Selected topics in lattice dynamics: a critical review (including breathers)

We present a review of lattice dynamics to provide the underpinnings for the study of nonlinear localized modes, the so-called breathers. After a historical survey we address the following topics: harmonic theories, anharmonic perturbation theory, self-consistent theories, classical simulation techniques, path-integral theories, realistic crystal potentials, and intrinsic localized modes. We discuss both static and dynamic properties of crystals, e.g., neutron and x-ray scattering. We do not consider transport properties. Throughout, our emphasis is on discussing the major advances in the field and citing the appropriate references. Our aim is to achieve clarity and simplicity for readers who wish to move on to the study of breathers. We have made a special effort to set up the language and notation that is generally accepted in the field. In order to acquaint the reader with the techniques used in lattice dynamics we have analyzed a number of key problems in detail including a comparison with the available experimental data.