However there is a clearer, more definitive way to identify the magic numbers. First the binding energies are tabulated for the isotopes of an elemment. Second the increments in the binding energy are computed for each increase in the number of neutrons. For example here is the results of the this computation for the element Lead (Pb). (In the following the examples are for variations in the number of neutrons with the proton number fixed. This is simply because it is easier to discuss this case since a fixed proton number corresponds to an element. No such terminology exists for nuclides with a fixed neutron number.)
The sawtoothed pattern comes from the formation of neutron spin pairs. There is a gradual decline in the incremental binding energy until 126 neutrons is reached. For the 127th neutron there is a precipitous drop in the incremental binding energy. This corresponds to a new shell being started that involves the next neutron being farther from the center of the nucleus.
The display for Polonium confirms the magicality of 126.
The change in the pattern of the incremental binding energies may involve an abrupt change in the amplitude of the fluctuations, an abrupt change in the slope of the relationship or a change in the curvature of the relationship as well as an abrupt drop in the level of the incremental binding energy. This abrupt change in the pattern may be called the filled shell phenomenon. There can be gradual changes in those characteristics over the course of the filling of a shell.
The display for Neodymium confirms the magicality of the number 82.
The magicality of 50 is illustrated by the display for Selenium.
It is generally easier to see the filled shell phenomenon for the higher magic numbers.
The display for Titanium clearly illustrates the magicality of 28, and the display for Iron shows a change at 28 neutrons but it is less noticable because it occurs at the beginning of the range.
The Case for the Magicality of 14
The display for Silicon clearly shows an abrupt change in the level of incremental binding energy at 14 neutrons.
The corresponding display for the nuclides in which the number of neutrons is fixed at 14 and the number of protons vary is shown below.
As it happens this display also demonstrates the magicality of both 6 and 8 as well as 14.
The display for Oxygen shows an abrupt change at 14 neutrons.
The display for Phosphorus shows an abrupt change at 14 neutrons, but there is also an abrupt change at 20 neutrons. Thus both 14 and 20 are magic numbers.
Anomalous Variation in the Filled Shell Phenomenon
While there is ample confirmation for the magicality of 14, it must be revealed that there are anomalous cases.
The display of Magnesium shows an abrupt change in incremental binding energy at 12 neutrons rather than 14.
The display for Aluminum shows something happening at 14 neutrons, but also at 12 neutrons.
The display for Neon has the abrupt change occurring at 10 neutrons rather than 8.
Thus there are anomalous cases in which the abrupt change occurs at magic numbers plus or minus 2. Another example is Manganese where the change seems to come at 30 rather than 28.
Since the change at 14 neutrons occurs unambiguously for the other elements but changes at 12 neutrons do not occur for other elements 14 is a magic number but 12 is not. The effect for Aluminum reflects some sort of transitional effect. Likewise there is a case where the change comes at 16 rather than 14.
So there are anomalous cases in which the abrupt change occurs at magic numbers plus or minus 2. These are at two units away from the magic number instead of at the magic number. For the case of 6 and 8 the change occurs at 6 and 8 rather than at 6 instead of 8.
The Case for the Magicality of 6 and 8
The display for Carbon clearly shows an abrupt change at 6 neutrons.
The corresponding display for the nuclides with 6 neutrons also shows the magicality of 6.
The displays for Nitrogen and Fluorine also show the change at 6 neutrons but Fluorine also shows a similar abrupt change at 8 neutrons.
Conclusions
In addition to the conventional magic numbers of {2, 8, 20, 28, 50, 82, 126} the numbers 6 and 14 are also magic numbers at which the filled shell phenomenon occurs. There is some anomalous variations in which the filled shell phenomeno occurs at some magic numbers plus or minus 2.
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