Free electron energies in simple square and cubic lattices at the 1st Brillouin Zone boundaries

exercise 3_4957

(a) Show for a simple square lattice (i.e., in two dimensions which possesses a lattice constant a ) that the energy of a free electron at a corner of the first Brillouin Zone (BZ) is higher than that of an electron at midpoint of a side edge (i.e., a face) of the 1 st BZ by a factor of 2.

(b) What is the corresponding factor for a simple cubic lattice possessing a lattice constant a in three dimensions (3D)?

(c) What bearing might the result of (b) have on the conductivity of divalent metals? Explain.