As described in the last post, I want to understand this paper, “Including nuclear degrees of freedom in a lattice Hamiltonian” by Peter Hagelstein and Irfan Chaudhary. It’s an important ingredient in the lossy spin-boson model of cold fusion. I will summarize what I said in the last post and expand on it in light of ensuing discussion and clarification from PH.

(Thanks to PH for the help; all mistakes are my own.)

Last time I introduced some terminology for different states for a nucleus in vacuum.

Notice the length contraction on the right. Time dilation would also have a distorting effect (not shown).

As shown in this image, we have both “convoluted states” (I made up that terminology) and energy eigenstates.

• For a nucleus at rest, the convoluted states are the same as energy eigenstates.
• An energy eigenstate for a nucleus in motion is a Lorentz transform of an energy eigenstate for a nucleus at rest. That means, the nucleus is both put into motion and distorted by (for example) length contraction.
• To get a convoluted state, you start with a nucleus at rest in an energy eigenstate, and you put it into motion without distorting it.

For low velocity (not zero, but much less than the speed of light), the convoluted states are similar but not strictly identical to energy eigenstates. That means if there is a nucleus in motion in the ground convoluted state, and you measure its energy, there is some nonzero probability that it will be in an excited energy eigenstate. Conversely, if there is an excited convoluted state in motion, and you measure its energy, you may find it in the ground state.

When I look at this paper now, I think of it as an attempt to quantify this relationship, i.e. an attempt to write energy eigenstates as a sum of convoluted states or vice-versa. Then the paper makes a lot more sense to me.

So far this is just math. It might or might not be relevant in any real physical situations. So, which states will actually occur under different circumstances? Let’s take two extreme limits:

• If the nucleus starts at rest in the ground state, and you very very gradually accelerate it (by pushing with an electric field), it winds up in the ground energy eigenstate, by the adiabatic theorem.
• If we smash the nucleus with an impulsive force, that instantaneously accelerates it to some new momentum, then I think it would wind up in a convoluted state. (I am imagining that this impulse imparts the same extra velocity simultaneously to every proton and neutron in the nucleus.)

In the real world, we are always somewhere between these extremes. The forces are neither infinitely slow nor infinitely fast. Therefore you would presumably get some quantum superposition of an energy eigenstate and a convoluted state.

We usually think of phonon oscillations as being pretty slow and gentle, close to the adiabatic situation, in which case we would expect the adiabatic result to occur, with a small non-adiabatic correction. Really, we would expect a negligible non-adiabatic correction. But Hagelstein would say that weird situations can occur, and we should be open-minded to the possibility that the non-adiabatic correction may not always be negligible.

(Terminology note: I think that Hagelstein uses the term “adiabatic polarization” to describe the distortions of the nucleus as it moves back and forth in a phonon oscillation, under the conditions where the adiabatic theorem actually applies, and where cold fusion does not occur.)

Is there an easy way to compute the non-adiabatic correction?

I was hoping that there would be some simple calculation that relates the force on a single nucleus to the resulting nuclear transition rate, without getting into all the complexities of the lossy spin-boson model. Therefore, I started to read some papers about “zener tunneling theory” in semiconductors, which seems mathematically analogous to this. My idea is: A semiconductor electron wavefunction has a unit-cell-periodic part and a plane-wave part, which I think is analogous to a nucleus having a core-state part and a center-of-mass-motion part. Zener tunneling theory describes how non-adiabatic forces on the electron can induce band-to-band transitions, which I think is analogous how non-adiabatic forces on the nucleus can induce nuclear transitions.

Well, I don’t know whether this foray into zener tunneling theory will lead me anywhere. I’ll work on it more if I get a chance. However, when I asked PH about this, he seemed sorta pessimistic. He suggested that the only way to calculate transition rates in this context is to use his lossy spin-boson model.

Big picture

Remember, the big picture is that this paper is describing an interaction between nuclear transitions on the one hand, and the external forces on a nucleus and its resulting motion on the other hand. This interaction is supposed to allow energy to be converted from nuclear energy (i.e. the energy released by Deuterium + Deuterium → Helium-4) into phonon energy. The effects of the interaction are supposed to be amplified a gazillion times by the various effects (like superradiance) that make up the “lossy spin-boson model“. Altogether, this is supposed to explain all the mysteries of cold fusion.

I am now maybe 80% confident that the authors are describing a real interaction in this particular paper, and 60% confident that they have calculated it correctly. I’m deeply skeptical that the rest of the lossy spin-boson model works as advertised, but I am reserving judgment until I understand it better. More blogging ahead!