Aluminium is face-centred cubic with lattice constant

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

Solid State Physics, Ch Tight binding and the band structures of real materials

Problem

a=4.05\,\text{\AA}

. A fit to measured properties of aluminium's electrons gives the Fourier component of the crystal potential at the

(200)

reciprocal lattice vector as

|U_{200}|=0.76\,\mathrm{eV}

. ((a)) Using the free-electron formula for the energy at the corresponding Bragg zone face,

E_{\rm zone}=\hbar^2k_{\rm zone}^2/2m_e

with

k_{\rm zone}=2\pi/a

(half the magnitude of

\mathbf{G}_{200}

), evaluate

E_{\rm zone}

in eV. ((b)) Form the ratio

|U_{200}|/E_{\rm zone}

. ((c)) State, using the criterion set out for choosing a starting point, whether nearly-free electrons or tight-binding orbitals are the appropriate description of aluminium's conduction band.

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