Free electron energies of an fcc lattice in the reduced zone scheme

exercise 3_4958

Consider the free electron energy bands of an fcc crystal lattice in the approximation of an empty lattice (i.e., without the periodic potential), but in the reduced zone scheme in which all k 's are transformed to lie 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 = (2 p / a )( 1/2, 1/2, 1/2). Let this energy at the zone boundary be 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 the crystal potential is taken into account.