Effective mass in a three-dimensional crystal (30 points) A three dimensional cubic crystal with lattice constant a has the following energy- wavevector relation: 2 kya 2 E(K) = E sin +sin. +sin ka 2 ka 2 (1) (a) Identify the points in the first Brillouin zone where the energy reaches a minimum or a maximum (b) Determine the effective mass (m*) along the x-direction at the band extrema. Calculate m* using Eo = 0.2 eV and a = 5 . (c) Is the effective mass dependent on the crystal directions x, y, z? How would you change Eq. (1) in order to obtain different effective masses in different directions?