2. (i) Show that in the nearest neighbour hopping model, the tight-binding s-band energies for bee and fee lattices of lattice parameter a are given by (a) fcc lattice: Ek kyk₂)=(+4B ( COS ka 2 kya ka k.a kya ka COS +cos COS +cos COS 2 2 (b) bec lattice: E(ka, ky, ks) (0) +8B cos B(Cos ka 2 kya ka Cos COS 2 2 (c) Find the maximum and minimum energies and the bandwidth in each case. Explain qualitatively how the band width will change as the atoms are brought closer to each other or taken further apart. (a-lattice parameter, B-nearest neighbour hopping element) (4,4,4 marks) (ii) For the bcc case, show that the above result explicitly satisfies the condition E(k) E(k+G), by considering one of the shortest nonzero reciprocal lattice vectors G. Show also that the results satisfy the rotational symmetry of a cubic lattice, by considering the rotation by 7/2 along one of the axes in k-space, say k. Show that the energy values remain unchanged under such a rotation. (4,4 marks)