San José State University
Thayer Watkins
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& Tornado Alley

The Binding Energy of Nuclides in Relation
to the Filling of Neutron/Proton Shells

In other work the question arose as to whether there might be substantial increases in binding energy as neutron and proton shells come close to filling. This was tested by constructing a table of the binding energies of all nuclides having equal numbers of neutrons and protons. From this table the incremental binding energies of additional neutron-proton pairs were calculated. The values are displayed in the graph below.

The sawtooth pattern arises from the formation of neutron and proton spin pairs. The sharp drops in incremental binding energy occur when the neutron and the proton shells are filled. These occur at 2, 6, 14 and 28. These are the nuclear magic numbers. The significance of 6 and 14 being magic numbers is that there exists a simple algorithm for a magic number sequence of {2, 6, 14, 28, 50, 82, 126},

The numbers 8 and 20 are also considered magic numbers but they are of a different nature than 2, 6, 14 and 28. There is something special about 8 and 20. The incremental binding energy drops to a marginally lower level and the values after that in the shell are somewhat different than the ones which came before. The number 42 is also of this nature. The numbers 8, 20 and 42 represent the filling of subshells. They also have the numerical characteristic that they are the sums of the previous two shell numbers; i.e., 8=6+2, 20=14+6 and 42=28+14.

There is a slight linear increase in the increamental binding energy for neutron-proton pairs as the shell fills up, more so for the lower and higher shells than for the middle shells.

Shell or
Increase in
binding energy
per n-p pair
7 to 140.2803
15 to 200.0572
21 to 280.0779
29 to 420.02633
43 to 500.1167

If the incremental binding energy is of the form

IBE = c1 + c2Q
BE = c0 + c1Q + ½c2

where Q is the number of objects in the shell. For shells with a large capacity the quadratic component could become significant.

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