A solid with electrons localized on lattice sites is often modeled by so- called "tight-binding" model. In this 1D model the dispersion relation is E(k) = −2E cos(ka), where a is the lattice constant. Starting from the effective equation of motion, describe how the electron moves in such a model under the influence of the constant electric field. Assume you measure electrical current as a function of time without any scat- tering. You will find an oscillatory behavior for the current. What is the frequency of this current? This is not observed in practice because scattering will exist.

The net potential energy E_N between two adjacent ions is sometimes represented by the expression: E_N = -C/r + D times exp (-r/rho) in which r is the interionic separation and C, d, and rho are constants whose values depend on the specific material. Derive an expression for the bonding energy E_0 in terms of the equilibrium interionic separation r_0 and the constants D and rho. Derive another expression for E_0 in terms of r_0, C and rho.