Note on the Slow Electrons in a Polar Crystal

A new perturbation technique is developed to investigate the properties of a slow electron in a polar crystal at absolute zero. The low-lying energy levels in a conduction band are calculated up to the order of K2 by the second order perturbation calculation, where ℏK is the total momentum of the system, i.e. that of the polaron. For K = 0 the energy agrees nearly with that obtained by the variational method. In the neighbourhood of K = 0 the effective mass becomes m(1-a/6)-1, a being the coupling constant between an electron and the longitudinal optical modes of lattice vibrations. It is also shown that for K = 0 the eigenfunction and the eigenvalue of the total Hamiltonian obtained by the variational procedure coincide with those in the zero-th approximation of the present perturbation method. The second order wave function is constructed which enables us to calculate the electro-static potential and the charge density due to the ionic polarization around the electron.