Figure 1.2 A plot of the current abundance of an element against its atomic number.
Nitrogen-14 has an abnormally high abundance, as its formation in stars is part of the CNO nucleosynthesis cycle [9]. The slowest step in the cycle is the proton capture by a nitrogen-14 nucleus. As a result, the cycle often terminates at this step, resulting in an excess of nitrogen atoms compared to the neighboring carbon and oxygen nuclei.
The Cobalt–Nickel and Tellurium–Iodine Atomic Mass Anomalies
Mendeleev organized his Periodic Table according to increasing atomic weight (mass). As Scerri has discussed, there were problems with this rigidity [10]:
The strictest criterion Mendeleev employed was that of the ordering of elements according to increasing atomic weight.... He would occasionally seem to violate this principle, however, in cases where the chemical characteristics of an element seemed to demand it. An example is his placement of tellurium before iodine, as the atomic weight of tellurium has the higher value of the two elements. But while making this reversal, Mendeleev did not just disregard the issue of atomic weight, but rather insisted that the atomic weight of at least one of these elements had to have been determined incorrectly, and that future experiments would eventually reveal an atomic weight ordering in conformity with the placement of tellurium before iodine.
The Cobalt–Nickel Anomaly
In this presumption, Mendeleev was wrong. The measured atomic weights were correct. Generally, as the number of protons in a nucleus increased, so did the number of neutrons in the common isotopes. This accounted for the general correctness of Mendeleev’s Periodic Table format. But it is the stable isotopic distribution that could explain the anomalies [11]. The first of these anomalous order pairs was cobalt (58.93) and nickel (58.69). In fact, the Mattauch Isobar Rule applies beautifully as we see in Figure 1.3 for the iron–cobalt–nickel isobars. The only feasible stable isotope of cobalt being cobalt-59.
Figure 1.3 Percentages of stable isotopes for the iron–cobalt–nickel sequence.
In fact, the average atomic mass of nickel is less than that of cobalt because of the high proportion of the nickel-58 isotope. The “missing” radioactive even–odd iron-59 isotope decays to the stable odd–even cobalt-59. The cobalt-59 nucleus must represent an energy minimum for the three isobars, as “missing” radioactive even–odd nickel-59 also decays to odd–even cobalt-59 but, in this case, by electron capture.
The Tellurium–Iodine Anomaly
The Mattauch Isobar Rule can be seen for Mendeleev’s other example of tellurium (127.6) and iodine (126.9), as shown in Figure 1.4. In this case, iodine-127 is the only possible stable isotope of iodine with a reasonable P:N ratio. The average atomic mass of tellurium is higher than that of iodine because of the high proportions of the tellurium-128 and tellurium-130 isotopes. The radioactive even–odd tellurium-127 isotope decays to the stable odd–even iodine-127 isotope. Similar to the cobalt-59 situation, the iodine-127 isotope must represent an energy minimum for the three isobars, as radioactive even–odd xenon-127 decays to odd–even iodine-127 by electron capture.
Figure 1.4 Percentages of stable isotopes for the tellurium–iodine–xenon sequence.
Nuclear Shell Model of the Nucleus
There are two common models of the nucleus, the liquid drop model (which is 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 and the other with spin Pauli’s exclusion principle also applies to nucleons.
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 synthes
ize 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, as early as 1949, it was predicted theoretically that the isotope could not be stable. It was not until 2003 that the radioactive decay of the “stable” isotope of bismuth was observed [17], and its half-life has now been determined as 1.9 × 1019 years. Beyond the “magic number” of 126 protons of lead, the number of positive charges in the nucleus becomes too large to maintain infinite nuclear stability, and the repulsive forces prevail.
Two postlead elements for which only radioactive isotopes exist, uranium and thorium, are found quite abundantly on Earth because the half-lives of some of their isotopes — 108 to 109 years — are almost as great as the age of Earth itself.
Synthesis of New Elements
A goal of both chemists and physicists has been the synthesis of atoms of new chemical elements. Such atoms generally have very short half-lives. In fact, in order to claim synthesis of a new element, the isotope must have a half-life longer than 10−14 seconds, thus excluding “quasi-atoms,” briefly existing species formed during nuclear collisions [18]. Seaborg designed a plot of the stability of elements using a geographical analogy of islands in a sea [19]. The goal was to reach the “island of stability” (Figure 1.7). Many variations and updates have appeared since this first one, all based upon the same theme.
Figure 1.7 The original geographic plot by Seaborg of isotope stability (from Ref. [19]).
To accomplish such syntheses, a target of a high atomic number element is bombarded with atoms of a neutron-rich element whose combined atomic number is that, or greater than that, of the desired element. Up to p = 112, the more common route for the synthesis of postactinoid elements was the use of “doubly magic” lead-208 or “singly magic” bismuth-209 as targets and stable neutron-rich nickel-64 or zinc-70 as projectiles [20]. However, to synthesize the “doubly magic” hessium-270, californium-248 was bombarded with magnesium-26 [21].
Beyond P = 112, a different route was chosen. This method involved taking atoms of one of the later actinoid elements and bombarding them with calcium-48 as a projectile. About 0.2% of natural calcium is neutron-rich “doubly magic” calcium-48 (20P, 28N). With a neutron–proton ratio of 1:1.4, calcium-48 has been the key to synthesizing many new elements. Using calcium-48 nuclei as projectiles, nuclear physicists have claimed the synthesis of isotopes of element 114 (Fl) from plutonium-244; element 115 (Mc) from americium-243; element 116 (Lv) from curium-248; element 117 (Ts) from berkelium-249; and element 118 (Og) from californium-249.
Now the aim is to make the first elements of the next period [22]. “Quo Vadis?” stated Karol, as he reviewed the nuclear challenges [23]. There are no long-lived target isotopes with even higher atomic number, while the most probable higher atomic number projectile would be titanium-50. Titanium-50 (abundance 5.2%) has the same magic number of neutrons as calcium-48 with two more protons. Therefore, impacting californium-249 should enable the synthesis of an isotope of element 120. Unfortunately, with a P:N ratio of only 1:1.27, it is less likely that long half-life atoms of the desired atomic number would be produced. Similarly, an isotope of element 119 might be expected from the impact of titanium-50 on berkelium-249. But to raise the probability of success, the focus is on the even-proton-numbered element 120.
Island of Stability
Much of the interest in the synthesis of new elements is the belief that, approaching the next set of magic numbers of protons and neutrons, the trend for ever-shorter half-lives will be reversed. As mentioned earlier, this P-N region has been called the island of stability. There have been predictions that the island is centered around P = 114 and N = 184 and encompass combinations of proton and neutron values around that. Many proposals have been made about the island’s precise location. The major difficulty is in the synthesis of nuclei with high enough numbers of neutrons to confer longer half-lives. Some of the more optimistic calculations have suggested half-lives in days, years, and even millions of years [24].
It has been claimed that other islands of stability may exist. The first of these would be around P = 126, N = 216 or 228; and the second near P = 164, N = 308 or 318. Only time and nuclear experimentation will tell whether these nuclei are forever beyond the limits of synthesis. The key to reaching any of the islands is, of course, adding enough neutrons to generate a high enough P:N ratio [25].
Commentary
Chemists so often overlook the fascinating world of nuclear structure. So much can be explained. In fact, knowing about P:N ratios and magic numbers is key to understanding the difficulties of synthesizing new elements.
References
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irsching, “Anomalies in the Periodic Table,” J. Chem. Educ. 58(6), 478–479 (1981).
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