Germanium has a 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

\Theta_D=374\,\mathrm{K}

. An Umklapp process needs phonons with wavevector of order half a zone dimension, hence energy of order

k_B\Theta_D/b

with

b\approx2

, so the number of phonons available to Umklapp-scatter carries the factor

e^{-\Theta_D/bT}

. Take

b=2

for germanium. ((a)) Estimate the value of this freeze-out factor at

300\,\mathrm{K}

and at

77\,\mathrm{K}

(liquid-nitrogen temperature). ((b)) Give the factor by which the Umklapp rate is estimated to fall on cooling from

300\,\mathrm{K}

77\,\mathrm{K}

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