Judged by calculation, not by opinion ── and the decisive argument turns out not to be about barriers at all
On 23 March 1989, Fleischmann and Pons of the University of Utah held a press conference announcing that they had produced nuclear fusion in a test tube. The announcement did not wait for peer review. More than thirty years later the topic has not gone away ── failed replications, retractions, lawsuits, a serious re-examination funded by Google at a scale of about ten million dollars from 2015 (result: negative), a new US Department of Energy ARPA-E programme begun in 2023, ongoing work at a Japanese university and a start-up. This bonus episode judges the matter by calculation, not by declaration. The tools built in the main series apply directly ── work out how many orders the Gamow factor is short, see how much screening in a metal fills in, and then settle it for an entirely different reason having nothing to do with barriers. After that, we look at what is genuinely interesting about this material (palladium hydride). Something genuinely interesting is there. It just is not fusion.
| Year | Event |
|---|---|
| 1989 | Fleischmann–Pons: electrolysing heavy water to load deuterium into a palladium cathode produced "excess heat" and neutrons, announced by press conference. The neutron claim was later withdrawn. Replication was attempted worldwide and almost all attempts failed |
| 1989 | The first US Department of Energy (ERAB) review: negative |
| 1990s | Piantelli and others claimed anomalous heat in nickel plus ordinary hydrogen. Independent replication was not established |
| 2004 | The second Department of Energy review: still not convincing, but concluded that funding a few narrowly defined questions would be acceptable |
| 2011 | Rossi's "E-Cat." The 2014 Lugano report was criticised for a serious error in the treatment of emissivity in its infrared thermometry. Litigation with the investing company ended in a 2017 settlement |
| 2015–18 | The Google-funded re-examination (about 30 researchers, three years, of order ten million dollars). Reported in Nature in 2019 ── no evidence of cold fusion was found. But the by-products (methods for high loading of metal hydrides, precision calorimetry) are genuine scientific results |
| 2023 | US ARPA-E launched an LENR (low-energy nuclear reaction) programme (8 projects, of order ten million dollars). Framed as "settling the question" |
| ongoing | In Japan, Iwamura and colleagues at Tohoku University with a start-up report anomalous heat in nickel-based nanocomposites. Independent confirmation outside the collaboration is not, at present, established |
Use Episode 2's Gamow factor directly. The partners are two deuterons, with Gamow energy \(E_G\approx986\) keV.
Insert room-temperature thermal energy \(k_BT=25\) meV \(=2.5\times10^{-5}\) keV:
$$\sqrt{\frac{E_G}{E}}=\sqrt{\frac{986}{2.5\times10^{-5}}}\approx 6280 \qquad\Longrightarrow\qquad e^{-6280}\approx10^{-2728}$$Out of the question. But this is for free deuterons colliding thermally. Inside a metal the deuterons are bound in the lattice, have zero-point motion and are screened by electrons. The correct calculation is the overlap of the nuclear wavefunctions, and gives a much larger value. So dismissing it with the naive thermal estimate is not fair.
Proper calculations were done by several groups in 1989. A well-known one is the Nature paper of Koonin and Nauenberg. The d–d fusion rate inside a deuterium molecule (D₂) is of order \(10^{-64}\) events per second per pair.
To produce one watt from d + d fusion (about 4 MeV = \(6.4\times10^{-13}\) J per event):
$$\frac{1\ \mathrm{W}}{6.4\times10^{-13}\ \mathrm{J}}\approx 1.6\times10^{12}\ \text{events per second}$$With perhaps \(\sim10^{23}\) deuterons in a few grams of palladium cathode, that comes to about \(10^{-11}\) events per second per pair.
Required: \(10^{-11}\). Calculated: \(10^{-64}\). 53 orders of magnitude short.
"Inside a metal the electrons screen the barrier, so it must be easier" ── a legitimate point, and screening does exist. What is more, the measured values exceed the theoretical ones.
There is one indicator that makes the answer clearest: how many times heavier the electron would have to be.
Screening working means, in effect, that the electron cloud shrinks and the deuterons get closer. Heavier electrons make a smaller cloud (the Bohr radius goes as \(1/m\)). So the question can be written:
"How many times heavier would the electron have to be for cold fusion to become observable?"
This is exactly the question Koonin and Nauenberg posed, and the answer is about 5 to 10 times.
For reference ── the muon is 207 times heavier than the electron, and that really does work (bonus ②). The lattice's screening, meanwhile, supplies an effective enhancement of at most a few percent.
The upper panel of the figure gives the d–d fusion rate against an effective electron mass (a scaling that joins the electron's \(10^{-64}\)/s/pair to the muon's \(10^{12}\)/s/pair). The line needed for one watt is crossed at a mass of about 8. The band within reach of lattice screening is far to the left of it.
The lower panel is a cross-check from a completely different direction. It converts a claimed excess heat into a number of neutrons. d + d has known branching ratios, so if heat is coming out then neutrons must be too ── and the panel shows what that would mean for the experimenter.
Everything so far has been an argument about quantity. There is in fact a more decisive problem ── and it is not even about barriers.
d + d fusion has well-known exits.
| Reaction | What comes out | Branching ratio |
|---|---|---|
| d + d → ³He + n | a 2.45 MeV neutron | about 50% |
| d + d → t + p | tritium and a proton | about 50% |
| d + d → ⁴He + γ | a 23.8 MeV gamma ray | about 10⁻⁷ |
What cold fusion claims, meanwhile, is this ── heat comes out. The ash is ⁴He. Almost no neutrons. No tritium. No gamma rays.
For that claim to hold, two things must be true at once:
① the branching ratio must invert by seven orders of magnitude (the \(10^{-7}\) channel becoming 100%)
② the 23.8 MeV gamma ray must vanish and turn into lattice heat
Neither can be explained by "the lattice assists tunneling," because ──
Tunneling is about the "entrance." It cannot change the "exit."
How you got through the barrier only sets how often you arrive at the nucleus. What comes out after arrival is decided by the compound nucleus (²He*) and its decay channels ── that is nuclear physics. The nucleus does not know whether you came from an accelerator or by slipping through a lattice.
One can of course claim "an unknown mechanism changes the exit too." But that is no longer a story about tunneling; it is entirely new nuclear physics. "The barrier is lowered because we are inside a metal" is a modest claim; "the nuclear branching ratios change by seven orders and a 23.8 MeV gamma ray disappears" is not. The two demand utterly different amounts of novelty.
Producing 1 W from d + d at the normal branching ratios means \(1.6\times10^{12}\) fusions per second, about half of which emit a neutron ── \(8\times10^{11}\) per second.
At 1 m the neutron flux is \(8\times10^{11}/(4\pi\times10^4\,\mathrm{cm^2})\approx6\times10^6\) /cm²/s. Multiply by the fluence-to-dose factor for 2.45 MeV neutrons (about \(4\times10^{-10}\) Sv·cm²):
A certainly lethal dose within one hour. Yet the neutrons reported in the 1989 experiments were only about \(10^{-9}\) times what the heat demanded.
That nine-order discrepancy is, on its own, the heaviest piece of evidence. "Heat but no neutrons" is incompatible with known nuclear physics. And the experimenters being unharmed is itself a datum.
Many recent claims use nickel plus ordinary hydrogen rather than palladium plus deuterium. It is said to be cheaper and more practical; as physics it is orders of magnitude worse.
| Reaction | Z₁Z₂ | E_G | ratio of √(E_G) |
|---|---|---|---|
| d + d | 1 | 986 keV | 1 |
| p + ⁵⁸Ni | 28 | about 760 MeV | about 28× |
The exponent is 28 times larger. Starting from a d–d case already 53 orders short, an exponent 28 times bigger ── this is no longer even a matter of orders.
A claim that "nickel is easier" would need an explanation that overturns this point, and none has been given.
Everything above has been negative. There are points to defend as well.
Excess-heat claims are typically a few percent of the input power. In open electrolytic cells, recombination of hydrogen and oxygen, uneven stirring, thermal lag and ambient drift can easily manufacture an apparent excess of a few percent. The fair reading is not "the claimants were dishonest" but "measuring a few percent of heat correctly is genuinely hard."
This is what makes the 2019 Nature re-examination exemplary: they spent three years not to refute but to measure properly. The result was negative, but the high-loading techniques, precision calorimetry and materials synthesis developed along the way remain as genuine results. That posture is the best available response to this topic.
Finally, the things this material really is remarkable for. Ironically, they sit at the intersection of Episodes 1 and 5 of the main series.
| Phenomenon | What it is | Relation to the main series |
|---|---|---|
| quantum diffusion of hydrogen | Hydrogen in palladium moves between interstitial sites by tunneling. At low temperature the diffusion coefficient stops depending on temperature | Episodes 1 and 3, exactly |
| PdH is a superconductor | \(T_c\approx9\) K. And PdD has a higher \(T_c\) (about 11 K) ── an inverse isotope effect, the opposite of BCS's naive prediction | Episode 5's isotope effect, running backwards |
| extraordinary hydrogen loading | D/Pd ratios above 1. As a density of hydrogen in a metal, higher than liquid hydrogen | the material's genuine peculiarity |
The firm parts: the Gamow energy for d–d fusion (about 986 keV) and the conclusion that fusion does not occur at room temperature; the known branching ratios of d + d (about 50% each for the n and t channels, about \(10^{-7}\) for ⁴He + γ); the conversion showing that 1 W from normal-branching d–d implies of order \(10^{12}\) neutrons per second and a lethal dose rate; that the 1989 reports had neutrons orders of magnitude below what the heat demanded; that nickel-based systems are far worse than d–d on Gamow grounds; the conclusions of the 1989 and 2004 Department of Energy reviews; and the negative result of the 2019 Nature re-examination. The point that "the entrance can change but the exit cannot" is the most robust of all, since it does not depend on the details of the barrier at all.
The softer parts and caveats: (1) The figure's "fusion rate versus effective electron mass" is an interpolation by scaling between two points ── the electron (\(10^{-64}\)/s/pair) and the muon (\(10^{12}\)/s/pair) ── not a first-principles calculation. Read it as a way to get a feel for the orders of magnitude. (2) The \(10^{-64}\) figure itself depends on the assumed configuration (a D₂ molecule, or an interstitial site) and varies by several orders across the literature. But no estimate changes the conclusion that it falls vastly short. (3) That measured screening potentials in metals exceed theoretical ones is a genuine open problem, and this episode does not deny it. What it denies is the claim that this fills 50 orders of magnitude. (4) "Independent confirmation is not established" does not mean anyone is committing fraud; it is a statement about reproducibility. (5) That ARPA-E is funding work is not evidence that it happens, but a judgement that the question is worth settling. Conversely, "funded, therefore suspect" is also wrong. (6) The body treats d–d and p–Ni. Claims involving other reaction systems (deuterium with heavier nuclei, say) need case-by-case examination, but the larger the charge the worse the Gamow factor.
The quantitative argument: the d–d fusion rate at room temperature is \(10^{-64}\)/s/pair; one watt needs \(10^{-11}\)/s/pair ── 53 orders short. Filling that would require the equivalent of "an electron about 8 times heavier," while lattice screening supplies a few percent. Nickel systems are a further 28 times worse.
The clincher: firmer still is the branching ratio. The exits of d + d are neutrons (50%) and tritium (50%), with ⁴He + 23.8 MeV gamma at \(10^{-7}\). The claim requires that to invert by seven orders and the gamma ray to disappear. Tunneling changes the entrance; it cannot change the exit ── because the nucleus does not know how you arrived. And 1 W of heat means \(8\times10^{11}\) neutrons per second, a lethal dose within an hour, whereas the neutrons reported were \(10^{-9}\) of that.
In fairness: measuring a few percent of excess heat correctly is genuinely hard, and calling the claimants dishonest would be wrong. The 2019 Nature re-examination spent three years not "to refute" but to measure properly, and left real by-products along with its negative result. And palladium hydride itself is genuinely interesting ── quantum diffusion of hydrogen, superconductivity in PdH and its inverse isotope effect. A real meeting place of tunneling and superconductivity. It just is not fusing.
Print / make a PDF: ⌘+P (Ctrl+P on Windows). On screen, move the effective electron mass to see where the required level is reached. The lower panel computes how many neutrons and how much exposure a claimed excess heat would imply. "See the answer" opens each solution.