An electron at the Fermi surface moves at a speed of some 1,000 km per second, while the speed of sound is only 5 km/s.
A typical thermal phonon at 1 kelvin has a wavelength of 50 nanometers and a frequency of 10¹¹ Hz.
A phonon has a very small momentum in its energy transport, but much larger in the movement of atoms in the lattice
A 0.4 meV phonon at 1 kelvin carries a tiny amount of energy at the speed of, say, 5 km/s. The momentum associated with the energy transport is very small.
An electron at the Fermi surface has a mass-energy 511 keV and moves 200 times faster. Its momentum is 250 billion times larger than that of the phonon.
But a phonon makes atoms in the lattice to vibrate. The momentum in the movement of the atoms is much larger than the momentum in the energy transport.
Electron scattering from a phonon
Let us assume that there is a macroscopic vibration of the lattice. The electron moves 200 times faster than the wave in the lattice. The electron feels a periodic potential which may scatter it.
charge distribution by phonon
+ + + + +
Λ = 50 nm
^
|
e- electron
de Broglie λ = 1 nm
The phonon concentrates the positive charge of the ions in the lattice at periodic locations.
The phonon makes a "diffraction grating" to the lattice, and the electron scatters from this grating. The spacing of the grating is typically Λ = 50 nm. The de Broglie wavelength of the electron is typically λ = 1 nm.
The first diffraction maximum is at the deflection angle
λ / Λ radians.
The momentum of the electron is
p = h / λ.
The change of the electron momentum in diffraction is
Δp = p λ / Λ
= h / Λ.
We may imagine that h / Λ is some kind of a "pseudomomentum" associated with the phonon. If the electron is scattered, it receives this pseudomomentum as real momentum from the lattice.
Question. The phonon momentum in energy transport is irrelevant. But is the momentum of atoms in the vibration relevant? Or can we treat the scattering like it would happen from a fixed diffraction grating in the lattice?
Rolf Heid (2017) writes about electron-phonon coupling. He says that the cloud of conducting electrons will screen the charge distribution caused by a phonon in the lattice. Thus, the scattering process is quite complicated. We will not try to calculate the effect, and trust the literature that scattering by phonons is the reason for most of resistivity at temperatures higher than 10 K.
Conclusions
Our time crystal model on April 8, 2022 was based on a misunderstanding of how electrons are scattered by phonons. We thought that it is like electron scattering by photons in otherwise empty space. But phonons can give a much larger momentum to the electron because the momentum comes from the lattice.
We will next try to tune our time crystal model to accommodate our better understanding of scattering.
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