Question

How does band theory explain why a crystal made of carbon atoms (i.e. diamond) is transparent, while a crystale made of aluminum atoms is not?

Answer 1

The answer lies in the band structure of the two materials. The band structure describes how the electrons in a solid are bound, and what other energy states are available to them.

the band gap for transparent diamonds is very wide as shown

Normally, diamond is not a conductor: all the electrons live in the "valence band", and you need a photon with at least 5.4 eV of energy to push an electron into the conduction band. In the process, that photon would be absorbed. A photon with less energy cannot give its energy to an electron, because that electron "has nowhere to go". And since visible light has energies of between 1.65 and 3.1 eV, only UV photons have enough energy to be absorbed by pure diamond.

Normally, diamond is not a conductor: all the electrons live in the "valence band", and you need a photon with at least 5.4 eV of energy to push an electron into the conduction band. In the process, that photon would be absorbed. A photon with less energy cannot give its energy to an electron, because that electron "has nowhere to go". And since visible light has energies of between 1.65 and 3.1 eV, only UV photons have enough energy to be absorbed by pure diamond.

In the case of aluminium the gap between balance band is less or overlapped some time.

And visible light has much more more energy to get trasition of electron and which cause photon to absorb and make it not to appear transparent.