Free electron energies in reduced zone. Consider the free electron energy bands of an fcc crystal lattice in the approximation of an empty lattice, hut in the reduced zone scheme in which all k’ s are transformed to lie in (he 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) (½, ½, ½). Let this he the unit of energy. This problem shows why band edges need not necessarily be at the zone center. Several of the degeneracies (band crossings) will be removed when account is taken of the crystal potential.