The free electron model
The free electron model of Arnold Sommerfeld describes the electron gas of the conducting electrons in a metal.
Electrons fly almost like non-interacting particles of a gas, but collide with phonons, defects and borders of crystals, and impurities once for every tens of nanometers of free flight.
The resistivity of copper is almost linearly dependent on the temperature at 20 K - 1,200 K. At less than 20 K, copper retains some resistivity which seems to come from impurities and defects in the crystal structure. Copper does not become a superconductor at low temperatures.
Why electrons do not bump into atoms in the crystal lattice or into each other?
Bloch's theorem explains why a single electron moving in a lattice potential does not get scattered. Its wave function is a plane wave modulated by a periodic function. We showed in our previous blog post that in such a situation, there is an almost total destructive interference of scattered waves. Thus, there is essentially no scattering.
However, in a metal there is a huge number of free electrons which could bump into each other. Bloch's theorem is an inaccurate description of the system.
The analogy of a many electron atom and a crystal of a metal
It is an empirical fact that electrons seem to fly freely and do not bump into atoms of a perfect crystal of a metal, or into each other. As far as we know, there exists no adequate theoretical model presently.
A similar empirical fact is that the electrons in an arbitrary atom seem to fill orbitals which are similar to the orbitals of hydrogen. For an unknown reason, we can solve the system for just one electron (i.e., hydrogen), and we obtain a qualitatively correct solution for a many electron system.
The "Fermi sea" of electrons in an atom consists simply of the orbitals of the electrons. In a metal, electrons seem to settle to different energy levels in a manner similar to a single atom. The orbitals in a metal might look like the solutions in Bloch's theorem.
Conclusions
The free electron model of Arnold Sommerfeld is an empirical result.
Empirically, resistance is caused by:
1. thermal vibrations of the lattice (but not by zero-point vibrations?);
2. defects and borders of crystals;
3. impurities in a crystal.
We have no adequate theoretical model for resistivity.
To explain superconductivity, we need to explain why items 1, 2, and 3 above do not cause resistance in a superconductor.
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