Clusters magic numbers

France M R, Buchanan J W, Robinson J C, Pullins S FI, Tucker J T, King R B and Duncan M A 1997 Antimony and bismuth oxide clusters growth and decomposition of new magic number clusters J. Phys. Chem. A 101 6214... 

The metal nuclei of the so-called full-shell clusters, also referred to as magic number clusters, generally exhibit hexagonal or cubic close packing (ccp). Figure 6 shows the stepwise formation of the first magic-number clusters built by 13, 55, 147, and 309 atoms. 

Figure 22. Series of mass spectra showing progression of the etching reaction of aluminum anions with oxygen. Note that magic number clusters corresponding to electron shell closings for 40 and 70 electrons (AI13 and AI23) appear as the terminal product species of reactions with oxygen at flow rates of (a) 0.0, (b) 7.5, and (c) 100.0 standard cubic centimeters per minute (seem). Taken with permission from ref. 92.

C.A. Schalley, P. Weis, Unusually stable magic number clusters of serine with a surprising preference for homochirality. Int. J. Mass Spectrom. 221, 9-19 (2002)... 

The fullerenes were first discovered by laser ablation cluster ion formation in a mass spectrometer.89 Soon after the identification of C60 as a magic number cluster, [LaC60]+ was also identified.90 Later, macroscopic quantities of fullerenes were produced by the carbon arc method91 soon followed by the macroscopic production of the lanthanum endohedral fullerenes.92 Fullerene chemistry has developed rapidly since the preparation of macroscopic quantities of these compounds. The development of the metallofullerenes has been hampered by the low yield... 

HqO This cluster has not been the source of special theoretical difficulties. All studies agree that the Eigen structure is very stable, corresponding to some "magic number" cluster features. It is illustrated by the first shell of Figure la. 

Figure 2. Magic-number clusters M obtained by surrounding a given atom by successive shells of atoms (the illustration is for cuboctahedral packing).

Figure 4. The 3x3x3 and 3 X 3 X 4 fee erystallite structures proposed by Pilgrim and Duncan for intermediate sized Ti/C and Zr/C magic number clusters and the structure of the Tii4Ci3 neutral cluster optimized by Dance from DFT calculations (Ti -C = 2.13 A, Ti5-C = 2.17 A, 713-0 = 2.05 A Ti -Ti = 3.02 A, Ti3-Ti3 = 2.89 A the superscripts are the coordination numbers, that is the number of bonds with atoms of the other element). Reproduced with permission from the authors.

Figure 3.1 Hexagonal close-packed full-shell magic number clusters of spherical atoms. The series starts as a single atom and then 12 others can be placed around it aU touching the central atom, thus giving 12-fold coordination. This is the first shell. The number of atoms in the nth shell is lOn + 2. The number of atoms in the total cluster is (5/3)n(n -I- 1) (2n + 1) + 2n 1.

Clusters with full shell atomic lattices are known as magic number clusters they have an extra stability due to this packing nature. The number of surface atoms Sq in nth shell for different crystal systems is as follows ... 

Section 11.2 includes a rough calculation to estimate the maximmn coordination number of a set of identical spheres. The result is that one sphere may be immediately smroimded by up to 12 spheres, giving a magic number cluster size of 13. Estimate the next magic number cluster size by this method. THINKING AHEAD [As the cluster grows from 13, where will the next spheres be added ]... 

Alternatively, one may compare the total energies for the energetically two lowest isomers of the same cluster size, i.e., tot(N, 2) - Etot N, 1). This has peaks for clusters that are particularly stable compared to the next isomer of the same cluster size. As seen in O Fig. 26-3, many of the magic-numbered clusters are also particularly stable according to this second criterion. This suggests that the characterization of certain clusters to be particularly stable is very robust. 

---