Primer: Paul M. Brown and the "Resonant Nuclear Battery"

Study guide for a self-taught reader at undergraduate level. It is built on the Step 3 arbiter's verdicts (arbiter.md, including its QE revision), the three reviews, and a deep read of the patent front page and figure sheets (deep/telsasymposium1990paperbypaulbrown-74ab16.md). Every number in the worked example was recomputed for this primer (example.py).

Source tags. [read] = taken from the documents in the project folder. [read: page image p.N] = read from a rendered page image. [web] = Internet source, URL given. [memory] = recalled from training, with confidence high/medium/low. "Calculation" = arithmetic done here from tagged inputs.

Source gap (repeated from the earlier steps). The folder holds only two documents: Brown's 1990 Tesla Symposium paper (with the front page of US Patent 4,835,433, figure sheets and a later CV) and a short 1999 Wired news item by Charles Platt. No document in the folder contains a measured power, efficiency or data table. The one numerical performance statement in the whole project comes from the full patent text on Google Patents [web: https://patents.google.com/patent/US4835433A/en], not from the folder. Brown's other papers (1987 to 1993) were not available. The waste-transmutation claims (4 and 5 below) rest only on the Wired article plus standard physics. Where a conclusion rests on [memory] alone, this primer says so.

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1. The idea in one paragraph

Paul Brown (died 2002 [web: https://www.power-eng.com/nuclear/nuclear-solutions-inc-regrets-loss-of-dr-paul-m-brown/]) said you could make a long-lived battery by putting a radioactive material inside a tuned electrical circuit. Radioactive atoms throw out fast charged particles. When those particles stop in a wire they knock electrons loose, and Brown argued that these freed electrons add to the current already flowing. If the circuit is "resonant" (it naturally rings back and forth at one frequency, like a bell), then, he argued, a small push from the radiation on every cycle could make up for the circuit's losses, keep it ringing forever after one start-up kick, and let you draw extra power off through a transformer. His patent says such a device delivered thousands of watts from a pinch of radium, uranium and thorium. Later (1999) he proposed using gamma rays to turn long-lived nuclear waste into short-lived or stable atoms quickly and cheaply, with the process heat running the plant. Brown placed himself in the free-energy tradition: his 1987 paper was titled "The Moray Device and the Hubbard Coil were Nuclear Batteries" [read p.3, reference list], so he read two famous claimed free-energy devices as hidden radioactive batteries. The starting facts are real: radiation really does produce electric charge, people have built small "nuclear batteries" since 1913, and gamma rays really can knock neutrons out of nuclei. What fails is the size of the claims. When you add up how much energy the radioactive material actually releases, the claimed output is about fifty million times too large.

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2. What you need to know first

2.1 Radioactivity, activity and the curie. An unstable nucleus decays at random. The decay constant λ (lambda, units 1/s) is the probability per second that one nucleus decays; half-life T½ = ln 2 / λ. The activity A = λN (N = number of nuclei) is decays per second, measured in becquerels (Bq, 1 decay/s) or curies (Ci). 1 Ci = 3.7 × 10¹⁰ Bq exactly, originally chosen as the activity of about 1 gram of radium-226 [memory, high]. Each decay releases a fixed amount of energy, the Q-value, shared among the emitted particles (alpha, beta, gamma, neutrino). Nuclear energies are given in MeV (mega-electron-volts); 1 MeV = 1.602 × 10⁻¹³ J. Read: Krane, Introductory Nuclear Physics (Wiley, 1988), the chapters on radioactive decay and on alpha and beta decay; or any modern-physics text such as Taylor, Zafiratos and Dubson, Modern Physics for Scientists and Engineers.

2.2 Power and energy conservation. Power is energy per second (watts = J/s). A device can never deliver more power than flows into it from all sources combined; that is the first law of thermodynamics. For a radioactive power source the input is (decays per second) × (energy per decay). This one product is the most important number in this whole primer. Read: Schroeder, An Introduction to Thermal Physics, early chapters on energy and the first law.

2.3 The LC tank and resonance. An inductor L (a coil, units henry H) and a capacitor C (units farad F) connected together pass energy back and forth between the magnetic field of the coil and the electric field of the capacitor, at the natural angular frequency ω₀ = 1/√(LC) (radians per second), or f = ω₀/2π in hertz. Every real circuit has resistance R (ohms), which turns some of the energy into heat on every cycle, so the ringing dies away unless something keeps adding energy. The quality factor Q tells how slowly it dies (high Q = low loss). Read: Griffiths, Introduction to Electrodynamics (for inductance and energy in fields); Horowitz and Hill, The Art of Electronics (for practical RLC circuits and oscillators); any introductory circuits or physics text covering "RLC circuits" (for example Halliday, Resnick and Walker, Fundamentals of Physics, the chapter on electromagnetic oscillations).

2.4 What an oscillator needs. A circuit that keeps oscillating by itself (an oscillator) needs (i) a power supply and (ii) a gain element (a transistor, a vacuum tube, or a "negative resistance" device) wired with feedback, so that energy is fed in in step with the oscillation, at the right moment of each cycle. Random pushes at random moments do not build up a clean oscillation; they make noise. Read: Horowitz and Hill, the sections on oscillators.

2.5 How radiation makes charge (ionisation and the betavoltaic effect). A fast alpha or beta particle loses its energy by knocking electrons off atoms as it slows. In a gas it takes on average about 25 to 35 eV to make one ion pair [memory, high]; in a semiconductor about 3 to 10 eV per electron-hole pair [memory, medium]. In a gas between two charged plates (an ionisation chamber) or in a semiconductor junction (a diode), those freed charges are swept to the electrodes and make a measurable current. A diode used this way with a beta source is a betavoltaic cell: real devices give microwatts, at single-digit percent efficiency [web: https://en.wikipedia.org/wiki/Betavoltaic_device]. The energy comes entirely from the radiation. Read: Knoll, Radiation Detection and Measurement (ionisation chambers and semiconductor detectors).

2.6 Conduction in metals. A metal already has an enormous number of free electrons, about 8.5 × 10²² per cubic centimetre in copper [memory, high]. Resistance R = ρℓ/A (ρ = resistivity, ℓ = length, A = cross-sectional area); copper ρ ≈ 1.68 × 10⁻⁸ Ω·m at room temperature [memory, high]. At high frequency, current crowds near the wire surface within a "skin depth" δ = √(2ρ/(ωμ₀)), where μ₀ = 4π × 10⁻⁷ H/m, which raises the effective resistance. Read: Kittel, Introduction to Solid State Physics (the free-electron model); Griffiths (skin depth, in the chapter on electromagnetic waves in conductors).

2.7 Photonuclear reactions (for claims 4 and 5). A gamma ray (high-energy photon) of roughly 8 to 30 MeV can be absorbed by a nucleus and knock out a neutron: a (γ,n) reaction. To do this the photon must supply at least the neutron separation energy S_n, the energy that binds the last neutron, typically 6 to 10 MeV for medium-heavy nuclei [memory, medium]. The absorption is strongest in a broad peak called the giant dipole resonance, about 7 to 40 MeV [web: https://en.wikipedia.org/wiki/Giant_resonance]. Read: Krane, the chapter on nuclear reactions.

2.8 Heat engines. Turning heat into electricity always wastes some of it. Real steam plants convert about 33 to 40 percent of heat into electricity [memory, high]; the Carnot limit is the theoretical maximum. Read: Schroeder, the chapter on engines and refrigerators.

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3. The textbook view

3.1 Radioactive power: the ceiling

For a pure nuclide of half-life T½ (seconds) and molar mass M (g/mol):

• decay constant λ = ln 2 / T½
• number of atoms per gram N/m = N_A / M (N_A = 6.022 × 10²³ per mole, Avogadro's number)
• specific activity a = λ N_A / M (decays per second per gram)
• maximum power P_max = A × E, where A = activity (decays/s) and E = energy released per decay (joules).

Nothing in the circuit around the source can change λ or E in any measurable way; decay rates have been checked against temperature, pressure and chemical state and change by at most about a part in a thousand (in special cases such as electron capture) [memory, medium]. So P_max is a hard ceiling on what a radioactive power source can give. A converter of efficiency η (0 < η < 1) gives P_out = η A E.

Useful handles (all calculation, from the inputs tagged):

Real radioisotope thermoelectric generators (RTGs, used on spacecraft) turn about 5 to 7 percent of the decay heat into electricity [memory, medium]; betavoltaics reach single-digit percent [web: betavoltaic page above].

3.2 The series RLC circuit

Symbols: L = inductance (H), C = capacitance (F), R = total series resistance (Ω), i = current (A), q = charge on the capacitor (C), U = stored energy (J).

Kirchhoff's voltage law around the loop gives the damped-oscillator equation

L d²q/dt² + R dq/dt + q/C = 0.

• Natural frequency: ω₀ = 1/√(LC), so f = 1/(2π√(LC)).
• It oscillates (rather than just decaying) when it is underdamped: (R/2L)² < 1/(LC), which is the same as R < 2√(L/C).
• The current amplitude decays as e^(−Rt/2L), so the stored energy decays as U(t) = U₀ e^(−Rt/L).
• The rate of energy loss is dU/dt = −i²R (averaged over a cycle, −I_rms² R).
• Quality factor Q = ω₀L/R = (1/R)√(L/C). The power needed to keep the oscillation steady equals the loss: P_loss = I_rms² R = ω₀ U / Q.
• At resonance the inductor's and capacitor's reactances (their "AC resistances", ωL and 1/(ωC)) are equal and cancel in a series circuit, so the circuit's impedance is just R. This minimises the loss for a given drive; it does not remove it, and it does not create energy.

A large circulating current is not output power. In a high-Q tank, a large current can slosh back and forth between L and C while very little real power is used. Real power delivered to a resistor is P = I_rms V_rms only when voltage and current are in phase; for a reactive element they are 90° out of phase and the average power is zero. Measuring "amps" and "volts" separately with two meters does not tell you the real power unless you also know the phase.

3.3 Ionisation current

If a source emits A particles per second and each makes n charge pairs, the most charge you can collect per second is I_max = A n e (e = 1.602 × 10⁻¹⁹ C). The energy for this comes from the particles: n × w = E, where w is the average energy per pair. So the ionisation "current" carries at most the decay power; a field applied from outside does work on these charges, but that work is drawn from the external battery, not created.

3.4 Energy balance of a (γ,n) reaction

For one photon of energy E_γ absorbed by a nucleus that emits a neutron:

E_γ = S_n + (kinetic energy of neutron and recoiling nucleus).

The reaction uses up at least S_n. Some of that comes back later (the neutron is eventually captured elsewhere, releasing roughly its binding energy, and the new nucleus may beta-decay), but the total heat you can collect cannot exceed the beam energy plus a few MeV from these later steps. Converting heat to electricity then loses another 60 percent or so.

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4. Where Brown departs from the textbook

Brown starts from real physics and says so in his own words. The departures come later.

The step to notice: Brown writes down the right inequality (source power ≥ i²R) and then never evaluates the left-hand side.

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5. Worked example: the patent's energy budget

Question. The patent says a core containing a 1 mCi radium needle, 200 g of uranium and 100 g of thorium gave 23 A at 400 V into a resistive load at 86 kHz [web: patent URL above]. Can those materials supply that much power?

Reproduce with: .venv/bin/python review/Brown-Paul/example.py

Nuclear data below are [memory, high] for half-lives and alpha energies and [memory, medium] for the chain Q-values; the answer does not change if any of them is off by a factor of two.

Step 1. The radium needle. "1 millicurie" is its activity: A_Ra = 10⁻³ × 3.7 × 10¹⁰ = 3.7 × 10⁷ decays/s. In a sealed needle, Ra-226 is followed quickly by its daughters Rn-222, Po-218, Pb-214, Bi-214 and Po-214, each decaying at the same rate. Two ways to count the energy per Ra-226 decay: - alpha kinetic energies only: 4.78 + 5.49 + 6.00 + 7.69 = 23.96 MeV - generous ceiling, total Q-values of all six decays (counting even the share that leaves with neutrinos): 4.871 + 5.590 + 6.115 + 1.024 + 3.270 + 7.833 = 28.70 MeV.

P_Ra = 3.7 × 10⁷ × 28.70 MeV × 1.602 × 10⁻¹³ J/MeV = 1.70 × 10⁻⁴ W (alphas only: 1.42 × 10⁻⁴ W).

That is 0.17 milliwatts.

Step 2. 200 g of uranium (assumed natural uranium, 99.27 % U-238). U-238: λ = ln 2 / (4.468 × 10⁹ y × 3.156 × 10⁷ s/y) = 4.92 × 10⁻¹⁸ s⁻¹; atoms per gram = 6.022 × 10²³ / 238.05 = 2.53 × 10²¹; so a = 12.4 kBq per gram of U-238, 12.35 kBq per gram of natural uranium. 200 g gives 2.47 × 10⁶ decays/s of U-238, the same of U-234 (in equilibrium), and a little U-235. Energy per decay: about 4.27 MeV (U-238 alpha) + 0.27 + 2.27 MeV (the two quick beta decays that follow) and 4.86 MeV for U-234. P_U ≈ 4.7 × 10⁻⁶ W.

Step 3. 100 g of thorium-232. a = 4.06 kBq/g, so 4.06 × 10⁵ decays/s. Even assuming the entire chain to lead-208 is in equilibrium (about 42.6 MeV per Th-232 decay; freshly purified thorium would have far less): P_Th ≈ 2.8 × 10⁻⁶ W.

Step 4. Total and comparison. P_available ≤ 1.70 × 10⁻⁴ + 4.7 × 10⁻⁶ + 2.8 × 10⁻⁶ = 1.78 × 10⁻⁴ W, assuming every joule is captured and turned into electricity at 100 %. P_claimed = 23 A × 400 V = 9,200 W. Ratio = 9,200 / 1.78 × 10⁻⁴ = 5.2 × 10⁷.

The claimed output is about fifty million times the most the core can release. The radium does almost all the work; the 300 g of uranium and thorium contribute about 4 % (example.py prints 4.2 %). This also answers the patent abstract's claim that a three-nuclide mixture gives "a greater flux" than one nuclide: the total is just the sum of the parts, and here the extra parts are almost nothing.

Turn it round: to get 9,200 W from the full radium chain at 100 % conversion would need 9,200 / (28.70 × 1.602 × 10⁻¹³) = 2.0 × 10¹⁵ decays/s = 54,000 Ci, about 55 kg of radium-226 (1 Ci of Ra-226 is about 1.01 g) instead of a 1 mCi needle (about 1 mg).

Step 5. The kindest reading: maybe 23 A was only circulating in the tank. In a high-Q tank, large currents can slosh between L and C without much real power (Section 3.2). Could "23 A" be such a circulating current, with tiny real output? Use Brown's own Fig. 7 condition, source power ≥ I²R. With I = 23 A rms and source power at most 1.78 × 10⁻⁴ W:

R_max = 1.78 × 10⁻⁴ / 23² = 3.4 × 10⁻⁷ Ω.

Compare one metre of 1 mm diameter copper wire: - at DC, R = ρℓ/A = 1.68 × 10⁻⁸ × 1 / (π × (0.5 × 10⁻³)²) = 0.021 Ω, already about 64,000 times too large; - at 86 kHz, the skin depth is δ = √(2 × 1.68 × 10⁻⁸ / (2π × 86,000 × 4π × 10⁻⁷)) = 0.22 mm, and modelling the current as flowing only in the outer 0.22 mm ring gives about 0.031 Ω, about 90,000 times too large.

A coil with many turns, plus a capacitor and transformer, has more resistance still. So even the circulating-current reading fails Brown's own condition: the source cannot even keep 23 A ringing in a real tank, let alone deliver it to a load. (Steps 4 and 5 are calculations made for this primer; Step 4 agrees with the arbiter's 4 × 10⁷ to 6 × 10⁷.)

What this example teaches. Before arguing about mechanisms (flux cutting, resonance, avalanche electrons), write down the energy input. If the ceiling is microwatts and the claim is kilowatts, no mechanism can close the gap. Brown wrote the right inequality in Fig. 7; this example just puts numbers into it.

A second quick check you can do yourself (Kr-85, claim 3). Kr-85: T½ = 10.728 y, mean beta 0.251 MeV [web: Kr-85 page above]. Specific activity = ln 2 × 6.022 × 10²³ / (10.728 × 3.156 × 10⁷ × 85) = 1.45 × 10¹³ Bq/g = 392 Ci/g; power 1.49 mW per Ci. For 0.25 W/cm³ at 100 % conversion you need 168 Ci/cm³. Pure Kr-85 gas at 0 °C and 1 atm (85 g / 22,414 cm³ = 3.8 × 10⁻³ g/cm³) holds 1.49 Ci/cm³, so you need about 113 atmospheres of isotopically pure Kr-85 even at perfect efficiency. At a betavoltaic's realistic 10 % or less, you would need more Kr-85 per cm³ than liquid krypton can hold (about 940 Ci/cm³ at about 2.4 g/cm³ [memory, medium]). (Calculation; matches the arbiter.)

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6. Evidence table

Verdicts are the arbiter's final verdicts (arbiter.md, Section 4, plus its QE revision). "Tested?" means a neutral, controlled test on record.

No claim is "Tested, not supported": no controlled experiment on any Brown device appears in any source. No source documents suppression or a dismissal without examination [arbiter, Section 5].

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7. What a learner should take away

Solid. - Radiation makes charge, and charge collected across a junction or electrode gap is real electricity. Moseley's 1913 cell, ionisation chambers, betavoltaic cells and RTGs all work, at the power their sources supply. - Brown's circuit formulas (f = 1/(2π√(LC)), R < 2√(L/C), dU/dt = −i²R) and his statement that the energy comes from the decay are correct. - Gamma rays of 8 to 30 MeV really can turn Cs-137 into Cs-136 (half-life about 13 days, then stable Ba-136) and Sr-90 into Sr-89 (about 50 days, then stable Y-89) [memory, medium to high]. The Wired article quotes mainstream physicists saying the idea is "not far-fetched" [read: Wired], and similar proposals appeared independently around the same time (Eremeev, 1999 [web: https://www.osti.gov/etdeweb/biblio/20064919, via the contrarian review]).

Wrong, and why the mistake is instructive. - Resonance stores energy; it does not make it. A high-Q circuit lowers the power needed to keep it going, but that power must still come from somewhere, and here it is capped at about 0.18 mW. Many "free energy" designs share this mistake: they treat a high circulating current as if it were output. - Write down the budget first. Brown wrote "β Flux ≥ i²R" and stopped. Putting numbers into it (Section 5) settles the strong claim in five lines. This is the most useful habit to take from this project. - Counting carriers. "Knocking electrons loose" sounds like it should increase current, but a metal already has about 10²³ free electrons per cm³. Always compare the added quantity with what is already there. - Fixed power vs gain. A source that delivers a fixed power and an amplifier whose gain scales with current behave completely differently; mixing them up hides the energy problem. - Check the headline example against the mechanism. Brown's own example of dangerous waste ("half-lives of 24,000 years") matches plutonium-239. Knocking one neutron out of Pu-239 gives Pu-238, which has an 88-year half-life and gives off a lot of heat, so it is not "safe within a few months". With actinides, gamma rays can also split the nucleus (photofission), making new fission products [memory, medium]. The idea works best for Cs-137 and Sr-90 and worst for the very example he quoted. - "Most acceptable" is not the same as "practical". The paper calls Kr-85 "environmentally the most acceptable radioisotope available for power production" [read p.3]. Yet its own 0.25 W/cm³ target needs about 113 atmospheres of pure radioactive krypton gas even at 100 % conversion (Section 5). A pressurised radioactive gas is hard to call the easy option. - Mixing sources does not multiply them. The patent says a mixture of three nuclides gives "a greater flux" than one. Each nucleus decays on its own clock, so activities simply add. Section 5 shows the uranium and thorium add about 4 % to the radium. - Heat from a beam is the beam's own energy. A photonuclear plant cannot run itself on heat that the beam put in, because turning heat back into electricity loses most of it.

Genuinely open (small, but real). - What did Brown actually measure to get "23 A at 400 V"? A start-up transient, a reactive current, a meter error at 86 kHz, or something else? The patent gives no method [web]. The paper's figures show only a voltmeter across the load and an ammeter in series, after a filter, with no phase or wattmeter reading [read: page image p.5], so a reactive volt-ampere reading cannot be ruled out; but neither can anything else be confirmed. The patent body, Brown's 1991 IECEC paper and any INEL report would help. - Does a radioactive source inside a very high-Q coil do anything measurable (noise, core charging, a small DC offset)? Physics predicts at most microwatts and no resonant benefit, but nobody has published a careful test. - Could a photonuclear plant be economic for a selected stream of Cs-137 or Sr-90? This is an engineering and cost question that energy conservation alone does not settle.

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8. Check yourself

1. A 10 Ci tritium source (mean beta energy 5.7 keV [memory, high]) is used in a betavoltaic cell with 5 % efficiency. What is the electrical output power?
2. A series LC tank has L = 100 μH and C = 34 nF. What is its resonant frequency? If R = 0.5 Ω, what is Q, and is the circuit underdamped?
3. In the same tank, a current of 2 A rms circulates. How much power must be supplied to keep it going? Could a 1 mCi radium source do it?
4. Brown says a single Sr-90 beta makes 80,000 ions in copper. If each takes about 25 to 30 eV, how much energy is that, and which beta in the Sr-90/Y-90 pair could supply it? What does this tell you about "amplification"?
5. A (γ,n) reaction on Cs-137 needs at least about 9 MeV. Afterwards the heat released by the neutron and the daughter's decay adds at most about 12 MeV of extra heat [assumption for this question]. If heat becomes electricity at 35 %, what is the largest photon energy E_γ for which the plant would make net electricity? Is that possible?
6. Brown's core holds 1 mCi of Ra-226, about 67 μCi of U-238 (200 g of natural uranium) and about 11 μCi of Th-232 (100 g of thorium), counting the parent nuclides only (no daughters). What is the total activity in decays per second? Can mixing the three nuclides make the total larger than that? Why or why not?

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9. Further reading

Mainstream. - K. S. Krane, Introductory Nuclear Physics (Wiley, 1988): radioactive decay, alpha and beta decay, nuclear reactions. - G. F. Knoll, Radiation Detection and Measurement (Wiley, several editions): ionisation chambers, semiconductor detectors, energy per ion pair. - D. J. Griffiths, Introduction to Electrodynamics: inductance, energy in fields, skin depth. - P. Horowitz and W. Hill, The Art of Electronics: RLC circuits, Q, and how real oscillators work. - C. Kittel, Introduction to Solid State Physics: free-electron model of metals. - Betavoltaic devices overview [web: https://en.wikipedia.org/wiki/Betavoltaic_device]. - Giant dipole resonance overview [web: https://en.wikipedia.org/wiki/Giant_resonance]. - H. G. J. Moseley, "The attainment of high potentials by the use of radium", Proc. Royal Society A 88 (1913) [memory, high for title and year; medium for volume]. - P. Rappaport, "The electron-voltaic effect in p-n junctions induced by beta-particle bombardment", Physical Review 93 (1954) [memory, high for title and year; medium for volume]: the founding paper of semiconductor betavoltaics.

Brown's own works. - P. M. Brown, "Tesla Technology and Radioisotopic Energy Generation", 1990 Tesla Symposium, Colorado Springs (in the project folder) [read]. - US Patent 4,835,433, "Apparatus for direct conversion of radioactive decay energy to electrical energy", filed 8 Feb 1988, granted 30 May 1989, assignee Nucell, Inc. [read: page image p.4 (front page); full text web: https://patents.google.com/patent/US4835433A/en]. - Brown's cited papers (not in the folder): "The Moray Device and the Hubbard Coil were Nuclear Batteries", Magnets in Your Future 2(3), March 1987; "Resonant Nuclear Power Supply", Raum & Zeit 1(2), 1989; "Resonant nuclear battery may aid in mitigating the greenhouse effect", American Nuclear Society Winter Meeting, 1989; "The beta voltaic effect applied to radioisotopic power generation", ANS Annual Meeting, 1990 [read p.3, reference list]. - Related prior art: Burke patents US 3,409,820 and 3,530,316 (cited on the patent front page [read: page image p.4, as read by the reviewers]). - C. Platt, "Nuking Nukes", Wired 7.02, February 1999 (in the project folder) [read].

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Answers to "Check yourself"

1. Activity 10 × 3.7 × 10¹⁰ = 3.7 × 10¹¹ decays/s. Power = 3.7 × 10¹¹ × 5.7 × 10³ eV × 1.602 × 10⁻¹⁹ J/eV = 3.4 × 10⁻⁴ W (0.34 mW). At 5 %: about 17 μW. That is the scale of real nuclear batteries.
2. f = 1/(2π√(100 × 10⁻⁶ × 34 × 10⁻⁹)) = 1/(2π × 1.84 × 10⁻⁶) ≈ 86 kHz (Brown's frequency). √(L/C) = √(100 × 10⁻⁶ / 34 × 10⁻⁹) = 54 Ω, so Q = 54 / 0.5 ≈ 110. Underdamped because 0.5 Ω < 2 × 54 = 108 Ω.
3. P = I²R = 2² × 0.5 = 2 W. A 1 mCi radium needle supplies at most 1.7 × 10⁻⁴ W, about 12,000 times too little, even at 100 % conversion.
4. 80,000 × 25 to 30 eV = 2.0 to 2.4 MeV. Sr-90's own beta has at most 0.546 MeV [web: https://en.wikipedia.org/wiki/Strontium-90], so only the Y-90 daughter's beta (up to 2.28 MeV [web: Y-90 page]) could do it. The ions use up the whole beta energy, so there is nothing left over to "amplify" anything; the ionisation is where the energy goes, not a multiplier.
5. Net gain needs 0.35 × (E_γ + 12 MeV) > E_γ, i.e. 4.2 > 0.65 E_γ, so E_γ < 6.5 MeV. But the reaction needs at least 9 MeV. Impossible, even with this generous 12 MeV of extra heat and before counting the energy lost making the gamma beam. (With the arbiter's tighter estimate of about 2.5 MeV extra heat the limit is 1.3 MeV.)
6. Activities add: (1000 + 67 + 11) μCi = 1078 μCi = 1.078 × 10⁻³ × 3.7 × 10¹⁰ ≈ 4.0 × 10⁷ decays/s. Mixing cannot raise it. Each nucleus decays at random on its own, at a rate set by its own decay constant λ, whatever atoms are next to it (Section 3.1), so the total is just the sum of the separate rates. Here U and Th add about 7 % of the parent decays (and about 4 % of the energy, Section 5, because the radium chain releases more energy per decay), so the patent's "greater flux from a mixture" adds almost nothing.