Number magic

Mass spectra from a carbon cluster ion source show strong magic numbers at C, and C d[136]. This led to the... 

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... 

Lindgard P-A and Bohr FI 1996 Magic numbers in protein structures Phys. Rev. Lett. 77 779-82... 

Comphcated theoretical calculations, based on filled shell (magic number) and other nuclear stabiUty considerations, have led to extrapolations to the far transuranium region (2,26,27). These suggest the existence of closed nucleon shells at Z = 114 (proton number) and N = 184 (neutron number) that exhibit great resistance to decay by spontaneous fission, the main cause of instabiUty for the heaviest elements. Eadier considerations had suggested a closed shell at Z = 126, by analogy to the known shell at = 126, but this is not now considered to be important. 

It is most important that TLV data be correctly used. Misuse can occur when uninformed individuals view these levels as magic numbers, below which workers are... 

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. 

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.

Nuclei with even numbers of both protons and neutrons are more stable than those with any other combination. Conversely, nuclei with odd numbers ot both protons and neutrons are the least stable (Fig. 17.12). Nuclei are more likely to be stable if they are built from certain numbers of either kind of nucleons. These numbers—namely, 2, 8, 20, 50, 82,114, 126, and 184—are called magic numbers. For example, there are ten stable isotopes of tin (Z = 50), the most of any element, but... 

Madclung constant, 63 magic number, 704 magnesium, 593 burning, F2... 

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. 

The packing diagram given in Figure 2 provides a simple structural basis of the magic numbers.1... 

Magic number Mantle Core or outer core Inner core... 

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... 

The structural interpretation of the principal quantum number of nucleonic orbital wave functions and the structural basis provided by the close-packed-spheron theory for the neutron and proton magic numbers are discussed in notes submitted to Phys. Rev. Letters and Nature (L. Pauling, 1965). 

The close-packed-spheron theory leads to a simple structural interpretation of the magic numbers (16) they are the numbers at which each layer of the nucleus achieves completion of... 

The magic number 20 corresponds to the KM structure, which is also the structure of the core for magic number 82. This structure, shown in Fig. 5 (left), involves nine spherons ligated about a smaller central spheron. Its stability may be attributed to its double completed-shell character. 

Two limiting structures with four spherons as core or inner core are shown in Figs. 6 and 7. The structure shown in Fig. 6 has the central tetrahedron of four spherons surrounded by a larger tetrahedron of four and a truncated tetrahedron of 12, a total of 16 spherons in the outer layer. The packing is triangular. This is the structure of the cpre for magic number 126. It has double completed-shell character, LN. 

The arrangement of 22 spherons around an inner tetrahedron of four spherons shown in Fig. 7 involves icosahedral packing each of the four inner spherons is surrounded by an icosahedron of 12, three of which are the three other inner spherons. This structure (26 spherons, 52 neutrons) with one spheron missing may be assigned to magic number 50. The complete structure, with 26 spherons, corresponds to the stable nucleus as discussed in the following section. 

The close-packed-spheron theory of nuclear structure may be described as a refinement of the shell model and the liquid-drop model in which the geometric consequences of the effectively constant volumes of nucleons (aggregated into spherons) are taken into consideration. The spherons are assigned to concentric layers (mantle, outer core, inner core, innermost core) with use of a packing equation (Eq. I), and the assignment is related to the principal quantum number of the shell model. The theory has been applied in the discussion of the sequence of subsubshells, magic numbers, the proton-neutron ratio, prolate deformation of nuclei, and symmetric and asymmetric fission. 

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... 

Miller, G. A. The magical number seven, plus or minus two some limits on our capacity for processing information. Psych Rev 1956 63 81-97. 

Slovokhotov, Yu.L. and Struchkov, Yu.T (1984) X-ray crystal structure of a distorted tetrahedral cluster in the salt [(Ph P)4Au4N] BF4 . Geometrical indication of stable electronic configurations in post-transition metal complexes and the magic number 18-e in centred gold clusters. Journal of Organometallic Chemistry, 177, 143-146. 

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. 

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