Phonon scattering cross section of polydispersed spherical nanoparticles

An approximate analytical solution is proposed to estimate the phonon scattering cross section of polydispersed spherical nanoparticles. Using perturbation of the Hamiltonian due to differences in mass and bond stiffness between a host medium and a spherical nanoparticle, an analytical solution is obtained for the scattering cross section in the Rayleigh limit when the size parameter approaches zero. In the geometrical scattering limit, when the size parameter approaches infinity, the van de Hulst approximation for anomalous diffraction is used to estimate the scattering cross section as a function of acoustic impedance mismatch between the host medium and the spherical nanoparticle. Finally, these two limiting cases are bridged by a simple expression to estimate the scattering cross section for intermediate values of the size parameter. Using this, the scattering cross section for a polydispersed distribution of spherical nanoparticles was also estimated as a function of the parameters defining the statistical size distribution.