Consider a square lattice that has got one atom of mass m at one lattice point. It interacts with 1st nearest neighbours only and phonon dispersion relation varies sinusoidally. (a) In the long-wavelength limit, obtain the density of phonon states D(ω)=dN/dω, i.e., the number of lattice-vibration modes per frequency.  (b) At high temperature (kT >> ħω), find the mean square displacement of an atom from its equilibrium position, and comment on the stability of twodimensional crystals interval dω. Berief answer for below question: (c) What is the specific heat at constant volume for a monatomic gas?

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