A rock-salt crystal has been cooled far enough that phonon-phonon scattering has…

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 rock-salt crystal has been cooled far enough that phonon-phonon scattering has almost died out, so its mean free path is limited by the crystal's own size instead: take ℓ=2.0 mm\ell = 2.0\,\mathrm{mm}, fixed, over the range considered here. Rock salt's Debye temperature is ΘD=321 K\Theta_D = 321\,\mathrm{K} (Chapter 6), its molar volume is Vm=13.5×10−6 m3 mol−1V_m = 13.5\times10^{-6}\,\mathrm{m^3\,mol^{-1}} (Section 7.3), and its sound speed is v=3400 m s−1v = 3400\,\mathrm{m\,s^{-1}} (Section 7.6). Using Chapter 6's T3T^3 law, CV=1944(T/ΘD)3 J mol−1K−1C_V = 1944(T/\Theta_D)^3\,\mathrm{J\,mol^{-1}K^{-1}}, predict the lattice conductivity at T=10 KT = 10\,\mathrm{K}, and compare it with the room-temperature value of 6.4 W m−1K−16.4\,\mathrm{W\,m^{-1}K^{-1}}.

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