Farnsworth-PT primer: the Radio Frequency Multipactor Amplifier (US 2,172,152)

Project folder: one document, US Patent 2,172,152, "Radio Frequency Multipactor Amplifier", Philo T. Farnsworth, filed 21 Feb 1938, granted 5 Sep 1939 [read; dates confirmed [web] https://patents.google.com/patent/US2172152A/en]. The text layer is noisy OCR but readable. For example, the drawing sheet says "1958" where it means 1938.

Source note. There is no source gap for the patent itself. However, the folder holds nothing else by Farnsworth. Everything here about how his multipactor tubes performed, what happened to them, and his other work comes from [web] or [memory] sources, and each such claim is tagged. The patent has no equations, no dimensions, no voltages and no measurements [read]. Every number in section 5 is therefore an illustration I calculated, not a figure from the patent. After quality engineering (QE), a deep read of PDF pp.2-3 checked the key passages against a page image of p.3 [read: page image p.3]. This revision follows that deep read and the arbiter's "QE revision" section.

Tag key: [read] = from the patent. [web] = Internet source, URL given. [memory] = recalled from training, with confidence. [calc] = my own calculation, reproduced in example.py. The web-search budget ran out while I wrote this primer, so the textbook and review-paper citations in section 9 are [memory] and were not re-checked online.

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1. The idea in one paragraph

When a fast electron hits certain surfaces, such as oxidised silver, it knocks out more than one new electron. Farnsworth's tube uses this to amplify a radio signal. A cylinder of fine silver gauze (the "screen cathode") surrounds a wire coil (the "helix"). The incoming radio-frequency (RF) signal makes the helix voltage swing up and down relative to the screen. Electrons leaving the screen at the right moment are pulled in through the helix, cross the inside, and strike the screen on the far side. Each strike releases several new electrons, which can repeat the trip. So one electron becomes many, in step with the radio wave. Of the electrons that reach the gauze from the helix, most strike it, but about one in four pass straight through its holes to an outer collecting cylinder. Secondaries that leave the gauze at the wrong moment are also drawn out to that collector. The current reaching the collector is the amplified output. A small electron gun at one end feeds in starting electrons. The patent says starting electrons could also come from stray ionised particles or from light shining on the gauze, but it prefers the gun "since it possesses advantages in regard to modulation". Varying the gun's current changes how many electrons each cycle starts with, so the same tube both amplifies the carrier wave and puts a signal (modulation) onto it. The physics behind this, secondary emission and the "multipactor" effect, is real and well established. The weakness is in the evidence: the patent gives no measurements, and no neutral test of this tube is on record.

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2. What you need to know first

Electric potential and energy (electrostatics). An electron (charge −e, where e = 1.602×10⁻¹⁹ C) that moves from a point at potential V₁ to a point at potential V₂ in a field that does not change with time gains kinetic energy e(V₂ − V₁). One electron-volt (eV) is the energy gained across 1 volt, which is 1.602×10⁻¹⁹ J. A key consequence: in a static field, an electron that starts and ends at the same potential gains no net energy, whatever happens in between. Read: Griffiths, Introduction to Electrodynamics, the chapter on electrostatics, specifically the sections on potential and on work and energy.

Charged-particle motion (Newton's second law with a force that varies in time). An electron in a uniform field E feels force F = −eE. If E varies as sin(ωt), where ω = 2πf is the angular frequency and f the frequency, you integrate the acceleration twice to get position. This is just first-year calculus. Read: Taylor, Classical Mechanics, the opening chapters on Newton's laws and one-dimensional motion.

Secondary electron emission. A "primary" electron hitting a surface releases on average δ (delta) "secondary" electrons. δ depends on the impact energy. It is below 1 at very low energies, rises to a peak (for metals and oxides, typically at a few hundred eV), and falls again at high energies [memory, high]. Secondaries leave the surface slowly, with a few eV each [memory, high], and in all directions. Photomultiplier tubes chain about ten such surfaces ("dynodes") to multiply one electron by about a million [memory, high]. Read: Bruining, Physics and Applications of Secondary Electron Emission (1954) [memory, medium on the exact title and year]. Any modern-physics text's section on photomultipliers also covers this.

The photoelectric effect. Light of high enough frequency releases electrons from a surface, and the number released is proportional to light intensity until the current is limited by space charge [memory, high]. Read: Serway, Moses and Moyer, Modern Physics, the photoelectric-effect section, or any intro modern-physics text.

Space charge. A dense cloud of electrons repels the next electrons and partly cancels the applied field. This puts a ceiling on how much current a tube can carry (Child–Langmuir law) [memory, high]. All you need here is that space charge makes electron avalanches level off, or saturate.

Vacuum-tube amplifiers and modulation. A tube amplifies when a small input voltage controls a large current drawn from a direct-current (d.c.) supply. The extra output power comes from that d.c. supply, not from the signal. "Modulating a carrier" means making the amplitude of a high-frequency wave follow a slower signal (amplitude modulation, AM, as in AM radio). Read: Spangenberg, Vacuum Tubes (1948) [memory, high that the book exists; medium on chapter contents], or any older electronics text.

Exponential growth and feedback. If each round multiplies a number by g, then N rounds give gᴺ. If g > 1 the number runs away until something limits it. If g < 1 and you keep adding a steady input, the total settles at input/(1 − g). The same algebra describes feedback amplifiers.

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3. The textbook view

3.1 Secondary emission yield

Let δ(K) be the average number of secondaries per primary at impact kinetic energy K. Multiplication needs δ > 1. Values cited by our reviewers: air-exposed silver peaks at δ ≈ 2.6 near 300 eV, and caesium oxide reaches δ ≈ 10 for low-energy electrons [web, cited by the mathematician reviewer, not re-checked by me or the arbiter: https://www.sciencedirect.com/science/article/pii/S2211379722000407 and https://www.sciencedirect.com/topics/engineering/secondary-electrons]. Below a few tens of eV, δ < 1 for these surfaces [memory, medium].

3.2 The multipactor effect

"Multipactor" (Farnsworth's word, from multiple impact) is a resonant electron avalanche in an RF field between surfaces [web: https://en.wikipedia.org/wiki/Multipactor_effect]. The classic case is two parallel plates a distance d apart, with voltage V₀ sin(ωt + φ) between them. Here V₀ is the peak RF voltage, ω = 2πf, and φ is the RF phase when the electron leaves. The uniform field gives the electron (mass m = 9.109×10⁻³¹ kg) the acceleration

a(t) = (e V₀ / (m d)) sin(ωt + φ).

Two conditions make an avalanche [web, Wikipedia URL above]: 1. Timing: the electron's transit time must be an odd number of half RF periods (T/2, 3T/2, ..., where T = 1/f). It then lands just as the field reverses, so its secondaries are pushed back the other way. 2. Yield: the impact energy must be in the range where δ > 1.

When both hold, the electron count grows as δᴺ after N impacts until space charge or loss of timing limits it. Today multipactor is mostly a nuisance that engineers design against in satellite RF hardware, accelerator cavities and waveguides [web, Wikipedia URL above]. Farnsworth studied it from 1934 and tried to use it as an amplifier [web, same URL].

3.3 Energy bookkeeping

Electrons gain energy only from fields. In a static field, energy conservation means kinetic energy + (−e)·V(position) is constant. So the net energy gained between two points depends only on their potential difference. With a time-varying potential, the electron can gain net energy even between two points at the same potential, because the potential has changed while it was in flight. Section 5 shows this is exactly how Farnsworth's tube must work.

3.4 Amplifier gain and linearity

For a multiplier, call the injected electrons per cycle n_in. If the output is M·n_in with M fixed, the device is linear. If M changes with signal level, the device distorts the signal. Avalanche devices saturate, meaning M falls once space charge builds up. A device with round-trip growth above 1 is an oscillator, not an amplifier, unless something holds it below threshold.

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4. Where this researcher departs

The departures are not from physics but from conventional tube design of 1938. Side by side:

The exact points where the patent asserts rather than shows something [read; all three reviewers and the arbiter agree]: - Gain: "greatly amplified" is supported only by a pointer to prior art, Farnsworth's earlier patents 2,071,515 and 2,071,516, which are not in the folder. - Linearity: stated explicitly only for the light-seeded case ("in its maximum value a linear function of the illumination intensity up to the point of space charge saturation"), then extended to the gun "as stated with reference to the photosensitive cathode". - Ten megacycles: "When the carrier frequency is raised above ten megacycles it is found necessary to provide a source of electrons". This is empirical, and no data are given. - One in four: the gauze's open-area fraction (about 1:4, p.2) is equated with the fraction of electrons arriving from the helix that pass through the holes; the rest strike and make secondaries [read: p.3, image-checked in the deep read]. For electrons arriving roughly radially, transmission equal to open area is a fair first approximation, modified by the field leaking through the holes, which bends electron paths like a weak lens [memory, medium]. It is still an assertion with no measurement. (An earlier version of this primer, following the contrarian reviewer and the arbiter, said the ratio applied to secondaries. That was a misreading; the arbiter's QE revision corrects it.) - Seed preference: the patent's preferred seed is the electron gun, for modulation. The light route is optional and not drawn [read: p.3]. - No focusing: listed as an object, not demonstrated.

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5. Worked example: where does the impact energy come from, and how much gain is safe?

All numbers below are reproduced by example.py:

.venv/bin/python review/Farnsworth-PT/example.py

It ran cleanly in about 1.5 s. Constants: e/m = 1.759×10¹¹ C/kg.

Step 1. Textbook two-plate multipactor (checks the arbiter's numbers)

An electron leaves plate 1 at rest at phase φ. Integrate a(t) from section 3.2 once, with v(0) = 0:

v(t) = (eV₀ / (m d ω)) [cos φ − cos(ωt + φ)]

Integrate again, with x(0) = 0:

x(t) = (eV₀ / (m d ω²)) [ωt cos φ + sin φ − sin(ωt + φ)]

Resonance: set x = d at t = π/ω (half a period). Since sin(π + φ) = −sin φ:

d = (eV₀ / (m d ω²)) (π cos φ + 2 sin φ) ⇒ V₀ = m ω² d² / (e (π cos φ + 2 sin φ))

Impact speed: cos(π + φ) = −cos φ, so v_imp = 2 eV₀ cos φ / (m d ω). Substituting V₀ gives v_imp = 2 ω d cos φ / (π cos φ + 2 sin φ).

Numbers for φ = 0.3 rad: cos φ = 0.9553, sin φ = 0.2955, so π cos φ + 2 sin φ = 3.592. - f = 100 MHz, d = 1 cm: ω² = (6.283×10⁸)² = 3.948×10¹⁷ s⁻². V₀ = 3.948×10¹⁷ × 10⁻⁴ / (1.759×10¹¹ × 3.592) = 62.5 V. v_imp = 2 × 6.283×10⁸ × 0.01 × 0.9553 / 3.592 = 3.34×10⁶ m/s, so K = ½ m v² = 31.7 eV.

[calc] These match the mathematician's and arbiter's figures. A fourth-order Runge–Kutta (RK4) numerical integration of Newton's law independently lands the electron at exactly 0.50000 periods with 31.749 eV.

Lesson: both V₀ and the impact energy scale as (f·d)². At 10 MHz and cm sizes, a resonant electron arrives far too gently to have δ > 1. Getting near the silver peak (about 300 eV) needs f·d around 300 MHz·cm. This "f·d" product is the standard way multipactor susceptibility is charted [memory, high].

Side check (contrarian's estimate). Accelerated from rest through 100 V, an electron reaches v = √(2 × 1.759×10¹¹ × 100) = 5.93×10⁶ m/s. Under uniform acceleration from rest, the transit time is 2d/v = 3.37 ns, not d/v = 1.69 ns, so the half-period match is at 148 MHz, not 300 MHz [calc; agrees with arbiter].

Step 2. Farnsworth's geometry: the d.c. bias cannot supply impact energy

The reviewers' parallel-plate picture is not the patent's geometry. In the patent, an electron leaves the screen, is pulled inward through the positive helix, crosses the inside, and strikes the same electrode (the screen) on the far side [read]. Flatten one diameter into a line, using illustrative dimensions because the patent gives none:

• x = 0: screen, potential 0
• x = g = 1 mm: helix, potential V_h(t) = V_dc + V_rf sin(ωt + φ)
• 1 mm to 19 mm: inside the helix, no field
• x = D − g = 19 mm: helix again
• x = D = 20 mm: screen, potential 0

(a) d.c. only. The electron starts and ends at potential 0. By section 3.3, it must arrive with exactly the energy it left with, however large V_dc is. Numerically, with a 2 eV launch energy (typical for a secondary): V_dc = 50, 300 and 1000 V all give impact 2.0000 eV. Only the transit time changes: 4.99, 2.11 and 1.16 ns [calc].

(b) The exact rule with RF. The electron's total energy is K − e V(x, t). It changes only because V changes with time, so integrating over the flight gives:

K_impact − K_launch = e (⟨V_h⟩_entry − ⟨V_h⟩_exit)

Here ⟨V_h⟩_entry is the helix voltage averaged over the time the electron crosses the entry gap (weighted by distance), and ⟨V_h⟩_exit is the same for the exit gap. The d.c. part cancels exactly. The electron gains energy only if the helix voltage is higher when it goes in than when it comes out. The most it can gain is e(V_max − V_min) = 2 e V_rf [calc]. Numerical check at V_dc = 100 V, V_rf = 150 V, f = 100 MHz, launched at rest:

The identity holds to every printed digit.

(c) The best case the patent describes. The patent says the returning electrons are the ones that leave "when the anode 30 is most positive" [read]. The best energy gain is then when the electron arrives half a period later, with the helix at its minimum. With V_dc = V_rf = 150 V (helix never negative), that happens at f = 243 MHz for this 2 cm geometry. The impact energy is then 296 eV, close to the 300 eV bound and near the silver yield peak [calc]. Scaling all lengths by k at fixed voltages scales all times by k, so f·D ≈ 487 MHz·cm. At 10 MHz the same voltages would need a screen about 49 cm across [calc].

What this changes. The mathematician wrote that "the DC bias on the helix, not the RF, must supply the impact energy" [read: mathematician.md], and the arbiter first endorsed that "the DC bias must matter" for energy. For screen-to-screen trips in the patent's geometry, that is backwards. The d.c. bias sets the timing, because it controls how fast the electron crosses. The energy comes from the RF swing, so ultimately from the RF drive. The arbiter's QE revision now records this. The screen sits at d.c. ground through the grounded side of the input coil [read: p.2-3], so the same rule applies to gun electrons if the gun cathode is also near ground, which the patent does not state.

A puzzle the patent leaves open. In case (c), the secondaries from that energetic impact leave the screen when the helix is at its minimum. The patent says that at this phase "the potential on collecting anode 25 will be controlling, and the electrons will be drawn thereto and collected" [read]. So the best-energy trip ends in a phase that sends the next generation to the output rather than round again. How the patent's "repeated ... trips" keep the phase and the energy right together is not explained. It may be in the earlier patents it cites. A full answer would need a 2-D cylindrical model with the real helix pitch and real emission angles. This 1-D model cannot settle it [calc, interpretation].

Step 3. Gain budget: why "greatly amplified" and "linear" pull against each other

Follow the patent's wording. Count the electrons arriving at the gauze from the helix. A fraction t ≈ 1/4 passes straight through the holes to the collector (the patent's figure). The other (1 − t) strike the wires, and each releases δ secondaries. A fraction r of those secondaries leaves in the right phase (helix most positive), is pulled back through the helix and arrives at the gauze again. The round-trip growth factor is

g_rt = (1 − t) δ r.

With δ = 2.6 and t = 0.25, (1 − t)δ = 1.95 secondaries are made per arrival, so the runaway threshold g_rt = 1 is at r = 1/1.95 = 0.513 [calc].

Runaway (g_rt > 1). If every secondary returned (r = 1), g_rt = 1.95. A millionfold rise takes ln(10⁶)/ln(1.95) = 20.7 trips, which is 1.03 µs at 10 MHz even at one trip per half period. With r = 0.75, g_rt = 1.46 and the rise takes 36.3 trips, or 1.82 µs [calc]. Growth then stops at space-charge saturation, where the output no longer depends on the seed. That makes the tube an oscillator or limiter, not a linear amplifier. (For scale: δ¹⁰ = 1,024, 59,049 and 9,765,625 for δ = 2, 3, 5, confirming the reviewers.)

Below threshold (g_rt < 1) with a steady seed n_in per cycle. Arrivals per cycle A satisfy A = n_in + g_rt A, so A = n_in/(1 − g_rt). The collected current is t A, so the current gain is G = t / (1 − (1 − t) δ r) = t / (1 − g_rt). G does not depend on n_in, so the output is linear in the seed. This is the patent's claim 3, and it holds only in this regime. Note that G > 1 only when 1 − g_rt < t, that is g_rt > 0.75: the tube gives no current gain at all unless it runs fairly close to threshold. And δ depends on impact energy, which depends on the RF amplitude:

[calc; r values are illustrative.] This model counts only the electrons that pass through the holes as output. The patent also says secondaries launched "when the anode 30 is least positive" go to the collector [read: p.3]. Counting those too adds (1 − t)δ(1 − r) to the numerator of G, which raises G but leaves the threshold and the sensitivity, both set by the denominator, unchanged [calc, reasoning]. High gain needs g_rt close to 1, and there the gain becomes extremely sensitive to the carrier amplitude, temperature and surface condition. That is the quantitative reason all three reviewers asked for a distortion and stability measurement that the patent does not give.

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6. Evidence table

The only performance report found is A. H. Halloran, RADIO magazine, Oct 1934. It reports "an undistorted output of 25 watts was obtained from an input of 25 watts", 100 W from a small tube "before the cathodes were destroyed by heat", and oscillation from 200 kHz to 60 MHz [web: https://borderlandsciences.org/journal/vol/44/n05/Farnsworth_Multipactor_Tube.html]. It is inventor-side and promotional-era. It predates this patent and may describe a different geometry. "25 W from 25 W" is ambiguous: it could be power gain, drive power or d.c. input (arbiter).

No source shows a neutral test that failed. No source shows the tube was dismissed without testing, or that work on it was suppressed. Why the multipactor amplifier vanished from practice is undocumented in any source the reviewers or arbiter found. The only documented limit is cathode heating (Halloran) [web, URL above]. The claim that it was "overtaken by triodes and pentodes" is [memory, medium] and unverified, so it is not used.

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7. What a learner should take away

Solid. Secondary emission with δ > 1, resonant multipactor multiplication, photoemission linear in light below saturation, and the energy bookkeeping (output power from the d.c. supplies, with RF timing the process). Nothing in the patent conflicts with physics, and it makes no energy-from-nothing claim. Farnsworth correctly saw, in 1934, a real avalanche mechanism that engineers today spend effort suppressing.

Weak, and instructive. - A patent is a disclosure, not a measurement. "As is now well known in the art" is a pointer, not data. Learn to separate what a document shows from what it asserts. - "Greatly amplified" and "linear" pull against each other. An avalanche is linear only below threshold, and there the gain is fragile (section 5, step 3). Any claim of a high-gain linear avalanche amplifier should come with a distortion measurement. - Energy bookkeeping exposes the physics. In the screen-to-screen geometry the d.c. bias cannot provide impact energy, and seeing why (section 5, step 2) is the most useful single idea in this project. Two expert reviewers missed it, because they reasoned from the textbook two-plate model rather than the patent's own geometry. - The f·d rule: resonant multipactor at 10 MHz needs large structures or weak impacts. This makes the patent's frequency remarks hard to check without dimensions.

Genuinely open. - What gain, distortion and noise did Farnsworth's multipactor tubes actually deliver? What did Halloran's 25 W / 25 W mean? - Was this screen-cathode/helix geometry ever built? - How does a single-electrode return geometry keep both phase and impact energy right over many trips (the puzzle in step 2c)? This could be studied with a 2-D simulation, which you could write yourself by extending example.py. - Why the device was abandoned. No source exists either way.

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8. Check yourself

1. In Farnsworth's geometry, an electron leaves the screen at 2 eV with a steady 500 V on the helix and no RF. What energy does it have when it hits the screen on the far side, and why?
2. Using the (f·d)² scaling, what peak RF voltage and impact energy does the two-plate resonance give at 200 MHz, 1 cm, φ = 0.3 rad?
3. With δ = 2.6 and t = 0.25, using G = t/(1 − (1 − t)δr), what returned fraction r gives a current gain G = 5 below threshold? Why would a designer hesitate to push G much higher?
4. Halloran reports "25 watts out from 25 watts in". Give two readings of this sentence and say what extra information would tell them apart.
5. The patent says a seed source is needed above about 10 MHz. Suppose stray electrons appear at a steady rate R per second. How does the number of stray seeds per RF cycle change with frequency, and how does that support the patent's remark?

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9. Further reading

Mainstream - Multipactor overview, history and the resonance conditions: https://en.wikipedia.org/wiki/Multipactor_effect [web]. - J. R. M. Vaughan, "Multipactor", IEEE Transactions on Electron Devices 35 (1988) [memory, high; web check not possible, budget exhausted]. This is the classic engineering treatment, including the δ(K) fitting formula. - R. A. Kishek, Y. Y. Lau et al., "Multipactor discharge on metals and dielectrics: historical review and recent theories", Physics of Plasmas 5 (1998) [memory, medium on volume and author list]. This review covers the history from Farnsworth onward. - D. J. Griffiths, Introduction to Electrodynamics: electrostatic potential and energy. - K. R. Spangenberg, Vacuum Tubes (McGraw-Hill, 1948) [memory, high]: tube amplifiers, space charge and secondary emission in the era's terms. - H. Bruining, Physics and Applications of Secondary Electron Emission (Pergamon, 1954) [memory, medium].

The researcher - P. T. Farnsworth, US 2,172,152, "Radio Frequency Multipactor Amplifier" (1939). This is the document reviewed: https://patents.google.com/patent/US2172152A/en [web]. - The prior Farnsworth patents it relies on: US 2,071,515 "Electron multiplying device" and US 2,071,516 "Oscillation generator" [read: cited in the patent]. Also US 2,091,439 "Multipactor oscillator and amplifier": https://patents.google.com/patent/US2091439A/en [web, via mathematician]. These are not in the folder (cited, not obtained), and they are the most important thing to obtain next. - A. H. Halloran, "The Farnsworth Multipactor Tube", RADIO, Oct 1934, reprinted at https://borderlandsciences.org/journal/vol/44/n05/Farnsworth_Multipactor_Tube.html [web]. - Later patents that cite this one (Google Patents citing list): US2925528A (1960, Hartnell-Beavis), US3096490A (1963, Raytheon), and US11588421B1 (2023) and US12136824B2 (2024), both by Robert M. Lyden [web: https://patents.google.com/patent/US2172152A/en, via arbiter]. None has been read. The two recent ones are an energy-harvesting context, so check whether they rely on the tube doing more than this patent claims.

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Answers

1. 2 eV. The screen is at the same potential at both ends of the trip, and a static field does no net work between two points at the same potential. The 500 V only shortens the transit time.
2. V₀ and K scale as (f·d)², and doubling f gives a factor of 4: V₀ ≈ 4 × 62.5 = 250 V and K ≈ 4 × 31.7 = 127 eV. That is still below silver's ~300 eV peak.
3. G = 0.25/(1 − 1.95r) = 5, so 1 − 1.95r = 0.05, giving g_rt = 1.95r = 0.95 and r ≈ 0.487 (threshold is 0.513). Near g_rt = 1, a 1 % change in δ (from carrier amplitude, temperature or surface ageing) changes G by tens to hundreds of percent. That means distortion and instability, and at g_rt ≥ 1 the device oscillates.
4. (a) A power gain of 1: 25 W of RF drive gives 25 W of RF output, with no amplification. (b) A conversion efficiency: 25 W of d.c. input power gives 25 W of RF output. Read literally, that would be 100 % efficiency, itself implausible, so the wording is probably loose. You would need the separately stated drive power, d.c. supply power and output power, measured with a calibrated load.
5. The number per cycle is R/f, so it falls as frequency rises. At high frequency many cycles start with no stray electron at all, so the avalanche starts erratically. A deliberate seed (light or the gun) fixes the number per cycle. This is my interpretation [calc, reasoning]. The patent gives only "it is found necessary ... in the time available" [read]. The reviewers' alternative wording, that build-up takes N/f seconds, actually implies faster build-up at higher f, so it does not by itself explain the remark. The arbiter's QE revision adopts the R/f argument.