Problem: Energy Gap

     

In the presence of a weak periodic potential, , the energy band of the one-dimensional free-electron gas will develop an energy gap of magnitude . The gap will open at wavenumbers , and around the energy . Calculate the density of states, assuming that the gap is much smaller than . (For the purpose of this calculation the two branches of the unperturbed energy spectrum can be approximated by , where is the group velocity of the electrons. The unperturbed density of states is constant, .)