Alkali metal clusters, magic numbers

The jellium model of the free-electron gas can account for the increased abundance of alkali metal clusters of a certain size which are observed in mass spectroscopy experiments. This occurrence of so-called magic numbers is related directly to the electronic shell structure of the atomic clusters. Rather than solving the Schrodinger equation self-consistently for jellium clusters, we first consider the two simpler problems of a free-electron gas that is confined either within a sphere of radius, R, or within a cubic box of edge length, L (cf. problem 28 of Sutton (1993)). This corresponds to imposing hard-wall boundary conditions on the electrons, namely... 

The study of bare metal clusters is central to the understanding of the links between solid state chemistry and that of discrete molecular species. Alkali metal clusters have been studied in molecular beams [12, 13], and the theoretical models proposed have attempted to interpret the abundances observed in the mass spectra of these clusters. These spectra show large abundances for specific numbers of metal atoms (N), the so-called magic numbers . Neutral alkali metal mass spectra show peaks at N = 2, 8, 20,40, 58, whereas cationic species show large abundances at N = 19, 21, 35, 41. The theoretical study of alkali metal clusters is simplified by the presence of only 1 valence electron per atom. 

Indeed it has been found [22] that Q30 (which follows ug(3) D sog(3) symmetry) yields correctly all magic numbers experimentally observed for alkali metal clusters up to 1500, the expected limit of validity for theories based on the filling of electronic shells [4], This indicates that u,(3), which is a nonlinear deformation of the u(3) symmetry describing the 3-dimensional isotropic (spherical) harmonic oscillator, is a good candidate for describing the symmetry of alkali metal clusters. 

It was shown in Ref. [22] that the Q30 model correctly reproduces all magic numbers observed for alkali metal clusters up to N = 1500, using for the model parameters the values r = 0.038, A = 0.39. But there, these values were chosen more or less by trial and error, so as to fit the experimental data. With the renormalization of the energy expression described in the previous section, we now have a natural way for determining the deformation parameter r, which is connected with the cluster stability. 

This effect on the stability of clusters is also experimentally observed for alkali (earth) metals (see Figure 10.2) and, in general, extends up to larger cluster sizes than the electronic magic numbers. 

Particularly exciting in cluster science has been how the simple jellium model has been able to in first order explain many quantities for clusters of alkali and noble metals as the appearance of magic numbers. Examples were given in Fig. 3. We have in connection with our studies of the reactivity of... 

The other kind of general behavior observed for both metallic and non-metallic clusters, is that A.fi cohj I and A all change in a non-uniform way consistent with the existence of shell structure within the cluster. That is, there is evidence for extra stability for certain values of N, called magic numbers. The evidence often comprises increased intensity in the mass spectra for the magic numbers. This has been seen for the alkali metals and the noble metals of Group 11. 

In quite another area of physics, the discovery of magic numbers in alkali and other metallic clusters [7] has provided a fresh example of the significance of electronic shell closure. These much larger shells have been shown to oscillate collectively, and the resulting oscillations are of great significance as an example of a many-body resonance. They are discussed at some length in chapter 12. 

When such clusters are formed in beams, essentially by many slow collisions of alkali atoms within a nozzle, it is found experimentally (by analysing the mass distribution using a time of flight spectrometer) that there are discontinuities in the intensity distribution of the peaks from which abundances can be computed for each cluster size. These discontinuities correspond to enhanced stability of metallic clusters around specific sizes (8, 20, 40, 58, etc). They are the same for all the different alkali metals, and are therefore referred to as magic numbers see fig. 12.5. They also turn out to be the same (at least, for the first few magic numbers) as those observed in nuclear physics. This similarity has led to an explanation based on the shell model and to the suggestion that the jellium model can be used to account for the properties of metallic clusters [683]. 

Experiments on noble metal clusters (Cun, AgN, Aun) indicate the existence of shell-effects, similar to those observed in alkali clusters. These are reflected in the mass spectrum [10] and in the variations of the ionization potential with N. The shell-closing numbers are the same as for alkali metals, that is N = 2,S,20,40, etc. Cu, Ag and Au atoms have an electronic configuration of the type nd °(n + l)s so the DFT jellium model explains the magic numbers if we assume that the s electrons (one per atom) move within the self-consistent, spherically symmetric, effective jellium potential. 

Experimental work on clusters containing similar amounts of two alkali-metal species is scarce. The main repre ntative work has been done by Kapp et al [123]. These authors performed supersonic expansions of a mixture of lithium and sodium vapors. The most salient r ulte are a) mixed Na-Li clusters are produced b) the magic numbers, revealed by the abundance in mass spectra, are the same as for the two pure spedes, that k, JV = 2,8,20,40,..., where indi-... 

It is possible to show that our Pythagorean approach can also be used to rationalize the magic numbers discovered in the analysis of the stability properties of clusters formed by either inert-gas [16] or alkali-metal [17] atoms. 

In all the models discussed above, the spin-plus-orbital degeneracy ensures that there will be filled shell electronic configurations at certain sizes. Atoms that have filled shell elecfronic configurations tend to be more stable. Extending that idea to clusters, one can claim that clusters with filled elecfronic shells will be more stable. The exact number of valence electrons that leads to filled elecfronic shells depends on the model, but most of the major peaks in the abundance spectra of the alkali clusters are contained in all the models we have discussed so far. One can, therefore, conclude that the enhanced stability of small metal clusters at specific sizes is due to electronic shell closure effects. These extra-stable clusters have been termed magic clusters in the literature. 

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