Response of a semi-infinite cubic lattice to uniform electric fields

Self-consistent field equations for the dipole moments of point polarizable atoms in slabs of cubic lattices with a uniform applied electric field are constructed. Results of Toeplitz operator theory are used to characterize the ranges of atomic polarizability for which there are unique solutions to the equations. A normal mode analysis of the frequency spectrum of the coupled dipole lattice is given and is used in the interpretation of the results for the simple cubic lattice. Approximate solutions of the self-consistent field equations for the semi-infinite lattice are constructed which display the exponential approach of the atomic dipoles to their infinite lattice value with increasing penetration into the lattice.