8 - Localized States and the Approach to Localization

Abstract

This chapter focuses on the self-consistent theory of localization. The C factor involves a 2d-fold integral for the maximally crossed (MC) diagram contributions. To locate the mobility edge and to calculate the localization length in d = 3 and 2, the integral is difficult to evaluate because of the nonspherical Brillouin zone. An important point about the (MC) contribution is that it does not obey the particle-hole symmetry about the center of the band. The localization is very unlikely in the long-wavelength limit because the Rayleigh frequency dependence tells that the scattering must be weak in the long-wavelength limit, in contrast to the quantum case. At high frequencies, geometric optics becomes a good description for classical wave propagation. In that regime, the effect of scattering, as expressed by the mean free path, is expected to saturate at a scale comparable to the particle size or interparticle separation.