Protons magic numbers

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). 

Because it was assumed that the next protonic magic number was 126 (by analogy with the neutrons), early studies of possible superheavy elements did not receive much attention [12—15), since the predicted region was too far away to be reached with the nuclear reactions available at that time. Moreover, the existence of such nuclei in nature was not then considered possible. The situation changed in 1966 when Meldner and Roper [16, 17) predicted that the next proton shell closure would occur at atomic number 114, and when Myers and Swiatecki [18) estimated that the stability fission of a superheavy nucleus with closed proton and neutron shells might be comparable to or even higher than that of many heavy nuclei. 

In addition to the proton magic number 114, a second superheavy magic proton number was investigated M. Z —164 [23, 28). Although the realization of such a nucleus seems to be far from any practical possibility at the moment, one should bear this region in mind because many most interesting questions could be answered if it were possible to produce these elements. One way to actually proceed... 

It has long been recognized that the liquid-drop model semi-empirical mass equation cannot calculate the correct masses in the vicinity of neutron and proton magic numbers. More recently it was realized that it is less successful also for very deformed nuclei midway between closed nucleon shells. Introduction of magic numbers and deformations in the liquid drop model improved its predictions for deformed nuclei and of fission barrier heights. However, an additional complication with the liquid-drop model arose when isomers were discovered which decayed by spontaneous fission. Between uranium and... 

Structural basis of neutron and proton magic numbers in atomic nuclei. Nature 208 (1965) 174. 

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 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. 

Elucidating the origin of magic numbers has been a problem of long-standing interest, made accessible through the use of the laser-based reflectron TOF technique and evaporative ensemble theory. Three test cases are considered, first protonated ammonia clusters where (NH3)4 NHj has been found to be especially prominent, and then two other cases are considered, one involving water cluster ions and another rare gas clusters. 

Figure 28. (a) Mass spectrum of protonated water clusters H+(H20) (n = 4-45) at 119 K and 0.3 torr He in a flow tube reactor. Note the prominence of H3O+(H2O>20 even under quasi-equilibrium conditions, (b) Mass-spectrometric abundance of OH-(H20)n produced under thermal conditions. Note a magic number at n = 20, though not as prominent as for the case of H30+ hydrates. Taken with permission from ref. 92. 

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. 

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).

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... 

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... 

Fmmi Magic numbers for nuclei are /1 / analogous to noble-gas electron configurations for atoms. A nucleus with 2,8,20, 28, 50, 82, or 126 protons or neutrons is particularly stable, just as an atom having a noble-gas electron configuration with 2,10,18,36,54, or 86 electrons tends to be stable. 

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