Harmonic Vibrational Excitations in Disordered Solids and the “Boson Peak”

We consider a system of coupled classical harmonic oscillators with spatially fluctuating nearest-neighbor force constants on a simple cubic lattice. The model is solved both by numerical diagonalization and by applying the single-bond coherent potential approximation. The results for the density of states $g\left(\omega \right)$ are in excellent agreement with each other. If the system is near the borderline of stability a low-frequency peak appears in the quantity $g\left(\omega \right)/{\omega }^2$ as a precursor of the instability. We argue that this peak is the analogon of the “boson peak,” observed in structural glasses and other disordered solids. By means of the level distance statistics we show that the peak is not associated with localized states.