Low-frequency Raman scattering from small spherical particles is analyzed. Frequencies of vibrational modes are calculated in elastic continuum approximation, which considers one nanoparticle as homogeneous elastic sphere and no force acting on the surface (so called stress-free boundary conditions). Procedure, similar to one presented in this paper, but for different boundary conditions, can also be done.

            Vibrational modes are classified according to the symmetry group of the sphere by the sphere angular momentum quantum number l. Index n (n=0,1,...) labels the sequence of eigen modes, in increasing order of frequency. On the basis of symmetry consideration, the only Raman-active modes of a sphere are the symmetric l=0 and quadropolar l=2 spheroidal modes.

            Parameters of this model are transverse (v_t) and longitudinal (v_l) sound velocities of material i.e. elastic properties of bulk material. Calculated vibrational frequencies for nanospheres of dimension d, on the basis of bulk material parameters, are usually used to verify w~1/d relationship and to compare nanoparticles dimension determined by this method (experimental low-frequency Raman spectra combined with calculations within this model) and other (XRD,  optical phonon region Raman, etc)  experimental and calculation methods.

 In this paper, vibrational modes frequencies are scaled as function of mentioned bulk parameters for l=0 and l=2 spheroidal modes up to n_max=4, to establish relations between vibrational modes frequencies in the case of stress-free boundary conditions. Calculated relations are compared with low-frequency Raman experimental results from literature (Ge, Si, CdS, CdSe,...) and also with our results for CeO₂ nanoparticles.

 These calculated relations can be practically used to examine nanoparticles of unknown bulk parameters material. One procedure, how to establish v_l and v_t of material from low-frequency Raman spectra of dimension d particles, is presented.

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