# Widom-Larsen 2: The meaning of enhanced mass

As described in the last post, Widom-Larsen theory states that the electron-capture process $e^- + p^+ \rightarrow n + \nu_e$ (electron plus proton turns into neutron plus electron neutrino) can and does happen on the palladium hydride surface. (Discussed in Sections 1-3 of the paper.)

Now, if you compare the energy of the two sides in $e^- + p^+ \rightarrow n + \nu_e$, you’ll see that this would work if the electron mass is at least 1.3 MeV/c², rather than the usual 0.51 MeV/c². Well, that’s exactly what Larsen and Widom are arguing! They say that the environment at the surface of a metal hydride has properties which dramatically increases the electron mass.

They actually say that the electron mass is increased not just to 1.3MeV but way beyond that, up to 10.5 MeV/c², twenty times higher than the textbook value. (eq 6 and 27).

I want to say immediately that this claim is crazy and I don’t believe it for a second. But that’s a story for a future blog post. For today, I will assume for the sake of argument that Widom and Larsen calculated the mass increase correctly. I’ll focus instead on understanding the mass increase and its consequences.

A changing electron mass may sound weird and abstract. But don’t worry! I’m going to try to explain it intuitively.

What does enhanced mass mean? It’s not necessarily obvious. For example, in solid-state physics, there are a bunch of different (non-equivalent) definitions of electron “effective mass”: there’s the “density-of-states effective mass”, and the “transport effective mass”, and the “cyclotron effective mass”, etc. The same electron might simultaneously have a large transport effective mass and a small cyclotron effective mass. I bring this up not because Widom-Larsen theory is talking about any of these kinds of traditional solid-state-physics effective mass (I’m quite sure that they’re not), but to emphasize that “mass” manifests in many different ways, and we need to know exactly what aspects of the mass get enhanced.

I think that the candidate definitions of mass in Widom-Larsen theory is:

1. Mass as rest energy [including electromagnetic interaction energy] over c²: In other words, calculate the total system energy with the electron present, versus the total system energy with the electron absent. The difference is the electron’s “rest energy”, which in this context includes not only the normal electron rest energy but also the extra energy related to its electromagnetic interactions. That’s the rest energy, and then the rest mass is related to it by E=mc².
2. Mass as inertia (i.e. the relation between velocity and kinetic energy, or between force and acceleration): In other words, momentum p and kinetic energy KE are related by $KE = p^2/(2m)$ for nonrelativistic motion, or $KE = \sqrt{(mc^2)^2 + (pc)^2} - mc^2$ for relativistic motion. The “mass” in these formulas might be what Widom-Larsen theory is talking about.

So what are Widom and Larsen talking about? My answer is: They are definitely talking about #1 (rest energy), and sorta talking about #2 (inertia).

How do I know that they are talking about #1? It’s just energy conservation. They are saying that you can conserve energy by turning a proton and heavy electron into a neutron. So there has to be a huge amount of energy associated with the presence of the heavy electron. When the heavy electron disappears, this energy becomes available for other uses, and it contributes towards the rest energy of the neutron.

My feeling about #2—and I’m not 100% sure—is that you can correctly plug the heavy mass into formulas like F=ma, KE=p²/(2m), etc., only if you are describing phenomena that happen much slower than the electric field oscillations that cause the mass increase, and you time-average all the quantities over many oscillation cycles. (Sorry, you may find this comment confusing, but it will make more sense after you read the next post.)

For the rest of this post, I will forget about #2 and just explore the consequences of #1.

Point 1: According to Widom-Larsen theory, In the high-electron-mass region, the electromagnetic interaction between an electron and its environment is ridiculously strong, and it is repulsive rather than attractive.

It is ridiculously strong because a single electron causes >0.8MeV of electromagnetic interaction energy. It is repulsive because the interactions raise the system’s energy, whereas an attractive interaction would lower it.

Incidentally, you might remember from high school chemistry that the interaction between a charged particle and a polarizable medium is attractive, not repulsive. Why is the interaction in Widom-Larsen theory repulsive?

Pondering this question, my first guess was that they made a sign error. But I changed my mind. It is repulsive. The repulsive interaction isn’t related to polarizability, but rather to the ponderomotive force. I will discuss this in depth in the next blog-post.

Point 2: If you want to move an electron from vacuum to a high-mass area, you need energy E=(Δm)c² to overcome the repulsive electromagnetic interactions. Correspondingly, an electron experiences a force F = -(∇m)c² that pushes it away from the high mass region.

That’s just freshman physics, nothing fancy.

Point 3: Related to that, “A region of high electron rest mass” is also “a region of high potential energy for electrons”.

(Again, this is freshman physics.)

Point 4: Finally, the main point. If there is a region where the electron mass is >0.8MeV higher than usual, i.e. where the electron has a potential energy of >0.8MeV, then you would never ever find any electrons in that region!!

There is an extraordinarily strong repulsive force pushing electrons away from the high-mass region. An electron just don’t have that much energy at room temperature. It might have 0.1 eV of kinetic energy. This is 0.000001% as much as it needs to overcome the repulsive force and enter the high-mass region. If it was heading towards the high-mass region, it would bounce off of it rather than entering it. The surface plasmons (or whatever else is responsible for creating this crazy electromagnetic environment) would “guard the gates” of the high-mass region, repelling any electron that approached.

Of course, if the region appeared sufficiently quickly, maybe there could be an electron in it that hasn’t had a chance to escape. (The electron didn’t climb the mountain, it stood still while a mountain rose beneath its feet, lifting it to the summit!) This loophole almost seems to save the day for Widom-Larsen theory, until you spend a minute trying to think through the specifics about how quickly the region would have to appear, how big it would have to be, and what all that would entail. I can’t see any way it would happen.

To summarize, here is my big-picture impression of (one aspect of) Widom-Larsen theory:

Back when cold-fusion was first announced, it presented a mystery:

Version 1 of the mystery: How do two D’s get enough energy (~10keV) to overcome the Coulomb barrier to fuse?

Widom and Larsen swapped that out for a different but equally vexing mystery

Version 2 of the mystery: How does an electron get enough energy (~800keV) to enable electron capture, $e+p \rightarrow n + \nu_e$?

Widom and Larsen answer this as follows:

• Step 1: The electron accumulates at least 800keV of potential energy in order to become a heavy electron
• Step 2: The heavy electron uses that potential energy (a.k.a. extra mass) to enable electron capture.

That sounds good, except that it leads to…

Version 3 of the mystery: How does an electron get enough energy (~800keV) to become a heavy electron?

I don’t see anything in the Widom-Larsen papers that answers this question. They just assume that the electrons are already in the region where they would be heavy.

(And don’t say “the electrons get the energy from surface plasmons”. The surface plasmons are part of the electromagnetic environment that I’ve been talking about this whole time. They are what’s pushing the electron away from the high-mass region in the first place. That means, if an electron is already in the high-mass region, and is exiting that region, then it will wind up with a large kinetic energy, and it got that energy from the surface plasmons. But an electron entering the high-mass region would lose kinetic energy to surface plasmons, not gain it.)

So, it seems to me that Widom-Larsen theory has gotten us nowhere! It traded one mystery for another.

Or am I misunderstanding something?