Potassium crystallises in a body-centred cubic Bravais lattice with conventional…

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 Tight binding and the band structures of real materials

Problem

Potassium crystallises in a body-centred cubic Bravais lattice with conventional cubic lattice constant

a=5.23\times10^{-10}\,\mathrm{m}

and contributes one valence electron per atom. The shortest reciprocal lattice vector of a body-centred cubic lattice has magnitude

|\mathbf{G}|=2\pi\sqrt2/a

. A pseudopotential calculation gives the Fourier component of the crystal potential at this

\mathbf{G}

|U_{\mathbf G}|=0.14\,\mathrm{eV}

. ((a)) Find the free-electron kinetic energy at the corresponding Brillouin zone face,

k=|\mathbf G|/2

. ((b)) Form the ratio

|U_{\mathbf G}|

to this zone-face energy, and use it, alongside the equivalent ratio for aluminium (

0.083

), to decide whether potassium is better started from free electrons or from atomic orbitals. Justify the choice.

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