A single-crystal sapphire ( ) rod of diameter has Debye temperature

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

Solid State Physics, Ch Anharmonicity: expansion, phonon lifetimes and thermal conductivity

Problem

A single-crystal sapphire (Al2O3\mathrm{Al_2O_3}) rod of diameter w=1.0 mmw=1.0\,\mathrm{mm} has Debye temperature ΘD=1035 K\Theta_D=1035\,\mathrm{K}, mass density ρm=3980 kg m−3\rho_m=3980\,\mathrm{kg\,m^{-3}}, molar mass M=101.96 g mol−1M=101.96\,\mathrm{g\,mol^{-1}} and five atoms per formula unit. Treat all phonon branches as travelling at a single representative speed v=6800 m s−1v=6800\,\mathrm{m\,s^{-1}}. Its measured thermal conductivity at T=20 KT=20\,\mathrm{K} is κ=4.0×103 W m−1K−1\kappa=4.0\times10^{3}\,\mathrm{W\,m^{-1}K^{-1}}. ((a)) Using the low-temperature Debye heat capacity per unit volume, CV=12π45nkB(TΘD)3C_V=\dfrac{12\pi^4}{5}nk_B\left(\dfrac{T}{\Theta_D}\right)^3, with nn the number density of atoms, calculate CVC_V at 20 K20\,\mathrm{K}. ((b)) Use the kinetic formula κ=13CVvℓ\kappa=\tfrac13C_Vv\ell to extract the phonon mean free path ℓ\ell at 20 K20\,\mathrm{K}, and compare it with the rod diameter ww. State what regime of κ(T)\kappa(T) this places the rod in at 20 K20\,\mathrm{K}.

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