A polycrystalline alumina ( ) ceramic 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 polycrystalline alumina (Al2O3\mathrm{Al_2O_3}) ceramic 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}}, five atoms per formula unit, and is treated as having a single representative phonon speed v=6800 m s−1v=6800\,\mathrm{m\,s^{-1}} for all branches. Its manufacturer quotes an average grain size d=50 μmd=50\,\mu\mathrm{m}. The measured thermal conductivity at T=15 KT=15\,\mathrm{K} is κ=15 W m−1K−1\kappa=15\,\mathrm{W\,m^{-1}K^{-1}}, far below that of a single-crystal sapphire rod of the same material at the same temperature. ((a)) Calculate the Debye heat capacity per unit volume CVC_V at 15 K15\,\mathrm{K}. ((b)) Extract the phonon mean free path ℓ\ell at 15 K15\,\mathrm{K} from κ=13CVvℓ\kappa=\tfrac13C_Vv\ell and compare it with the quoted grain size dd. State what the comparison implies about whether the quoted grains, or some finer feature, limit κ\kappa in this ceramic.

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