Propagation and localisation of vibrational modes in 3-dimensional disordered systems: the binary force constant model

We consider a system of coupled harmonic oscillators on a cubic lattice. Th e force constants are supposed to take two distinct values at random accord ing to a bond concentration x. The density of states (DOS) is evaluated bot h by numerical diagonalisation and in coherent-potential approximation (CPA ). There is excellent agreement between the results of the two methods. Nea r the concentration, where the bonds with the larger force constants percol ate, the DOS differs appreciably from the crystalline one and is anomalousl y enhanced at low frequencies as compared to Debye's omega(2) law ("boson p eak"). These features are shared with models with continuous distributions of force constants. The mean free path near the strongest anomaly is compar able to the wavelength of the excitations. However, the level distance stat istics reveals that the corresponding states are delocalised.