There is exactly one cold fusion that genuinely happens at room temperature ── it works, and still it does not pay
Bonus ① said "if the electron were about 8 times heavier, cold fusion would happen." So what if you actually use a heavier particle? The answer has been known for a long time ── it happens. The muon is 207 times heavier than the electron; put one into a deuterium molecule and the molecule shrinks by a factor of 207, bringing the nuclei within 500 fm of each other, and they fuse within picoseconds. And this is not theory: Alvarez found it by accident in a bubble chamber in 1956. Fusion at the temperature of liquid hydrogen, i.e. 20 K ── cold fusion in the literal sense. And yet, seventy years on, it is not generating power. What stops it is not the barrier but something far more mundane. This episode is about that remaining factor of two.
| electron | muon | |
|---|---|---|
| mass | 0.511 MeV | 105.7 MeV (206.8×) |
| lifetime | stable | 2.197 microseconds |
| charge and spin | −e, 1/2 | −e, 1/2 (same as the electron) |
Since its charge and spin match the electron's, a muon can stand in for an electron inside an atom. But it is heavy. The Bohr radius is inversely proportional to mass, so ──
| Molecule | internuclear distance |
|---|---|
| ordinary D₂ (electrons) | 74,000 fm (0.74 Å) |
| ddμ (a muon) | about 500 fm |
| the range of the nuclear force | a few fm |
Still more than a hundred times the range of the nuclear force ── but for tunneling that is plenty. Episode 1's \(S_E=\int\sqrt{2m(V-E)}\,dx\) is an integral over distance, so cutting the distance by a factor of 150 shrinks the exponent dramatically.
| System | fusion rate | Meaning |
|---|---|---|
| a D₂ molecule (electrons) | \(10^{-64}\) | would not happen if you waited \(10^{47}\) times the age of the universe |
| ddμ | \(\sim10^{9}\) | nanoseconds |
| dtμ (deuterium + tritium) | \(\sim10^{12}\) | picoseconds. Effectively instantaneous |
From \(10^{-64}\) to \(10^{12}\) ── a leap of 76 orders of magnitude. Bonus ① calculated that 53 orders were needed. The muon overshoots the requirement by 23 orders. Which is why it happens with room to spare.
It is called catalysis because the muon is not consumed and can be used repeatedly.
One cycle takes about \(10^{-8}\) s, so within the muon's 2.2 μs lifetime it could in principle go round about 220 times.
① α-sticking, \(\omega_s\) ── the helium (\(\alpha\)) produced by fusion has charge +2 and attracts the muon strongly. With a probability of about 0.45–0.6% the muon is captured by the α and carried away. Some are stripped back off in flight (reactivation); this figure is the net.
→ On its own that caps things at \(1/0.0045\approx220\) cycles.
② the muon lifetime \(\tau_\mu=2.2\) μs ── at a cycling rate \(\lambda_c\approx10^8\)/s, that is \(\lambda_c\tau_\mu\approx220\) cycles within the lifetime.
Remarkably, these two are almost the same size. Together they give \(N\approx110\). The best experimental value is about 150 cycles (with density and temperature optimised). By coincidence, nature has placed the two limits at the same height.
Making one muon costs, realistically, about 5 GeV in an accelerator (about 50 times the muon's rest mass of 106 MeV, because production is inefficient).
One d+t fusion yields 17.6 MeV. Therefore:
And since converting heat back to electricity is only about 40% efficient, running an actual power plant in the black needs of order 700 cycles.
Achieved: 150. Needed: 300–700. Not orders of magnitude ── a factor of two to five. That distance has not closed in seventy years.
The figure is that balance sheet. Move the sticking probability, the cycling rate and the muon production cost and it shows how many times one muon works and how the books come out.
Look at the distance between the current state of the art (the blue point) and break-even (the green line). Whichever knob you turn, the line is hard to cross ── halving the sticking would do it, but that is a number fixed by nuclear physics and not ours to move.
Failing to become a power source does not diminish the phenomenon. Muon-catalysed fusion is the demonstration that "thin the barrier and fusion happens even at room temperature."
| Method | effective mass | fusion rate | Verdict |
|---|---|---|---|
| an ordinary electron | ×1 | \(10^{-64}\) | does not happen |
| screening in a metal lattice | ×1.0–1.1 | of order \(10^{-60}\) | not enough |
| (needed for 1 W) | ×8 | \(10^{-11}\) | ── |
| a muon | ×207 | \(10^{12}\) | it happens (demonstrated) |
Because muon catalysis exists, bonus ①'s argument can be stated in the far stronger form "not impossible in principle, merely insufficient in quantity." "Thin the barrier and it happens" is correct. The problem is that a lattice does not have anything like the muon's power.
And ironically ── the real one, having completely solved the barrier problem, is stopped for an entirely different reason (sticking and lifetime). Bonus ①'s lesson, "solve the entrance and you get stuck somewhere else," repeats here too.
Established: the muon's mass ratio of 206.77 and lifetime of 2.197 μs; that muon substitution shrinks the molecule by about a factor of 207; that the dtμ fusion rate is of order \(10^{12}\)/s; the predictions of Frank (1947) and Sakharov (1948) and the observation by Alvarez and colleagues (1956); the structure of the catalytic cycle and a net α-sticking probability of about 0.4–0.6%; that cycling rates of order \(10^8\)/s are reached at high density; that experiments observe of order 100–150 fusions per muon (the series of experiments by Jones and others); and the 17.6 MeV released by d+t. All established experimental facts.
Ranges and caveats: (1) The 5 GeV muon production cost is approximate. Depending on accelerator design it ranges from about 2 to 10 GeV, and that directly sets the break-even estimate; moving the slider shows the effect. (2) The sticking probability depends on temperature and density; the "effective" value used here includes reactivation by stripping in flight. (3) The break-even cycle count depends on assumptions about heat-to-electricity conversion and how the neutrons are used (breeding blankets and so on). The 284 in the body is the minimum line for recovering the muon production energy alone; an actual plant needs more margin. (4) The cycling rate and the sticking probability cannot in fact be varied independently (both depend on density and temperature); the figure simplifies them into independent knobs. (5) Concepts other than direct power generation exist (hybrid reactors triggered by muon catalysis to drive fission, for instance), but none has been made practical.
The muon is 207 times heavier than the electron and shrinks the molecule by the same factor. The internuclear distance becomes 500 fm and the fusion rate leaps from \(10^{-64}\) to \(10^{12}\)/s ── 76 orders. Fusion in picoseconds. A real cold fusion, discovered by accident by Alvarez in a bubble chamber in 1956.
What stops it is not the barrier. ① the muon sticks to the helium it just made (about 0.45%, capping at 220 cycles) and ② the muon's 2.2 μs lifetime (also capping at about 220 given the cycling rate) ── two limits at almost the same height, giving 100–150 in practice. Meanwhile a muon costs 5 GeV to make, so break-even is about 284 cycles and a power plant needs of order 700. Not orders of magnitude but a factor of two to five. And it has not closed in seventy years.
As science it is a complete success, and it also provides the baseline for bonus ① ── "thin the barrier and it happens" is correct; a lattice simply does not have the muon's power. And the real one, too, is stuck somewhere else after solving the entrance. That repeated shape is the real lesson of this field.
Print / make a PDF: ⌘+P (Ctrl+P on Windows). On screen, move the three knobs to see which of them, improved how far, would reach break-even. The sticking probability is fixed by nuclear physics and not ours to move. "See the answer" opens each solution.