Zero Point Lattice Displacement And Strain

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.

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