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of atoms with the same nucleon (mass) number but differing numbers of protons and neutrons, known as isobars, an interesting pattern emerges, known as the Mattauch Isobar Rule:

      The Mattauch Isobar Rule states that: if two adjacent elements in the Periodic Table have isotopes of the same nucleon number, then at least one of the isobars must be a radionuclide (i.e., radioactive).

      This phenomenon is illustrated by the “triplet” isobars, argon-40, potassium-40, and calcium-40, where the argon and calcium isotopes are both stable, while the intervening isobar of potassium is radioactive.

      The lack of any stable isotopes of technetium and promethium have always been a notable feature of the Periodic Table. Johnstone et al. have used the Mattauch Isobar Rule as a justification of the instability of all technetium isotopes [2]. The neighbors on either side, molybdenum and ruthenium, have six and seven stable isotopes, respectively. These isotopes span the range of “normal” P:N ratios, thus precluding any technetium isotope having a possibility of existence within that range.

The underlying phenomenon was discussed by Suess. He accounted for the instabilities for both technetium and promethium as follows [5]:

      After the filling of the 50- and 82-neutron shell [see discussion below], an upward shift in the β decay energies occurs equivalent to the drop in the binding energy of the last neutron. This shift is somewhat larger, however, for the odd Z than for the odd N nuclei, indicating that the drop is not equal for paired and for unpaired neutrons.... Thus, for a given I [mass number], the isobars with odd numbers of neutrons become stable at a lower mass number than those with an odd number of protons. This difference is large enough to cause the β-instability of all nuclei with a certain odd number of protons, incidentally those of Z = 43 and 61.

       Even Numbers of Nucleons

Elements with even numbers of protons tend to have large numbers of stable isotopes, whereas those with odd numbers of protons tend to have one or, at most, two stable isotopes. For example, cesium (55 protons) has just one stable isotope, whereas barium (56 protons) has seven stable isotopes. The greater stability of even numbers of protons in nuclei can be related to the abundance of elements on Earth. As well as the decrease of abundance with increasing atomic number, we see that elements with odd numbers of protons often have an abundance about one-tenth that of their even-numbered neighbors (see Figure 1.2). This observation is known as the Oddo–Harkins Rule [6]:

      The Oddo–Harkins Rule states that an element with an even atomic number is more abundant than either adjacent nucleus with an odd number of protons.

At the end of the curve, there is a “drop-off.” With only radioactive isotopes, the abundances of thorium and uranium have diminished with time. The reduction in abundance over time is also true for other radioactive isotopes, especially potassium-40 [7].

There are two notable exceptions to the Oddo–Harkins Rule. Beryllium would be expected to be much more abundant than it is, while nitrogen would be expected to be significantly less [8]. One might expect beryllium-8 with its 1:1 P:N ratio to be common. However, this nucleus has an extremely short lifetime, splitting into two helium-4 nuclei (helium-4 is “double magic” as we will discuss in the following).

      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

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