Question 28: (Bands in 3D) Recall the following general expression for the band structure of a tight-binding model: E(k) = = ε0-t Σε where the sum runs over all the lattice points, which are nearest neighbors of the origin. The parameters t and ε were explained in the lectures. (a) Show that the band structure E(k) for an fcc lattice with lattice constant a is: E(K) = 80- 4t {cos kxa 2 kya COS + cos 2 kxa kza 2 2 COS +cos⋅ kya cos za 2 2 (b) Close to the origin I in k-space, one can make a Taylor-expansion of E(k) in orders of K. The resulting second-order term may be mapped on a free-electron type model with: E(k) = const. + h²k² 2m* where m* is the effective electron mass. Carry out that Taylor expansion, and determine m*(t, a).