The Nature of Vibrations of a 2-D Disordered Lattice Model with   Nano-scale Order
  Motivated by the unusual soft mode anomalies in the relaxor ferroelectrics, we present a numerical study of vibrational excitations of a disordered lattice model with chemical order on the nano-scale. We find an Ioffe-Regel crossover in acoustic modes separating propagating low-frequency modes from the intermediate frequency diffusons, and localized modes at the high-frequency end of the spectrum. At a fixed degree of disorder, we find increasing uncertainty in the momentum of long wave-length optical modes with decrease in the size of nano-ordered regions. We determine the cause for this to be a strong mixing between acoustic and optical modes, believed to be central to the model of waterfall phonon anomalies. 
Prasenjit Ghosh and Umesh V. Waghmare