A model of the stretching vibrations of a polyethylene chain may be made by considering a linear chain of identical masses M connected by springs of alternating force constants C1 and C2. Calculate the characteristic frequencies of such a chain is the repeat distance is a and the force constants are related according to C1 = 5C2. Obtaining values for w as k -> 0 and k -> pi/a, sketch the dispersion curves for the optical and acoustic branches of the vibration spectrum.