A short theoretical introduction of the Lamb theory on the vibrations of elastic spheres will be described for the free nanoparticles as well as for the nanoparticles embedded in matrix. The spherical case is well understood – the normal modes of the sphere are divided into torsional and spheroidal and, experimentally, have been measured in various situations. Here, the application of the Lamb theory will be illustrated by the low frequency Raman scattering (LFRS) measurements on different powder samples: TiO₂, SnO₂, and CdS. The effects of sintering of nanoparticles will be described with the ZrO₂ nanoparticles doped with Sn, Ge and Cu. For the nanoparticles embedded in matrix a theoretical treatment of polarization-dependent low-frequency off-resonant Raman scattering which establishes a relation between the particle size, the frequencies, and the widths of various phonons. The selection rules for the Raman scattering of the spherical particles and of the particles of irregular shape will be discussed. Polarized and depolarized scattering from confined acoustic phonons were distinctly resolved near the laser line. A good agreement between the experimental and the theoretical results will be illustrated by TiO₂, CdS_xSe_1-x, and HfO₂ nanoparticles in glass matrix. The nanoparticle sizes and even sizes distribution obtained from Raman scattering agree well with those obtained from transmission electron microscope proving the LFRS to be a simple, fast and reliable method for the size distribution measurements. By inverse procedure, starting from the Raman spectra and known particles size distribution, the mean sound velocities of longitudinal and transverse phonons of nanoparticles could be deduced, providing LFRS to be unique technique for determination of the sound velocities in nanoparticles. The possibility of coherent Raman scattering will be illustrated by the LFRS on the self-organized Ge superlattice.

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