San José State University
Thayer Watkins
Silicon Valley
& Tornado Alley

The Existence of Subshells Within
the Neutron Shells of Nuclei

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 isotones. They can also be established in terms of sharp drops in the incremental binding energies. This test however also establishes that 6 and 14 are magic numbers.

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

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. 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 results in a change in the pattern of the incremental binding energies.

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 51 through 82 Neutrons

As an illustration of what is involved consider the data for the incremental binding energies of the neutrons in isotopes of the element Niobium (atomic number 41).

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.

Thus there are changes of the slope of the pattern after 56 neutrons and after 64 neutrons. This would indicate a subshell being filled at 56 neutrons and another one at 64 neutrons. The numbers of neutrons in the shell involving 51 through 82 neutrons for the filling of the subshells are then 6 and 14.

The pattern of the data for Strontium (atomic number 38) repeats the pattern observed for Niobium.

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

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

The same phenomena is revealed in the data for elements with atomic numbers 33 through 37.

There are slight changes of pattern after 72 and 76 neutrons as is illustrated in the data for Silver (atomic number 47).

The Shell Containing 83 through 126 Neutrons

The evidence for subshells is available for the next higher shell.

Thus, for Terbium (atomic number 65) the numbers of neutrons in the shell where there is a change in the pattern are 6, 10 and 14. Both 6 and 14 are magic numbers.

The data for Gadolinium, atomic number 64, shows the same subshell phenomenon.

The data for Mercury (atomic number 80) illustrates a change in pattern after 114 neutrons.

The Shell Containing 127 or more Neutrons

The data for Thorium (atomic number 90) provides evidence for changes of the pattern at 132 and 140 neutrons.

There is a change in the pattern after 152 as illustrated in the data for the following elements.

The data for Mendelevium (atomic number 101), Nobelium (atomic number 102), Lawrencium (atomic number 103) and Rutherfordium (atomic number 104) show a similar change of pattern after 152 neutrons.

When the data for Berkelium, Californium, Einsteinium and Francium are put together, as shown below, they indicate subshells being filled at 146 and 152 neutrons.

The Shell Containing 29 through 50 Neutrons

Manganese and the group of elements associated with Iron (Cobalt and Nickel) make interesting cases to look at for evidence of subshells.

There appears to be something in the nature of a subshell being filled at 36 or 38 neutrons. The consistency of the pattern is revealed when the data for the four elements are plotted in the same graph.

The Shell Containing 15 through 28 Neutrons

In the lower shells the pattern is sometimes distorted by the n=p effect but some of the graphs are quite striking.

The examination of the data for subshells in this shell can start with Phosphorus.

The sharp drop after 20 neutrons would represent the filling of a subshell.

The data for Sulfur, atomic number 16, gives a similar picuture.

The sharp drop after 16 neutrons is just the n=p effect.

The sharp drop after 18 neutrons is again the n=p effect.

When the data for the above elements are put together in the same graph it is clear there is a subgroup involved for 20 neutrons and 24 neutrons.

The Shell Containing 7 through 14 Neutrons

In the this shell as well as the one above the pattern is sometimes distorted by the n=p effect but the graphs are still quite striking.

When the data for these elements are put together it is clear that 8 is the significant number in the 7 through 14 shell and represents 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.

When the data for these elements are put together in one graph, as shown below, it is clear that 4 neutrons is a significant number in the shell. Four neutrons represents the formation of a subshell of two.


The results indicate that subshells are formed within shells. When a subshell is filled there often results in a change in the pattern of the incremental binding energy of neutrons.

HOME PAGE OF applet-magic
HOME PAGE OF Thayer Watkins