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) = 3hw²_D/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.