  A polaron is an electron interacting with a polar crystal, which is able to
form a bound state by using the distortions of the crystal induced by its own
density of charge. In this paper we derive Pekar's famous continuous model for
polarons (in which the crystal is replaced by a simple effective Coulomb
self-attraction) by studying the macroscopic limit of the reduced Hartree-Fock
theory of the crystal. The macroscopic density of the polaron converges to that
of Pekar's nonlinear model, with a possibly anisotropic dielectric matrix. The
polaron also exhibits fast microscopic oscillations which contribute to the
energy at the same order, but whose characteristic length is small compared to
the scale of the polaron. These oscillations are described by a simple periodic
eigenvalue equation. Our approach also covers multi-polarons composed of
several electrons, repelling each other by Coulomb forces.
