This proposal applies diffraction analysis to the local atomic structure of crystalline solids, in particular superconducting oxides and related materials. Initial studies include the effects of dynamics on the pulsed neutron powder diffraction (Plazcek correction) and the direct observation of lattice dynamics using pulsed neutrons. This research is followed by diffraction studies of single crystals using a novel technique of two-dimensional pair distribution function analysis. The observations are examined in the context of the mechanism of superconductivity based upon carrier-induced local lattice deformation (polaron formation). In the process of these studies the real space method of diffraction analysis is further developed. %%% The data derived from these experiments are pertinent to furthering the understanding of the microscopic mechanisms of high temperature superconductivity.