This is usually identified in a reduced DOS representation, g ( ω ) / ω 2 vs ω, which shows a peak which can be detected experimentally by methods like inelastic neutron scattering or nuclear inelastic scattering. In contrast, amorphous materials show an excess contribution at low frequencies. In crystals, the low frequency DOS, g ( ω ), follows the so-called Debye model well that is, by simple counting of possible sound waves compatible with the boundary conditions one obtains the frequency dependence of the DOS, g ( ω ) ∝ ω 2. Researchers have known for more than four decades that the density of vibrational states (DOS) of amorphous materials differs in a characteristic way from that of crystalline ones. The modes lying closest to the transverse acoustic Van Hove singularity are marked in red to allow easier tracing. To avoid crowding of the picture, the system size is only 5 × 5 × 5 here. The middle panel shows the evolution of energy levels for interpolations of the dynamical matrix between the two systems. The histogram is established from numerical calculations of eigenvalues in 25 instances of a 23 × 23 × 23 lattice. The right panel shows that after introduction of disorder in the force constants, the reduced DOS exhibits a boson peak (pink curve). The blue curve shows the classic Van Hove singularities (the discontinuities in the DOS). The left panel shows the DOS of a simple cubic crystal, analytically calculated, with transverse force constants being a fourth of the longitudinal. Show moreįigure 1: Evolution of the vibrational DOS in a model with disorder in the force constants. Evolution of the vibrational DOS in a model with disorder in the force constants. Figure 1: Evolution of the vibrational DOS in a model with disorder in the force constants.