A diamond anvil in a high-pressure cell has a lower-grade conductivity than the…

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

A diamond anvil in a high-pressure cell has a lower-grade conductivity than the purest bulk diamond,

\kappa = 1500\,\mathrm{W\,m^{-1}K^{-1}}

300\,\mathrm{K}

, with

c_V = 1.8\times10^6\,\mathrm{J\,m^{-3}K^{-1}}

v = 1.3\times10^4\,\mathrm{m\,s^{-1}}

and

a = 3.57\,\mathrm{\mathring{A}}

as for bulk diamond. ((a)) Estimate its phonon mean free path, in nanometres and in lattice constants. ((b)) The anvil is

t = 1.5\,\mathrm{mm}

thick and carries

Q = 2.0\,\mathrm{W}

of heat away from the sample through a cross-sectional area

A = 0.50\,\mathrm{mm^2}

. Using the definition of thermal conductivity,

J_x = -\kappa\,\mathrm{d}T/\mathrm{d}x

, in steady state with a uniform gradient, estimate the temperature difference across the anvil's thickness.

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