Solution to Set 5

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Diatomic harmonic chain

Problem 1

Consider a chain of atoms with alternating masses $m_{1}\;$ and $m_{2}\;$, connected with elastic springs with constant $K\;$, moving only in the x-direction. Derive the dispersion relation $\omega ^{\alpha }(k)\;$ for this chain, with the index $\alpha =1,2\;$ coresponding to the acoustic and the optical branch, respectively. 

Problem 2

Determine the speed of sound for this chain. What is the lowest frequency of long-wavelength sound corresponding to the optical branch?

Problem 3

Sketch the motion of the atoms corresponding to the edge of the Brillouin zone, both for the optical and the acoustic branch.

Problem 4

Determine the Debye temperature for this system, and determine the form of the specific heat $c_{V}(T)$ in the limits of high and low temperatures. 

Problem 5

Consider low temperatures ($T\ll T_{D}\;$) and determine the wavelength of the most abundant phonons $\lambda _{max}$ (Hint: note the analogy with Wien's Law!)