A rock-salt rod of diameter is cooled to a temperature low enough that phonon-phonon…

PHYSICS · Multi-part · 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 rock-salt rod of diameter d=2.0 mmd = 2.0\,\mathrm{mm} is cooled to a temperature low enough that phonon-phonon scattering has essentially stopped, so the mean free path is fixed by the rod's own diameter, ℓ=d\ell = d, independent of temperature. Rock salt's Debye temperature is ΘD=321 K\Theta_D = 321\,\mathrm{K}, its molar volume is Vm=13.5×10−6 m3 mol−1V_m = 13.5\times10^{-6}\,\mathrm{m^3\,mol^{-1}} and its sound speed is v=3400 m s−1v = 3400\,\mathrm{m\,s^{-1}}. ((a)) Starting from κ=13cVvℓ\kappa = \tfrac13c_Vv\ell and the low-temperature T3T^3 law for CVC_V, show that in this boundary-limited regime κ∝T3\kappa \propto T^3, and write the constant of proportionality in terms of dd, vv, VmV_m and ΘD\Theta_D. ((b)) Evaluate κ\kappa at T=8 KT = 8\,\mathrm{K}. ((c)) A second rod is cut ten times thinner, d=0.20 mmd = 0.20\,\mathrm{mm}. Find its κ\kappa at 8 K8\,\mathrm{K}, and say whether its conductivity peak sits at a higher or a lower temperature than the thick rod's, and whether it is higher or lower.

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