A 2D square lattice possessing a weak periodic potential

exercise 3_4953

Consider a square lattice in two dimensions having a lattice constant of a with a crystal potential U ( x , y ) = -4 U cos(2 p x / a ) cos(2 p y / a ). Using Fourier analysis and by determining which reciprocal lattice vectors fulfill the condition for Bragg reflection of electron waves, find approximately to first order the energy gap ( E g ) at the corner point ( p / a , p / a ) of the First Brillouin Zone.