Introduction to Solid State Physics · Problem 1

PHONONS II. THERMAL PROPERTIES — Charles Kittel

Physics — Solid State · singularity-in-density-of-states

Problem

Singularity in density of states. (a) From the dispersion relation derived in Chapter 4 for a monatomic linear lattice of

N

atoms with nearest-neighbor interactions, show that the density of modes is

D(\omega) = \frac{2N}{\pi} \cdot \frac{1}{(\omega_m^2 - \omega^2)^{1/2}}

where

\omega_m

is the maximum frequency. (b) Suppose that an optical phonon branch has the form

\omega(K) = \omega_0 - AK^2

, near

K = 0

in three dimensions. Show that

D(\omega) = (L/2\pi)^3 (2\pi/A^{3/2})(\omega_0 - \omega)^{1/2}

for

\omega < \omega_0

and

D(\omega) = 0

for

\omega > \omega_0

. Here the density of modes is discontinuous.