Nuclei are composed of protons and neutrons. About 1949 the German physicists Maria Goeppert Mayer and Hans Jensen proposed that these nucleons are separately organized in shells. Based upon the relative numbers of stable isotopes and isotones they asserted that the number of nucleons in these filled shells are 2, 8, 20, 28, 50, 82 and 126. These became known as nuclear magic numbers. See Magic Numbers for details.

Relative numbers of stable isotopes and isotones is a rather crude criterion for identifying the filled shells numbers. It is difficult to justify the exclusion of some numbers that seem as qualified for inclusion as ones that were included. The data on the binding energies of nuclei provides a more precise method of filled shell identification. The incremental binding energy for a nucleon type drops sharply when a shell is filled and the next one has to go into a higher shell.

This method identifies 6 and 14 as magic numbers. If 6 and 14 replace 8 and 20 in the sequence of magic numbers there is a simple algorithm that generates the sequence {2, 6, 14, 28, 50, 82, 126}. The algorithm is of the sort that explains the occupancies of electron shells in atoms in terms of four quantum numbers. But there is something special about the numbers 8 and 20. They appear to represent the filling of subshells within shells. Note that 8 is 6+2 and 20 is 14+6. The occupancies of the subshells appear to replicate the occupancies of the shells; i.e., the first subshell of a shell contains 2, the first and second contain 6, the first, second and third contain 14 and so forth.

In the case of subshells the filling of a subshell is indicated by a change in the relationship between the incremental binding energies of a nucleon type and the number of that nucleon type in the nuclide. That change may be a drop in the level but it can also be a change in the slope of the relationship or a change in the amplitude of the odd-even fluctuations.

Based upon this scheme the number of nucleons in each subshell are given below.

Here is an illustration of the definite drops for the numbers 56 and 64.

Various investigators have proposed new magic numbers such as 34, 42, 78 and 96. The above table indicates that these are the numbers in filled subshells. They can be called submagic numbers for being subshell magic numbers. It is quite likely that other proposed new magic numbers will in fact be submagic numbers leaving the term magic number strictly for those in the sequence {2, 6, 14, 28, 50, 82, 126}. (The next number in the sequence is 184.)

Double Magic

It has been long noted that nuclides whose proton number and neutron number are both magic numbers are exceptionally stable. For example, He₄ with p=2 and n=2. These are calld doubly magic nuclides.

Nuclides which are doubly submagic also are notably more stable than other nuclides. For example Ca₄₀ with p=20 and n=20. Likewise for Ca₅₄ with p=20 and n=34. But this is not necessarily always the case. For example O₂₈ with p=8 and n=20 is unstable. Likewise Mg₃₂ with p=12 and n=20 is unstable. This may be due to too large of a deviation between p and n. Stability requires a proper balance between n and p.

What is true for doubly submagic nuclides also applies for nuclides having one nucleon number magic and the other one submagic.

The Evidence Concerning
Stable Isotopes and Isotones

The table below shows the magic and submagic numbers along with the stable nuclides of the number and for the two adjacent nucleon numbers. The triad for neutron numbers is above and that for the proton numbers below.

At the higher proton numbers it is difficult to find even one stable isotope.

It is also difficult to see on the basis only of stable isomers how 126 could be designated as a magic number. But it is one as shown in the following graphs of incremental binding energies.

Conclusion

A set of significant numbers of nucleons can be identified that reflect the filling of subshells within nuclear shells. While not as significant as the nuclear magic numbers the submagic numbers are significant for matters of nuclear stability.