The energy of electrons at the top of the Fermi sea is typically 7 eV, which corresponds to the energy of a particle at the temperature 81,000 K. The energy which binds a Cooper pair is 1 meV, which corresponds to 11.6 K.
Conduction of electricity
Suppose that we have a block of metal at 0 K. The electrons in the Fermi sea have the lowest possible energy, and they cannot lose any of their kinetic energy to the nuclei. The electrons are, in a sense, in a highly ordered state. It is some kind of a "crystal".
Then we heat up the metal to the room temperature. The electrons bump into each other and into nuclei and radiate photons. The state of the electrons is no longer highly ordered. Outgoing radiation is a symptom of high entropy.
Then we apply an electric voltage over the block. Electrons get accelerated by the voltage. Electrons bump into nuclei and radiate away some of their kinetic energy. That is, resistance is producing heat from an electric current.
Superconductivity occurs at a low temperature. Presumably, the ordered state of electrons can somehow make electrons avoid bumping into nuclei when we apply a voltage over the block.
In BCS theory it is Cooper pairs which bring order to electrons.
Time crystals
A time crystal is a system where the lowest energy state contains a "repetitive motion" of particles. Frank Wilczek proposed time crystals in 2012.
Every atom or a molecule is, in a sense, a time crystal. There is a repetitive motion of electrons. A permanent magnet is a time crystal which produces a macroscopic magnetic field.
If a superconductor can make electrons to move without radiating anything, then, if we have a current flowing through the superconductor, it is a time crystal where new crystallization happens on that side where electrons enter, and melting happens on the other side.
We may define an ordinary crystal as an ordered structure which is static in some frame (static except of zero-point oscillations of particles). If electrons are flowing in a superconductor, then it is not an ordinary crystal. Nuclei are static but electrons flow.
What do we know empirically about superconductors?
We know at least the following things:
1. A superconductor stops being a superconductor above a transition temperature T which may vary between 4 K and 120 K.
2. A superconductor can move a surplus of electrons without radiating heat, or it radiates very little heat.
3. The specific heat of the material is high around the transition temperature. As if we were melting a "crystal" at that temperature.
4. The transition temperature is lower with heavier isotopes of the nuclei. To form, the "crystal" apparently requires nuclei which can oscillate wide enough.
5. The energy of a particle at the temperature 4 K ... 120 K is 0.3 meV ... 13 meV.
The facts above do not necessarily mean that electrons are ordered into pairs, or that they would form a Bose-Einstein condensate. Is there evidence that such a condensate really is formed?
The "crystal" of electrons at 0 K
Since a block of metal, or any material, does not radiate at 0 K, it has very low entropy, and its electrons and nuclei form some kind of a "crystal". We do not know much about what that crystal is like. Apparently, the Fermi sea is filled up to some surface. But what are the wave functions of the electrons or nuclei like?
If the block is a superconductor, it can move a surplus of electrons from one side to the other with zero or very small radiation. The system acts as a time crystal up to some transition temperature T, or a current density.
In the update today to our previous block entry (March 12, 2022) we remarked that in a highly ordered system, scattering from collisions of particles may have very low power because there may be almost total destructive interference in the scattered wave.
Classically, a crystal of electrons might slide past the lattice of the nuclei without touching. There would be no energy loss.
Another way to avoid scattering is to use a material which cannot be excited at the available energies. Could it be that the system below the transition temperature somehow becomes unable to be excited? We can warm up the block of metal to, say, 1 mK. The system can absorb very low-energy photons. Thus, there are excited states which could be produced by bumping electrons.
But could it be that a bumping electron cannot find a free state for itself?
Let us assume that the cloud of electrons stays static, and the nuclei are slowly dragged through the cloud. In this frame, it is a nucleus which bumps into an electron. If the electron cloud is a kind of a "crystal", then it probably can start to vibrate, and be excited. The lattice of the nuclei can do the same.
Conclusions
The (almost) zero resistance of superconductors may be due to:
1. A band gap: there is no free state for the system, such that kinetic energy of the flowing electrons could be converted to vibration. Is that really possible? Crystals tend to have very many excited states, and the states can have very small energies.
2. Electrons and nuclei form a time crystal. There is no scattering, that is, electrons and nuclei do not bump at all. The time crystal probably is bound by vibrations of the lattice of nuclei. Heavier isotopes damp vibrations, and the time crystal melts at a lower temperature if the nuclei are of a heavier isotope.
We need to investigate what evidence do we have of a Bose-Einstein condensate in a superconductor.
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