Gold is face-centred cubic with lattice constant and one conduction electron per atom

PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Free electrons, energy bands and electronic structure · Electronic transport in solids

Solid State Physics, Ch The Boltzmann equation and electrical transport

Problem

Gold is face-centred cubic with lattice constant

a=4.08\,\mathrm{\AA}

and one conduction electron per atom, giving

n=5.90\times10^{28}\,\mathrm{m^{-3}}

. Treat the conduction electrons as a free, isotropic Fermi gas of electron mass

m_e

. ((a)) A gold wire carries a steady current density

j=5.0\times10^{6}\,\mathrm{A\,m^{-2}}

. Estimate the drift velocity

v_d=j/ne

and compare it with the Fermi velocity

v_F=\hbar k_F/m_e

, where

k_F=(3\pi^2 n)^{1/3}

, to estimate the fractional shift

\delta k/k_F

of the Fermi sphere. ((b)) The screened-Coulomb estimate for copper gives a transport cross section

\sigma_{\rm tr}=3.2\,\mathrm{\AA^2}

for a solute with

\Delta Z=1

. Take the same

\sigma_{\rm tr}

for a

\Delta Z=1

solute in gold at one atomic per cent,

n_{\rm imp}=0.01n

. Using

1/\tau_{\rm imp}=n_{\rm imp}v_F\sigma_{\rm tr}

and

\rho_0=m_e/ne^2\tau_{\rm imp}

, find

\tau_{\rm imp}

and the residual resistivity per atomic per cent in gold, and say which property of the host the result depends on.

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