ABSTRACT
Clusters are aggregates of a nite number of atoms or molecules [1], bridging the gap between the isolated atom/molecule and the macroscopic solid state of matter, as schematically shown in Figure 7.1. For large clusters, the development of properties is typically a smooth change with the size, o en connected to the surface/volume ratio. For small clusters, this smooth development is replaced by increasingly strong variations in the property. is is the so-called quantum size regime in which each atom makes a di erence. In some sense, such small clusters can be regarded as molecules, with each size being a unique entity. If a cluster is larger than just a few atoms, it o en adopts the geometry that minimizes the surface-atom fraction: the surface atoms generally have lower coordination (number of nearest neighbors) than the inner, bulk atoms, and the cluster thus reaches the lowest energy structure by maximizing the bulk-atom fraction. is has been shown to lead to the peculiarly stable geometric shell structures, rst observed as “magic numbers” in mass spectra for rare gas clusters [2]. In the cluster formation process (discussed in more detail below), those aggregates are more likely to be created which have enclosed, onion-like, close-to-spherical layers, or shells. Such geometry gives certain well-de ned numbers of atoms per cluster in a stable con guration-the “magic”
numbers. e phenomenon of magic numbers is already observed in the case of a cluster containing 13 atoms, for which the most stable geometry is o en an icosahedron-a geometric structure with all the vertices on a spherical surface. e 13th atom is in the center of such a spherical cluster. e number of atoms N in a cluster with m enclosed completed icosahedral shells can be calculated exactly:
− −
21 = (2 1)(5 5 +3) 3
mN m m m
(7.1)
with the number of atoms Nm in each mth shell de ned by the formula
= + > 210 – 20 12, 1mN m m m (7.2)
us, the magic numbers for the icosahedral geometry are 13, 55, 147, 309, 561, 923, etc. It is the clusters with these amounts of atoms per unit that are more abundant in the mass spectra of inert gas clusters. It should be noted here that the same formulas are valid for the shells of cuboctahedral geometry in which case shells are not any more spherical.
