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In recent years, the quasi-particle spectra of various condensed systems, crystalline as well as disordered and amorphous, became also the “object” of applications and technical developments and not only of fundamental research. This led to the interest in the theoretical and experimental study of the quasi-particle spectrum of such compounds, which are among the most popular and advanced structural materials. Most of these substances have heterogeneous structure, which is understood as a strong spatial heterogeneity of the location of different atoms and, consequently, the heterogeneity of local physical properties of the system, and not as the coexistence of different phases (i.e. heterophase). To these structures belong disordered solid solutions, crystals with a large number of atoms per unit cell as well as nanoclusters. This chapter is devoted to the study of vibration states in heterogeneous structures. In such systems, the crystalline regularity in the arrangement of atoms is either absent or its effect on the physical properties of the systems is weak, affecting substantially the local spectral functions of different atoms forming this structure. This effect is manifested in the behavior of non-additive thermodynamic properties of different atoms (e.g. mean-square amplitudes of atomic displacements) and in the contribution of individual atoms to the additive thermodynamic and kinetic quantities. The most important elementary excitations appearing in crystalline and disordered systems are acoustic phonons. Moreover, in heterogeneous nanostructures the application of the continuum approximation is significantly restricted; therefore we must take into account the discreteness of the lattice. This chapter contains a theoretical analysis at the microscopic level of the behavior of the spectral characteristics of acoustic phonons as well as their manifestations in the lowtemperature thermodynamic properties. The chapter consists of three sections. The first section contains a detailed analysis at the microscopic level of the propagation of acoustic phonons in crystalline solids and disordered solid solutions. We analyze the changes of phonon spectrum of the broken crystal regularity of the arrangement of atoms in the formation of a disordered solid

The Features of Low Frequency Atomic Vibrations and Propagation of Acoustic Waves in Heterogeneous Systems