Introduction to Solid State Physics · Problem 3

PHONONS II. THERMAL PROPERTIES — Charles Kittel

Physics — Solid State · zero-point-lattice-displacement-and-strain

Problem

Zero point lattice displacement and strain. (a) In the Debye approximation, show that the mean square displacement of an atom at absolute zero is

\langle R^2 \rangle = 3\hbar\omega_D^2/8\pi^2\rho v^3

, where

v

is the velocity of sound. Start from the result (4.29) summed over the independent lattice modes:

\langle R^2 \rangle = (\hbar/2\rho V)\Sigma\omega^{-1}

. We have included a factor of

\frac{1}{2}

to go from mean square amplitude to mean square displacement. (b) Show that

\Sigma\omega^{-1}

and

\langle R^2 \rangle

diverge for a one-dimensional lattice, but that the mean square strain is finite. Consider

\langle (\partial R/\partial x)^2 \rangle = \frac{1}{2}\Sigma K^2 u_0^2

as the mean square strain, and show that it is equal to

\hbar\omega_D^2 L/4MNv^3

for a line of

N

atoms each of mass

M

, counting longitudinal modes only. The divergence of

R^2

is not significant for any physical measurement.