(a) Explain the concept of a phonon and the phonon dispersion curve. (b) (5 marks) A two dimensional square lattice has one atom of mass m per lattice point. The interatomic forces are only between nearest neighbours and have a force constant K. The phonon dispersion curve is taken to be 4K w(q): = -sin m In the long wavelength limit, obtain the density of phonon states g(w) = dN/dw i.e. the number of lattice vibration modes per frequency interval dw. (10 marks) (c) At high temperatures (kgT » hw), find the mean square displacement of an atom from its equilibrium position and comment on the stability of two dimensional crystals. (10 marks)