lylos said:
Alright, so for a diatomic chain there would be two key normal mode frequencies. Those two frequencies are determined by the incident wavelength. There's the optical branch and the acoustical branch?
Generally speaking a dispersion relation just relates the kinetic energy of some wave-like excitation to the momentum of it. Monatomic and diatomic chains are basic models for phonon dispersion relations, so I suppose these are meant here. This means, that there is not necessarily light involved.
As an easy model, which is used in most basic courses, just imagine your atoms as little balls, model the forces between one atom and its next neighbours as a spring and ignore all other forces. Now you got a long chain off balls connected with springs. You can easily imagine, that an atom behaves like a harmonic oscillator, if it is moved out of its equilibrium position and can therefore oscillate. As you have a long chain, you can also imagine a collective oscillation of all atoms around their equilibrium position. In this basic model, this is a phonon.
If you go to a diatomic chain, you can imagine, that there are several basic modes. All atoms can move in phase or the first kind of atoms can oscillate in antiphase to the second kind, for example. If you have charged particles instead of neutral atoms the second mode produces an oscillating dipole moment. Therefore this is called the optical branch.
However, the whole topic is more complicated - you can also take transversal modes into account and more than just 1D chains and such stuff, but for the moment, it should be sufficient to know, that the dispersion relation relates the kinetic energy of this excitation of all atoms to its momentum.
If you need a more technical approach, a google search for phonon dispersion relations might provide you with some info. The books of Kittel and Ashcroft/Mermin will work as well.