The density of electrons in a perfect or imperfect lattice

The spatial distribution of electronic charge is calculated exactly for three simple cases. These are the perfect lattice, perfect lattice with a single vacancy and perfect lattice with a single substitutional impurity. The potential is assumed to be of muffin tin form. The results are similar in appearance to the equations for the Kohn-Rostoker band structure calculations and appear ideally suited to numerical computation. Examples are given to illustrate the results. These are rather simple, but two interesting conclusions can be drawn: the charge at a vacancy is not directly proportional to the density of states of the host lattice and an impurity can cause two bound states to appear from a narrow band.