PHZ 6426 (Solid State I) Fall 1997

Homework 4, due October 3

1. Construct the "empty lattice" band structure of an fcc Bravais lattice along the (111) reciprocal lattice direction. Draw the energy bands along this direction within the reduced zone scheme. Show all levels up to an energy six times that of the lowest band at the zone boundary. Indicate the precise energy of each level at the zone center and at the zone boundary.
2. Suppose that the crystal has a weak periodic potential having the following nonzero Fourier components:
   • V( K in {111}) = 0.3.
   • V( K in {200}) = 0.4.
   Recall that {hkl} means "(hkl) and all directions related to it by crystal symmetry". All other Fourier components are negligibly small.
   Consider each of the empty-lattice bands from part 1. Calculate to first order in the lattice potential the energy of each of these bands at the zone center and at the zone boundary along the (111) direction. Sketch the (111) energy bands across the entire zone.