Problem 1 (25 points) Band structure of aluminum. Aluminum is an fcc crystal with cube length a = 4.05A (see A +M, Fig.4.8, 4.9 for a picture of the lattice, Fig. 5.2(b) for a picture of the Brillouin zone). The following high symmetry points in the Brillouin zone have special names: I = (0, 0, 0), X = 2" (1, 0, 0), L = " (1, 1, 1), W = " (1, 2, 0). (Note that X, L, W are on the zone boundary). The Fourier transform of the lattice can be modeled by V (q) = - 4Te' Z cos(qRuc) (1) where Z = 3, S is the volume of the primitive cell in real space, and Rc = 0.6A. Part A (10 points) Working within the nearly free electron approximation, sketch the bandstructure from I to X, from X to W, from W to L, and from L to I. Label any degeneracies. If there are any avoided crossings, compute the gap (in eV). Part B (5 points) Near point W, four place waves are nearly degenerate. Take the Schrodinger equation written in momentum space, and identify the 4 * 4 part of the Hamiltonian matrix that acts on the nearly degenerate space spanned by (Ck, Ck_G, , Ck_G2) Ck_G3), Where Gi = 2 (1, 1, 1), G2 = 2" (1, 1, -1), G3 = 4 (1, 0, 0). Part C (10 points) Show that the following symmetry operators commute with the Hamiltonian from part (b) 0 0 0 S1 = S2 = 0 (2) 0 0 0 01 Show that the eigenvectors of the Hamiltonian have eigenvalues +1 under S, and $2. Now consider the band structure from X to W and from W to L. Does the S, and/or S2 symmetry survive along these lines? Hence show how the bands are connected between X, W and L