A 1D lattice with a basis composed of two atoms and weak delta-function periodic potential

exercise 3_4956

Consider a one dimensional solid of length L = Na , which possesses a basis composed of two identical atoms. The distance between the atoms in the basis is b ( b < a /2). The centers of adjacent pairs of atoms are a distance a apart, which is the 1D lattice constant. We represent the potential energy as a sum of delta functions on each atom, as shown below.

(a) If A is sufficiently small, the nearly free electron model can be used with 1 st order perturbation theory to calculate the band gaps. Calculate the Fourier components U G where G m = 2 p m / a and m is a positive or negative integer. Find the magnitude of the energy gaps ( E gm ) at the m th Brillouin zone boundary, i.e., where k m = G m /2.

(b) If b = a /2, what happens to the energy gaps calculated in (a)? Give a qualitative explanation.