1. = ... CH = CH CH = CH . n-1 n A polyethylene chain may be considered as a linear chain of atoms with mass M connected by springs with alternating force constants K1 and K2, where K1(K2) represents the double (single) bond, and K1 > K2. The repeat distance of the chain is a. a) Write down the equations of motion for the nth and (n-1)th atoms. [2 marks] b) The dispersion relation for this chain of atoms is given by 1/2 4K_K2sin? (ka) w2 K1+K2 M 1+(1- 2)) (K1+K2)2 where w is the angular frequency and k is the wavevector. Sketch the dispersion curves for the optical and acoustic branches of the vibration spectrum in the range -I/a