Anharmonicity, vibrational instability, and the Boson peak in glasses

We show that a vibrational instability of the spectrum of weakly interacting quasilocal 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 ${\ensuremath{\omega}}_{c},$ 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 ${\ensuremath{\omega}}_{b}$ which is a function of interaction strength. The excess of the DOS over the Debye value is $\ensuremath{\propto}{\ensuremath{\omega}}^{4}$ at low frequencies and linear in $\ensuremath{\omega}$ in the interval ${\ensuremath{\omega}}_{b}\ensuremath{\ll}\ensuremath{\omega}\ensuremath{\ll}{\ensuremath{\omega}}_{c}.$ Our results are in an excellent agreement with recent experimental studies.