Two rock-salt crystals

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

Two rock-salt crystals, one very pure and one deliberately doped with impurities, are cooled to 15 K15\,\mathrm{K}. 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}}, unchanged from the room-temperature value used elsewhere in this chapter. At 15 K15\,\mathrm{K} the pure crystal's measured conductivity is κpure=600 W m−1K−1\kappa_{\rm pure} = 600\,\mathrm{W\,m^{-1}K^{-1}} and the doped crystal's is κdoped=90 W m−1K−1\kappa_{\rm doped} = 90\,\mathrm{W\,m^{-1}K^{-1}}. ((a)) Find the heat capacity per unit volume at 15 K15\,\mathrm{K} from the T3T^3 law. ((b)) Find the phonon mean free path in each crystal, in micrometres and in lattice constants (rock salt's lattice constant is a=5.64 A˚a = 5.64\,\mathrm{\mathring{A}}). ((c)) Explain, in terms of what limits ℓ\ell in each crystal, why the two mean free paths differ so much while cVc_V and vv are the same for both.

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