In Widom-Larsen theory, they say that the surface of a metal hydride has very energetic (“heavy”) electrons that can undergo electron capture (electron plus proton turns into neutron plus electron neutrino).
A central claim in the theory is that the electron capture process creates extremely slow neutrons. And I mean extremely slow! The neutron kinetic energy is supposed to be as low as electron-volts, or even less! (Room-temperature thermal neutrons have millions of times higher kinetic energy. Even “ultracold neutrons” are fast by comparison.)
This claim is important because they are trying to explain cold fusion experiments in which neutron detectors register very few if any neutrons. So any theory involving neutron production would seem to be ruled out. However, Widom and Larsen say that a super-slow-moving neutron would be very likely captured by a nucleus within a very short distance, so it would have negligibly probability of reaching a neutron detector a few centimeters away. Thus, the theory is not ruled out by the experiments after all!
(Incidentally, I am deeply skeptical that slow neutrons would actually solve this problem. Even if the neutron is ultra-slow, the relative velocity between the neutron and a nucleus is not ultra-slow, because the other nucleus is shaking around randomly at room temperature. And if the ultra-slow neutron is elastically scattered from a moving nucleus, it would stop being ultra-slow! But that’s a story for another day. For this post I want to investigate whether there are slow neutrons in the first place.)
At first you may wonder: Isn’t the final neutron momentum determined by energy-momentum conservation?? If the electrons have up to 10.5MeV of energy, as stated in the papers, then the final neutron momentum would be typically huge, not super-low. This seems like an obvious question, and I have no idea how Widom and Larsen would answer it. Maybe there is a distribution of electron energies, and only the electrons with just the right energy do this reaction? But that would be a negligible fraction. The energy window is way too narrow. You could argue that there’s something like the Mössbauer effect (this would be recoil-less neutrino emission, whereas the Mössbauer effect is recoil-less photon emission). But the neutron isn’t bound into a lattice, like nuclei are, so I don’t think it works.
I am strongly inclined to believe that this already proves that Widom and Larsen cannot be right about the neutron momentum. But anyway, I’ll leave that point aside and press on. Let’s assume for the sake of argument that energy and momentum conservation is not an issue. It happens magically! I want to look at the papers and figure out why Widom and Larsen think that the neutrons are very likely to be created with an extremely low momentum. Then I can decide whether the argument is any good.
Here is the big picture of their argument:
- Step 1: Surface plasma wave has long wavelength
- Step 2: ???
- Step 3: Neutron has long wavelength
(Remember that long wavelength corresponds to low momentum, by the de Broglie wave equation.)
The big problem is step 2, which (I will argue) does not and cannot work. I think I know what they were trying to get at, so as background I’ll present…
An example where this kind of argument actually works: A photon is absorbed by an electron in a semiconductor. The important part is that when this process really happens, the electronic excitation wave lines up perfectly with the photon wave! They have the same wavelength and direction, the same peaks and troughs! (In an introductory semiconductor physics course, they would describe this fact in a way that sounds different, but is actually the same: .)
Why do the waves match up? The way to think of it is: The electron-photon interaction can happen at any location. When the waves match up, the quantum transition processes all act coherently (“in phase”). The process “the photon interacts with the electron over here” adds coherently (in phase) with the process “the photon interacts with the electron over there”. Therefore the total transition probability is dramatically higher when the waves match up.
Widom-Larsen is trying to make a similar argument. When the plasmon wave matches up with the neutron wave, then the quantum transition processes all act coherently (according to this argument). The process “Electron capture occurs over here” would be coherent with the process “Electron capture occurs over there”. So the transition probability would be dramatically higher when the waves match up.
But it doesn’t work. Those transition processes cannot add coherently because they have different final states.
If proton A is converted to a neutron, then there is no more proton where proton A used to be. If proton B is converted to a neutron, then there is no more proton where proton B used to be. You cannot have quantum interference between two processes unless the final states are identical.
Widom and Larsen are arguing that the final neutron state can be identical in both of these processes, if the neutron is delocalized. OK, sure, that’s possible. But that doesn’t help, because the proton configuration is different!
The wavelength of the isotopic spin wave has no relation to the wavelength of the final neutron.
The isotopic spin wave is:
Nothing special happens when . Those two parameters are unrelated. It is only k, not k‘, which is related to the surface plasmon wavevector.
Summary: Even if everything else in Widom-Larsen is true, i.e. there is really electron capture producing neutrons, I think that Widom and Larsen are wrong to say that those neutrons would have an unusually low momentum.
(Or am I misunderstanding something?)