  Resonantly enhanced Raman scattering in dielectric nanostructures has been
recently proven to be an effcient tool for developing nanothermometry and
experimental determination of their mode- composition. In this paper, we
develop a rigorous analytical theory based on the Green's function approach to
calculate the Raman emission from crystalline high-index dielectric
nanoparticles. As an example, we consider silicon nanoparticles which have a
strong Raman response due to active optical phonon modes. We relate enhancement
of Raman signal emission to Purcell effect due to the excitation of Mie modes
inside the nanoparticles. We also employ the numerical approach to the
calculation of inelastic Raman emission in more sophisticated geometries, which
do not allow a straightforward analytical form of the Green's function
description. The Raman response from a silicon nanodisk has been analyzed
within the proposed method, and the contribution of the various Mie modes has
been revealed.
