3. Free electron energies in reduced zone. Consider the free electron energy bands of an fee crystal lattice in the approximation of an empty lattice, but in the reduced zone scheme in which all k's are transformed to lic in the first Brillouin zone. Plot roughly in the [111] direction the energies of all bands up to six times the lowest band energy at the zone boundary at k = (2π/a), ). Let this be the unit of en- ergy. This problem shows why band edges need not necessarily be at the zone cen- ter. Several of the degeneracies (band crossings) will be removed when account is taken of the crystal potential.