Trying to get a physical understanding of a Fermi gas

[MarkL]

TL;DR Summary

trying to get a physical understanding of fermi gas

I would like to get a more physical interpretation of conduction electrons (fermi gas) in a metal. I imagine ionized valence electrons close to the ions, with the fermi level (highest energy electrons) of the gas participating in conduction. A point of confusion for me...the first ionization energy for most metals are always higher than the fermi level, i.e. wouldn't the electron want to combine with the ions rather than conduct?

Also, I have some confusion with quantum states. Textbooks usually demonstrate this with $λ_n$'s in a potential well. To understand the actual position of electrons (fermions), I give the well zero potential. This is just a box. At low density this is a classical gas. At higher densities, where the space between electrons is less than DeBroglie wavelength, this would be quantum. By Pauli exclusion, one electron (λ) per well. So, N electrons in N identical states (wells) and identical fermi energies. Is this correct?

 

[Drakkith]

A point of confusion for me...the first ionization energy for most metals are always higher than the fermi level, i.e. wouldn't the electron want to combine with the ions rather than conduct?

If I'm not mistaken, the first ionization energy is only for single atoms, not atoms that are bound into the metallic lattice of a bulk material. When bound, the valence states overlap and allow for the delocalization of the electrons occupying those states. So the valence electrons are not bound to a single atom, despite not being ionized.

 

[hutchphd]

Summary:: trying to get a physical understanding of fermi gas

So, N electrons in N identical states (wells) and identical fermi energies. Is this correct?

If I understand you, this is very wrong. The electrons (particularly conduction electrons) are all in one "box" which is the chunk of solid matter. The different states are quantized because the wavelength must match the box boundary conditions and Fermi exclusion prevails. The Fermi level is where you are in wavenumber when you put in the final electron. This is basic solid state theory a la Ashcroft and Mermin.