The electronic structure of divided metals

These expected changes in magnetic behavior are just one example of so-called quantum-size effects arising from situations in which the Kubo gap , d, is comparable with, or greater than, the thermal energy, kT, at any particular temperature. 

A schematic representation of the emerging discretization of electronic energy levels with decreasing metallic particle size is shown in Fig. 2. The cluster size appropriate for a SIMIT can thus be readily calculated. Assuming the cluster or particle is approximately spherical, it can be shown that the diameter, D, of a particle containing N atoms is given by  

For a prototypical i -band metal such as sodium with Ep = 3.2 eV at room temperature kT 0.025 eV), the condition 3 kT (and presumed metallic conduction on the basis of Eq. 1), would then be realized in a metal cluster or particle containing more than 200 atoms, or a diameter of ca 20 A. Of course, this also leads to the inescapable conclusion that a sodium cluster with a diameter below ca 20 A would be insulating at room temperature  

Of course, the finite size of the system also leads to a significant number of atoms being located on the surface of the cluster or particle. We can estimate the percentage of atoms, P, N), which lie on the surface of a cluster of nuclearity N. For 

We also show in Fig. 3 the elfective percentage of surface atoms for sodium particles as a function of particle diameter. Ranges of Ps N) values for the various cluster size regimes are also given in Table 1, from which it is apparent that clusters of as many as 10000 atoms still have nearly 20% of their atoms on the surface. In fact, Ps N) only drops below 1% for a system with 6.4 x 10 atoms (corresponding to a diameter of approximately 0.16 pm for sodium clusters). 

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