Topics in the Physics of Granular Materials.

In this dissertation, we treat several problems in systems consisting of two kinds of materials with very different physical properties. Examples are random mixtures of normal metal and insulator (N/I), and random mixtures of superconductor and normal metal (S/N). After a review of the basic ideas of percolation theory, the problem of anomalous frequency dependent transport in the presence of a singular distribution in the normal metal conductivity in a N/I mixture is studied. In particular, if the conductivity follows the distribution P(sigma)~ sigma^{-alpha}, then in the dilute limit of normal metal and in the low frequency limit, the real part of the conductivity behaves as omega^{2-alpha}. Analogous problem in a S/N mixture is also studied. We also treat the problem of nonuniversal behaviors in the critical current density J_{c} in a S/N mixture, and in the critical applied field E_ {c} in a N/I mixture. Near the percolation threshold p_{c}, J _{c}~ (p - p_{c})^{v} and E_{c}~ (p_ {c} - p)^ {y}. The exponents v and y are obtained in both lattice and continuum systems. In a S/N mixture in which the conductance of the normal component is fluctuating, the relative fluctuation in the conductance of the whole mixture diverges with an exponent k^' as the threshold is approached. We calculate the exponent k^' using the Migdal-Kadanoff renormalization technique. In two dimensions, k^' = 1.339; and in three dimensions, k^' = 0.660. Two problems in dilute composites are treated. The effects of clustering of small metal particles in a composite on the absorption is studied using a differential effective medium approximation. For fractal clusters with radius R and fractal dimension d_{f}, the enhancement factors in the electric dipole and magnetic dipole absorptions are found to be (R/a) ^{5/2(3-d_{f})} and (R/a)^{1/2(1 + d_{f })} respectively, where a is the radius of the small particles. The effective dielectric tensor in a mixture consists of a small fraction of Faraday-active component is obtained using the Maxwell-Garnett approximation. The off-diagonal part of the dielectric tensor and the angle of Faraday rotation per unit length of sample is calculated.