A synthetic diamond plate of thickness 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 synthetic diamond plate of thickness w=8.0 mmw=8.0\,\mathrm{mm} has Debye temperature ΘD=2230 K\Theta_D=2230\,\mathrm{K}, mass density ρm=3515 kg m−3\rho_m=3515\,\mathrm{kg\,m^{-3}}, molar mass M=12.01 g mol−1M=12.01\,\mathrm{g\,mol^{-1}} and one atom per formula unit (carbon). Treat all phonon branches as travelling at a single representative speed v=1.20×104 m s−1v=1.20\times10^{4}\,\mathrm{m\,s^{-1}}. Its measured thermal conductivity at T=25 KT=25\,\mathrm{K} is κ=3.0×104 W m−1K−1\kappa=3.0\times10^{4}\,\mathrm{W\,m^{-1}K^{-1}}. ((a)) Calculate the Debye heat capacity per unit volume CVC_V at 25 K25\,\mathrm{K}. ((b)) Extract the phonon mean free path ℓ\ell at 25 K25\,\mathrm{K} from κ=13CVvℓ\kappa=\tfrac13C_Vv\ell, and compare it with the plate thickness ww. State what this comparison tells about which scattering mechanism limits κ\kappa at 25 K25\,\mathrm{K}.

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