Transparency of solids using bandgaps and relation to conduction and valence bands I think I understand how a solid can appear transparent as long as the energy of the photons travelling through it are not absorbed in the material's bandgap. But how does this band gap relate to conduction and valence bands which explain insulators, semconductors, and conductors as described in the 'band theory for solids' (say) here: http://hyperphysics.phy-astr.gsu.edu/hbase/solids/band.html ?
Specifically I wish to use answers to this question to construct a useful visualisation of this aspect - trying to explain to lay people why some solids are transparent and some are semiconductors (or both) and using the bandgap to do so. See a basic, perhaps rushed, explanation of transparency started here: http://www.youtube.com/watch?v=Omr0JNyDBI0
There are two good reference questions here:
Transparency of materials
and followup here:
Why isn't light scattered through transparency?
So of course its not just about bandgap but also how classical waves are built up from a quantum substrate and materials with structures smaller than the wavelength of light.
That's my visualisation challenge...
I intend to show:


*

*high energy X-rays make many solids appear to be transparent because
they are not absorbed vs the energies in IR, visible, and UV light
for some materials.

*photons vs electrons being fired into a material

*glass structure vs crystal lattics or random structure solids

*opaque thin carbon vs transparent thick diamond and glass - why is paper opaque at visible light frequencies

*QED to classical wave showing how one leads to the other and how each
model explains transparency

*Conduction, semiconduction, insulation and bandgaps and their relation to visible light energy transparency

*and probably some other aspects.


To reiterate my question:
How does the bandgap theory of solids link transparency to conduction and valence bands (or other bands) ?
Feel free to suggest mechanisms for visualisation or any other contributing factors to explain this property of QED.
 A: Beginner's guide to band structure follows. I've taken considerable liberties with the details to simplify this so don't take it too literally!
This is going to seem an odd place to start, but consider filling up the atomic orbitals in an atom with electrons. If you take a noble gas, e.g. Xenon, you'll find each orbital is filled completely with two electrons and this is why Xenon is inert. If you take Potassium instead you find all the lower orbitals are filled with two electrons, but the outmost orbital contains only one electron so the orbital is only half full. This is why Potassium is very reactive.
When you clump atoms together into a solid, the interactions between the atoms spread the sharp atomic orbitals into energy bands. Suppose our solid contains $n$ Xenon atoms, then each band can contain 2$n$ electrons. But each Xenon atom contributes 2 electrons to each band, so the energy bands in solid Xenon are all full. That's why solid Xenon is an insulator. In the case of Potassium all the lower energy bands are full up with 2$n$ electrons, but the top band contains only $n$ electrons, i.e. it is only half full, because each Potassium atom has only 1 electron left over to put into this band. That's why solid Potassium is a conductor.
The position of an electron in an energy band doesn't just determine its energy, it also determines its momentum. If you want to make an electron move so it can conduct electricity you need to change it's momentum, and therefore you need to change its position in the energy band. But when bands are full you cannot change an electron's energy/momentum because there are no free spaces in the band for the electron to move into. That's why filled bands are insulating and part filled bands are conducting.
Now, if you imagine taking your solid and filling up the energy bands with electrons there is going to be a highest occupied band and a lowest unoccupied band. Now the nomenclature can be a little confusing. If the highest occupied band is full (like solid Xenon) we tend to refer to it as the valence band, and the lowest unoccupied band as the conduction band. The energy difference between the bands is the band gap. The reason why we call the lowest unoccupied band the conduction band is because any electrons that get excited into it will conduct; electrons in the valence band won't conduct (because the valence band is full).
But, if the highest occupied band is only part full (like solid Potassium) we call this band the conduction band because the electrons in it can conduct. Strictly speaking the highest band is both the valence band and conduction band, but convention dictates we call it the conduction band. In metals we're usually not fussed about the lowest unoccupied band and the band gap because they aren't involved in conduction of electricity.
Now, on to transparency. When a photon interacts with an electron it transfers it's momentum to the electron i.e. it changes the momentum of the electron. But if you recall from above, you can't change the momentum of an electron in a full band. The only way to change the electron momentum is to hit it hard enough, i.e. with enough energy, to make it jump over the band gap into the lowest unoccupied energy band. So if you measure the optical absorption as a function of energy you find there's little absorption until the photon energy matches the band gap, and the absorption suddenly rises. For many materials the band gap energy corresponds to ultra-violet light, so the solid doesn't absorb visible light i.e. it's transparent. As you say, these solids are also insulators because the same mechanism (change of electron momentum) determines both conductivity and optical absorption.
In metals the lowest occupied band (the conduction band) is only partially full so electron momentum can be changed by any amount you want. That's why metals absorb light (and radio waves etc) very strongly and are opaque.
Incidentally you do get borderline cases. Pure silicon is an insulator, but the band gap is only about 1.12 eV and this is less than the wavelength of red light. So silicon absorbs light even though it's an insulator. Well, it's an insulator in the dark. As soon as you shine light on it the electrons you excite over the band gap conduct electricity, so silicon conducts when you shine light on it.
I hope all this helps. If you want to clarify any of the above please comment.
