Natural germanium has isotopes ( ), ( ), ( ), ( ) and ( )

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

Natural germanium has isotopes

^{70}\mathrm{Ge}

(

20.5\%

^{72}\mathrm{Ge}

(

27.4\%

^{73}\mathrm{Ge}

(

7.8\%

^{74}\mathrm{Ge}

(

36.5\%

) and

^{76}\mathrm{Ge}

(

7.8\%

), with mass numbers taken as the atomic masses in u. An isotopically enriched sample has mass variance

g_{\rm iso}=1.0\times10^{-5}

. ((a)) Compute the mass variance

g_{\rm iso}=\sum_if_i(1-M_i/\bar M)^2

of natural germanium. ((b)) Assuming that near its conductivity peak the mean free path is set by isotope scattering alone, both before and after enrichment, estimate the largest factor by which enrichment could raise the peak conductivity, using

\kappa\propto\ell_{\rm iso}\propto1/g_{\rm iso}

. ((c)) Explain, without further calculation, why the gain measured at room temperature is expected to be much smaller than this factor. ((d)) Enriched germanium is measured to reach only about eight times its natural conductivity at its peak. Explain why this is so much smaller than your estimate in (b).

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