Primer: Langmuir-Irving (Tonks and Langmuir, "Oscillations in Ionized Gases", 1929)

Study document for the reader (Step 4). It makes no new verdicts; the verdicts are the arbiter's (arbiter.md). Source tags: [read] from the project document; [read: page image p.N] where a reviewer read a rendered page; [web] with URL; [memory] with confidence; [calculation] for numbers recomputed in example.py.

Project document: Lewi Tonks and Irving Langmuir, "Oscillations in Ionized Gases", Physical Review 33, 195-210 (February 1929), one 17-page scan [read: page image p.1]. The paper is signed "Research Laboratory, General Electric Company, Schenectady, N.Y., November 20, 1928" [read: p.16]. The authors thank H. M. Mott-Smith Jr. for help with the theory (including his unpublished theory of infinite plane trains of oscillations, which shows that a stationary observer sees no frequency below the plasma frequency ν_p) and S. Sweetser for help with the experiments [read: pp.15-16].

Source gap (repeated from claims.md and arbiter.md). The project holds just this one document, and its text layer is 100% OCR (optical character recognition), which garbled every equation. The theory pages (PDF pp.1-9) were transcribed from page images and re-derived by four separate readers. The experimental pages were read from page images by the reviewers (pp.11-17). In the QE (quality engineering) round, two further deep reads were done from page images: pp.5-6 (the authors' thermal-electron and beam-resonance arguments) and pp.13-15 (Figs. 8-10, with the 15 points of Fig. 10 read off and fitted). Pp.9-12 (apparatus, detector calibration, Figs. 2-7) were checked by the QE agent against the text layer only; no page-image deep read of them was done, and Figs. 4-7 have not been read as data. Details from those pages are tagged [read: p.N, QE text check]. Nobody on the review read the authors' 1928 PNAS (Proceedings of the National Academy of Sciences) paper, their June 1929 "Note" (Phys. Rev. 33, 990), Tonks's 1931 papers, or the full text of any later paper. Everything said about later work comes from abstracts or titles, and it is labelled that way.

Run the numbers yourself: .venv/bin/python review/Langmuir-Irving/example.py

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1. The idea in one paragraph

An ionized gas, such as the glowing inside of a mercury-vapour arc tube, contains free electrons and positive ions in nearly equal numbers. Tonks and Langmuir asked what happens if the light, mobile electrons are pushed a little to one side of the heavy ions. A thin layer of uncovered positive charge then pulls them back. They overshoot, are pulled back again, and the whole electron cloud rings like a mass on a spring, at a frequency set only by how many electrons there are per cubic centimetre. The same reasoning applied to the ions, with the hot electrons acting as a springy background, gives slower "electric sound waves". The authors gave the quasi-neutral glowing region its modern name, "plasma". They measured radio-frequency noise coming from mercury arcs and argued that it fits these oscillations. They also suspected (while admitting that the oscillations' voltage "appears to be very small"), without being able to prove it, that the oscillations explain a long-standing puzzle: in low-pressure arcs, electrons share energy far faster than ordinary collisions allow (later called "Langmuir's paradox"). The theory became the foundation of plasma physics. The experiments were honest but weak, and the puzzle is still argued about today.

2. What you need to know first

Gauss's law in one dimension. Gauss's law says that electric field lines start on positive charge and end on negative charge: ∇·E = 4πρ in Gaussian units (ρ = charge per unit volume). If everything depends on one coordinate x only, it becomes dE/dx = 4πρ, so a slab of charge makes the field jump across it. This is all the electricity the main derivation uses. Read: Griffiths, Introduction to Electrodynamics, the chapter on electrostatics (ch. 2).

Gaussian versus SI units. The paper uses Gaussian (cgs) units: lengths in cm, mass in g, charge in statcoulombs (esu, electrostatic units), field in statvolt/cm (1 statvolt ≈ 300 V). Coulomb's law is F = q₁q₂/r², with no 1/(4πε₀), so where SI writes ρ/ε₀, Gaussian writes 4πρ. Translation rule for this primer: 4πne² (Gaussian) ↔ ne²/ε₀ (SI). Both give the same frequency (example.py checks this). Read: Griffiths, Appendix C (units); Jackson, Classical Electrodynamics, appendix on units.

Simple harmonic motion (SHM). If a mass m feels a force −Kx proportional to its displacement x and pointing back, then m ẍ = −Kx (the dots mean time derivatives), and x oscillates as cos(ωt) with angular frequency ω = √(K/m) in radians per second. The ordinary frequency is ν = ω/2π in cycles per second (Hz). The paper always quotes ν, which is why its formulas show π where modern books show 4π. Read: any first-year mechanics text, e.g. Kleppner and Kleppner, An Introduction to Mechanics, the chapter on harmonic oscillators; Taylor, Classical Mechanics, ch. 5.

Waves and dispersion relations. A plane wave varies as exp[i(kx − ωt)], with wavenumber k = 2π/λ (λ = wavelength). Putting this into a linear equation turns each ∂/∂x into ik and each ∂/∂t into −iω, giving an algebraic relation between ω and k called the dispersion relation. From it come the phase velocity ω/k (how fast the crests move) and the group velocity dω/dk (how fast energy and a wave packet move). If ω does not depend on k, the group velocity is zero: the medium rings in place but carries nothing along. Read: Griffiths, ch. 9 (electromagnetic waves, including dispersion); French, Vibrations and Waves.

Boltzmann distribution. In thermal equilibrium at temperature T, the density of particles with potential energy U is proportional to exp(−U/kT), where k is Boltzmann's constant. For electrons (charge −e) in electric potential V, U = −eV, so n_e = n exp(eV/kT_e): electrons crowd into regions of higher potential. For small eV/kT this is approximately n(1 + eV/kT). Read: Schroeder, An Introduction to Thermal Physics, the chapter on the Boltzmann factor (ch. 6).

Debye screening. A charge placed in a plasma attracts opposite charges around it, which cancel its field beyond a distance called the Debye length, λ_D. Its potential becomes (q/r)·exp(−r/λ_D) instead of q/r. The Debye length for electrons is λ_De = (kT_e/4πne²)^{1/2}. Plasma behaves collectively over distances larger than λ_D. Read: Chen, Introduction to Plasma Physics and Controlled Fusion, ch. 1.

Electron-volts and temperatures. 1 eV (electron-volt) ≈ 11,600 K. The paper's typical electron temperature T_e = 10⁴ K is about 0.86 eV.

3. The textbook view

This is the physics every plasma textbook now teaches, mostly in a form that descends from this paper. Symbols:

• n = number of electrons per cm³, equal to the number of singly charged ions (quasi-neutrality)
• e = magnitude of the electron charge = 4.803×10⁻¹⁰ esu
• m (the paper's m_e) = electron mass = 9.109×10⁻²⁸ g
• m_p = positive-ion mass (the "p" means positive ion, not proton); for mercury, m_p = 200.6 atomic mass units = 3.33×10⁻²² g
• k = Boltzmann constant = 1.381×10⁻¹⁶ erg/K; T_e = electron temperature; T_i = ion temperature
• c = speed of light = 3.00×10¹⁰ cm/s
• ω = angular frequency (rad/s); ν = ω/2π = frequency (Hz); λ = wavelength; k_w = 2π/λ = wavenumber (written k in physics books; here k_w where it might be confused with Boltzmann's k)

(a) Electron plasma oscillation. Hold the ions fixed and shift the electrons by a small displacement ξ(x). Gauss's law gives a restoring field E = 4πneξ, so each electron obeys

m ξ̈ = −eE = −4πne² ξ.

This is SHM with

ω_pe² = 4πne²/m (Gaussian) = ne²/(ε₀m) (SI), so ν_pe = ω_pe/2π = (ne²/πm)^{1/2} ≈ 8980 √n Hz, with n in cm⁻³.

This is the "plasma frequency" in the NRL (US Naval Research Laboratory) Plasma Formulary, f_pe = 8.98×10³ n^{1/2} Hz [memory, high].

(b) Light in a plasma. A transverse electromagnetic (EM) wave in a cold plasma obeys

ω² = ω_pe² + c²k_w², or in the paper's form ν² = ne²/πm + c²/λ².

Waves below ν_pe cannot propagate. This is why the ionosphere reflects AM radio but not FM [memory, high].

(c) Thermal correction (Bohm-Gross, 1949). With finite electron temperature the electron oscillation does propagate:

ω² = ω_pe² + 3k_w² kT_e/m, with group velocity dω/dk_w = 3k_w(kT_e/m)/ω.

Bohm and Gross, Phys. Rev. 75, 1851 (1949) [web] https://doi.org/10.1103/PhysRev.75.1851. A pressure correction was already published by Linder, Phys. Rev. 49, 753 (1936) [web, title only via Crossref].

(d) Ion-acoustic waves. Let the ions move, and let the electrons be hot enough to stay in Boltzmann equilibrium. The result is

ω² = k_w² c_s² / (1 + k_w² λ_De²), where c_s = (kT_e/m_p)^{1/2} is the ion sound speed.

For long wavelengths it is a sound wave, ω = c_s k_w, whose speed comes from electron temperature and ion mass. For short wavelengths ω approaches the ion plasma frequency, ω_pi = (4πne²/m_p)^{1/2} = ω_pe (m/m_p)^{1/2}. This requires T_i ≪ T_e; otherwise the wave is strongly damped (Landau damping, a collisionless energy exchange between the wave and particles moving at about its speed) [memory, high]. Ion-acoustic waves were observed in the laboratory by Alexeff and Neidigh, Phys. Rev. 129, 516 (1963), whose summary calls them the "ionic sound waves predicted by Tonks and Langmuir" [web] https://journals.aps.org/pr/abstract/10.1103/PhysRev.129.516 (abstract).

(e) Debye length. λ_De = (kT_e/4πne²)^{1/2} ≈ 6.90 (T_e/n)^{1/2} cm (T_e in K, n in cm⁻³).

4. Where the authors depart (and where they do not)

This project is unusual in the review series. In its theory the "inventors" are the textbook: the textbook view above is largely what this paper founded. The point of departure is therefore not a contradiction with physics. It is (i) a few approximations and conventions that later work refined, and (ii) an experimental interpretation that goes further than the data could carry.

Two misprints in the paper do not affect any result: the screened potential is printed q/(Kr)·exp(+r/λ_D) where it should be exp(−r/λ_D), and the first occurrence of the crossover length is printed 2(2π)^{1/2}λ_D where every later occurrence, and the number used, is 2√2·π·λ_D [read: page image p.7, deep read]. Whether the June 1929 Note corrects them is unknown.

The real physics behind the inference about scattering. The motivating puzzle is real. Langmuir found in 1925 (Phys. Rev. 26, 585) that in low-pressure mercury arcs with a hot filament, many primary electrons "rapidly acquire velocities whose voltage equivalent is greater than the total drop across the tube", while others lose energy [read: p.1]. Binary (two-particle) Coulomb collisions are far too rare at these pressures to do this. Baalrud, Callen and Hegna (2009) call it a "serious discrepancy in the kinetic theory of gas discharges" with no "definitive answer" [web, open copy p.1] https://plasmatheory.engin.umich.edu/wp-content/uploads/sites/519/2021/01/BaalrudPRL2009a.pdf. Collective fields can exchange energy with electrons, and step 1 of the worked example shows the fields are large: shifting all electrons by one spacing makes 8.4 V/cm. So the idea is physically reasonable. The step that does not follow from the data is that these particular oscillations do cause the scattering. Dittmer (Phys. Rev. 28, 507, 1926) had already argued that amplitudes of "20 volts or more" would be needed [read: p.1-2], and the paper never shows its oscillations reach that. The authors knew it: "although their voltage-amplitude appears to be very small we believe that they play a fundamental role in high current gaseous discharges" [read: p.2]. The voltages they actually measured at the crystal detector were about 9 mV (p.10), 30-50 mV (p.11), at most about 7 mV in Fig. 9 (p.14) and about 90 mV (p.16) [read, QE text check; Fig. 9 read: page image p.14]. Against Dittmer's 20 V that is a gap of two to three orders of magnitude. One caution, from the arbiter: a voltage at the detector is not the oscillation voltage inside the plasma. The coupling between them is unknown, and the detector had no high-frequency calibration. So the gap is a real problem only if the coupling is of order one. The paper gives no number that reaches 20 V, and its authors held their view anyway. The reader weighs both facts.

How the authors thought energy could move (pp.5-6). The paper names three "factors … which may result in a transfer of energy by plasma-electron oscillations" [read: page image p.5]: 1. The oscillating electrons may move as a body, like the beam from the hot filament. A beam has lower density, so its own oscillation is slower. For the beam in their 72.6 V, 25 mA arc they compute a beam density of 3.8×10⁶ cm⁻³ and a beam plasma frequency of 1.8×10⁷ Hz. The deep read checked this arithmetic. 2. An unsymmetrical oscillation has a field "unlimited in extent", which reaches outside the tube. The authors think this is probably how the oscillations were detected at all. 3. Fast electrons of fairly uniform speed passing through a local oscillation are alternately sped up and slowed down. This "rythmical" modulation "will excite neighboring portions of the plasma into oscillation".

On p.6 they add a "very suggestive possibility". Waves on the moving beam strike the nearly stationary "ultimate" electrons (the slow, thermalised plasma electrons). The Doppler shift raises the frequency, and it may match the ultimates' natural frequency "and give rise to resonance" [read: page image p.6]. The deep read compared this with later physics [memory, medium for the history]. The idea is the matching condition of what is now called the beam-plasma (two-stream) instability, where a beam feeds energy into plasma waves. For the paper's beam, that theory gives a resonant wavelength of about 0.6 cm and a growth rate of about 0.05 ω_pe, so a wave grows by a factor e in about three plasma periods [calculation, deep read]. The paper does not say that the wave grows at the beam's expense. On the same page it says that streaming electrons "withdraw energy" from the oscillation. Two cautions from the arbiter: - A 0.6 cm wave is far shorter than the 27-81 cm waves the authors detected. If this mechanism acts, it acts on waves they did not measure. - The beam-plasma instability as the mainstream explanation of Langmuir's paradox is [memory, medium] and was not web-checked. Baalrud et al. (2009) propose a different instability (ion-acoustic). Both stay open.

Looney and Brown, "The Excitation of Plasma Oscillations", Phys. Rev. 93, 965 (1954), later reported beam-excited plasma oscillations [web, title and bibliographic data only] https://api.crossref.org/works?query.bibliographic=Looney+Brown+excitation+of+plasma+oscillations+1954.

5. Worked example: a mercury arc at n = 10¹⁰ cm⁻³ and T_e = 10⁴ K

These are the paper's own "typical" values [read: page image pp.2, 7]. Every number below is reproduced by example.py [calculation]; the paper's printed value is given in brackets.

Step 1. How strong is the restoring field? The average spacing between electrons is ξ = n^{−1/3} = (10¹⁰)^{−1/3} = 4.64×10⁻⁴ cm [paper: 4.6×10⁻⁴]. Shift every electron in a slab by this distance. Integrating Gauss's law dE/dx = 4πρ across the uncovered layer gives

E = 4πneξ = 4π × 10¹⁰ × 4.803×10⁻¹⁰ × 4.64×10⁻⁴ = 0.0280 statvolt/cm.

Multiply by 299.8 V per statvolt: E = 8.4 V/cm [paper: 0.028 esu, 8.3 V/cm]. The energy density is E²/8π = 3.1×10⁻⁵ erg/cm³ [paper: 3.0×10⁻⁵]. The lesson: moving the electrons by a single inter-particle gap produces volts per centimetre. A plasma fights charge separation very hard, which is why it stays neutral.

Step 2. The frequency. The force on each electron is −eE = −(4πne²)ξ. This is a spring with K = 4πne², so

ω_pe = (4πne²/m)^{1/2} = (4π × 10¹⁰ × (4.803×10⁻¹⁰)² / 9.109×10⁻²⁸)^{1/2} = 5.64×10⁹ rad/s, ν_pe = ω_pe/2π = 8.98×10⁸ Hz [paper: 9.0×10⁸].

The coefficient: e/√(πm) = 8978.7, so ν = 8980√n [paper: 8980]. A radio wave of this frequency has wavelength in air λ = c/ν = 3.00×10¹⁰/8.98×10⁸ = 33.4 cm [paper: 33 cm], inside the 27-81 cm band the authors detected [read: page image p.4].

SI check: n = 10¹⁶ m⁻³, e = 1.602×10⁻¹⁹ C, ε₀ = 8.854×10⁻¹² F/m, m = 9.109×10⁻³¹ kg give ω² = ne²/(ε₀m) = 3.18×10¹⁹ s⁻², so ν = 8.98×10⁸ Hz. Same answer.

Step 3. Read the experiment backwards. If the formula is right, a measured frequency tells you the density: n = (ν/8980)². The two "scattering states" of the arc gave [read: page image p.14]:

The frequency ratio is 1080/660 = 1.64. The paper says "almost twice"; the earlier claims list rounded this to "1:2", which is wrong as a number. Since ν ∝ √n, the densities differ by 1.64² = 2.68, not 4. The two runs were also at different anode voltages, so the change in state and the change in voltage are mixed together. Note that this step assumes the formula to infer n; it cannot also test the formula, because n was never measured independently. That circularity is the core weakness of the 1929 experiments.

Step 3b. What the paper's Fig. 10 actually shows. Fig. 10 is the paper's only quantitative check of ν ∝ √n inside the plasma. It plots wavelength in air against a relative ionization scale, on log-log axes. The QE deep read took the 15 points off the page image [read: page image p.15, deep read]. On log-log axes, λ ∝ n^{−1/2} is a straight line of slope −1/2. Fitting log λ against log(relative ionization) gives [calculation, example.py step 3b]:

What this means: - A slope of 0 (no dependence) is ruled out: −0.60 is 3.7 standard errors from 0. - −1/2 fits well. - The authors say the error is in the ionization values ("too much reliance is not to be placed on them") and give no uncertainty for them [read: p.15]. When the error is in x, the ordinary fit is pulled toward 0. The honest range is then about −0.6 to −1.2, and −1/2 cannot be told apart from −1. - The ionization spans a factor 3.4 (0.54 decade, 1.19 to 4.09). The wavelength spans 26.4-81.4 cm, a factor 3.1. - The line drawn on the figure is labelled "SLOPE = −0.5", but it measures about −0.67 against the figure's own gridlines. The paper does not explain this. - The two outliers, #4 (369 MHz) and #5 (1079 MHz), sit at the same relative ionization, 1.47. One density cannot give two frequencies under ν = 8980√n. #5's 1079 MHz matches the low-scattering state's 1080 MHz, and two other points match the high-scattering 660 MHz. So the outliers may be the two-state arc of Step 3. That is the deep reader's inference, because the paper does not say which point came from which run.

The lesson is about how to read a fit: the slope's uncertainty depends on which variable you think carries the error.

Step 4. The ion branch. Mercury ions are √(m_p/m) = 605 times more sluggish in frequency [paper rounds to 600]. So

ν_pi = 8.98×10⁸/605 = 1.48×10⁶ Hz [paper: 1.5×10⁶],

the highest frequency an ion wave can have at this density. The authors once found a resonance at 1.38 MHz, but it "had become so elusive" that it could not be measured again [read: page image p.16]. Under the formula, 1.38 MHz would correspond to n = 8.6×10⁹ cm⁻³, which was not measured, so the near-match is not a test.

At long wavelengths the ion wave is a sound wave: c_s = (kT_e/m_p)^{1/2} = (1.381×10⁻¹⁶ × 10⁴ / 3.33×10⁻²²)^{1/2} = 6.44×10⁴ cm/s [paper: 6.5×10⁴]. That is about 640 m/s, close to twice the speed of ordinary sound in air, but set by the electron temperature and the ion mass.

The changeover happens near k_wλ_De = 1. With λ_De = 6.90×10⁻³ cm, the crossover wavelength is 2πλ_De = 0.0434 cm [paper: 4.4×10⁻² cm, written 2√2πλ_D with its 8π Debye length]. Evaluating the dispersion relation:

This reproduces the paper's statement that a factor √10 either side of the crossover, each limiting formula is "only 5 percent out" (exactly 4.7%) [read: page image p.8].

Step 5. How good is "zero group velocity"? Add the Bohm-Gross thermal term 3k_w²kT_e/m. At the paper's wavelength of 33 cm, the fractional change in ω² is 5×10⁻⁶ and the group velocity is 1.5×10⁵ cm/s, which is 1.5 km/s, about 5×10⁻⁶ of the speed of light. At λ = 0.1 cm the group velocity reaches 4×10⁷ cm/s, about the electron thermal speed. So the cold-plasma statement is essentially exact for the paper's experiment and fails only at millimetre scales.

Step 6. The bulb resonance the authors left open. Treat the 18 cm bulb as half a wavelength (λ = 36 cm) of an EM wave in the plasma: ν = [(8.98×10⁸)² + (3×10¹⁰/36)²]^{1/2} = 1.22×10⁹ Hz, λ in air = 24.5 cm [paper: 12.2×10⁸, 24.6 cm]. The observed band 27-81 cm is 370-1110 MHz. The prediction lies just outside its short end, and the authors wrote that the decision "must be reserved" [read: page image p.4]. They were right not to claim it.

6. Evidence table

Other observations reported in the paper and kept in the record [read: page images pp.11-16, per reviewers; pp.10-12 and p.16 details read, QE text check]: - A critical oscillation near 230-263 MHz existed only when the mercury reservoir was between 19 and 21 °C. Its frequency did not rise steadily with temperature (263, 231, 250 MHz), which the theory does not explain. - That oscillation vanished when filament g emitted just 2 mA, or when the filament heating current was reversed [read: p.11]. - A 115 MHz wave appeared at 30-50 mV on the detector. Cooling the mercury bath from 20 °C to 19.5 °C or below made its peak drop sharply and its frequency jump to about 900 MHz [read: p.11]. A magnetic field strong enough to deflect the electron beam made the 115 MHz wave vanish and a 730 MHz wave appear in its place [read: p.12]. Reversing the heating current made the 115 MHz wave disappear [read: p.11]. - The authors tentatively placed the 115 MHz wave among the beam oscillations, adding that "the other evidence is not clean cut". It rose only 10% for a 60% increase in emission, where √n scaling predicts 26%. The authors stated this discrepancy [read: p.12]. - The abstract and the body disagree about 55 MHz. The abstract says: "Another observed frequency of 50 to 60 megacycles may correspond to oscillations of the beam electrons" [read: p.1]. The body tests that idea with a Doppler argument and rejects it: "it seems to be impossible to identify the 55 with the theoretical 18 megacycle oscillation" [read: pp.15-16]. The abstract was not updated, so a summary taken from the abstract repeats a reading the authors themselves withdrew. The abstract also says "the theory offers no explanation of the cause of the observed oscillations" and calls the correlation with scattering "tentative until simpler experimental conditions can be attained" [read: p.1]. - Controls for plasma (not circuit) origin in the 300+ MHz range: - amplitude depends on position on the wall; - a two-electrode interference test showed a phase change of "considerably more than 180 degrees"; - frequency and amplitude were "independent of the potential of the receiving collector over the whole range explored, namely from −80 v. with respect to the cathode to 5 v. above anode potential" (up to about 38 V at the 33 V anode setting) [read: p.13]; - the same frequency appeared with the beam on the collector, deflected past it, or with the collector out of the beam; - unlike the 115 and 230 MHz waves, reversing the heating current changed nothing [read: p.13, deep read]. - The 1.38 MHz line was found with a 2 mH coil at 25 mA emission and 63 V anode, "practically the condition for the 230 or 115 megacycle oscillation", with a peak of about 90 mV. A later check used a Radiola 17 amplifier (550-1450 kc), but by then the line "had become so elusive" that it could not be found again; the amplifier showed a broad noise continuum instead. The authors add that "frequencies higher than this were not sought with an amplifier", so whether 1.38 MHz was the ion short-wave limit (1.5 MHz) is unknown [read: p.16]. - Between 26 and 42 V on the anode there were oscillations of about 90 mV whose wavelengths could not be found [read: p.16]. - The detector caveat: amplitudes in different runs differed "several times", "as if a frequency conversion factor for the crystal varied from time to time or with the crystal setting" [read: p.10]. Amplitudes from different runs therefore cannot be compared closely.

What happened to these observations (record that goes quiet). The two-state arc, the 1.38 MHz line and the 19-21 °C window do not reappear in anything the Investigator found. The arbiter weighs these explanations: (a) they were not robust phenomena (tube-specific marginal oscillators), for which the 2 °C window and the irreproducibility are evidence, while the wall-position and interference controls argue for a real plasma origin; (b) they were absorbed into later literature on striations, double layers and hysteretic discharge modes under other names, which is not confirmed by any citation found; (c) they were covered in the authors' own follow-ups, which nobody has yet read (the June 1929 Note; "A General Theory of the Plasma of an Arc", Phys. Rev. 34, 876, 1929); (d) the raw data stayed in Langmuir's laboratory notebooks at the Library of Congress (mm77029413, unexamined) [web] https://www.loc.gov/item/mm77029413/. No evidence of suppression or secrecy was found; the work was published openly and its main line was followed up continuously (1931-2015). Reading the Note, the Collected Works vol. 5 and the notebooks would decide between them. For the two-state arc, a replication now has precise targets: 20 mA emission, 27 V anode, the emission ranges of each state, and the bar-magnet switch [read: page image p.14, deep read].

7. What a learner should take away

Solid. The electron plasma frequency, the EM cutoff and the ion-acoustic dispersion relation are correct. They were derived here from nothing more than Gauss's law, Newton's law and the Boltzmann factor. Every numerical example in the theory section checks to rounding. If you can reproduce steps 1, 2 and 4 of the worked example, you understand the core of linear plasma physics. The name "plasma", and the idea that a plasma behaves collectively over distances larger than a Debye length, come from this line of work.

Wrong, and instructive. Nothing in the theory is wrong within its stated approximations. The instructive limits are: - "Zero group velocity" is a property of a model (cold electrons), not of nature. Step 5 shows how to check when a neglected term matters: compare its size at the wavelengths you actually care about. - An experiment that uses a formula to infer the quantity it needs to test the formula (step 3) is circular. Without an independent density measurement, the 1929 data could only be consistent with ν ∝ √n, not prove the coefficient 8980. Most of the later confirmation came from other people's experiments. - Reported rounding matters: "almost twice" became "1:2" in a summary, and the real ratio is 1.64. - Read the body, not just the abstract. The abstract still offers the 50-60 MHz beam-oscillation reading that the body withdraws (§6). - A fit's error bar depends on which variable carries the error. Fig. 10 gives a slope of −0.60 ± 0.16 if the wavelengths carry the error, and about −1.16 if the ionization values do (step 3b).

Box: the authors' own thermal argument, a right instinct with the wrong sign (pp.5-6). Tonks and Langmuir did not ignore electron temperature. Their argument [read: page image pp.5-6]: 1. Call an electron "fast" if it crosses one wavelength in less than one period (speed above v = λν), and "slow" otherwise. 2. Fast electrons pass through the disturbance before they can take part, so only slow ones count in the density n that sets ν = 8980√n. 3. A shorter wavelength lowers v, so fewer electrons count, n falls, and ν falls. The predicted sign is that frequency decreases as wavelength decreases. 4. A lower ν lowers v* again. This "cumulative action … may make oscillations below a certain minimum size impossible". They mention unpublished calculations and give no number.

What later physics says (Bohm and Gross 1949 [web]; Landau 1946 [memory, high]): - Sign: wrong. Thermal motion acts like an extra pressure that adds to the restoring force, so ω² = ω_pe² + 3k_w²kT_e/m. Frequency rises as wavelength shortens. example.py step 5 shows ν/ν_pe = 1.0000 at 10 cm, 1.0028 at 1 cm and 1.25 at 1 mm. - Fast/slow split: right idea. Dividing electrons at λν is dividing them at the wave's phase velocity ω/k. That is the dividing line in Landau damping, where electrons near the phase speed trade energy with the wave. The authors said each fast electron "either contributes energy to it or draws energy from it" but did not say which wins. For a thermal (Maxwellian) distribution the net effect is damping. - Minimum size: right place. Thermal electrons become "fast" when λ < v_th/ν_pe. At n = 10¹⁰ cm⁻³ and T_e = 10⁴ K, that length is 0.0434 cm, exactly 2πλ_De, the wavelength (k_wλ_De = 1) where Landau damping kills the wave in about one period [calculation, example.py step 5; deep read].

None of this could be tested with their 27-81 cm waves, where the thermal shift is below 10⁻⁵. The lesson: a physical instinct (thermal electrons limit how small an oscillation can be) can be right while the mechanism built on it predicts the wrong trend. Only the full calculation tells them apart.

Genuinely open. - What actually thermalizes the electrons in low-pressure arcs (Langmuir's paradox)? Candidates: plasma-electron oscillations (this paper; Merrill and Webb 1939), beam-plasma (two-stream) instability, whose matching condition the authors' p.6 resonance idea anticipates ([memory, medium] as an explanation of the paradox; §4), sheath oscillations (Gabor 1955), ion-acoustic instability in the presheath (Baalrud 2009, which uses the "electric sound wave" branch of this very paper), or non-local kinetics that dissolve the paradox (Godyak and Alexandrovich 2015, in argon). The decisive experiment has the 1939 design: a movable probe measuring local oscillation amplitude and the electron energy distribution at the same point, checked against Dittmer's 20 V requirement. - Was the magnet-switchable two-state arc real and general? A modern tube at fixed anode voltage (27 V, 20 mA emission) with a calibrated receiver, recording the frequency of each state, would settle it. - What do the June 1929 Note and Langmuir's notebooks say about the elusive 1.38 MHz line and the 19-21 °C window? - What exponent does the arc really follow? Fig. 10 excludes no dependence but cannot separate −1/2 from −1. It would take an absolute density measurement and an ionization scale with a stated uncertainty, followed by an errors-in-variables fit (one that allows error in both variables). - Were the two Fig. 10 outliers the two scattering states? Repeating the two-state arc at the same anode voltage and recording each state's frequency would answer this.

8. Check yourself

1. Why does the paper's formula ν = (ne²/πm)^{1/2} contain π, while modern books write ω² = 4πne²/m (Gaussian) or ne²/(ε₀m) (SI)? Are they the same physics?
2. An ionospheric layer reflects radio waves up to 9 MHz at vertical incidence. Using ν = 8980√n, what is its peak electron density in cm⁻³?
3. In the ion-acoustic wave, why does the speed depend on the electron temperature but the ion mass?
4. The authors measured 660 MHz and 1080 MHz in two states of the arc. Explain why this cannot test the coefficient 8980, and what extra measurement would.
5. At n = 10¹⁰ cm⁻³ and T_e = 10⁴ K, is "zero group velocity" a good approximation for a 33 cm oscillation? For a 1 mm one? Give the numbers.

9. Further reading

Mainstream: - F. F. Chen, Introduction to Plasma Physics and Controlled Fusion (Springer, 3rd ed. 2016). Ch. 1 (Debye shielding, plasma parameter) and ch. 4 (waves: plasma oscillations, Bohm-Gross, ion-acoustic waves) [memory, high for chapter content]. - D. J. Griffiths, Introduction to Electrodynamics: ch. 2 (Gauss's law), ch. 9 (EM waves and dispersion, including plasma). - NRL Plasma Formulary (free PDF from the US Naval Research Laboratory): plasma frequency, Debye length and ion sound speed in practical units. - D. Bohm and E. P. Gross, "Theory of Plasma Oscillations. A. Origin of Medium-Like Behavior", Phys. Rev. 75, 1851 (1949) [web] https://doi.org/10.1103/PhysRev.75.1851. - Alexeff and Neidigh, Phys. Rev. 129, 516 (1963), observation of ion sound waves [web] https://journals.aps.org/pr/abstract/10.1103/PhysRev.129.516. - S. D. Baalrud, J. D. Callen and C. C. Hegna, "Instability-Enhanced Collisional Effects and Langmuir's Paradox", Phys. Rev. Lett. 102, 245005 (2009) [web] https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.102.245005 (title and reference read on the APS page by the Investigator); open author copy at the URL in section 4 (only p.1 read). - V. Godyak and B. Alexandrovich, "Langmuir paradox revisited", Plasma Sources Sci. Technol. 24, 052001 (2015) [web] (abstract). - Malmberg and Wharton, "Collisionless Damping of Electrostatic Plasma Waves", Phys. Rev. Lett. 13, 184 (1964): the measurement of Landau damping [web, Crossref bibliographic data only]. - Looney and Brown, "The Excitation of Plasma Oscillations", Phys. Rev. 93, 965 (1954): beam-excited plasma oscillations [web, Crossref bibliographic data only].

Earlier and later experimental work by others on the same oscillations: - L. A. Pardue and J. S. Webb, Phys. Rev. 31, 1122 (1928), and "Ionic Oscillations in the Glow Discharge", Phys. Rev. 32, 946 (1928): earlier, independent experimental work on the ion branch, which Tonks and Langmuir compare with their sound-wave range [read: page image p.8; web] https://api.crossref.org/works?query.bibliographic=Pardue+Webb+1928. Not read. - H. J. Merrill and H. W. Webb, "Electron Scattering and Plasma Oscillations", Phys. Rev. 55, 1191 (1939): the closest independent follow-up [web abstract] https://journals.aps.org/pr/abstract/10.1103/PhysRev.55.1191. The printed initials differ from the 1928 J. S. Webb; whether they are the same person is unchecked. - K. G. Hernqvist (RCA), "Plasma Ion Oscillations in Electron Beams", J. Appl. Phys. 26 (1955), DOI 10.1063/1.1722036: ion plasma oscillations observed in beams (claim 3) [web, Crossref title only].

The authors' own key works: - L. Tonks and I. Langmuir, "Oscillations in Ionized Gases", Phys. Rev. 33, 195 (1929): the project document. - I. Langmuir, "Oscillations in Ionized Gases", PNAS 14, 627 (1928), free at https://www.pnas.org/doi/10.1073/pnas.14.8.627 [web]: the earlier announcement. - L. Tonks and I. Langmuir, "Note on 'Oscillations in Ionized Gases'", Phys. Rev. 33, 990 (1929) [web] https://journals.aps.org/pr/abstract/10.1103/PhysRev.33.990: content unknown. - L. Tonks and I. Langmuir, "A General Theory of the Plasma of an Arc", Phys. Rev. 34, 876 (1929). - I. Langmuir, "Scattering of Electrons in Ionized Gases", Phys. Rev. 26, 585 (1925): the paradox [cited in the document, p.1]. - L. Tonks, "Plasma-Electron Resonance, Plasma Resonance and Plasma Shape", Phys. Rev. 38, 1219 (1931): bounded plasmas, where a sphere resonates at ν_pe/√3 rather than ν_pe [memory, high for the factor]. - L. Tonks, "Ionisation Density and Critical Frequency", Nature 132, 101 and 710 (1933): applies the plasma frequency to the ionosphere's critical frequency (an extension of claim 1) [web, Crossref title only]. - L. Tonks, "The Birth of 'Plasma'", Am. J. Phys. 35 (1967): first-hand history ("The author was there") [web abstract]. - The Collected Works of Irving Langmuir, vol. 5 (Pergamon, 1961), scanned at https://archive.org/details/the-collected-works-of-irving-langmuir-vol.-5 [web; contents not verified].

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Answers

1. They are the same. The paper quotes ν in cycles per second; ν = ω/2π, so ν² = ω²/4π² = 4πne²/(4π²m) = ne²/(πm). SI replaces Gaussian 4π e² with e²/ε₀. Only the bookkeeping differs, and example.py gets 8.98×10⁸ Hz both ways.
2. n = (ν/8980)² = (9×10⁶/8980)² = (1002)² ≈ 1.0×10⁶ electrons/cm³ (10¹² m⁻³), a typical daytime F-layer peak density [memory, high for "typical"].
3. The ions carry almost all the inertia (mass) of the moving fluid. The restoring "pressure" comes from the electrons: they are hot and light, so they rearrange instantly into a Boltzmann distribution, and the electric field they set up pushes on the ions with a strength proportional to kT_e. Inertia from ions plus springiness from electrons gives c_s = (kT_e/m_p)^{1/2}.
4. With no measured density, each frequency only tells you what n would have to be if the formula were true (1.45×10¹⁰ and 5.4×10⁹ cm⁻³). Any coefficient would give some density, so nothing is tested. An independent density measurement in the same tube, such as microwave interferometry or a calibrated probe, at several densities across more than a decade, would test it.
5. At 33 cm: the Bohm-Gross correction to ω² is 5×10⁻⁶ and the group velocity is 1.5×10⁵ cm/s, so "zero" is an excellent approximation. At 1 mm (0.1 cm): the correction is 0.56 (comparable to ω_pe² itself), and the group velocity is 4×10⁷ cm/s, about the electron thermal speed. There, the cold model fails.