  We have used density functional perturbation theory to investigate the
stiffness of interatomic bonds in small clusters of Si, Sn and Pb. As the
number of atoms in a cluster is decreased, there is a marked shortening and
stiffening of bonds. The competing factors of fewer but stiffer bonds in
clusters result in softer elastic moduli but higher (average) frequencies as
size is decreased, with clear signatures of universal scaling relationships. A
significant role in understanding trends is played by the coordination number
of the bulk structure: the higher this is, the lesser is the relative softening
of elastic constants, and the greater the relative damping of vibrational
amplitudes, for clusters compared to the bulk. Our results could provide a
framework for understanding recent reports that some clusters remain solid
above the bulk melting temperature.
