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The Existence of Subshells Within
the Neutron Shells of Nuclei

The Nature of Nuclei

A nucleus is composed of protons and neutrons, collectively called nucleons. The mass of a nucleus is generally less than the combined masses of its nucleons. This mass deficit when expressed in energy units via the Einstein formula of E=mc² is called binding energy. The Binding Energy (BE) of a nuclide represents the energy required to break a nucleus up into its constituent nucleons. Binding energy is usually expressed in terms of millions of electron volts (MeV).

Let p and n be the numbers of protons and neutrons, respectively, in a nucleus. Then the binding energy of a nucleus BE is a function of only its p and n. The incremental binding energy of a neutron (IBEn) is defined as

IBEn(p, n) = BE(p, n) − BE(p, n-1)

The incremental binding energy of a proton is defined analogously.

The Nature of Nuclear Shells

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. The conventional magic numbers are {2, 8, 20, 28, 50, 82, 126}. These values were established by examining the relative numbers of stable isotopes and isotopes. They can also be established in terms of sharp drops in the incremental binding energies. For example, consider the incremental binding energies of neutrons as a function of the number of neutrons in the nuclide for the isotopes of Tin.

The sharp drop after 82 neutrons establishes that 82 is a magic number; i.e. that a shell is completely full. It was also found to be a magic number by Goepert Mayer and Jensen. The reason for the drop is that a neutron in a higher shell is at a greater average distance from the other nucleons than one in a lower shell This test however also establishes that 6 and 14 are magic numbers.

Here there are sharp drops in incremental binding energy at 6, 8 and 14 neutrons The sharp drop at 8 neutrons is, at least in part, due to that being the number of protons.

It is a very remarkable fact the filled shell numbers are the same for neutrons as for protons.

The So-Called Nuclear Magic Numbers

Maria Goeppert Mayer and Hans Jensen established that there are certain numbers of neutrons and protons for which there are an unusually large number of stable nuclides. This was taken to be evidence of the complete filling of nucleonic shells. Eugene Wigner coined the term magic number for them and unfortunately he name stuck.

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 wrong with the conventional sequence of 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}. Double these 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 but 6 and 14 included. Magic numbers 8 and 20 are the sums of the two previous magic numbers in the sequence. This suggest that within a shell there are subshells and the filling of a subshell also results in a change in the pattern of the incremental binding energies.

Subshells within Shells

The subject of this webpage is the investigation of the existence of subshells within the neutron shells of nuclei. The principles involved in this investigation are:

The Shell Containing 29 through 50 neutrons

As an illustration of what is involved consider the data for the incremental binding energies of neutrons in the isotopes of Niobium (which contain 41 protons).

The sharp drop after 50 neutrons indicates that a shell is filled at 50 neutrons and any additional neutrons must go into a higher level shell. The odd-even fluctuations are due to the formation of neutron-neutron spin pairs.

Ignoring the odd-even fluctuations the pattern is as below.

In addition to the sharp drop after 50 there is a distinct change in the pattern after 56 neutrons. This indicates a subshell of 6 neutrons in the neutron shell that comes after 50 neutrons.

The pattern of the data for the isotopes Yttrium with 39 protons repeats the pattern observed for p=41; with a change in the pattern after n=50 and another one for n=56. Here the change is a decrease in the amplitude of the odd-even fluctuations associated with the formation of neutron-neutron spin pairs. There is also something happening at 58 or 59 neutrons.

The data for the isotopes of Strontium with 38 protons shows a similar pattern, but the change is more subtle at n=56.

The sharp drop after 38 neutrons is the n=p effect. This did not show up in the case of n=39 because effect of the formation of a neutron-neutron pair nearly offset the non-formation of a proton-neutron pair.

The data for the isotopes of Zirconium with 40 protons more or less duplicates the graphs for the above cases.

The data for the elements with proton numbers 38 through 41 are shown below.

This display confirms that there is a completion of a shell at n=50 and something similar at n=56 which is probably the filling of a subshell.

The Shell Containing 51 through 82 neutrons

First consider the incremental binding energies of neutrons in the isotopes of Neodymium and Samarium.

There is of course the drop due to the full shell at 82 neutrons, but also a changes in the pattern at 88 and 92 neutrons which might indicate subshells. There is also something occurring at 74 and 78 neutrons although it is less definite.

When we look at the display for Promethium (p=61) we find a definite changes of pattern at 74, 78, 88 and 92 neutrons.

It is notable that 88 is 82 plus 6 and 78 is 50 plus 28.

For Tungsten (p=74) there are no sharp drops in incremental binding energy and the there may not be any changes of pattern either. But if one follows the lines through the data points there seems to be changes of slope at 92, 100, 108 and 114 neutrons.

The Shell Containing 83 through 126 neutrons

The data for Thorium displays some definite changes in the pattern. There is of course the sharp drop at 126 neutrons for the filled shell, but there are also changes of pattern at 132 and 140 neutrons and perhaps also at 124 neutrons. It is notable that 132 is 126 plus 6.

The display for the isotopes of Uranium (p=92) suggest there is some change near 134 neutrons and near 138 neutrons.

The display for Proactium (p=91) confirms there is something occurring at about 132 neutrons and also near 142 neutrons.

Again it is to be noted that 132 is 126 plus 6.

The Shell Containing 15 through 28 neutrons

The case of Argon (p=18) is an interesting one.

The display shows a sharp drop in the incremental binding energy of neutrons after 14 neutrons, which identifies 14 as a magic number corresponding to a filled shell. But there is also a sharp drop after 20 neutrons as well as 28 neutrons, conventional magic numbers. The sharp drop after 18 neutrons is just the n=p effect and does not indicate a shell or subshell.

The Shell Containing 7 through 14 neutrons

Consider the data for Nitrogen (p=7) and Neon (p=10).

The n=p effect nearly offsets the binding energy due to the formation of a neutron-neutron spin pair.

It is clear that 8 is the significant number in the 7 through 14 shell. It can be considered to represent the formation of a subshell of two neutrons.

The Shell Containing 3 through 6 Neutrons

Even the data for the next-to-smallest shell has some bearing on the matter of subshells.

Here the n=p effect is combined with the drop due to the filling of a neutron shell.

It is clear that 4 neutrons is a significant number in the shell and that 4 neutrons represents the formation of a subshell of two.


It was found that there are changes in the pattern of incremental binding energy that prevail over a sequence of proton numbers. This result is interpreted as an indication that there are subshells formed within shells.

(To be contined,)

For more on nuclear subshells see Subshells.

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