I know that the band gap is related to conductivity. What I'm wondering is what it is like for an experimentalist who is trying to figure out what an unknown material, a black box, is doing. The only tool the experimentalist has is a beam of light at various wave lengths and a photo-detector which he/she can place at various positions. From the scattering data, the experimentalist is tasked to predict what the electrical properties of the material are, i.e. current as a function of the history of the voltage. I say history because in the case of things like integrated circuits, the way it responds to voltages depends on the way that past voltages were applied. So that is my question, how can I predict the electrical properties of a material using scattering data? Equally interesting, how do we do the converse, predict the scattering data using the electrical properties?
**EDIT:**People talk about band gaps and how they are related to conductivity. That you need to knock an electron into a conduction band. Is that how it works? Photo-diodes work like this(I think) and also the higher the temperature the more a semi-conductor will conduct; this is consistent with the energy level picture. It has been my understanding that this energy band is described by eigenstates of a Hamiltonian. Yet as some have pointed out, strictly speaking, the Hamiltonian can have any spectrum and the conductivity, where the frequency $\omega=0$ has nothing to do when $\omega \neq 0$. However, actual Hamiltonians can only have local interactions(I assume). So that constrains things a bit. So here is my question. How do energy levels of a material, as determined by scattering experiments, correspond to a band-structure/conductivity and vice-versa?