One of the elements of the physics of nuclei is the matter of magic numbers. They represent a shell being completely filled so additional nucleons have to go into a higher shell. A higher shell involves a greater separation from the other nucleons and lower interaction energy. The conventional magic numbers are {2, 8, 20, 28, 50, 82, 126}. These numbers were found in the case of protons by comparing the number of stable isotopes for different proton numbers. For neutrons the magic numbers were found by comparing the number of stable nuclides with the same neutron numbers.

A stronger indication of magicality of a number is in terms of the incremental binding energies (IBE).

Let BE(n, p) indicate the binding energy of a nuclide with n neutrons and p protons. The incremental binding energy of the n-th neutron in that nuclide is

IBEn = BE(n, p) − BE(n-1, p)

For example, the IBEn for the isotopes of Strontium are:

The sharp drop after 50 neutrons is evidence of a shell being filled. There are 50 neutrons in all of the shells up to that point. The odd-even sawtooth pattern is an indication of the formation of neutron-neutron spin pairs. The amplitude of the fluctuation associated with the formation of neutron-neutron spin pairs also includes the effect of the adjustment to the spin pair. The sharp drop after 38, 38 being the atomic number of Strontium, is a result of there not being any additional formation of neutron-proton spin pairs after 38 neutrons. This will be referred to later as the n=p effect.

The examination of the incremental binding energies reveals the magicality of the conventional nuclear magic numbers, but it also reveals that 6 and 14 are magic numbers.

It is a very remarkable fact the filled shell numbers are the same for protons as for neutrons. The data for protons is not included here simply in order to keep the details of this topic manageable.

If only the conventional magic numbers {2, 8, 20, 28, 50, 82, 126} are considered the shell capacities are {2, 6, 12, 8, 22, 32, 44}. Thus there is the anomaly of the shell capacity decreasing from 12 to 8 rather than increasing for each higher shell number as occurs for all of the other cases. This suggests that there may be something not quite right with the conventional sequence of magic numbers.

The Nuclear Magic Numbers

Consider the following algorithm. Take the number sequence {0, 1, 2, 3, 4, 5, 6} and generate the cumulative sums; i.e., {0, 1, 3, 6, 10, 15, 21}. Now add 1 to each of these numbers to get {1, 2, 4, 7, 11, 16, 22}. Now take the cumulative sums of that sequence to get {1, 3, 7, 14, 25, 41, 63}. These are doubled because there are two spin orientations for each nucleon. The result is {2, 6, 14, 28, 50, 82, 126} which is just the magic numbers with 8 and 20 left out.

This algorithm can be justified in terms of there being nucleonic states characterized by sets of four quantum numbers. The algoritm can be translated into a formula.

Let n be the shell number, the principal quantum number for a nucleon. The sum of the integers m for m=0 to (n−1) is n(n−1)/2=n²/2−n/2. When 1 is added to each of these terms the result is n²/2−n/2+1. This series summed from 1 to s is

s(s+1)[(2s+1)/12 − 1/4] + s = s(s+1)(2s −2)/12 + s
= s(s+1)(s-1)/6 + s

A formula for the nuclear magic numbers M(s) as a function of the shell number s taking into account the two spin states is then

M(s) = 2[s(s²-1)/6 + s] = 2s[(s² + 5]/6 = s(s² + 5)/3

The last one is the most succinct. Here is a tabulation of the results of computation with it.