How do we predict electrical properties of a material using scattering data and vice-versa?

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?

• The simple answer is that you can't. The behavior of a material at dc bears absolutely no useful relationship to its behavior at 1e15Hz. Maybe you are trying to ask a different question, and I just don't understand you correctly? – CuriousOne Sep 15 '14 at 21:30
• Well, Kramers & Kronig have a little something to add to the discussion. If you know $\epsilon$ as a function of all $\omega > 0$ you can determine $\epsilon(\omega=0)$. But that is being a little picky... – Jon Custer Sep 15 '14 at 21:40
• @JonCuster: In practice Kramers-Kronig doesn't do anything for you, because you would have to know the whole continuum of frequencies to determine the value at a single frequency. It's a very interesting mathematical theorem, though. – CuriousOne Sep 15 '14 at 23:49
• I probably should write an answer, because I work on Monte Carlo simulation of electron scattering in matter, and we do use the optical properties of the materials to determine the characteristics of slow secondary electron generation. That's not necessarily the main effect relevant here, but it can be important for modeling the behavior of the (slow secondary) electron detectors. – Thomas Klimpel Sep 16 '14 at 9:25
• Thanks for the replies. They do not quite answer my question, though they are very informative. I have edited my post to be more clear. – sebastianspiegel Sep 18 '14 at 22:55