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The Incremental Binding Energy versus Proton Number
at the Transition to a Higher Neutron Shell

The ionization energies of electrons in atoms and ions displays very regular patterns as functions of the proton number of their nuclides. The analogous relation of incremental binding energies of nuclides as functions of their proton number also have a degree of regularity. Shown below are those relations for the 27th, 28th and 29th neutrons. At 28 neutrons one shelll is filled and the 29th neutron starts a new shell.

There are several notable features of this display:

The fact that the incremental binding energy levels are lower for the 29th neutron is consistent with that neutron being located at a greater distance from the center of the nucleus and thus requiring less energy to dislodge it for every proton number. This evidence confirms the shell modell of nuclear structure. The fact that the incremental binding energy levels are higher for the 28th neutron than the 27th reflects the fact that each additional neutron interacts with the nucleons already in the nucleus.

When the relations are shifted so that the points of the sharp increases coincide the graph is as follows.

When the differences in incremental binding energies are computed from that rearrangement of the data the result is as shown below.

The next shell is filled at 50 neutrons. The displays shown below show the same linearity and parallelism as at the 28th neutron, but the sharp jumps where the proton number equals the neutron numbers do not occur because there are no stable nuclides where the proton number is equal or larger than the neutron number.

The above procedure can now be used to verify that neutron number 14 is indeed a magic number; i.e., a number after which a new higher neutron shell begins to fill. The corresponding graphs showing the data for the 13th, 14th and 15th neutrons are shown below.

As in the case of the 27th, 28th and 29th neutrons there are sharp increase at the points where the proton number is equal to the neutron number. When the data is plotted versus the difference between the proton and the neutron number the result is:

These results would appear to confirm the magicality of 14, but the procedure must be applied to a case where there is no question of magicality. Below are two graphs displaying the information for neutrons 29, 30 and 31.

As can be seen all of the phenomena which were displayed for the magic numbers 28 and 50 occur also for neutron 30. Thus the results have no relevance for magicality. What is involved is that when the neutron number is even the incremental binding energy is enhanced due to the formation of neutron spin pairs. This is illustrated when the incremental binding energy is plotted versus the neutron number as in the case of lead, shown below.

The analysis was worthwhile in identifying the relative large increases in incremental binding energy which occur when the proton number becomes equal to the neutron number.

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