Raman Scattering from Acoustic Phonons Confined in Spherical Nanoparticles

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. Parameters of this model are transverse $(v_T)$ and longitudinal $(v_L)$ sound velocities of material, i.e. elastic properties of bulk material. Frequencies of vibrational modes are scaled as function of mentioned bulk parameters for symmetric l=0 and quadrupolar l=2 spheroidal modes, in the case of stress-free boundary conditions. Calculated values are compared with the low-frequency Raman experimental results from literature (Ge, Si, CdS, CdSe, $CeO_2$, ...). These calculated relations can be practically used to examine nanoparticles of any bulk material. We presented also a procedure how to establish $v_L$ and $v_T$ of material from low-frequency Raman spectra and dimension d of particles.