A hydrogen molecule consists of two protons held together by a covalent bond. One way to think of such bonds is to imagine that there is only one electron is directly involved as shown below. The bonding electron only spends 1/7 of its time between the protons, making the bond, the other 6/7 of the time the electron shields the two protons by 3/7 e^– each, reducing the effective charge of each proton to 4/7e⁺.

We see that the two shielded protons will repel each other with the force of F_R = Ke (16/49 e² /r²) where e is the charge of an electron or proton, r is the distance between the protons (r = 0.74Å = 0.74×10^-10m), and Ke is Coulomb’s electrical constant, Ke ≈ 8.988×10⁹ N⋅m²⋅C⁻². The attractive force is calculated similarly, as each proton attracts the central electron by F_A = – Ke (4/49) e²/ (r/2)². The forces are seen to be in balance, the net force is zero.

It is because of quantum mechanics, that the bond is the length that it is. If the atoms were to move closer than r = 0.74Å, the central electron would be confined to less space and would get more energy, causing it to spend less time between the two protons. With less of an electron between them, F_R would be greater than F_A and the protons would repel. If the atoms moved further apart than 0.74Å, a greater fraction of the electron would move to the center, F_A would increase, and the atoms would attract. This is a fairly pleasant way to understand why the hydrogen side of all hydrogen covalent bonds are the same length. It’s also a nice introduction to muon-catalyzed cold fusion.

Most fusion takes place only at high temperatures, at 100 million °C in a TOKAMAK Fusion reactor, or at about 15 million °C in the high pressure interior of the sun. Muon catalyzed fusion creates the equivalent of a much higher pressure, so that fusion occurs at room temperature. The trick to muon catalyzed fusion is to replace one of the electrons with a muon, an unstable, heavy electron particle discovered in 1936. The muon, designated µ-, behaves just like an electron but it has about 207 times the mass. As a result when it replaces an electron in hydrogen, it forms form a covalent bond that is about 1/207^th the length of a normal bond. This is the equivalent of extreme pressure. At this closer distance, hydrogen nuclei fuse even at room temperature.

In normal hydrogen, the nuclei are just protons. When they fuse, one of them becomes a neutron. You get a deuteron (a proton-neutron pair), plus an anti electron and 1.44 MeV of energy after the anti-electron has annihilated (for more on antimatter see here). The muon is released most of the time, and can catalyze many more fusion reactions. See figure at right.

While 1.44MeV per reaction is a lot by ordinary standards — roughly one million times more energy than is released per atom when hydrogen is burnt — it’s very little compared to the energy it takes to make a muon. Making a muon takes a minimum of 1000 MeV, and more typically 4000 MeV using current technology. You need to get a lot more energy per muon if this process is to be useful.

You get quite a lot more energy when a muon catalyzes deuterium fusion or deuterium- fusion. With these reactions, you get 3.3 to 4 MeV worth of energy per fusion, and the muon will be ejected with enough force to support about eight D-D fusions before it decays or sticks to a helium atom. That’s better than before, but still not enough to justify the cost of making the muon.

The next reactions to consider are D-T fusion and Li-D fusion. Tritium is an even heavier isotope of hydrogen. It undergoes muon catalyzed fusion with deuterium via the reaction, D+T –> ⁴He +n +17.6 MeV. Because of the higher energy of the reaction, the muons are even less likely to stick to a helium atom, and you get about 100 fusions per muon. 100 x 17.6 MeV = 1.76 GeV, barely break-even for the high energy cost to make the muon, but there is no reason to stop there. You can use the high energy fusion neutrons to catalyze LiD fusion. For example, 2LiD +n –> 3⁴He + T + D +n producing 19.9 MeV and a tritium atom.

With this additional 19.9 MeV per DT fusion, the system can start to produce usable energy for sale. It is also important that tritium is made in the process. You need tritium for the fusion reactions, and there are not many other supplies. The spare neutron is interesting too. It can be used to make additional tritium or for other purposes. It’s a direction I’d like to explore further. I worked on making tritium for my PhD, and in my opinion, this sort of hybrid operation is the most attractive route to clean nuclear fusion power.

Robert Buxbaum, September 8, 2022. For my appraisal of hot fusion, see here.