Neutrons magic numbers

Variations in star formation history should be imprinted on the s- and r-process ratios as well, however their interpretation can be more complicated because of uncertainties in their exact sources (and thus yields). Y and Ba trace the first and second peak in neutron magic number, respectively, and can be used to examine r-process yields in very metal-poor stars. However, they also have a significant contribution from the s-process in AGB stars, which dominates their production with increasing metallicity. Since AGB s-process yields are thought... 

Figure 4. The initial steps of the s-process from 56Fe through to the neutron magic number N = 50. s-only and r-only nuclei are labelled. Possible branches in the s-process path occur at nuclei identified by the shaded boxes. The branch at 85Kr is discussed in the text. Several proton-rich nuclei (open boxes to left of the s-process path) are present these are products of neither the s- not the r-process. The inset shows the standard abundance distribution with the iron-group peak, and r- and s-process peaks identified. After Kappeler et al. (1989).

The structure of the nucleus plays a part in determining the magnitude of the cross-section. This is especially the case at the neutron magic numbers where the cross-section experiences a marked decline. For example, the cross-section at 30 keV for 139La at N = 82 is about 40 milli-barn but the typical cross-section for other odd Z nuclei... 

For exposures equivalent to about 20 neutrons per Fe seed nucleus (r 0.5), the seed nuclei are converted principally to elements of atomic mass A from 70 to 85 with very little synthesis of nuclei beyond the neutron magic number N = 50. At about 50 neutrons per Fe seed nucleus, the Fe seeds have been flushed to nuclei between the magic numbers N = 50 and N = 82. And at exposures of 130 neutrons per Fe seed nucleus, the greatest crmnm occur between N = 82 and Pb. Finally for higher exposures the Fe seeds are predominantly converted to Pb nuclei. A recent discovery of lead stars shows that a high neutron to seed ratio is possible for metal-poor stars (Van Eck et al. 2001). 

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. 

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... 

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... 

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 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. 

Alpha (a.) decay. As we shall see later, the alpha particle, which is a helium nucleus, is a stable particle. For some unstable heavy nuclei, the emission of this particle occurs. Because the a particle contains a magic number of both protons and neutrons (2), there is a tendency for this particular combination of particles to be the one emitted rather than some other combination such as s3Li. In alpha decay, the mass number decreases by 4 units, the number of protons decreases by 2, and the number of neutrons decreases by 2. An example of alpha decay is the following ... 

The magic numbers which impart stability to a nucleus are 2, 8, 20, 28, 50, 82 or 122. The isotope, 39K, has a magic number equal to its number of neutrons, so it is probably stable. The others have a larger neutron-to-proton ratio, making them neutron-rich nuclei, so 40K and 41K might be expected to decay by beta emission. In fact, both 39K and 41K are stable, and 40K does decay by beta emission. 

Scientists have known that nuclides which have certain "magic numbers" of protons and neutrons are especially stable. Nuclides with a number of protons or a number of neutrons or a sum of the two equal to 2, 8, 20, 28, 50, 82 or 126 have unusual stability. Examples of this are He, gO, 2oCa, Sr, and 2gfPb. This suggests a shell (energy level) model for the nucleus similar to the shell model of electron configurations. 

Neutron capture processes give rise to the so-called magic-number peaks in the abundance curve, corresponding to closed shells with 50, 82 or 126 neutrons (see Chapter 2). In the case of the s-process, the closed shells lead to low neutron-capture cross-sections and hence to abundance peaks in the neighbourhood of Sr, Ba and Pb (see Fig. 1.4), since such nuclei will predominate after exposure to a chain of neutron captures. In the r-process, radioactive progenitors with closed shells are more stable and hence more abundant than their neighbours and their subsequent decay leads to the peaks around Ge, Xe and Pt on the low-A side of the corresponding s-process peak. 

Fig. 2.2. Energy levels for ISW and 3DHO potentials. Each shows major gaps corresponding to closed shells and the numbers in circles give the cumulative number of protons or neutrons allowed by the Pauli principle. In a more realistic potential, the levels for given (n, T) are intermediate between these extremes, in which the lower magic numbers 2, 8 and 20 are already apparent. Adapted from Krane(1987).

Fig. 6.2. Neutron-capture cross-sections at energies near 25 keV. Very large dips occur at the magic numbers. After Clayton (1984). Copyright by the University of Chicago. Courtesy Don Clayton.

From time to time, a -decay occurs, increasing Z by 1 unit this leads to an increase in Q (corresponding to the increased distance above the neutron drip line) and consequently to further neutron captures until Q is again reduced to the appropriate value and a further fi-decay occurs. At the magic numbers, this leads to a vertical zig-zag track, paralleling the rise in the neutron drip line. Along this track,... 

Because element 117 (ununseptium) has not yet been produced, its properties are not known. This does not hinder speculation as to what some of the properties and characteristics of jj Uus will be when it is discovered and where it will fit into the scheme of what is known of elements 116 and 118—if they are confirmed. One thing that can be assumed with a high degree of probability is that Uus does not have a magic number of protons and neutrons, and thus is not included as one of the elements in the island of stability. ... 

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... 

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