2.7. For a certain three-dimensional simple cubic lattice, the phonon spectrum is independent of polarization X and is given by v(kr, ku, kz) = w0 * sqrt(kx^2 + ky^2 + kz^2).

1. Sketch a graph of w vs. k for 1. ky = kz = 0 and 0 < kx < sqrt(2) (i.e., along T - X).
2. Draw the w vs. k curve for the Debye approximation to these dispersion curves as dashed lines on the diagram used in part (a).
3. What are the critical points of this phonon spectrum? How many are there?
4. Make a rough sketch of the Debye density of states g(w). How will the actual density of states differ from the Debye approximation?
5. Using this example, discuss the shortcomings and the successes of the Debye model in predicting the thermodynamic properties (like specific heat) of solids.

Views: 5,368 students