We show that a vibrational instability of the spectrum of weakly interacting quasi-local harmonic modes creates the maximum in the inelastic scattering intensity in glasses, the Boson peak. The instability, limited by anharmonicity, causes a complete reconstruction of the vibrational density of states (DOS) below some frequency ωc, proportional to the strength of interaction. The DOS of the new harmonic modes is independent of the actual value of the anharmonicity. It is a universal function of frequency depending on a single parameter -the Boson peak frequency, ω b which is a function of interaction strength. The excess of the DOS over the Debye value is ∝ ω 4 at low frequencies and linear in ω in the interval ω b ≪ ω ≪ ωc. Our results are in an excellent agreement with recent experimental studies.

We consider diffusion of vibrations in 3d harmonic lattices with strong force-constant disorder. Above some frequency ωIR, corresponding to the Ioffe-Regel crossover, notion of phonons becomes ill defined. They cannot propagate through the system and transfer energy. Nevertheless most of the vibrations in this range are not localized. We show that they are similar to diffusons introduced by Allen, Feldman et al., Phil. Mag. B 79, 1715(1999 to describe heat transport in glasses. The crossover frequency ωIR is close to the position of the boson peak. Changing strength of disorder we can vary ωIR from zero value (when rigidity is zero and there are no phonons in the lattice) up to a typical frequency in the system. Above ωIR the energy in the lattice is transferred by means of diffusion of vibrational excitations. We calculated the diffusivity of the modes D(ω) using both the direct numerical solution of Newton equations and the formula of Edwards and Thouless. It is nearly a constant above ωIR and goes to zero at the localization threshold. We show that apart from the diffusion of energy, the diffusion of particle displacements in the lattice takes place as well. Above ωIR a displacement structure factor S(q, ω) coincides well with a structure factor of random walk on the lattice. As a result the vibrational line width Γ(q) = Duq 2 where Du is a diffusion coefficient of particle displacements. Our findings may have important consequence for the interpretation of experimental data on inelastic x-ray scattering and mechanisms of heat transfer in glasses.

We show that the same physical mechanism is fundamental for two seemingly different phenomena such as the formation of two-level systems in glasses and the boson peak in the reduced density of low-frequency vibrational states g͑͒ / 2 . This mechanism is the vibrational instability of weakly interacting harmonic modes. Below some frequency c Ӷ 0 ͑where 0 is of the order of Debye frequency͒, the instability, controlled by the anharmonicity, creates a new stable universal spectrum of harmonic vibrations with a boson peak feature as well as double-well potentials with a wide distribution of barrier heights. Both are determined by the strength of the interaction I ϰ c between the oscillators. Our theory predicts in a natural way a small value for the important dimensionless parameter C = P ␥ 2 / v 2 Ϸ 10 −4 for two-level systems in glasses. We show that C Ϸ͑W / ប c ͒ 3 ϰ I −3 and decreases with increasing interaction strength I. The energy W is an important characteristic energy in glasses and is of the order of a few Kelvin. This formula relates the two-level system's parameter C with the width of the vibration instability region c , which is typically larger or of the order of the boson peak frequency b . Since ប c տប b ӷ W, the typical value of C and, therefore, the number of active two-level systems is very small, less than 1 per 1 ϫ 10 7 of oscillators, in good agreement with experiment. Within the unified approach developed in the present paper, the density of the tunneling states and the density of vibrational states at the boson peak frequency are interrelated.

The vibrational properties of model amorphous materials are studied by combining complete analysis of the vibration modes, dynamical structure factor and energy diffusivity with exact diagonalization of the dynamical matrix and the Kernel Polynomial Method which allows a study of very large system sizes. Different materials are studied that differ only by the bending rigidity of the interactions in a Stillinger-Weber modelization used to describe amorphous silicon. The local bending rigidity can thus be used as a control parameter, to tune the sound velocity together with local bonds directionality. It is shown that for all the systems studied, the upper limit of the Boson peak corresponds to the Ioffe-Regel criterion for transverse waves, as well as to a minimum of the diffusivity. The Boson peak is followed by a diffusivity's increase supported by longitudinal phonons. The Ioffe-Regel criterion for transverse waves corresponds to a common characteristic mean-free path of 5-7Å (which is slightly bigger for longitudinal phonons), while the fine structure of the vibrational density of states is shown to be sensitive to the local bending rigidity.

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