PHZ 6426 (Solid State I)
Sample Exam 2

Consider in-plane vibrations of identical atoms of mass which lie at the vertices of a 2D square lattice with lattice constant a. Assume harmonic nearest-neighbor interactions only, with a force constant K. Let u_l,m denote the displacement from equilibrium of the atom in the l'th column and m'th row.

(a) Write down the equation of motion for u_l,m.

(b) Calculate the dispersion relation omega(q).

(c) Sketch the dispersion relation for the (10) and (11) directions, all the way from q=0 to the edge of the first Brillouin zone.

(d) Calculate the limiting form of the dispersion for |qa|<<1, and hence determine the sound velocity, c(q). Is the velocity isotropic?

(e) Write down an expression for the lattice heat capacity, and the form to which it reduces in the low-temperature limit.