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You started above with lead -> conductor and then jumped to
"reduction in separation" causing a "split into band gaps," as if you
would see a similar effect in lead if only you could get the atoms
closer together. I recommend stepping back a bit.
What you find specifically though, about a crystalline collection of
Pb atoms, as opposed to that of Si, is that there is always *some*
direction (actually, most) in k-space where you can find a band
intersection, whereas in Si, there is a region in the E(k) family of
curves where there are no intersections at all in any direction. Aka
THE band gap.
nothing per se about conductivity. You have your band structure. You
then pour in all the electrons to fill the bands as they can.
Conduction, semiconduction, or insulation, then is a function of
electron population and mobility in those bands. In the lowest
approximation, valence bands exhibit limited mobility, conduction
bands do not. However, mobility alone doesn't do much good as a
concept without considering population.
*Metals are relatively boring at this level - pour the electrons in
and given the free-electron-like intersecting bands, you're pretty
well good to go. High mobility, high population.
*Semiconductors are the canonical example of interest. Pour the
electrons in, then encounter a rich world, frameworked by band
structures, wherein population mechanisms (eg absorption of a
radiation quantum, temperature, donors via dopants) and mobility lead
to a plethora of interesting effects.
*Depopulate and/or demobilize, and you have an insulator. As was
mentioned, there's more to this than band structure.
Ashcroft and Mermin has been highlighted as the standard text. To me
it's a lot like Jackson - all the basic answers are there if only you
can see them. Allow me to suggest another book (though not undergrad
level): Wave Mechanics of Electrons in Metals, by Stanley Raimes. Out
of print, and although obviously concerned with metals, a really
clear exposition of many solid state concepts - one of the better
pedagogical physics texts I've read.
Stefan Jeglinski
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