  In order to clarify the relationship between the phonon spectra of
nanoparticles and their melting temperature, we studied in detail the
size-dependent low energy vibration modes. A minimum model with atoms on a
lattice and harmonic potentials for neighboring atoms is used to reveal a
general behavior. By calculating the phonon spectra for a series of
nanoparticles of two lattice types in different sizes, we found that density of
low energy modes increases as the size of nanoparticles decreases, and this
density increasing causes decreasing of melting temperature. Size-dependent
behavior of the phonon spectra accounts for typical properties of
surface-premelting and irregular melting temperature on fine scales. These
results show that our minimum model captures main physics of nanoparticles.
Therefore, more physical characteristics for nanoparticles of certain types can
be given by phonons and microscopic potential models.
