1 2 1 Q.5 A linear chain consists of 2N atoms of mass M, with equilibrium spacing -a. Each atom interacts only with its nearest neighbors. The force constant for the interaction between the atom at na and the atom at na+a is y, while the force constant for the interaction between the atom at na and the atom at na-a is Y2. Take Y > Y2. Each cell of length a contains two nonequivalent atoms. (a) Derive an expression for the normal mode angular frequency as a function of propagation constant, K. (b) Find the normal mode frequencies for K = 0 and for K = Indicate which modes belong to the acoustical branch and which belong to the optical branch. (c) Sketch the dispersion curves. (d) How does the answer to part (b) change if Y2 > V1?