For fcc aluminium ( )

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

For fcc aluminium (

a=4.05\,\mathrm{\AA}

), the empty-lattice construction gives a three-fold degenerate level at the symmetry point

K=(2\pi/a)(3/4,3/4,0)

. Its crystal potential has only

U_{111}

and

U_{200}

nonzero (all Fourier coefficients belonging to the same

\{hkl\}

family taken equal, real and positive, with the origin on an atom). ((a)) Verify that the plane waves labelled

\mathbf G_1=(2\pi/a)(0,0,0)

\mathbf G_2=(2\pi/a)(1,1,1)

and

\mathbf G_3=(2\pi/a)(1,1,-1)

are exactly degenerate at

K

in the free-electron energy, and give their common energy as a multiple of

E_0=\hbar^2(2\pi/a)^2/2m

. ((b)) For each pair among

\mathbf G_1,\mathbf G_2,\mathbf G_3

, identify whether the difference

\mathbf G_i-\mathbf G_j

belongs to the

\{111\}

or the

\{200\}

family, and hence write the

3\times3

matrix of the central equation truncated to this basis, in terms of

\varepsilon^0

U_{111}

and

U_{200}

. ((c)) Using

E_0=9.17\,\mathrm{eV}

U_{111}=0.24\,\mathrm{eV}

and

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

, write out the matrix with numbers and find its three eigenvalues.

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