Natural carbon is 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 carbon is

98.93\%

^{12}\mathrm{C}

and

1.07\%

^{13}\mathrm{C}

. An isotopically enriched diamond sample is

99.9\%

^{12}\mathrm{C}

and

0.1\%

^{13}\mathrm{C}

. ((a)) Compute the isotopic mass variance

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

for both the natural and the enriched sample. ((b)) Assuming that, near its conductivity peak, diamond's phonon 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. ((c)) State, without calculating a number, whether the gain at room temperature should be closer to this factor or much smaller than it, and why.

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