Phonon and fracton vibrational modes in disordered harmonic structures: A self-consistent theory

The density of vibrational modes N(ω) of a harmonic lattice with bond or site disorder is calculated. We predict the existence in three dimensions of a crossover frequency ωc~|c-c*|3/2, where c* is the critical concentration for percolation. For c>c*, N(ω)~ω² if ωωc (phonon regime), and N(ω)~ω1/3 if ωωc (fracton regime). This is the first analytical theory to predict a value of 4/3 for the spectral dimension of a percolation cluster in three dimensions, in agreement with the conjecture of Alexander and Orbach and with recent simulations.