UPDATE April 19, 2022: If we have a closed electric circuit where a part of the circuit is superconducting, and there is a potential wall which prevents changes in the current in the superconductor, then we can build a perpetuum mobile. Just put a little resistor into the loop, and it will keep producing energy for ever.
Superconductivity cannot be based on such a potential wall.
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UPDATE April 13, 2022: Leggett's argument below is based on a superconducting current in a closed superconducting loop. Superconductivity happens also in a linear wire which is just a part of of an ordinary, non-superconducting, current loop. Why would an argument for a closed loop apply to the linear case?
Also, suppose that we have a superconducting wire and add a few electrons to one end. If there would be a large potential wall which blocks changes in the current in the wire, then the electrons would not be able to move. The superconductor would appear as an insulator for small amounts of charge.
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Why the time crystal of the electron cloud in a single atom cannot vibrate?
In an earlier blog post we remarked about the analogy between a many electron atom and the Fermi sea in a metal.
In an atom, only the electrons at the top energy level scatter from a photon. The result is an excited atom or an ion plus an electron.
The electron cloud of an atom is a time crystal, too. Why a photon cannot make the time crystal of the electron cloud to vibrate? The orbitals of electrons would be in a collective oscillating motion.
The reason has to be that the resonant frequencies of the cloud are very high. The cloud rather breaks apart than starts to vibrate.
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Let us have two particles bound by a potential. The potential pit may be so shallow that the first excited energy level breaks the system apart.
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However, if we make a chain of many particles, the chain will have low-frequency resonant frequencies. The vibration for a single pair is so gentle that it does not break their binding.
A crystal made of many atoms does have excited states where the orbitals of the electrons vibrate.
The surface of the Fermi sea and scattering from phonons: BCS theory
An electron at the top of the Fermi sea can scatter from a phonon and move to a higher energy state. The analogue is a photon which hits an atom and moves a single electron to a higher energy state.
In BCS theory electrons can form bound Cooper pairs through an (obscure) interaction which is mediated by lattice vibrations.
How does BCS theory prevent electrons from scattering from phonons?
On pages 1177 - 1178 the authors write:
"Our theory also accounts in a qualitative way for those aspects of superconductivity associated with infinite conductivity and a persistent current flowing in a ring. When there is a net current, the paired states (k₁↑, k₂ ↓) have a net momentum k₁ + k₂ = q, where q is the same for all virtual pairs. For each value of q, there is a metastable state with a minimum in free energy and a unique current density."
"Nearly all fluctuations will increase the free energy; only those which involve a majority of the electrons so as to change the common q can decrease the free energy. These latter are presumably extremely rare, so that the metastable current carrying state can persist indefinitely."
The claims above remind us of the Landau model for superfluidity which we covered in our March 26, 2022 blog post.
Bardeen, Cooper, and Schrieffer describe a model where conducting electrons form an ordered structure. But they do not give any reason why this structure would be a superconductor. The sentence "These latter are presumably extremely rare" is their only comment on the subject.
Question. What kind of moving structures can survive a hit from a random phonon without losing kinetic energy of the structure?
Superconductivity lectures by Anthony J. Leggett: the proof of superconductivity breaks conservation laws
At the Physics Stackexchange, two users recommend the following lectures:
Anthony J. Leggett (2015) has written a course on superconductivity. Lecture 14 concerns the stability of the superconducting current. Let us look at it.
Leggett considers a superconducting current in a circular loop of a wire. He assumes that quantum mechanics dictates that the angular momentum of the rotating electron-phonon system must be quantized as an integer multiple of ħ.
"We are particularly interested in the probability of a fluctuation that takes us to the lowest saddle-point in the free energy barrier that separates states of different winding number."
On page 9 Leggett calculates that there is a huge potential barrier which prevents the rotating system from decaying to a lower angular momentum state.
He concludes that small quanta like phonons cannot move the system to a lower angular momentum state. The superconducting current keeps flowing almost indefinitely.
But Leggett's model, or calculation, cannot be correct if the rotating system can scatter quanta to the direction of the rotation. We can extract significant energy and angular momentum from the rotating system by letting it scatter a large number of quanta. If the system would keep rotating at the original speed, we would have a breach of conservation of energy and angular momentum.
The rotating system consists of moving electrons and lattice vibrations. It will certainly interact with, say, moving charges inside the superconductor. It would be very surprising if it cannot scatter any quanta.
Leggett's model or analysis is probably incorrect.
To prove that a superconducting current can flow for ever, we probably need to prove the claim in classical or semiclassical physics. Quantum mechanics does not help us because the rotating system is a macroscopic object, and the phonon gas, as a whole, is macroscopic, too.
UPDATE April 12, 2022: Our new blog post today shows that one really cannot say that the angular momentum of a macroscopic system is quantized in units of ħ.
Conclusions
The original paper by Bardeen, Cooper, and Schrieffer describes a model which might explain the phase change of the conducting electron system close to the critical temperature. That is, the model may explain the specific heat and some other properties of the low temperature phase.
Their paper does not try to prove that the model implies superconductivity. They simply assume that the low energy phase is superconducting.
At the Physics Stackexchange, various users give different explanations for superconductivity. None of the explanations is detailed enough to be convincing.
Anthony J. Leggett's lectures aim to prove superconductivity of the BCS model, but the proof breaks conservation of energy and angular momentum if the rotating system is able to scatter something. The model or the proof is probably incorrect.
We conclude that the proof of superconductivity is an open problem of theoretical physics.
For our own time crystal model we argued in an earlier blog post that it might let phonons go through without scattering. We need to elaborate our model and our analysis of it.
We need to check if the BCS model can superconduct.
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