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. The force constants are supposed to take two distinct values at random according to a bond concentration x. The density of states (DOS) is evaluated both by numerical diagonalisation and in coherent-potential approximation (CPA). There is excellent agreement between the results of the two methods. Near the concentration, where the bonds with the larger force constants percolate, the DOS differs appreciably from the crystalline one and is anomalously enhanced at low frequencies as compared to Debye’s ω law (”boson peak”). These features are shared with models with continuous distributions of force constants. The mean free path near the strongest anomaly is comparable to the wavelength of the excitations. However, the level distance statistics reveils that the corresponding states are delocalised.