Vibrational modes of free nanoparticles: from atomic to continuum scales

Vibration analysis of free standing silicon nanoparticles, with sizes ranging from 0.732 to 4.223 nm, are calculated using two different methods: molecular mechanics and classical continuum elasticity. Three different geometries are studied: cubes, spheres, and tetrahedrons. Continuum mechanics methods provide good estimates of the lowest natural frequency of particles having at least 836 (R>1.5 nm) and 800 (R>1.28 nm) atoms for cube- and tetrahedron-shaped nanostructures, respectively. Equations for vibrational frequencies of smaller particles as a function of size are proposed. The vibrational modes obtained from both methods were practically the same for the sphere- and tetrahedron-shaped particles with a large number of atoms. However, for the cube geometry only the shape of the modes corresponding to the lowest couple of frequencies occur in the same order. In general, vibrational modes shapes obtained using both methods are the same although the order in which they appear may be shifted.