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SOLVED:One-dimensional chain with one atom per unit cell: We have determined the phonon dispersion relation for an infinite chain of atoms with lattice spacing a and one atom per unit cell (mass M ). The result is (4.10). For longitudinal vibrations, sketch how the atoms would move for k close to 0(k \ll \pi / a) and for k at the Brillouin zone boundary (k=\pi / a).

SOLVED: Consider a square lattice that has one atom of mass m at each lattice point. The atom interacts with its 1st nearest neighbors only, and the phonon dispersion relation varies sinusoidally.(a) In the long-wavelength limit, obtain the density of phonon states D(w) dM/dw; i.e. the number of lattice-vibration modes per frequency. (10)(b) At high temperature (KT >> hw), find the mean square displacement of an atom from its equilibrium position and comment on the stability of two-dimensional c