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The Incremental Binding Energy of Small Nuclides

The pattern of changes in the binding energy of nuclides for increasing numbers of neutrons is very regular for the larger nuclides. For smaller nuclides the variation is irregular. The material here examines the pattern of incremental binding energy for the nuclides from Hydrogen to Carbon. Carbon can considered to be near the borderline between regular and irregular. Here is the data for Carbon.

There are two ranges over which the incremental binding energy declines linearly with odd-even fluctuations. One range is from 3 to 6 neutrons. The other range is from 9 to 16 neutrons. The fluctuations probably reflect the effect of the formation of neutron spin-pairs. The ranges indicate the filling of shells. When 6 neutrons are reached a shell is filled and the incremental binding energy drops sharply. The number 6 is said to be a magic number. The 9 to 16 range indicates filling (partially) of another shell. The values for 7 and 8 neutrons indicate the filling of a subshell. The number 8 is also considered a magic number. There is a drop in the incremental binding energy when 8 neutrons is reached but it is not as large as the drop at 6. The nature of the data for 7 and 8 is the major irregularity of the pattern. Otherwise the pattern is regular. The amplitude of the odd-even fluctuations is nearly constant within each shell. The downward slope is greater for the 3 to 6 shell than for the 9 to 16 shell.

In the case of Boron, shown below, the sharp drops in the incremental binding energy come at 4 and 6 neutrons instead of at 6 and 8, as in the case of Carbon.

For Berylium the sharp drop in incremental binding energy comes at two neutrons, as it should since 2 is a magic number representing filled shells. There is almost as large of a drop at 4 neutrons as at 2. The drop and change in the pattern at 6 neutrons is there but barely noticable.

For Lithium the sharp drop is at 2 neutrons. There are decreases at 4 and 6 neutrons but of much smaller magnitudes.

It is with Helium that serious deviations from the pattern appear.

The sharp drop in incremental binding energy comes at 2 neutrons, as it should. For higher numbers of neutron it appears as if the relationship might have a slight upward slope, contrary to the other cases.

And finally there is the problem case of Hydrogen.

The incremental binding energy rises for the second neutron and then falls sharply. This is as it should be. But the incremental binding energy does not increase for the fourth neutron which could create another neutron pair. There is a rise with the fifth neutron that puts the level above what it is for the third neutron.

What may be involved is that for H4 there is an excess excess of neutrons compared with the number of protons. It is understandable that an excess of proton compared to the number of neutrons would create instability. A large number of protons means the distance between the protons is large enough that the electrostatic repulsions are outweighing the strong force attraction. It is not known why an excess of neutrons compared with the number of protons should create nuclear instabilities but it clearly does. For each element there is a maximum number of neutrons it can have and still be stable enough to have its mass measured.

A bent-line regression of the incremental binding energies for the nuclides from Hydrogen through Magnesium has a coefficient of determination (R²) of 0.842.

(To be continued.)


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