pi lambda_D. I confirm all.
6. Digest p.8 "wavelength greater than ~10^-5 cm" should be about 10^-1 cm (Step 1 already says this; my recomputation agrees, upper 5 % point 0.137 cm).
7. One addition to Step 1's flag table. The statement "Zero group velocity" is correct for the cold-plasma model, and I quantify below how large the correction is for the paper's own parameters.
8. facts.json: I did not re-audit every entry; the entries I compared against the pages agree. The facts.json how_read for the experimental pages should read "digest + page images pp.13-17 (this review)" if anyone updates it.

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Verification of the mathematics (question a, common to claims 1-3)

I re-derived the central results independently.

Plasma frequency (Eqs. 1-6). Displace electrons by xi(x). Electron density changes by -n d(xi)/dx, so the net positive charge density is rho = e n d(xi)/dx. Gauss: dE/dx = 4 pi rho, so E = 4 pi n e xi. Newton for an electron (charge -e): m xi'' = -eE = -4 pi n e^2 xi. Simple harmonic motion with omega^2 = 4 pi n e^2/m, so nu = omega/2 pi = (n e^2/(pi m))^(1/2). Numerically e/sqrt(pi m) = 8979, matching the printed 8980. The odd-looking "pi" is just the 2 pi in omega = 2 pi nu absorbed into the 4 pi of Gaussian units: 4 pi/(2 pi)^2 = 1/pi. [read: page image pp.2-3, recomputed]

Worked example (the paper's): n = 1e10 cm^-3 gives nu = 8980 x 1e5 = 8.98e8 Hz, wavelength in air = 3e10/8.98e8 = 33 cm. Check passed.

Vector derivation (Thomson's method, Eqs. 5B, 5C). Maxwell's curl equations plus the electron equation of motion with v x H dropped, then take the divergence: d^2(delta n)/dt^2 + (4 pi n e^2/m) delta n = 0. This is rigorous for linearised, cold, collisionless, non-magnetised plasma and shows the oscillation frequency is independent of the shape of the displacement, because the equation contains no space derivatives. Hidden assumptions (not stated by the authors as such): (i) small amplitude (linearisation), (ii) cold electrons (no pressure term), (iii) no collisions, (iv) infinite uniform background ion "jelly", (v) no magnetic field. All five are standard and the first, second and fourth are acknowledged in the text ("rigid jelly", "xi small").

Electromagnetic dispersion. With div E = 0, curl curl E = -laplacian E, so omega^2 = 4 pi n e^2/m + c^2 k^2. With k = 2 pi/lambda and dividing by 4 pi^2: nu^2 = n e^2/(pi m) + c^2/lambda^2. This is the modern cold-plasma transverse dispersion relation. The bulb estimate (8.9e8)^2 + (3e10/36)^2 under a square root gives 12.19e8 Hz and 24.6 cm in air; matches the printed 12.2e8 and 24.6 cm. The use of lambda = 36 cm (bulb diameter 18 cm as half a wavelength) is a hand-waving boundary condition for a spherical bulb; the authors flag it, saying the identification "must be reserved".

Ion branch (Eqs. 7-12). Ion continuity gives delta n_p = -n d(xi)/dx; Boltzmann electrons give delta n_e = n(exp(eV/kT_e) - 1), linearised to n eV/kT_e; Poisson and the ion force equation give d^2/dx^2 (xi'' + (4 pi n e^2/m_p) xi) - (4 pi n e^2/kT_e) xi'' = 0. Fourier mode exp[i(kx - omega t)] gives omega^2 = k^2 omega_pi^2 /(k^2 + a), omega_pi^2 = 4 pi n e^2/m_p, a = 4 pi n e^2/kT_e, i.e. omega^2 = k^2 c_s^2/(1 + k^2 lambda_De^2) with c_s = (kT_e/m_p)^(1/2), lambda_De = (kT_e/4 pi n e^2)^(1/2). I confirmed this matches the printed form nu = (n e^2/(pi m_p + n e^2 m_p lambda^2/kT_e))^(1/2). Limits: short wavelength gives the ion plasma frequency (1.48e6 Hz for Hg, recomputed 8980/607 x 1e5 = 1.48e6; the paper's "1.5e6" is right); long wavelength gives the sound speed, 6.4e4 cm/s for Hg+ at T_e = 1e4 K. The 5 % transition statement at lambda = 2 sqrt(2) pi lambda_D x 10^(+-1/2) holds (4.7 %, recomputed).

Debye length. lambda_D = (kT_e/8 pi n e^2)^(1/2) = 4.90 (T_e/n)^(1/2) cm; this is the two-species Debye-Hueckel length (both ions and electrons at T_e), smaller by sqrt 2 than the modern electron Debye length (6.90 (T_e/n)^(1/2) cm). Convention, not an error, but it is why the "awkward" factor 2 sqrt(2) pi appears: 2 sqrt(2) pi lambda_D = 2 pi lambda_De, i.e. k lambda_De = 1, where the dispersion curve bends. Misprint: "exp(r/lambda_D)" for the screened potential should be exp(-r/lambda_D). For n = 1e10, T_e = 1e4: lambda_D = 0.0049 cm, matches the printed 0.005.

Self-consistency of the continuum treatment. The authors average fields over volumes holding many particles. Number of particles in a Debye sphere, n lambda_De^3 at the paper's values, is about 1.2e3 (1.2e3 [memory, high], matches Step 1's own figure). The averaging is justified. The paper itself notes the ion spacing 4.6e-4 cm is well below the transition wavelength 4.4e-2 cm.

Group velocity. Step 1 notes the zero group velocity is a cold-plasma result. I quantified it using the Bohm-Gross relation omega^2 = omega_pe^2 + 3 k^2 v_th^2 with v_th^2 = kT_e/m [memory, high; Bohm and Gross 1949; the Investigator's lead on Linder 1936 shows a pressure correction was published earlier, [web] Crossref DOI 10.1103/physrev.49.753, content not read]. At n = 1e10, T_e = 1e4 K the group velocity is 1.5e5 cm/s at lambda = 33 cm, 5.0e6 cm/s at lambda = 1 cm, and 4.0e7 cm/s at lambda = 0.1 cm (thermal speed 3.9e7 cm/s). For the long wavelengths the paper cares about (tens of cm), the correction is five orders of magnitude below c, so "zero" is an excellent approximation; but it is not exactly zero, and for wavelengths near the Debye length it is of the order of the electron thermal speed.

Ion sector, a hidden assumption that matters. The ion-wave derivation treats the ions as a cold fluid (no ion pressure) and electrons as isothermal Boltzmann. That is legitimate only if T_i << T_e (ion-acoustic waves are strongly Landau-damped when T_i is comparable to T_e) [memory, high]. The paper does not state T_i. In a mercury arc the ions are near the gas temperature (a few hundred K), and T_e ~ 1e4 K, so the assumption holds well in this setting. Also, the Boltzmann assumption requires wave phase speed << electron thermal speed (6.4e4 vs 3.9e7 cm/s: satisfied) and frequency << electron plasma frequency.

Energy-transfer numbers. The beam example (72.6 V, 25 mA, current density 3.1e-4 A/cm^2 gives velocity 5.05e8 cm/s, charge density 6.1e-13 C/cm^3, n = 3.8e6 cm^-3, nu = 1.75e7 Hz) was recomputed: correct. The 8.3 V/cm field for displacement by one electron spacing: correct (8.3 V/cm, 3.0e-5 erg/cm^3).

Doppler step (p.15). The relation nu_d = (v_w + v_e)/lambda with v_w/lambda replaced by the beam's natural frequency nu_e gives lambda = v_e/(nu_d - nu_e). This presumes a beam in vacuum oscillating at its own plasma frequency computed from the beam density alone. The authors themselves state why this fails inside the plasma ("impossible to think that these electrons do not modify the beam oscillations tremendously"; Mott-Smith's result that nu_e is the minimum for a stationary observer) and conclude it is "impossible to identify the 55 with the theoretical 18 megacycle oscillation" [read: page image pp.15-16]. This is honest self-refutation of that identification, not an error.

Numerics of the experimental section. The temperature-dependent frequencies (263, 231, 250 MHz at 19, 20, 21 C appendix temperature; 2 % accuracy) are not monotonic in temperature, although mercury vapour pressure rises monotonically with temperature, and the plasma-electron frequency should track density. Non-monotonic frequency with temperature is not explained by the paper [read: digest p.11-12, not rechecked on the page image, p.11-12 image not examined by me]. The 115 MHz beam-electron identification expects a 26 % rise for a 60 % rise in emission (sqrt 1.6 = 1.265, recomputed) but found 10 %; the authors state the discrepancy [read: digest p.13-14].

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Claim-by-claim review

Claim 1. Natural plasma-electron oscillation, nu = (n e^2/(pi m))^(1/2) = 8980 n^(1/2) Hz

a. Mathematically consistent? Yes. Derived two ways (one-dimensional and vector); each step checks; units consistent in Gaussian; the 8980 constant is correct. The cold, collisionless, linear, uniform-background assumptions are disclosed or standard.

b. Consistent with established experiments? Yes. The same formula (f_pe = 8.98e3 n^(1/2) Hz) is the standard plasma frequency in modern formularies [memory, high, NRL Plasma Formulary] and underlies ionospheric critical-frequency measurements and plasma reflection of radio waves [memory, high]. The Investigator's finding that Tonks himself applied the plasma frequency to ionospheric critical frequency (Nature 132, 101 and 710, 1933 [web] Crossref) is consistent. Merrill and Webb (1939) report that "the observed frequencies agreed well with the formula derived by Tonks and Langmuir" in a mercury arc [web] https://journals.aps.org/pr/abstract/10.1103/PhysRev.55.1191 (abstract only read by the Investigator).

c. Tested? By whom? Replicated? In the 1929 paper the experimental test was weak (see claim 4). Independent confirmations: Merrill and Webb 1939 (a different group, same detection method); later plasma-oscillation experiments by Armstrong 1947, Neill 1949, Wehner 1951, Emeleus 1955-56, Dawson 1959 listed by the Investigator [web] Crossref (none read by me). The formula as theory is checked far better by the large body of later work than by the 1929 data.

d. If true, what would change? Nothing; this is mainstream physics.

Credit: the 1929 paper is the first systematic statement of this formula for laboratory gas discharges, crediting J. J. Thomson's method (footnote 6) [read: page image p.4]. Thomson's 1928 paper (Phys. Soc. Proc. 40, 82) is cited. Earlier forerunners exist (Rayleigh, Lorentz theory of ionospheric/dielectric dispersion) [memory, medium]; I have not verified priority.

Verdict: Established (as the cold-plasma, linear result; the experimental support in the paper itself is weaker than the theory, but the formula is supported by a century of later work).

Claim 2. Zero group velocity of the oscillations; EM dispersion nu^2 = n e^2/(pi m) + c^2/lambda^2

a. Consistent? Yes. The EM dispersion is correct (modern omega^2 = omega_pe^2 + c^2 k^2). The zero group velocity follows exactly from Eqs. (5B)/(5C), which contain no spatial derivative. Note the claim mixes two kinds of waves: the longitudinal electrostatic oscillation (zero group velocity in this model) and transverse EM waves (non-zero group velocity c^2 k/omega; the paper says there is "no natural frequency" for them, only a cut-off at nu_e).

b. Consistent with experiment? The EM part: yes (plasma cut-off/reflection, standard [memory, high]). The zero-group-velocity part: true only in the cold limit. With finite T_e, the Bohm-Gross term gives a group velocity of order 3 k v_th^2/omega (my numbers above: negligible at tens of cm, ~ thermal speed near the Debye length). Langmuir waves with finite group velocity have been observed in later experiments [memory, high; Looney and Brown 1954 reported by memory, medium]. The 1929 paper is therefore right at the stated order of approximation.

c. Tested? No direct group-velocity test in the paper. Merrill and Webb's narrow, localised oscillating regions (a few tenths of mm wide, per the Investigator's abstract) are compatible with standing, non-propagating local oscillations but do not measure group velocity. Linder 1936 extended the theory to electron pressure [web] Crossref title only.

d. If true? Nothing; mainstream.

Verdict: split. The EM dispersion relation: Established. Zero group velocity: Historically superseded but instructive in its strong form (exactly zero), an excellent approximation for long wavelengths and cold electrons; the thermal correction is standard physics since the 1930s-1940s.

Claim 3. Plasma-ion oscillations with Boltzmann electrons; sound-like waves at v = (kT_e/m_p)^(1/2); ~1.5 MHz limit for Hg

a. Consistent? Yes. Derivation re-done and symbolically matched (see above). The two limits and the transition at k lambda_De ~ 1 are correct. The 1.5 MHz limit is arithmetically correct for n = 1e10. Misprint exp(r/lambda_D) (sign) and the first "2(2 pi)^(1/2) lambda_D" do not propagate. Unstated assumption: cold ions (T_i << T_e), which holds in the mercury arc.

b. Consistent with experiments? The formula is the modern ion-acoustic wave dispersion, with ion-acoustic speed (kT_e/m_i)^(1/2) [memory, high]. Ion-acoustic waves were later directly observed in laboratory plasmas (Alexeff and Neidigh 1962 [memory, medium], not verified by web). The Investigator found Hernqvist (RCA) 1955 observing ion plasma oscillations in electron beams [web] Crossref DOI 10.1063/1.1722036 (not read) and Peterson and Puthoff 1959 [web] Crossref. In the 1929 paper there is a loose comparison with Webb and Pardue's oscillations from a few hundred Hz to 240 kHz in an air discharge, which the authors themselves say allows "no quantitative comparison" [read: page image p.8].

c. Tested? In 1929 not tested quantitatively. The 1.38 MHz resonance found by a tuned circuit was close to the 1.5 MHz limit, but the authors could not repeat it: the wave "had become so elusive", the radio amplifier found "a continuous spectrum" of noise, with peaks that changed with slight tube changes [read: page image p.16]. So the 1.38 vs 1.5 MHz agreement is a coincidence of one unreproduced measurement with a loose upper bound (an upper limit does not predict a single frequency). Later: Baalrud, Callen and Hegna (2009) invoke convective ion-acoustic instabilities near discharge boundaries to explain Langmuir's paradox, which ties the ion branch of this paper to the present-day theory [web] https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.102.245005 (abstract and p.1 read by the Investigator).

d. If true? Nothing; it is mainstream.

Verdict: Established for the theory (the ion-acoustic wave is a verified phenomenon). Never properly tested in this paper for mercury specifically: the paper's own experimental "evidence" for the 1.5 MHz limit is a single irreproduced 1.38 MHz peak.

Claim 4. Mercury arcs emit RF oscillations ~1e7-1e9 Hz as noise; 300+ MHz wavelengths scale as (relative ionization)^(-1/2); two stable scattering states with frequencies near 1:2

Pieces:

4i. RF oscillations of 1e7-1e9 Hz exist. An observation, with crystal detector, galvanometer and Lecher wires. The authors call it "electrical noise" without clear resonances. Plausible and consistent with Penning 1926. The calibration (60-cycle, log-log slope 0.5, i.e. detector response proportional to the square root of the input power, as expected for a square-law detector) is documented, but the authors also say the absolute amplitude is unreliable between runs ("frequency variation factor noted", "relative amplitudes within run considered reliable" [read: digest p.10-11, not rechecked]). Detector frequency response at 1e9 Hz was not independently checked ("absence of high frequency checks on the crystal detector", [read: page image p.13]).

4ii. Wavelength proportional to (relative ionization)^(-1/2). The formula nu proportional to n^(1/2) implies wavelength proportional to n^(-1/2): a log-log slope of -1/2 [recomputed from Eq. 6]. Fig. 10 plots 15 points of wavelength (about 10 to 100 cm) vs a relative ionization scale (about 2 to 6, i.e. less than one decade) [read: page image p.15]. Mathematical assessment: with 15 scattered points over under one decade of the abscissa and an admitted unreliable x-axis ("partial failure of some internal checks"), a slope of -1/2 cannot be distinguished from a fairly wide range of slopes. A fit of -1/2 is a necessary consequence, not a discriminating test. Two points out of fifteen are off. This is the paper's strongest numerical test of the formula inside the discharge, and the authors themselves say "too much reliance is not to be placed on them." Absolute check of n: a frequency of 810-1080 MHz implies n = (nu/8980)^2 = 0.8-1.4e10 cm^-3 (my calculation), a physically plausible arc density, but the paper has no independent absolute density (edge effects at the electrodes).

4iii. Two stable states, frequencies ratio "almost twice" (1080 vs 660 MHz). Ratio is 1.64 (see correction 2). Under the plasma formula it needs a density ratio of 2.7. The paper does not measure the two densities, and the two frequencies were taken at different anode voltages. As a bistability of the discharge (hysteresis with collector voltage, magnet-triggered switching, state diagram Fig. 9: only low scattering at 10 and 15 mA, both at 20 and 25 mA, only high at 35 mA) it is a real observation worth keeping. The "1:2" is not an established feature.

a. Mathematically consistent? The formula tested is consistent. The comparison (slope -1/2 on a relative scale) is arithmetically correct but weak. The ratio 1.64 vs "almost twice" is a loose description, not a derivation.

b. Consistent with established experiment? Qualitatively: discharge plasmas do emit RF at the plasma frequency range, in line with known results (Merrill and Webb 1939; later: Armstrong, Neill, Wehner, Emeleus [web] Crossref titles only). Bistable scattering states are not a standard mainstream result; I know of no replication. Not contradicted, not confirmed.

c. Tested? Replicated? Merrill and Webb (1939) used a movable probe, crystal detector and Lecher wires, found "stable periodic oscillations of considerable magnitude" in narrow regions, in addition to "turbulent" disturbances with no measurable frequency; the frequencies "agreed well" with the formula [web] abstract only, https://journals.aps.org/pr/abstract/10.1103/PhysRev.55.1191. That is the nearest thing to a replication; I have only its abstract. No replication of the bistable states found by the Investigator.

d. If true? Nothing in mainstream physics would have to change; the formula is standard.

Verdict: split. RF noise 1e7-1e9 Hz and its general frequency range: Plausible, untested in the 1929 setting beyond the authors' own non-neutral data, later supported by Merrill and Webb. The n^(-1/2) law as a test of Eq. (6): Never properly tested (the 1929 paper's own test uses an unreliable relative scale; no absolute density; the authors themselves caution). The bistable-state observation: Never properly tested (single laboratory, no replication found; the "1:2" is really 1.64).

Claim 5. The oscillations explain the anomalous electron scattering / energy gain in low-pressure Hg arcs (Langmuir's paradox)

a. Mathematically consistent? The paper offers no calculation connecting oscillation amplitude to the observed energy transfer. It estimates only the field energy density (3.0e-5 erg/cm^3) for a one-spacing displacement and lists three qualitative mechanisms (p.5-6). There is no mathematics to be consistent or inconsistent. One quantitative reasoning point: Dittmer (cited on p.1-2) argued the oscillations would need amplitudes of "20 volts or more" to explain the scattering, and the authors do not show their observed amplitudes (millivolts on a crystal detector, uncalibrated at these frequencies) reach this. Beam-oscillation identification (the 18 MHz, 55 MHz, Doppler argument) fails by the authors' own account.

b. Consistent with established experiments? The anomaly is real as reported in the 1925 papers (rapid Maxwellisation at low pressure) and has been described as a "serious discrepancy" ever since (Baalrud et al. 2009, [web] open copy https://plasmatheory.engin.umich.edu/wp-content/uploads/sites/519/2021/01/BaalrudPRL2009a.pdf, page 1 only read by the Investigator). Mechanism candidates: bulk plasma oscillations (Tonks-Langmuir, Merrill-Webb 1939), sheath oscillations (Gabor, Ash and Dracott 1955, Nature 176, 916), ion-acoustic instabilities (Baalrud 2009), non-local kinetics with striations that deny the premise (Godyak and Alexandrovich 2015, argon, 1-3 mTorr, "essential deviation" from Maxwellian [web] https://iopscience.iop.org/article/10.1088/0963-0252/24/5/052001, abstract only). The last one is evidence against the premise in a different gas and geometry, not against the mercury observation in 1925-1929.

c. Tested? Merrill and Webb (1939): oscillation regions coincide with the electron-scattering regions in a movable-probe mercury-arc experiment and they conclude that "scattering is due to plasma oscillations which receive their energy from the fast electrons". That is a correlation experiment by an independent group, not an energy budget; I only have its abstract. Gabor's school argued for a different oscillation mechanism. No decisive controlled test is on record. It was not dismissed without testing: it was investigated from 1928 to 2015 and remains unsettled. Note that the structure of the paper's later experiments ("experiments under essentially simpler conditions", cylindrical tube, p.16) was announced and the follow-up Note (Phys. Rev. 33, 990) and the companion theory (Phys. Rev. 34, 876) were not available to me.

d. If true? Little: collisionless collective scattering by plasma waves is accepted plasma physics. What would change is the claim that Coulomb-collision theory alone fixes the electron distribution at low pressure.

Verdict: Never properly tested (not dismissed without testing; one supportive independent correlation, several rival mechanisms, one study that questions the premise; the authors' own caveat in 1929 was that the data "are inadequate").

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Investigator's findings I weighed (kept in the record whatever the verdict)

• Tonks and Langmuir, "Note on 'Oscillations in Ionized Gases'", Phys. Rev. 33, 990 (June 1929) [web] https://journals.aps.org/pr/abstract/10.1103/PhysRev.33.990: not readable (paywall). Unknown whether it corrects the misprints (sign of exp(r/lambda_D); the first 2(2 pi)^(1/2) lambda_D) that I found. Open item.
• Langmuir, PNAS 14, 627 (1928) [web] https://www.pnas.org/doi/10.1073/pnas.14.8.627, free PDF, not downloaded. The earlier announcement; the quickest route to see whether the experimental pages' weaknesses were stated more or less cautiously there.
• Tonks 1931 papers ("The High Frequency Behavior of a Plasma", "Plasma-Electron Resonance, Plasma Resonance and Plasma Shape") [web] Crossref: bear on whether the plasma-frequency independence of geometry (claim 1, 1-D versus bounded) survives. Mathematically, the infinite-medium result is exactly shape-independent; for a bounded plasma the resonance shifts (a uniform sphere resonates at nu_e/sqrt 3, a slab or plane-bounded geometry differs) [memory, high for the sphere result: Mie/Dattner-Tonks type results]. The paper's own statement of shape independence (p.3) is correct for the bulk oscillation and its wording for spherical/cylindrical shapes carries the caveat "unless xi/r remains small". I did not read the 1931 papers, so I do not know what Tonks wrote.
• Merrill and Webb 1939, Linder 1936, Hernqvist 1955, Gabor et al. 1955, Martin and Emeleus 1956, Baalrud et al. 2009, Godyak and Alexandrovich 2015: discussed above under claims 1-5; all from abstracts or titles except the first page of Baalrud (read by the Investigator).
• Historical record: the Investigator found no anomaly of disappearance: the work continues openly (Tonks 1931-34, Merrill-Webb 1939, Gabor 1955, Baalrud 2009). I agree: nothing for the mathematician to add. The open question is physical (the scattering mechanism). The Langmuir papers at the Library of Congress (lab notebooks 1894-1957, [web] https://www.loc.gov/item/mm77029413/) would hold the raw numbers behind the 1.38 MHz and two-state observations; unexamined.
• Tonks, "The Birth of 'Plasma'", Am. J. Phys. 35 (1967) [web] via Bing/ADS snippet only: a first-hand history; unread.
• A discrepancy for the Investigator's note: Baalrud et al. describe Langmuir's tube as about 3 cm in diameter, while this paper's first tube is an 18 cm bulb; the Investigator already suggests it is the 1925 apparatus.

Web checks

I did no new web calls. The [memory] facts on which my verdicts depend (plasma frequency 8.98e3 sqrt n; Bohm-Gross; ion-acoustic dispersion and speed; T_i << T_e requirement; NRL formulary coefficients) are high-confidence and standard. Medium-confidence items (Alexeff-Neidigh 1962; Looney-Brown 1954; priority before Thomson) do not decide any verdict and are marked as [memory] where used.

Summary of verdicts