513.001 Molecular and Solid State Physics

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Hall problem 2.3

We may make a model of the stretching vibrations of a polyethylene chain -CH=CH-CH=CH-CH=CH- by considering a linear chain of identical masses M with alternating force constants K1 and K2. Determine the frequency of vibration of the normal mode with wave number k. Let a be the repeat distance of the chain. Sketch the optical and acoustic branches of the dispersion curves for -π/a < k < π/a.

Solid State Physics, H. E. Hall, Wiley (1974).

This is essentially the same problem as Kittel problem 4.5. Write down Newton's laws in analogy to equations 4.18 in Kittel. Follow the same solution in Kittel until equation 4.22. Solve the equation analogous to 4.22 for ω². This will be the dispersion relation.