A perfect harmonic crystal of length sits between two reservoirs held at slightly…

PHYSICS · Subjective · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Lattice dynamics and thermal properties of solids · Lattice dynamics and thermal properties of solids

Condensed Matter Physics, Ch Thermal expansion and lattice thermal conductivity

Problem

A perfect harmonic crystal of length LL sits between two reservoirs held at slightly different temperatures, one at each end. Nothing inside a purely harmonic crystal scatters a wave packet. (a) Show that the only length available to stand in for the mean free path ℓ\ell in κ=13cVvℓ\kappa = \tfrac13c_Vv\ell is the crystal's own length LL, so κ=13cVvL\kappa = \tfrac13c_VvL. (b) Using rock salt's cV=1.85×106 J m−3K−1c_V = 1.85\times10^6\,\mathrm{J\,m^{-3}K^{-1}} and v=3400 m s−1v = 3400\,\mathrm{m\,s^{-1}}, evaluate κ\kappa for L=1 mmL = 1\,\mathrm{mm}, 1 cm1\,\mathrm{cm} and 1 m1\,\mathrm{m}. (c) Explain why no single finite number can be called "the" thermal conductivity of a harmonic crystal.

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