Consider a one-dimensional discrete chain of identical atoms. The chain has length L and lattice constant a. The atoms have mass m and the forces are characterized by the spring constant k. 1. What is the energy dispersion of the vibration normal modes? 2. Derive the density of states as a function of energy. 3. Compare your answers to the Debye model. What are the significant differences? 4. For a one dimensional crystal with 4 atoms in the unit cell, how many acoustic, and how many optical lattice vibration branches do you expect in the energy dispersion relation?