Magic numbers in clusters

The next magic number for jellium clusters is 40. This is a particularly important magic number in cluster chemistry, since numerous 40 valence electron bare clusters with 9 to 11 vertices of the post-transition elements in Groups 13 to 15 are known as isolable species in intermetallics or salts with suitable counterions. Examples of such species include lun, Geg", and Big, all of which have been isolated in intermetallics (for Inn ) or as stable salts with suitable counterions (Geg" and Big ) and characterized by X-ray crystallography. 

Work on clusters also had its origin in this decade. Initial work led in following years to the production of the well-known Stuttgart pseudopoten-tials, which enables the realistic calculation of systems containing heavier elements. It also prepared the route to many studies on magic numbers in cluster chemistry and eventually to fullerenes and nanotubes. 

Fig. 12. Mass spectra of singly charged clusters composed of a single Qo molecule coated with a large amount of Na (background subtracted). The even-odd alternation extends up to approximately x = 66. Note that x = 12 does not appear as a magic number in these spectra.

Pioneering work on the mass spectra of Na clusters by Knight et at.34 provided the first insight that clusters and nuclear physics have something in common. They observed that Na clusters consisting of 2, 8, 10 and 40 atoms were unusually stable and coincided with the magic number in nuclear physics where nuclei with the same numbers of protons and/or neutrons were known to be very stable.34... 

The first experimental evidence for the existence of magic numbers was reported by Recknagel and coworkers (Echt et al. 1981). Since then, a number of experimental and theoretical studies of magic numbers in rare gas clusters have been published (Buck and Meyer 1986 Carnovale et al. 1989 Castleman et al. 

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. 

The fact that the magic numbers are the same as for pure Li or Na clusters is easily understood. If we think again in terms of the jellium-on-jellium model of Sect. 6, w + (Li) — n+(Na) = 0.0029 a.u., which is a very small number. This indicates that a jellium model, in which the jellium density is an average of the Li and Na densities, will predict well the magic numbers in the mixed clusters. However this avera I jellium model would be unable to say anything about the distribution of atoms in the cluster, or about mixing properties. 

Sakurai M, Watanabe K, Sumiyama K, Suzuki K (1999) Magic numbers in transition metal (Fe, Tr, Zr, Nb, and Ta) clusters. J Chem Phys 111 235-238... 

FIGURE 13.11 Idealized nanoclusters of close-packed atoms with one to five shells of atoms, together with the numbers of atoms (magic numbers) in these clusters. (Reproduced with permission from Ref. 40c.)... 

Anagnostatos, G. S. (1987). Magic numbers in small clusters of rare-gas and alkali atoms. Physics Letters A, 124, 85. 

Thus a simple question arises Based on this model, at which size can we expect a stable alkaline-earth metal cluster As each alkaline-earth metal has two itinerant valence s electrons, stable clusters are expected for clusters constituting 4, 10, 20,... atoms, which was indeed confirmed by experiments. Although the configuration IS IP 1D ° corresponds to a closed electronic structure, 18 was not observed as a magic number in homogeneous clusters. This is because both ID and 2S levels... 

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