UPDATE April 16, 2022: The model of electron scattering from a phonon is erroneous in this blog post. The phonon makes a "diffraction grating" to the lattice and the electron scatters from the grating.
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1. phonons from thermal vibrations;
2. impurities in crystals;
3. borders of crystals.
Let us investigate how a time crystal model of superconductivity can overcome the obstacles 1, 2, and 3.
Phonons in a metal
Wien's displacement law states that the black body radiation at 1 K has its peak at the wavelength 3 mm, which corresponds to the frequency 10¹¹ Hz. The energy of a typical photon is h f = 6.6 * 10⁻²³ J, or 400 microelectronvolts.
Let us place a block of the metal in a cavity which contains the 1 K black body radiation. The metal interacts with the radiation primarily through oscillations of its electron content, just like in a radio antenna.
The oscillation of electrons presumably creates oscillation in the lattice of atoms. These oscillations are phonons.
The speed of sound in a typical metal is 5 km/s. Thus, the typical thermal phonon has a wavelength of 50 nm at 1 kelvin.
The typical spacing of metal atoms in a crystal lattice is 0.1 nm.
The lattice of the metal is distorted by a phonon, which in turn creates an electric field. A conducting electron can interact with this electric field and scatter.
In umklapp scattering, the phonon exchanges momentum also with the lattice.
The time crystal of electrons must let phonons pass through in superconductivity
In electric resistance, a phonon scatters from an individual electron. We can remove that resistance if the phonon can pass through the time crystal of electrons without scattering.
Actually, it is enough that the energy of the phonon does not change if it is scattered. If the electron drift velocity would slow down, the electrons would lose energy to phonons.
1. The structure of the time crystal must be so strong that a 400 microelectronvolt phonon cannot kick an electron out of the structure.
2. The time crystal must be ordered or uniform enough at the scale of 50 / 2 = 25 nanometers, so that destructive interference removes almost all of the scattered wave.
3. Phonons must not be reflected by the moving parts in the time crystal. If the spatial form of the time crystal is static, then this probably is the case.
4. The time crystal should not have a resonant oscillation frequency at the phonon frequency, so that the phonon cannot lose energy to the time crystal. How can we accomplish that?
Item 4 is the hardest part. Let us assume that the potential which binds together individual parts of the time crystal is not steep. However, the energy required to detach a part from the crystal is large. In BCS theory it is assumed that some collective effect keeps the "condensate" together and makes the dissociation energy large for any individual part.
For an ordinary crystal, the energy required to detach an inner atom is huge. We would need to split the crystal into pieces.
Since the potential is not steep, vibration frequencies of the time crystal may be very low, much lower than the 10¹¹ Hz of a phonon at 1 K.
With these assumptions we obtain: the phonon is not able to break the time crystal, nor is the phonon able to be absorbed by the time crystal. The phonon passes through the time crystal without a change in energy. The time crystal can keep transporting electrons at the original velocity.
An analogous setup is a block of glass moving in a bath of black body radiation. If the block is very small, photons will scatter from it and the block loses its momentum. But if the block is large, photons will pass through the block without losing energy.
The time crystal in a superconductor must be large compared to the phonon wavelength 50 nm.
The time crystal and impurities
Impurity atoms scatter electrons and make them to lose their drift velocity. How can a time crystal avoid this?
crystal
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surface with a spike
Let us have an ordinary crystal sliding over a solid surface with a one atom spike.
Since the crystal and the surface are macroscopic objects, we can measure very precisely the position of the crystal and the spike. Therefore, we can use classical physics, and in classical physics it is self-evident that the spike will create vibrations in the crystal.
We conclude that the time crystal must avoid impurity atoms. Otherwise, there will be power dissipation and resistance.
Can the time crystal avoid them? Crystallization means that a system falls to a lower energy state which is reached through creating order in the system. If there would be an impurity atom within the time crystal, and it would make electrons to scatter, it would presumably destroy order. That would require energy which is missing from the system.
Can impurities prevent the formation of a time crystal altogether? We have to check what empirical research says about this.
Quantum mechanics prevents the electron in the hydrogen atom from falling to a very low orbit. A short-wavelength orbit would possess so much kinetic energy p² / (2 m) that the energy cannot be recovered from the lower potential energy. There is a quantum force which stops the electron from falling.
Similarly, a quantum force may keep the time crystal away from impurities which cause scattering and break the time crystal.
Borders of crystals of the metal
This is a major obstacle to superconductivity. The time crystal can avoid impurity atoms, but it cannot avoid borders of the crystals in the metal.
The border means some kind of a potential hill, and it will scatter electrons back unless the electrons are bound in a time crystal.
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crystal obstacle
Let us study an ordinary crystal sliding over a line-like obstacle. The atoms on different rows in the diagram will generate sound waves or electromagnetic waves whose phase is
n α,
where n is the number or the row and 0 < α < 2 π.
We assume that
α > s / v * 2 π f,
where s is the spacing of adjacent atoms of the crystal meeting the obstacle, v is the speed of sound or of electromagnetic waves, and f is the frequency of the wave.
Then the phase of the waves, when observed from a faraway location, is
n β,
where 0 < β < 2 π. We can argue like in our April 1, 2022 blog post that destructive interference wipes out almost all of the waves.
If the analysis carries over to the time crystal, then borders of metal crystals do not prevent superconductivity, unless the border is normal to the flow of the time crystal. We may argue like in the previous section that the time crystal will avoid such normal borders.
Conclusions
We now have a time crystal model of superconductivity. The model addresses the problems of phonons, impurities, and crystal borders.
We will next compare our model to BCS theory.
The structure of the hypothetical time crystal is unclear. The crystal must include both lattice vibrations and conducting electrons. Electrons move much faster than lattice vibrations. How can the time crystal accommodate such different speeds of its components?
Cooper pairs in BCS theory might correspond to bindings between electrons in the time crystal model. The "condensate" of BCS theory corresponds to the time crystal itself. The concept of a Cooper pair as well as the condensate are fuzzy concepts in BCS theory.
We cannot calculate the orbitals of a many electron atom. It looks like we have the same problem in solid state physics. The details of the wave functions of electrons in the Fermi sea are unknown. To understand superconductivity well, we should know the wave functions.
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