This is an analysis of the properties of quasi-local vibrations, and the conditions of the formation thereof, in a realistic model of the crystal lattice on a microscopic scale. The evolution of quasi-local vibrations with an increase in the concentration of impurity atoms, is examined. It is shown that the formation of boson peaks occurs mainly due to the additional dispersion of high-velocity acoustic phonons (connected to the atomic vibrations of the main lattice), caused by the scattering of these phonons by the quasi-local vibrations localized at the impurities. We demonstrate a connection between the boson peaks in disordered systems, and the first van Hove singularity, in regular crystal structures. We analyze the manifestation of quasi-local vibrations and boson peaks, as it relates to the behavior of low-temperature heat capacity, and how it changes with an increasing impurity concentration.

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