Nuclear magic numbers

The natural appearance of nuclear magic numbers, and the golden-ratio limitation on nuclear distribution, indicate the development of an excess surface layer of protons, which correlates well with periodic variation of nuclear spin, and which may be an important parameter in the understanding of superconductivity. 

Table 2.3 shows the number of stable isotopes as a function of the even or odd nature of the number of nucleons. Of the 266 naturally occurring stable isotopes, more than half have an even number of both protons and neutrons. By comparison, there are only four stable isotopes having both an odd number of protons and neutrons. Furthermore, even a cursory inspection of the cosmic abundances of the elements shown in Figure 2.2 makes it clear that there are more elements having an even atomic number than there are those with an odd atomic number. In fact, there even seem to be certain magic numbers of nucleons that are consistent with the most stable nuclei. The nuclear magic numbers are analogous to the common observation... 

Spin-orbit interaction. A decisive step in the interpretation of the correct nuclear magic numbers was made by Goeppert Mayer (1949) andbyHaxel etaL (1949). They proposed that a new term be added to the potential energy in Eq. (2.20), the so-called spin-orbit term ... 

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 ... 

Fig. 12 - 3D representation of nuclear magic numbers 28, 50, 82, and 126. The number of spheres changing to each other is equal to the magic numbers 2, 8, 20, 28, 50, 82, and 126. The segments of dark blue lines bridge the numbers whose sum is equal to the next... 

Analytical representations of nuclear magic numbers can be found in the literature [13, 14]. A formula valid for all nuclear magic numbers is ... 

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. 

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. 

Representation of nuclear energy levels showing the energy levels of nucleons and the magic numbers corresponding to filled shells. Shaded areas represent the gaps between the shells. 

The isotope of element 114 with 184 neutrons is predicted to be especially stable, since it has magic numbers of both protons and neutrons in its nuclei. This element may sit atop an island of stability in the sea of possible combinations of subatomic nuclear particles. Other islands, here picked out in contours whose height denotes the degree of stability, occur for lighter elements such as some isotopes of lead and tin... 

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... 

Hamilton, J.H. Magic Numbers, Reinforcing Shell Gaps and Competing Shapes in Nuclen. Progress in Particle and Nuclear Physics. 15, 107-134 (1985). 

The correlation of nuclear stability with special numbers of nucleons is reminiscent of the correlation of chemical stability with special numbers of electrons— the octet rule discussed in Section 6.12. In fact, a shell model of nuclear structure has been proposed, analogous to the shell model of electronic structure. The magic numbers of nucleons correspond to filled nuclear-shell configurations, although the details are relatively complex. 

The reasons behind the existence of this correlation are considered in the next section. Before proceeding to the analysis of these reasons, let us formulate the major conclusion of Section 4 the magic number of nuclearity is a result of the optimal proportion between the energy and entropy factors in tetranuclear magnesium clusters. 

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