Maria Goeppert-Mayer and other physicists examining the properties of the isotopes of elements discerned that isotopes in which the proton and/or the neutron numbers were particular values have notable properties such as stability. These magic numbers are

It is profoundly significant that there are magic numbers for the proton and neutron numbers separately because that indicates that the protons and neutrons are organized in separate shells. In some respects a proton and a neutron appear to be simply different states of a basic particle called a neucleon. However there apparently is a distinct differentiation in the nucleus of the the proton and neutron nucleons.

The magic numbers for atomic electron structure are perfectly explainable in terms of a formula. The increments in the particular stable atomic numbers are equal to twice the square of an integer. These increments correspond to the maximum occupancy levels of shells. Therefore in looking for an explanation of the nuclear magic numbers it is reasonable to look at the increments in the magic numbers; i.e.,

The difference in consecutive magic numbers could be sums of shell occupancy levels. For example, the value of 44 might correspond to two shells of 22 each. Likewise a value of 22 might correspond to two shells, one of occupancy 20 and one of occupancy 2.

The maximum occupancy levels for the atomic electron shells being the square of an integer n corresponds to the sum of the first n odd numbers. Let look at the sum of the first even numbers.

The cumulative sum sum values show up in the set of increments of nuclear magic numbers and the increments can be represented as simple sum of these cumulative sum values, as is shown below.

If there is anything to this pattern the number 14=2+12 should be something like a magic number increment (rather than a magic number). This would mean that 22 and 34 should be nearly magic numbers. Titanium, atomic number 22, has five stable isotopes but scandium, the element with atomic number 21, has only one and vanadium, the element with atomic number 23 has only two. Selenium, atomic number 34, has six stable isotopes but bromine, the element with atomic number 35, has only two and arsenic, the element with atomic number 33 has only one. This is suggestive of the magic-ness of 22 and 34 and thus that 14=2+12 is a magic number increment.

These results look impressive but, as the table below shows, there is an even-odd alternation in the relationship between the number of stable isotopes and the atomic (proton) number.

The odd-even alternation indicates a pairing of protons within the nucleus; perhaps the existence of alpha particle subsystems.

The average number of stable isotopes increases with proton number reaching a peak for proton number 50 (Tin) and declines generally thereafter, as shown in the table below.

The real test of the magic-ness of a number is whether it appears so in terms of the neutron number. The number of stable isotopes as a function of neutron number are:

Note that Silicon is the element with atomic number 14 and it has three stable isotopes whereas the elements Aluminum and Phosphorus having atomic numbers 13 and 15, respectively, have only one stable isotope each. The isotope of silicon which is most prevalent has atomic weight 28=14+14. The binding energy per nucleon for silicon is however not exceptionally high, but it is the end result of a fusion process in stars called silicon burning, indicating a significant degree of stability.

(To be continued.)

For more on the nuclear shell model see Nuclear Shell Structure