Model lead as a tetravalent free-electron metal that is face-centred cubic with…

PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Crystal structure, bonding and diffraction · Free electrons, energy bands and electronic structure

Solid State Physics, Ch Bloch's theorem and the nearly-free-electron method

Problem

Model lead as a tetravalent free-electron metal that is face-centred cubic with conventional cubic lattice constant a=4.95 A˚a=4.95\ \mathrm{\AA} and four atoms per conventional cell, each contributing 44 free electrons. ((a)) Find the free-electron Fermi wavevector kFk_F in A˚−1\mathrm{\AA^{-1}}. ((b)) Compute the distances ΓX=2π/a\Gamma X = 2\pi/a, ΓL=(2π/a)(3/2)\Gamma L = (2\pi/a)(\sqrt3/2) and ΓW=(2π/a)(5/2)\Gamma W = (2\pi/a)(\sqrt5/2) of the fcc Brillouin zone, in A˚−1\mathrm{\AA^{-1}}. ((c)) Decide whether the free-electron sphere fills the first Brillouin zone completely, and give the ratio kF/ΓWk_F/\Gamma W.

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