# Widom-Larsen 4: The protons’ magical restoring force

As explained best in the 2008 paper, Widom-Larsen theory envisions a weird sort of surface plasma oscillation. Now, normal plasma oscillations (in solid-state physics) refer to a collective electron motion. But they’re talking about proton motion (the protons or deuterons embedded in the hydrated palladium). There is a vague suggestion that the electrons also move. But they certainly emphasize the proton motion by itself, and describe it in an explicit way here:

Suppose a proton of mass $M_p$ is embedded in a sphere with a mean electronic charge density $\rho_e= (-e)n$. If the proton suffers a small displacement u, then an electric field will be created $eE=-(\frac{4\pi e^2 n}{3}\mathbf{u}) = -M_p \Omega^2\mathbf{u}$

to satisfy Gauss’ law $\nabla \cdot \mathbf{E}= 4\pi \rho_e$. This electric field will try to push back the proton to the center of the sphere. The equation of motion of the proton $M_p\ddot{\mathbf{u}}=e\mathbf{E}=-\Omega^2M_p\mathbf{u}$ yields the oscillation.

This almost seems sensible, but what’s with the sphere of negative charge? The authors need to have that negative charge there: Without it, there would be no restoring force for the protons. But why would the electrons just be sitting there stationary?? If the electric field exerts a force on the protons, wouldn’t it also exert a force on the electrons? And since electrons are 2000X lighter than protons, wouldn’t the electric field create 2000X more electron motion than proton motion?? So why are they calculating proton motion while pretending the electrons are stationary?

One possible answer is: “The electrons are stationary because they can’t move, because of the Pauli exclusion principle.” Well, that could indeed prevent electron motion in an insulator subject to a modest electric field. But here, not only is the material not an insulator, but the electric field is supposed to be insanely high, easily high enough to allow electrons to move (even in a really good insulator) by jumping into higher-energy states. (The Widom-Larsen electric field is supposed to be ten times larger than the field in this paper, for example.) Anyway, if the electrons could not easily respond to the electric field, then they would not become “heavy” in the first place (see previous post), which means there would be no electron capture anyway.

Another possible response is: “The electrons are just sitting there because they’re so heavy.” However, they are only heavy insofar as they can move in response to the electric field (see previous post), and then they only act heavy in situations where you can average over many cycles of their oscillatory motion. They would not act heavy in how they respond to the oscillatory force itself.

So, as far as I can tell: The description in that excerpt above seems to be a model that cannot apply to the system they’re talking about. It’s not even the right starting point for a more fleshed-out model.

Does this matter? Yes, very much! In the paper, they infer oscillating electric field strength from an experimental value of proton displacement. That inference assumes that the electrons stay put. The dubiousness of this assumption may help explain how they wound up guessing that there is an insanely-high AC electric field.