Copper is face-centred cubic with lattice constant and one atom per primitive cell

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

Copper is face-centred cubic with lattice constant a=3.61 A˚a=3.61\ \mathrm{\AA} and one atom per primitive cell. In the empty-lattice (free-electron) approximation the energy of the level labelled by wavevector k\mathbf k and reciprocal lattice vector G\mathbf G is ε(k−G)=ℏ22m∣k−G∣2\varepsilon(\mathbf k-\mathbf G)=\dfrac{\hbar^2}{2m}|\mathbf k-\mathbf G|^2, and the allowed fcc reciprocal lattice vectors are G=(2π/a)(h,k,l)\mathbf G=(2\pi/a)(h,k,l) with h,k,lh,k,l either all even or all odd. Take the point L=(2π/a)(12,12,12)L=(2\pi/a)\big(\tfrac12,\tfrac12,\tfrac12\big) and use ℏ2/2m=3.81 eV A˚2\hbar^2/2m=3.81\ \mathrm{eV\,\AA^2}. ((a)) Find the reciprocal lattice vector G≠0\mathbf G\neq0 for which ∣k−G∣=∣k∣|\mathbf k-\mathbf G|=|\mathbf k| at this point. ((b)) Give the energy of the resulting level and its degeneracy.

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