Atomic magic numbers

In the atomic shell model, the shells are filled with electrons in order of increasing energy until they fill a closed shell, producing the inert core of noble gases. The atomic magic numbers 2, 10, 18, 36, 54, 86, correspond to the total number of electrons in filled shells [3, 8, 9]. 

Formula 7 generates the following atomic magic numbers ... 

Since Pascal s triangle has proven successful in determining atomic magic numbers, the question arises as to whether it can also help to identify nuclear magic numbers - can it provide a geometrical image and an analytical formula just as for atoms ... 

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.)... 

Novoselov, K. S.,Geim, A. K., Morozov, S.V., Jiang, D., Zhang, Y., Dubonos, S. V., Grigorieva, I. V., Firsov, A. A. (2004). Electric field effect in atomically thin carbon films. Science, 306, 666. Noya, E. G., Doye, J. P. K. (2006), Structural transitions in the 309-atom magic number Lennard-Jones cluster. Journal of Physical Chemistry, 124, 104503. 

The significance of the magic number 32 found in the experiment may also be stated in a different manner. If a cluster containing Ba and a fuUerene molecule will be stable and, thus, result in a clearly discernible structure in the mass spectra every time there is exactly one Ba-atom situated on each of the rings of the ful-lerene molecule, this property might be used to count the rings of a fullerene. Of course, such a proposal has to be verified using other fullerenes, for example, C70 which is available in sufficient quantity and purity for such an experiment. 

The Structural Basis of the Magic Numbers.—Elsasser10 in 1933 pointed out that certain numbers of neutrons or protons in an atomic nucleus confer increased stability on it. These numbers, called magic numbers, played an important part in the development of the shell model 4 s it was found possible to associate them with configurations involving a spin-orbit subsubshell, but not with any reasonable combination of shells and subshells alone. The shell-model level sequence in its usual form,11 however, leads to many numbers at which subsubshells are completed, and provides no explanation of the selection of a few of them (6 of 25 in the range 0-170) as magic numbers. 

Summary.—The assumption that atomic nuclei consist of closely packed spherons (aggregates of neutrons and protons in localized Is orbitals—mainly helions and tritions) in concentric layers leads to a simple derivation of a subsubshell occupancy diagram for nucleons and a simple explanation of magic numbers. Application of the close-packed-spheron model of the nucleus to other problems, including that of asymmetric fission, will be published later.13... 

A striking observation that lacks a satisfactory explanation is the existence of magic numbers, i.e. the fact that in a distribution of clusters some species with a certain number of carbon atoms are much more abundant than others. The exact clustering mechanisms are not completely understood, and, as noted by Rohifing et al.(IO), the origin of the observed distribution of clusters may depend upon instrumental factors. Accounting for this fact, however, there still seems to be a preference for clusters with certain numbers of atoms which cannot be explained solely as due to the experimental conditions. 

As holds for other cluster systems, certain magic cluster electron counts exist, which indicates for a certain cluster-halide ratio and interstitial present the filling of all bonding molecular orbitals and therefore the thermodynamically most stable situation. For main group interstitial atoms these are 14 cluster-based electrons whereas for transition-metal interstitials the magic number is 18 [1, 10-12]. All of these phases are synthesized by high-temperature solid-state chemical methods. A remarkable variety of different structure types has been... 

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. 

A mole of a substance is the number of elementary particles (atoms, molecules) found in the mass (in grams) of that substance that corresponds to its atomic or molecular weight. In molar volumes, which amount to 22.4 L for gases and are different from solid to solid, there are always the same number of atoms or molecules. This "magic number", Avogadro s number, is 6.022xl023 mol-1. 

Thus, as long as t0 o, is large, aN declines slowly with increasing atomic mass number k, but when o becomes very small at the magic neutron numbers, there is a sudden drop (Fig. 6.3). (Such behaviour is characteristic of any distribution... 

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