The Periodic Table. Geoff Rayner-Canham
useful in the context of nuclear fission) and the Meyer–Jensen shell model.
The Meyer–Jensen shell model is a model of the atomic nucleus that invokes quantum principles to explain the nucleon energy levels.
Such a model is of crucial importance in nuclear spectroscopy. In this chapter, we will look at the use of the Meyer–Jensen model to explain stabilities of isotopes and to show its application in element synthesis. For discussion of the deeper levels of the theory, the Reader must refer elsewhere to the realm of particle physics [12].
According to the nuclear shell model, the principal quantum number, n, for nucleons, like that for electrons, can have values of 1, 2, and so on. However, for nucleons, the angular momentum quantum number, l, is not bound by the principal quantum number (unlike the electron quantum model). That is, for n = 1, l can be 1, 2, 3, and so on, resulting in energy levels of 1s, 1p, 1d, and so on. The other two quantum numbers are consistent with the electron case. Thus, there is one 1s quantum state, three 1p quantum states, five 1d quantum states, and so on. Each quantum state can hold two nucleons, one with spin
There are two complicating factors for nucleon levels. First, for a polyprotonic or polyneutronic nucleus, as for a polyelectronic system, the energy level degeneracy is lost. That is, 1p will be higher in energy than 1s; 1d higher than 1p; 1f higher than 1d. As a result, 1f is higher in energy than 2s. There is a parallel with electron-level filling, where the 4s orbital can fill before the 3d; the 6s before the 4f; and so on.
Second, the phenomenon of spin–orbit coupling is a major secondary factor for nucleon energies. That is, splitting occurs with a specific level, for example, the 1p level is split into two sublevels, the lower holding a maximum of four nucleons and the upper, two nucleons. To incorporate spin–orbit coupling, a different quantum number, j, is necessary, the total angular momentum quantum number can be the positive values of (l ± ½). The j value is linked to a matching magnetic orbital quantum number, mj, where mj can have values of:
Here we are only interested in the results, not the detailed derivations.
Figure 1.5 shows the energy levels and sublevels up to 70 nucleons. For electrons, an energy “layer” is completed upon filling each np6. For nucleons, as shown, there is not the same consistency. Instead, it is essentially where there are the larger energy level “gaps.” These gaps correspond to filling a total of 2, 8, 20, 50, 82, and subsequently 126, nucleons. On progressing through the remainder of this chapter, the importance of these numbers will become apparent.
Figure 1.5 Nuclear shell energy levels, the single nucleon to the left, multinucleon to the right, up to 70 nucleons.
“Magic Numbers” and Element Isotopes and Isotones
We showed earlier that the nuclei with even numbers of protons had more isotopes than those with odd number of protons. The element with the most stable isotopes is tin, a total of 10. Tin has a “magic number” of protons: 50. These tin isotopes have nucleon numbers of 112, 114, 115, 116, 117, 118, 119, 120, 122, and 124. Not only do the majority of the stable isotopes have even numbers of neutrons, but in terms of abundance, 83.4% of naturally occurring tin has even numbers of neutrons.
Nuclei with the same number of protons and different numbers of neutrons are called isotopes, similarly, nuclei with the same number of neutrons and different numbers of protons are called isotones. The largest number of stable isotones are two of the magic numbers: those of N = 50 and for N = 82. For N = 50, the five stable isotones are krypton-86, strontium-88, yttrium-89, zirconium-90, and molybdenum-92. For N = 82, the six stable isotones are barium-138, lanthanum-139, cerium-140, praseodymium-141, neodymium-142, and samarium-144.
“Double-Magic” Nuclei
If the possession of a completed quantum level of one nucleon confers additional stability to the nucleus, then we might expect that nuclei with filled levels for both nucleons — so-called doubly magic nuclei — would be even more favored. This is indeed the case. In particular, helium-4 with 1s2 configurations of both protons and neutrons is the second most common isotope in the universe, and the helium-4 nucleus (the α-particle) is ejected in many nuclear reactions. Similarly, it is the next doubly completed nucleus, oxygen-16 (8P, 8N), which makes up 99.8% of oxygen on this planet. As we saw in Figure 1.1, the number of neutrons increases more rapidly than that of protons. Thus, the doubly stable isotope is lead-208 (82P, 126N). This is the most massive stable isotope of lead and the most common in nature.
Some doubly magic nuclei can be created whose P:N ratio is far from the usual range of values and physicists have sometimes set out specifically to synthesize the missing one of the pair. Tin provides one such example. Tin-132 (50P, 82N) is a long-known radioactive isotope with a half-life of 40 s. Then in 1994, highly neutron-deficient tin-100 (50P, 50N) was synthesized [13]. It has a short half-life of 1 s, but considering it has such an abnormal P:N ratio, it is surprising that it is that long-lasting.
Calcium is even more evidence of the doubly magic pair phenomenon. The only stable isotope of calcium is the significantly neutron-deficient calcium-40 (20P, 20N). The other doubly magic calcium isotope is the neutron-rich calcium 48 (20P, 28N), which has an exceptionally long half-life of almost 1020 years. A “doubly magic” trio is that of nickel [14]. Nickel isotopes are known for nickel-48 (28P, 20N); nickel-56 (28P, 28N); and nickel-78 (28P, 50N).
More “Magic Numbers”?
In recent years, there has come the possibility of synthesizing nuclei very far from the normal P:N ratio. In such circumstances, there seem to be additional closed shell values, resulting in new “magic numbers.” There is a specific interest in exotic “doubly magic” atoms. A definition of a doubly closed shell nucleus is a spherical shape and abnormally high energy needed to raise a nucleon to an excited state. One exceptionally neutron-rich nuclei that fit the criteria is oxygen-24 with a ratio of 1:2.0, indicating that 16 neutrons may provide a “magic number” in this circumstance [15]. Similarly, calcium-54 shows similar characteristics, suggesting another “magic number” of 34 under such a high ratio of 1:1.7 [16].
Limits of Stability
In the universe, there are only 80 stable elements (Figure 1.6). For these elements, one or more isotopes do not undergo spontaneous radioactive decay. No stable isotopes occur for any element after lead, and two elements in the earlier part of the table, technetium and promethium (both mentioned earlier) exist only as radioactive isotopes.
Figure 1.6 Periodic Table showing elements with one or more stable isotope.
Traditionally, bismuth, or more correctly bismuth-209, was considered the last stable isotope. However,