In our blog post on April 22, 2022 we tried to explain a superconducting current in a linear wire which forms a loop with an ordinary wire.
An energy gap model explains permanent currents in an isolated time crystal
Let us have a block of a metal at 0 K, and no external current flowing through the block. The block is at the lowest energy state and cannot radiate. The motion of electrons in the block forms a time crystal.
In this case we can appeal to an energy gap to explain the loss of resistivity. If an internal current inside the time crystal would slow down, that would require more energy.
This effect is true for all isolated systems at 0 K. There cannot be any resistance in the movement of electrons.
In a permanent magnet there is a considerable ordered current inside. We argued in our previous blog post that the magnetic field of the magnet must be static. If the magnetic field would change periodically, there would be electromagnetic radiation.
For a superconductor we have an additional requirement that we can add an arbitrary current which flows through the superconductor, and there is no resistance.
All energy gap or potential wall models of superconductivity are broken: a perpetuum mobile would exist?
In our previous blog post we assumed that a single electron joins a flow of many electrons which carries the new electron to the other end of the wire.
We ran into problems. The model would make a perpetuum mobile possible.
The perpetuum mobile problem may exist in all models which try to explain superconductivity through an energy gap or a potential wall. That is, there would be a potential wall which prevents the system from decaying from a nonzero current state to a zero current state.
Empty space has zero resistance
We can implement zero resistance by sending electrons through empty space. Then there is no potential wall between different magnitudes of the current - and no perpetuum mobile is possible.
To prevent the electrons from dispersing, we can load them on a "ship" which takes many electrons at a time through empty space.
A charged ship in empty space is a nice model for superconductivity. It certainly satisfies energy and momentum conservation.
The mystery in superconductivity is how this ship can sail through the lattice, impurities, crystal borders, and thermal phonons in a superconductor - and still feel (almost) no friction.
The smoothness of the electron cloud of a time crystal may explain superconductivity
The standard free path model of electrical resistivity suggests that it is the collisions of individual electrons to phonons, impurity atoms, and grain (crystal) boundaries which are responsible for resistance.
If we can eliminate these collisions, then the resistance can be extremely small.
For a molecule in the lowest energy state, a static observer cannot see any individual electron motion because if he did, he then the molecule would radiate.
What about a slowly moving observer? The molecule may be electrically polarized in a complicated pattern. A slowly moving observer can see the "bumps" in the electric field of the molecule. He probably cannot see any individual electrons.
Seeing an individual electron in a molecule requires considerable energy, because of uncertainty principles. Maybe they are invisible for a slowly moving observer?
Thus, the molecule appears like a smooth, elastic object to a slowly moving observer.
The problem of grain boundaries: a "flux" model cannot explain zero resistance
Conducting electrons in a superconductor have to cross grain boundaries to enable the electron drift, and the existence of an electric current.
Let us analyze this in a model where the time crystal of conducting electrons and lattice vibrations initially stays at the same place. Then we apply a voltage for a short time to make the time crystal to move.
The grain size in a typical metal is of the order 10 micrometers.
Fermi velocity
^ #
/ #
/ #
/ #
/ #
e- -->
drift grain boundary
The Fermi velocity is ~ 10⁶ m/s and the drift velocity is only ~ 10⁻⁶ m/s.
The de Broglie wavelength of a Fermi electron is
λ = h / p
~ 1 nm.
The "drift flux" of conducting electrons is ~ 10⁻¹² on top of the "Fermi flux" of electrons.
If the size of a grain is 10 micrometers, that corresponds to 100,000 atoms. Bloch's theorem suggests that the reflected flux of electrons at a boundary might be even much less than 1 / 100,000. A conducting electron passes
~ 10⁶ m / 10 μm = 10¹¹
grain boundaries per second. It might be reflected ~ 1 million times per second.
We calculated on April 25, 2022 that 10¹¹ reflections would drain the momentum in the magnetic field of the current.
That many reflections for each conducting electron might happen in 100,000 seconds, or in a day.
The "flux" model cannot explain a current which flows for 30,000 years without adding more energy.
The speed of sound in the time crystal
The time crystal forms at a temperature ~ 10 K, which suggests that the binding energy is around 1 millielectronvolt.
The binding energy of an atom of a metal in a crystal is 0.1 - 1 eV.
The rest mass of conducting electrons is ~ 1 / 10,000 of the atoms.
The speed of sound may be proportional to
sqrt(E / m),
where E is the binding energy and m is the mass.
We conclude that the speed of sound in the time crystal is ~ 30 km/s.
However, we know that the speed of a potential change in a wire is close to the speed of light. How can we reconcile this with the speed of sound?
The speed of sound assumes that the zones of extra negative charge move in unison with the zones of extra positive charge in the lattice. This is very different from just moving the electrons.
How does a system with the lowest resonant frequency f react to a smooth disturbance which lasts for a long time t?
The system may, for example, be point masses joined together with springs, or an elastic body.
Let us assume that the system moves slowly with a velocity v and meets a smooth potential pit or a smooth potential hill.
part of the system
v
● -----> time t
______ ______
\_____/
smooth potential well
If the part moves very slowly, the system has plenty of time to adjust to the extra force which is pulling or pushing the part.
Let t be the time under which the part passes the potential pit. If
t >> 1 / f,
then the potential pit cannot produce significant vibration to the system. The force field of the potential pit is in that case conservative, and kinetic energy is not lost to vibration.
An elastic time crystal meets a grain boundary
elastic time crystal of electrons
|/\/\/\/\/\/\/\/\/\/\/\| --> drift
● ● ● ●
grain boundaries
Let us consider the drift as a movement of the time crystal of electrons. We ignore the movement of individual electrons because we assume that the time crystal is at its lowest energy state in its comoving frame.
The elastic time crystal will be deformed as it slides past grain boundaries.
Deformations create vibrations of the time crystal. How much energy do these vibrations drain from the movement?
The de Broglie wavelength of Fermi electrons is ~ 10 nm. We may assume that the smallest "feature size" in the time crystal is of that order.
How long does it take for a 10 nm feature to slide over a grain boundary? The width of the boundary is one atom, or 0.1 nm.
The velocity of the time crystal is 1,000 nm/s. We conclude that the time of the transit is
t ~ 0.01 s.
Let the time crystal be of a size 1 meter. We calculated above that the speed of sound in the crystal is ~ 50 km/s, or might even be close to the speed of light. Then the lowest resonant frequency of the time crystal is
f ~ 50 kHz,
or higher.
We have
t >> 1 / f.
The time crystal cannot lose much of its kinetic energy to vibration because it moves so slowly.
Can we calculate an upper bound for the energy loss?
Almost all of the momentum of the time crystal is stored in its collective magnetic field. The magnetic field wants to keep the current constant. If some part of the time crystal slows down, other parts must make up for the lost current.
Let the part lose the momentum p in the collision. The magnetic field must put the momentum p back to the system. That should be easy.
We guess that the momentum p lost in the transit over the potential pit is much less than the momentum of the electrons in that part.
A part collides with a grain boundary every 10 seconds. The momentum p which it loses is much less than
10⁻¹¹
of the corresponding momentum in the magnetic field. The current will last much longer than
10¹² seconds
= 30,000 years
if we just look at the dissipation at grain boundaries.
There is a grain boundary for each 10 micrometers. But there might be an impurity atom for each 10 nanometers, or 100 atoms.
Our calculation may explain the insignicant dissipation at impurity atoms, too.
Phonons and the elastic time crystal
Let us then look at the interaction of phonons and the elastic time crystal.
On April 25, 2022 we calculated that the black body radiation power at 4 K is
1.4 * 10⁻⁵ J/m².
A typical "large energy" quantum at that temperature is
1 meV = 1.6 * 10⁻²² J.
Since the speed of sound is ~ 5 km/s, we have one quantum in each cube whose side is 40 micrometers.
The quantum is a phonon in the lattice.
We need to do a sanity check. The number of phonons grows as ~ T³ with the temperature. At 293 K we would have one phonon in a cube whose side is
40 micrometers / 73
= 550 nm.
In the link it is calculated that the free path in copper at 293 K is 39 nm.
Why is the free path so short? Does a conducting electron scatter many times from a single phonon? Or does it scatter from low-frequency phonons?
Phonons will certainly make the time crystal to vibrate. The time crystal will be in an equilibrium with its environment. But can phonons rob significant momentum and energy from the slow movement of the time crystal?
Let us write a new blog post about this difficult problem.
