Why is it that in stars undergoing gravitational collapse electron degeneracy kicks in? Why couldn't the electrons form energy bands like in semiconductors?

  • $\begingroup$ I'm not very sure, but I would think that the most likely reason is temperature. The collapsing star has a temperature in the million Kelvin range, whereas the energy bands are in the room temperature range. $\endgroup$ – Kyle Kanos Mar 29 '14 at 16:06

Quantised energy levels/bands arise from electrons moving in a potential well. In say a white dwarf, the degenerate electrons can have kinetic energies of order 100-keV to 1MeV and can usually be assumed to behave like (almost) free particles. In which case there are a set number of available momentum eigenstates per unit volume $$ g(p)\,dp = \frac{8 \pi p^2}{h^3}\,dp ,$$ where $p$ is the momentum. The electrons fill up all the available energy states (because of the Pauli exclusion principle) up to the Fermi momentum/energy.

There are small corrections made to account for the fact that positive charge is concentrated in nuclei whilst the electrons are (almost) uniformly distributed (electrostatic corrections). Below densities of $10^{7}$ kg/m$^3$ (brown dwarfs, planets) the assumption of a uniform electron distribution breaks down, the Fermi (kinetic) energies move to an atomic scale and further corrections need to be made (Thomas-Fermi corrections) which assume the electrons move in the spherically symmetric potential of a nucleus.

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