Zero point lattice displacement and strain
(a) In the Debye approximation, show that the mean square displacement of an atom at absolute zero is R) = 3hw2D/8π2 pv3, where v is the velocity of sound. Start from the result (4.29) summed over the independent lattice modes: <R2> = (h/2pV) Σw–1. We have included a factor of ½ to go from mean square amplitude to mean square displacement.
(b) Show that Σw–1 and (R2) diverge for a one-dimensional lattice, but that the mean square strain is finite. Consider <(∂R/∂x)2> = ½ ΣK2u20 as the mean square strain, and show that it is equal to hw2DL/4MNv3 for a line of N atoms each of mass M, counting longitudinal modes only. The divergence of R2 is not significant for any physical measurement.