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Is it an accurate statement to say that free electrons in a metal experience NO restoring force when they interact with electromagnetic waves? I understand that the electrons exist in a space filled with ions, and doesn't the cumulative potential that is present due to the presence of the ions exert an electric field on the electrons. Even in the case of simple metals, where you can say that the nucleus is shielded by the valence electrons, so called Coulomb shielding, how significant is the shielding. Naively, it seems to me that because the charge in the nucleus is not balanced by the charge in the bound electrons, there should be some net potential that the free electron sees.

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The free electron model is surprisingly good at predicting the properties of electrons in metals, and this implies that the electrons really are nearly free. However when you look more closely there is of course an interaction with the lattice. This is modelled using the (rather predictably named) nearly free electron model.

The conduction electrons are delocalised, so you shouldn't think of them as little balls bouncing off the ion cores. The spatial extent of their wavefunction is typically far greater than the lattice repeat, hence the relatively weak interactions. However interactions with the lattice are responsible for electrical resistance and thermal conductivity, and at very low temperatures for superconductivity. However note that these aren't interactions between a single electron and a single ion core, but rather interactions between electron waves and lattice waves (phonons).

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  • $\begingroup$ If we look more far (in energy scale), we may notice that electrons are not free but are in Bloch's states too. $\endgroup$ – Nogueira May 13 '15 at 20:02
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Since the electrons are in a periodic lattice, then you can think of an idealized single-electron picture which leads to the band structure of an electron. In this ideal picture a you can give the electron a little push, and it will travel indefinitely without scattering. Well, at least until it collides with the metal's surface.

In reality, the metal is not a perfect ideal lattice, and there is more than one electron. There are impurities, electron-phonon interactions, and electron-electron interactions. All of these lead to scattering (the electron's momentum is "dissipated" by being randomized). For very high frequency electromagnetic radiation (as in, visible light), you also need to worry about the plasma frequency.

However, the electrons are definitely still "free" in the sense that they can roam around the metal. It's just that they diffuse in a random walk, rather than flying ballistically in a straight line.

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If you imagine a metal where there is only one conduction electron per constituent atom, then the net charge of each ion sitting in a lattice situ is +1. The resulting periodic potential seen by the conduction electrons looks like this.

If this potential can be considered a weak perturbation to the case in which the conduction electrons are completely free, as it is the case for simple metals, then the electrons are highly delocalised in the solid, their mobility is very high and the current they produce in response to an electromagnetic perturbation has negligible retardation with respect to the driving field phase. In this sense is correct to say electrons in a metal experience no restoring force.

For what concerns the solid constitution, you have to bear in mind that the electric field produced by the presence of electrons and ions together is what keeps the whole thing together in this "condensed" state of matter. Without one of the two you would get fantastic Coulomb explosions!

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